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Computable Aspects of Closed Sets

Permanent Link: http://ufdc.ufl.edu/UFE0022003/00001

Material Information

Title: Computable Aspects of Closed Sets
Physical Description: 1 online resource (150 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: bounded, bt, capping, classes, closed, computability, continuity, continuous, degrees, effectively, enumerations, numberings, pi01, randomness, turing, wtt
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A closed set in 2^N may be viewed the set of infinite paths through a tree; a set A is computable if there is a computer program which halts and gives the correct answer on every query to the membership predicate for A. A numbering, or enumeration, is a map from the natural numbers, N, onto a countable collection of objects; if there is a computable numbering onto a subset A of the natural numbers, then we say that A is computably enumerable (or c.e.). The set of infinite paths through a computable, or equivalently a co-c.e., tree is called an effectively closed set. In this work, we investigate: (1) numberings for different families of effectively closed sets, (2) notions of randomness for nonempty closed subsets of 2^N, (3) notions of randomness for continuous functions from 2^N to 2^N, and (4) continuity properties of C_bT, the c.e. degrees under the Turing reduction that requires that each use be bounded by a computable function. (1) Numberings and effectively closed sets. We show that certain families (or classes of families) of effectively closed sets-- such as the decidable, homogeneous, thin, small or the entire family of effectively closed sets, or string verifiable families-- possess, or do not possess, (injective) computable or effective numberings. This works builds upon the seminal work by Friedberg 1958, who constructed an injective numbering of the c.e. sets. (2) Randomness of closed sets. In the space of closed sets, we give a probability measure and define a version of the Martin-Lof Test for randomness. We show that random closed sets are never effectively closed, but are, on the other hand, always perfect, have measure zero, and have box dimension log_2(4/3). Every random closed set contains random and non-random elements, but no n-c.e. elements. We also explore alternate notions for randomness, such as the problem of compressibility of trees. Finally, we consider the problem of when a randomly chosen closed set meets a closed Q; this is the study of capacities. (3) Randomness of continuous functions. As in (2), we give a probability measure and define a version of the Martin-Lof Test for randomness. We show that the image of a random continuous function is always non-injective and perfect, but not necessarily surjective. Furthermore, computable elements map to random elements. Also, random closed sets arise as inverse images of 0^N, but not, in general, as images. The former motivates a study of pseudo-distance functions. Finally, we consider our results in the context of n-randomness. (4) Continuity properties in C_bT. We show in C_bT that for any incomplete nonzero b, there is an a > b such that for any x, the meet of b and x equals 0 iff the meet of a and x equals 0. We prove this by first showing that that the Seetapun 1991 local noncappability theorem in the c.e. Turing degrees also holds in C_bT. This theorem demonstrates that every incomplete nonzero b is noncappable with any nontrivial degree below some a > b (i.e. if x < a and the meet of x and b equals 0 then x = 0.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Cenzer, Douglas A.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022003:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022003/00001

Material Information

Title: Computable Aspects of Closed Sets
Physical Description: 1 online resource (150 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: bounded, bt, capping, classes, closed, computability, continuity, continuous, degrees, effectively, enumerations, numberings, pi01, randomness, turing, wtt
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A closed set in 2^N may be viewed the set of infinite paths through a tree; a set A is computable if there is a computer program which halts and gives the correct answer on every query to the membership predicate for A. A numbering, or enumeration, is a map from the natural numbers, N, onto a countable collection of objects; if there is a computable numbering onto a subset A of the natural numbers, then we say that A is computably enumerable (or c.e.). The set of infinite paths through a computable, or equivalently a co-c.e., tree is called an effectively closed set. In this work, we investigate: (1) numberings for different families of effectively closed sets, (2) notions of randomness for nonempty closed subsets of 2^N, (3) notions of randomness for continuous functions from 2^N to 2^N, and (4) continuity properties of C_bT, the c.e. degrees under the Turing reduction that requires that each use be bounded by a computable function. (1) Numberings and effectively closed sets. We show that certain families (or classes of families) of effectively closed sets-- such as the decidable, homogeneous, thin, small or the entire family of effectively closed sets, or string verifiable families-- possess, or do not possess, (injective) computable or effective numberings. This works builds upon the seminal work by Friedberg 1958, who constructed an injective numbering of the c.e. sets. (2) Randomness of closed sets. In the space of closed sets, we give a probability measure and define a version of the Martin-Lof Test for randomness. We show that random closed sets are never effectively closed, but are, on the other hand, always perfect, have measure zero, and have box dimension log_2(4/3). Every random closed set contains random and non-random elements, but no n-c.e. elements. We also explore alternate notions for randomness, such as the problem of compressibility of trees. Finally, we consider the problem of when a randomly chosen closed set meets a closed Q; this is the study of capacities. (3) Randomness of continuous functions. As in (2), we give a probability measure and define a version of the Martin-Lof Test for randomness. We show that the image of a random continuous function is always non-injective and perfect, but not necessarily surjective. Furthermore, computable elements map to random elements. Also, random closed sets arise as inverse images of 0^N, but not, in general, as images. The former motivates a study of pseudo-distance functions. Finally, we consider our results in the context of n-randomness. (4) Continuity properties in C_bT. We show in C_bT that for any incomplete nonzero b, there is an a > b such that for any x, the meet of b and x equals 0 iff the meet of a and x equals 0. We prove this by first showing that that the Seetapun 1991 local noncappability theorem in the c.e. Turing degrees also holds in C_bT. This theorem demonstrates that every incomplete nonzero b is noncappable with any nontrivial degree below some a > b (i.e. if x < a and the meet of x and b equals 0 then x = 0.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Cenzer, Douglas A.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022003:00001


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COMPUTABLE ASPECTS OF CLOSED SETS


By

PAUL BRODHEAD





















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008






























2008 Paul Brodhead


































To my Family, the Artisans of Life







ACKNOWLEDGMENTS

I give thanks, first and foremost, to my Ph.D. advisor Douglas Cenzer, who must be

noted in this generation as among the most patient and irrevocably kind of men. His sense

of humor also trumps the cloudiest of di ,- be they even the clouds of hurricanes which,

in fact, occurred as my studies progressed in Florida. These characteristics, combined with

my determination, helped me gain an understanding, broad and in-depth, of computability

that allowed me to successfully undertake and develop the research presented here. The

road to this point began long ago, but reached its pinnacle at Florida under his direction.

The road began at the University of Wisconsin, in my first semester; I was first in-

spired to pursue mathematics while taking Algebra and Trigonom( I1: Professor Arnold

Johnson made the course enjo--1,bl-' and (! i11. ii,;.- making it unlike any mathematics

course I ever took. Him I thank, for as have I ahv--i- had high goals, his influence was

critical as it was at this point in life that I decided that I was to pursue a Ph.D. in math-

ematics. Later, in my last semester at Wisconsin, he helped me develop a mathematical

maturity beyond expectations, as he led me in an undergraduate research project in his

area, Algebra. Yet, between these experiences, many others must be thanked.

During my undergraduate years, many inspiring teachers pushed the mathematical

envelope, delivering the endeavor of the ages, the enjoi,--bl'-! mental fray of the breaking

of mathematical barriers. I thank: Steffen Lempp, Reed Solomon, H. Jerome Keisler,

Arnold Miller, Maury Bramson, Steve Bauman, Simon Hellerstein, Daniel Rider, Daniel

Shea, and Jennifer Ziebarth. In particular, I thank Steffen Lempp for introducing me to

computability theory while I was in his linear algebra class, as it was at this moment that

I determined my Ph.D. specialization. My mindset was ever afterwards directed this way.

For example, I thank Arnold Miller for his patience in listening to my numerous attempts

in using computability theory in his year-long abstract algebra class. I thank H. Jerome

Keisler for listening to me in his logic course. I also thank Reed Solomon for helping

me even further in his graduate logic course and his subsequent guidance in my first

undergraduate research project, the study of Hilbert's Tenth Problem. I thank for Daniel








Shea (also the undergraduate advisor) for accepting my destined course in mathematics as

I sat in his abstract analysis course, vigorously driven under a department mathematics

scholarship, when years prior I entered his office whilst still in Algebra and Trigonometry,

demanding help in setting up my undergraduate curriculum for an eventual Ph.D. in

mathematics.

Upon the closing of my Wisconsin experience, excellence in mathematics, a higher

level still, evolved at the University of Puerto Rico-Humacao during the immediate

summer. I participated in the NSF-funded undergraduate research experience, the

Summer Institute of Mathematics for Undergraduates. I thank: Reinhard Laubenbacher,

Abdul Jarrah, Rebecca Garcia, Malarie Cummings, Cora Seidler, Ivelisse Rubio, and

Herbert A. Medina. Herbert, Ivelisse, and Reinhard, in particular, helped me to see the

urgency in the tackling of the problems at hand. I am greatly appreciative of this help to

this dv.

My research rigor and core mathematical talent bloomed in graduate school at

Florida. I thank: Douglas Cenzer, Bill Mitchell, Jindrich Zapletal, Jean Larson, Alexandre

Turull, Paul Robinson, James Keesling, Jonathan King, Scott McCullough, Stephen

Summers, and Shari Moscow. I especially thank Alexandre Turull. I had the privilege

of having six semesters with him; he helped me to see a problem, its value, and its

complexities. I thank my dissertation committee: Douglas Cenzer, Rick Smith, Bill

Mitchell, Murali Rao, and Beverly Sanders. I also thank Bernard Mair, Rick Smith,

Murali Rao, and Juan Liu; my experiences with them convinced me to move in valuable

directions that took me around the globe and beyond.

My collaborators, near and abroad, you I thank: Douglas Cenzer, Angsheng Li,

Weilin Li, George Barmpalias, Seyyed Dashti, Rebecca Weber, Jeffrey Remmel, Rod

Downey, Noam Greenberg, Keng Meng Ng, and Bji, rn Kjos-Hanssen. Many many more

do I also thank I have collaborated with many people, if even for a short time, and their

contributions are appreciated. I thank: Peter Hinman, Denis Hirschfeldt, Carl Jockusch,

Russell Miller, Johanna Franklin, Bakhadyr Khoussainov, Jan Riemann, Ted Slaman,








Andr6 Nies, Robert Soare, Steve Simpson, Antonio Montalbdn, Barbara Csima, Joe Miller,

Ver6nica Becher, and Wolfgang Merkle.

I thank the people and institutions who contributed their support through funding:

the Southeast Alliance for Graduate Education and the Professoriate, the National Science

Foundation, the University of Florida, the Chinese A( i'I i,: of Sciences, the Association

for Symbolic Logic, and others.

Moii people deserve credit far beyond the brief, or even omitted, acknowledgement

here; I am extremely grateful to you. Of special consideration is my advisor. The mag-

nitude of his support is above any measurable scale. His support in all my endeavors,

professionally and beyond, will surely continue to help shape me. This dissertation and the

work therein, and the experiences associated with the attainment thereof are a witness to

this.

To the many people who helped me along this journey, which is actually just a

beginning, you I thank.







TABLE OF CONTENTS


page


ACKNOWLEDGMENTS .......................

A BSTR A CT . . . . . . . .

1 INTRODUCTION ........................

1.1 General Overview .. ..................
1.2 Classical Computability .................
1.3 Closed Sets in Computability ..............

2 EFFECTIVELY CLOSED SETS AND ENUMERATIONS ..


2.1 Introduction . . . . . . . . .
2.2 The Family of 1 Classes ............................
2.2.1 N 11i ili. ii,-; in the Literature . . . . . .
2.2.2 Equivalence of the Numberings .....................
2.2.3 Equivalence of the Numberings (Alternate Proof) .. ........
2.2.4 Injective Computable N, 11i11. i i-, .. . . .........
2.2.4.1 Original c.e. sets argument .. ...............
2.2.4.2 Ordered tuples of disjoint c.e. sets .. ............
2.2.4.3 Results for effectively closed sets .. ...........
2.3 String Verifiable Families of 1 Classes .. ................
2.3.1 Definition and Examples .. ....................
2.3.2 Computable and Effective Nu11.11 li., . ..........
2.3.3 Families Containing the Clopen Classes .. .............
2.4 N i,,, ,1 Fam ilies of 1 Classes . . . . . . .
2.4.1 Homogeneous Classes .. .....................
2.4.2 Decidable Classes . . . . . . .
2.4.2.1 An injective computable numbering (Alternate proof) .
2.4.2.2 Trees without dead ends: A numbering result .. .....
2.4.2.3 Trees with dead ends: A necessity .. ............
2.4.3 Thin and Perfect Thin Classes .. .................
2.4.3.1 The Martin-Pour El Construction .. ............
2.4.3.2 Non-existence of computable numbering .. ........
2.4.4 Small, Very Small, and Nondecidable Classes .. ...........
2.4.4.1 Numberings and high/noncomputable sets .. .......
2.4.4.2 Non-existence of effective numbering .. ..........

3 RANDOM CLOSED SETS .. . .......................

3.1 O verview . . . . . . . . .
3.2 Effective Randomness of Reals .. .....................
3.2.1 Introduction . . . . . . . .
3.2.2 Constructive Martingale Randomness .. ..............
3.2.3 Prefix-free Randomness .. .....................
3.2.4 Martin-Lof (n-)randomness .. ...................








3.3 Martin-Lof Randomness of Closed Sets
3.3.1 The Hit-or-Miss Topology on C
3.3.2 Toward a Measure ........
3.3.3 Canonical Coding and Measure
3.3.4 Ghost Coding .. ........
3.3.5 Coding Equivalance .......
3.3.6 Coding and Joins of Closed Sets
3.4 Members of Random Closed Sets .
32 4 1 Pnsitive RPsuilts


3.4.2 Negative Results ....................
3.5 Measure and Dimension .. ...............
3.5.1 M measure . . . . . .
3.5.2 D im ension . . . . . .
3.6 Prefix-Free Complexity of Closed Sets .. ..........
3.6.1 Lower Complexity Bounds .. ...........
3.6.2 Upper Complexity Bounds .. ...........
3.7 Other Notions of Randomness for Closed Sets .......
3.7.1 Randomness with Regular Probability Measures .
3.7.2 Randomness with the Inclusion of Trees with Deads
3.8 Random Closed Sets and Effective Capacity .. ......
3.8.1 Computable Capacities .. .............
3.8.2 Regular Measures and Capacities of Closed Sets .


Ends


4 RANDOM CONTINUOUS FUNCTIONS .. ...........

4.1 Overview .. .. .. .. ... .. .. .. .. . . .
4.2 Definining Randomness for Continuous Functions . ..
4.2.1 Representing Functions . ............
4.2.2 Representing Sequences . ............
4.2.3 A Sound Definition . ...............
4.3 Random Continous Functions and Images . ......
4.3.1 Perfect Images, in every instance . .......
4.3.2 Non-injective Images, in every instance . ....
4.3.3 Non-surjective Images, in instances . ......
4.3.4 Images of computable elements ....... . .
4.4 Random Closed Sets arising from random continuous functions
4.4.1 A Positive Result: Inverse Images of 0 . ....
4.4.2 A Negative Result: Images, in general . ....
4.5 Pseudo-Distance Functions . ..............
4.6 n-Random ness . ... .. .. .. .. ... .. .. .. .
4.7 Future W ork . . . . . . .

5 CONTINUITY OF CAPPING IN CBT .. . .........

5.1 Introduction . . . . . . .
5.2 Continuity Results . ..................
5.2.1 Continuity Results in C . ...........
5.2.2 Continuity Results in CbT and Main Result . ..


94
95
95
96
97
99
99
100
103
104
106
106
109
110
111
112


113
114
114
115








5.3 Requirements and Strategies ............... .... .. 118
5.3.1 The requirements ............... ....... ..118
5.3.2 A P-strategy .................. ............ .. 119
5.3.3 An R-strategy .................. ........... 120
5.3.4 An S-strategy .................. ........... .. 121
5.4 The Priority Tree .................. ........... .. 125
5.5 The Construction .................. ........... .. 128
5.6 The Verification .................. .............. 132

REFERENCES .................. ................ .. .. 144

BIOGRAPHICAL SKETCH ........... ........ . ... 150







Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

COMPUTABLE ASPECTS OF CLOSED SETS

By

Paul Brodhead

May 2008

C(!i r: Douglas Cenzer
Major: Mathematics

A closed set in {0, 1}" may be viewed the set of infinite paths through a tree; a set A

is computable if there is a computer program which halts and gives the correct answer on

every query to the membership predicate for A. A numbering, or enumeration, is a map

from N onto a countable collection of objects; if there is a computable numbering onto a

set A C N then we -iv that A is computably enumerable (or c.e.). The set of infinite paths

through a computable, or equivalently a co-c.e., tree is called an effectively closed set.

In this work, we investigate: (1) numberings for different families of effectively

closed sets, (2) notions of randomness for nonempty closed subsets of 2", (3) notions of

randomness for continuous functions from 2N to 2N, and (4) continuity properties of CbT,

the c.e. degrees under the Turing reduction
a computable function.

(1) Numberings and effectively closed sets. We show that certain families (or classes

of families) of effectively closed sets such as the decidable, homogeneous, thin, small

or the entire family of effectively closed sets, or string verifiable families possess, or do

not possess, (injective) computable or effective numberings. This works builds upon the

seminal work by Friedberg [46], who constructed an injective numbering of the c.e. sets.

(2) Randomness of closed sets. In the space of closed sets, we give a probability

measure and define a version of the Martin-L6f Test for randomness. We show that

random closed sets are never effectively closed, but are, on the other hand, alv-- i perfect,

have measure zero, and have box dimension log12 Every random closed set contains

random and non-random elements, but no n-c.e. elements. We also explore alternate








notions for randomness, such as the problem of compressibility of trees. Finally, we

consider the problem of when a randomly chosen closed set meets a closed Q; this is the

study of capacities.

(3) Randomness of continuous functions. As in (2), we give a probability measure

and define a version of the Martin-L6f Test for randomness. We show that the image of

a random continuous function is alv--i-i non-injective and perfect, but not necessarily

surjective. Furthermore, computable elements map to random elements. Also, random

closed sets arise as inverse images of 0", but not, in general, as images. The former

motivates a study of pseudo-distance functions. Finally, we consider our results in the

context of n-randomness.

(4) Continuity properties in CbT. We show in CbT that for any b / 0, 0', there is an

a > b such that for any x, b A x 0 iff a A x = 0. We prove this by first showing that

that the Seetapun local noncappability theorem in the c.e. Turing degrees [84] also holds

in CbT. This theorem demonstrates that every b / 0, 0' is noncappable with any nontrivial

degree below some a > b (i.e. if x < a and x A b = 0 then x = 0).







CHAPTER 1
INTRODUCTION

This thesis ia an accumulation of my work as a graduate student at the University

of Florida. Much of this work is joint and published, or to be published. The citations

are listed at the beginning of the appropriate chapters. In this chapter we introduce

various notions and expound upon these in later chapters. Each of C'! lpters 2-5 contains

a distinct topic from computability theory.

1.1 General Overview

Computability theory is a field of mathematical logic; the subject captures the precise

notion of an algorithmic process towards the study of decidable/undecidable problems

in mathematics and nature. Its most notable historical contribution to mathematics is

the disruption of Hilbert's Program by G6del's Incompleteness Theorem. More recent

work has shown that other mathematical problems are unsolvable, in the sense that

no computer algorithm can solve every instantiation of the problem. Examples include

Hilbert's Tenth Problem (to decide whether a given Diophantine equation has solutions),

the word problem for groups (to decide whether a given product of generators and their

inverses is the identity element of a group defined by a finite set of equations between

such products), and the homeomorphy problem (to decide whether the topological spaces

defined by a given pair of simplicial complexes are homeomorphic) [32]. Maturation

of computability, however, through applications and techniques, has broadened its

interactions with other fields, most notably computer science. Each of C'! Ilpters 2-5 is

a witness to this broadening.

In C'! lpter 2, we focus on the study of effectively closed sets of binary reals. Thought

of as a set representing the solutions to some problem, effectively closed sets characterize

many structures in mathematics and computer science. In algebra, for example, they

represent the prime ideals of a computable enumerable Boolean algebra or a commutative

ring with identity [47]. In graph theory, they represent the set of solutions to ri ,nr:

problems with computable graphs, such as Hamiltonian circuits or vertex partitions [20].

In computer science, effectively closed sets arise in the study of non-monotonic logics and








Lc-languages [25, 68, 70]. Given the wide variety of applications, the work in ('!i ipter 2

focuses on methods of enumerating various families of effectively closed sets. The idea

is to provide, based on desired properties, complete listings of the objects in this case,

effectively closed sets representing the set of solutions to problems of a certain type.

We show that there is an injective computable enumeration of the entire class of sets, of

certain families of string verifiable classes, and of the decidable and homogeneous classes.

We also show that no computable enumeration exists for thin, perfect thin, small, very

small, or nondecidable classes.

In ('!i lpters 3 and 4, we extend notions related to effective randomness for binary

reals, to closed sets (C'!i plter 3) and continuous functions (C'! plter 4); various global

properties are obtained. A binary real is effectively random if it is impossible for a

computer to find regularity or patterns in it. For a closed set, representing some set

of solutions, this means that it is difficult for a computer to precisely obtain or locally

describe this set of solutions, given the lack of pattern. In both chapters, we obtain

so-called basis and antibasis theorems. For instance, every random closed set contains

random and non-random elements, but omits various elements of computability-theoretic

interest, such as the properties of being f-c.e. (f a polynomial), 1-generic, or of incomplete

c.e. degree. We also show that concepts from both chapters are closed related; random

closed sets arise as inverse images of random continuous functions mapping to 0", but

not, in general, as images. Methods employ, ,1 in all of this work range from techniques

in computability and effective randomness, to techniques related to the study of effective

Hausdorff dimension and classical probability.

Finally, in connection with the roots of computability theory, in ('! Ilpter 5 we focus

on the classification of information content by means of Turing degree theory. Mathemati-

cal structures or objects are often encoded as sets of natural numbers. Reduction methods,

such as the Turing reducibility, allow the information content of these sets to be classified

into equivalence classes, called degrees. In Chapter 5, the focus is on the bounded Turing

reducibility [16]; we show that capping, the operation which takes the meet of a given








noncomputable incomplete c.e. degree with another noncomputable incomplete c.e. degree

such that the resulting meet is the degree 0 of the computable sets, is continuous. That is,

for any b / 0, O', there is an a > b such that for any x, b A x = 0 iff a A x = 0. As this

is a three-quantifier statement in the c.e. bT-degrees, this result gives insight into the three

quanifier theory of the same, whose decidability/undecidability is currently unknown. As

an aside, in recent work by Soare, the bounded Turing reducibility with the identity use

has led to applications of degree theory to differential geometry [72, 87].

The rest of this chapter is devoted to introducing to basic definitions, terminology,

and notations that will be used throughout this entire work. Section 1.2 covers basic no-

tions and notations for topics in classical computability (e.g. computable sets, computably

enumerable sets, partial computable functions, Turing degrees). Section 1.3 covers the ba-

sics of closed sets in -: As algorithmic randomness is a topic covered only in ('!i Ilters 3

and 4, we postpone a general introduction of this topic and provide it in Section 3.2.

1.2 Classical Computability

We generally follow the notation of Soare [86] for notions that arise from classical

computability. For an in-depth treatment of the basic foundations of the subject, we refer

the reader there.

A set A is computable if there is a computer program, or equivalently a computable

function, which halts and gives the correct answer on every query to the membership

predicate for A. It is computably enumerable (c.e.) if the computer program is required

to halt only on queries for elements in A; this gives rise to the standard example of a c.e.

set, namely the halting problem the set of Turing machines, officially encoded as a set of

natural numbers, that halt when given their own binary input. Computably enumerable

sets are the domains, therefore, of so-called partial computable functions; we index the

partial computable {0, 1}-valued functions as {(,}1E,. Partial computable functionals that

take natural number (m) and real (x) inputs are indexed as te; we will write K1(m) for

the result of applying 4~, to m and x.








Other related notations are standard: 0e,s denotes that portion e, defined by stage

s, and e(x) I means that e, is defined on x (and T means undefined). We also index

the primitive recursive functions, a smaller class of total functions, as {i7} ,Ew. (*,*)

2 -- wo is typically a computable bijection such that (0, 0) = 0. A and '(A) denote the

complement and power set of A, respectively. z = x E y is the coding together of two reals

x and y, so that z(2n) = x(n) and z(2n + 1) = y(n) for all n.

In computability, a reduction is a binary relation on subsets of N that captures a

relationship between the information content of two sets. The Turing reduction
main reduction used in computability theory. A is Turing reducible to B, written A
if membership in A can be determined by a computer algorithm that has full access to

the membership predicate for B. Intuitively, the information content of A is viewed as

computable, or recoverable, from B. Furthermore B is viewed as an oracle, in terms of

information content, for determining membership in A.

Various restrictions on how much oracle information is allowed to be used in deter-

mining the membership of a single element have given rise to different kinds of reductions.

For example, the bounded Turing reduction
determining membership of x in A use at most f(x) amount of B, where f, called the use,

is bounded by a computable function. The identity bounded Turing reductions insists that

the use f be bounded by the identity function.

Reductions often give rise to equivalence classes, called degrees, where two sets are

equivalent if they are mutually reducibile. The Turing degrees that contain c.e. sets are

called c.e. Turing degrees. We denote C and CbT as the structures of the c.e. degrees under

the Turing reductions and the bounded Turing reductions respectively. The study of the

Turing degrees has been one of the 1i i i"r themes in computability research; these degrees

capture the structure of the undecidable problems in arithmetic and nature.

1.3 Closed Sets in Computability

We generally follow the notation of Cenzer [19] for closed sets: For a finite string

a E aW, let 7o- = n. We let 0 denote the empty string, which has length 0. A word (a) of







length 1 is may be identified with the symbol a. For two strings ao, T, i- that r extends a

and write a c T if lal < |Ir and a(i) = r(i) for i < al. Similarly a C: x for x E 2" means

that a(i) = (i) for i < aol. Let a7- denote the concatenation of a and r. Given a finite

string a, let I(a), or alternatively [a], be the interval of all infinite sequences extending a,

i.e. I(ao) = {x E 2" : C x}. Each such interval is a clopen set and the clopen sets are just

finite unions of intervals.

A subset T of u<" is a tree if it is closed under initial segments. The set [T] of infinite

paths through T is defined by x E [T] => (Vn)x [n E T, where x [n = ((0),... ,x(n- 1)).

We -i- that a tree T C w< and set [T] are clopen if there is a nonempty finite S C c<'

so that T = 0 or T = {a : a C or 7- C o for some 7- E S}. P C w" is closed if and only

if P = [T] for some tree T. Now a nonempty closed set P may be identified with a unique

tree Tp {oa : P n I(a) / 0}. Tp has the distinct property of having no dead ends; that is,

if a E Tp then either a^0 E Tp or cral E Tp.

P is an effectively closed set, or IIo class, if P = [T] for some computable tree T Other

definitions are equivalent; in particular P is a fIo class if and only if P = [T] for some

primitive recursive tree T and also if and only if P [T] for some II tree T. Note that

if P is a IIo class, then Tp is a II set, but not in general computable. P is said to be a

decidable Io class if Tp is computable. P is said to be a strong HIo class, if Tp is a IIo set,

or equivalently if P [T] for some A tree; P is said to be a strong A class if Tp is A.

Thus any fII class is also a strong A class. Any decidable fIo class contains a computable

element (in particular the leftmost and rightmost paths) and similarly any strong A class

contains a A element. On the other hand, there exist fII classes with no computable

elements and strong Io classes with no A elements. A IIo class is said to be special if it

does not contain a computable member.

A c.e. open set is defined to be the complement of an effectively closed set. That is, if

P = [T], then ww P = U(<,)T I(a). There is a natural effective enumeration Po, P1,...

of the II classes and thus an enumeration of the c.e. open sets. Thus we can ;v- that a

sequence So, Si,... of c.e. open sets is effective if there is a computable function, f, such







that S, = 2'


Pf(,) for all n. For any c.e. set W, we define the c.e. open set generated by


W to be


O(W) U{I(o): () s w}.


Also let


O(W) [n = {x n : x e O(W) and (Vj < n) x(j) < n}.


For a detailed development of II classes, see [20] or [24].







CHAPTER 2
EFFECTIVELY CLOSED SETS AND ENUMERATIONS

The following chapter is joint work with Douglas Cenzer and has been submitted

as an article entitled Effectively Closed Sets and Enumerations [13]. A preliminary

version of this research was originally presented at the Third International Conference

of Computability and Complexity in Analysis in Gainesville, Florida in 2006 by P.

Brodhead. This preliminary work was published in the referred conference proceedings as

Enumerations of IH Classes: A il,.',ii.L i and Decidable Classes (P. Brodhead) in hard

copy and in Springer Electronic Notes in Theoretical Computer Science, Elsevier Science

167 (2007), 289-301 [12].

Portions of this work were also presented by P. Brodhead at the 2005 SACNAS

Conference (October 2005, Denver, CO), 7th Annual Graduate Student Conferences in

Logic (April 2006, Madison, WI), the 2007 Association for Symbolic Logic Annual Meeting

(\! I'ch 2007, Gainesville, FL), the 2nd New York Graduate Student Conference in Logic

(\! I'ch 2007, New York, NY), and the 8th Annual Graduate Student Conference in Logic

(April 2007, Chicago, IL). Due to inclimate weather, R. Miller presented in place of P.

Brodhead at the New York conference.

2.1 Introduction

The general theory of numberings was initiated in the mid-1950s by Kolmogorov, and

continued under the direction of Mal'tsev and Ershov [44]. A numbering, or enumeration,

of a collection C of objects is a surjective map F : w C. In one of the earliest results,

Friedberg constructed an injective computable numbering i of the Et or computably

enumerable (c.e.) sets such that the relation "n E c (e)" is itself E. More generally, we

will -iv that a numbering i of collection of objects with complexity C (such as n-c.e.,

E, or U11) is effective if the relation "x cE (e)" has complexity C. We will also consider

enumerations where the relation "x E Q(e)" has a different complexity than C. (For

example, there is a c.e., but not computable, numbering of the computable sets.)

A numbering p is acceptable with respect to a numbering v, denoted v < p, iff there

is a total computable function f such that v = o f. If p is acceptable with respect







to all effective numberings, then p is said to be acceptable. The ordering < gives rise to

an equivalence relation -, and two numberings in the same equivalence class are called

equivalent. Furthermore, the structure (C) of all numberings of C modulo forms an

upper semilattice under <. Here injective numberings occur only in the minimal elements

and acceptable numberings occur only in the greatest element. In this chapter, we study

effective numberings of families of effectively closed sets (i.e. II classes).

Enumerations of HI classes were given by Lempp [61] and Cenzer and Remmel

[23, 24], where index sets for various families of HI classes were analyzed. For a given

enumeration &(e) = P, of the II classes and a property R of sets, {e : R(P)} is said to

be an index set. For example, {e : P, has a computable member} is a E complete set. See

[20] for many more results on index sets.

Certain types of HI classes are of particular interest. Let P be a HI class. We will -

that P is thin if for every HI subclass Q of P, there is clopen set U such that Q U n P.

We -w that P is homogenous if, given distinct a, -r Tp of the same length,


a'i E Tp < rTi Tp.


For P C {0, 1}", P is homogeneous if and only if P is the class of separating sets S(A, B)

for two disjoint c.e. sets A, B, that is,


S(A,B)= {C C : A C C and B n C 0}.

P is small if there is no computable function Q such that, for all n, card(Tp n ;()) > n.

Let Qp(n) be the least k such that card(Tp n wk) > n; then P is very small if the function

Qp dominates every computable function g -that is, Qp(n) > g(n) for all but finitely

many n.

A numbering e v-> [T,] of HI classes is called a tree numbering and written e v-> Te.

Numberings based on primitive recursive trees and on Hn trees are both studied in the

literature (see [23, 24, 20]). If the set {(e, a) : a e T} is computable, then the numbering

(e) = [T,] is said to be a computable numbering.







We begin our study with the family of II classes in Section 2.2. In Sections 2.2.1

2.2.3, several commonly used numberings are studied and shown to be equivalent via a

computable permutation. In Section 2.2.4, we give give a Friedberg numbering of the II

classes; this motivates a study of a general class of families of II classes, called string veri-

fiable families in Section 2.3. In Section 2.4, we consider named families of II classes. We

obtain positive results for homogeneous can decidable classes in Sections 2.4.1 and 2.4.2.

We obtain negative results for thin, perfect thin, small, very small, and nondecidable

classes in Sections 2.4.3 and 2.4.4.

2.2 The Family of II Classes

2.2.1 Numberings in the Literature

In this section, we present several different computable numberings of II classes

that have appeared in the literature. We also present an effective, but not computable,

numbering. In each case we demonstrate that each provides a complete numbering of the

II classes.

Numbering 1: Primitive Recursive Functions [23]

For each e, let 7 be the eth primitive recursive function from u to U and let


7 G U, (VT C a) 7Te((T)) 1.


Then Ue is a (uniformly) primitive recursive tree for all e and if {a : 7((a)) = 1} is

any primitive recursive tree, then Ue is that tree. Therefore the sequence Uo, U1,...

contains all primitive recursive trees and hence the mapping li(e) = [Ue] is a computable

numbering of the II classes.

Numbering 2: Computably Enumerable Sets [20]

Let

2(e) O(We).

This is an effective numbering since the relation "x e Q2(e)" is II This can actually be

improved to a computable numbering, as follows.







For each e, recall that We,s is the set of elements enumerated into the eth c.e. set We

by stage s and let

a G Se (VT 7() { W,,\.

Then Se is a (uniformly) primitive recursive tree for all e. Let P = [T] be a II class,

where T is a computable tree. It follows that for some e,

a c T ^ (a) i We.

Then P = [Se]. It follows that the sequence [So], [Si],... contains all nIT classes and hence

the mapping Q(e) [Se] is a computable numbering of the II classes. It is easy to see

that in fact [Se] = 2(e).

Numbering 3: Universal IIt Relation [52, p.73]

There is a universal II relation U C w x 2" such that if Q(x) is a II0 class, then there

is an e e w such that Q {x : U(e, x)}. U is defined in terms of the Kleen T-predicate, so
that essentially

U(e,x) = V (0) .

Define (' (e) = {x : U(e, x)} to obtain an effective numbering.

To see that this is a computable numbering, let

a e Re 4) 1(0) T

so that ,' (e) = [Re] and the trees Re are uniformly primitive recursive.

Numbering 4: The Halting Problem [53]

Consider the mapping given by


64(6) -{X: 4)x(e)(e)T}.

This is a computable numbering, since 4(6) = [T,], where

a Te 4 ),(e) T.







For any computable tree T, choose a so that 1{(n) converges if and only if a c T. Then

a e T V (a)l

so that [T] = 4 ().

Numbering 5: Total Computable Functions

Here we will consider an effective, but not computable numbering i based on

computable trees. This numbering will be used in connection with string verifiable

families of classes in Section 2.3.

Let -(e) = [Te], where




This enumeration is uniformly HI, but is not computable, since the relation 0e(m) I is

c.e. non-computable. Clearly each -(e) is a II class. If e, is total and T is a tree such

that, for all a, we have a c T 0 ,e((a)) = 1, then Te = T and is a nI class. Hence

this enumeration has the crucial property that, for every computable tree T, there exists e

such that T =Te.

2.2.2 Equivalence of the Numberings

In this section we show that the computable numberings of section 2.2.1 are mutually

equivalent via a computable permutation. Each of these is equivalent to the effective

enumeration of section 2.2.1 via a A-permutation. We need the following proposition.

Proposition 2.2.1. (a) For each pair i,j with 1 < i < 5 and 1 < j < 4, there is a

computable function f such that bj = o f.

(b) For each j ( 5, there is a A function f such that -. = o f.

Proof. (Q1 < V2): Use the S m n Theorem to define f so that


Wf(e) = n : 7e(n) / e}.

Then a E U, a E Sf(,).







( ',_< .): Define f so that, for all m,


S(e) (m) = (least n) (x n) e We).

Then 2 (e) (f(e)).

( <'. 4): Define f so that KI(nu) +- (0) for all n. Then x e (e) <- x e

4 (f (e)).

( <4 Q i): Recall that 4)4(e) [{a : (e)T}].
Define the primitive recursive function g so that for each e,

'1, if I) (e)T;
TAf(e)(r7))
0, otherwise.

Then 1(e) = 4(9(e)).

(Q1 < .): Define the primitive recursive function f such that, for each e,


( ) if (VT C 7a)7((T)) 1,
0, otherwise.

Then 1l(e) (f(e)).
The rest of the proof follows by composition. O
Theorem 2.2.2. For any computable numbering o which is computably equivalent to )2,
there is a computable permutation p such that '_. o p.
Proof. The proof is a modification of an argument due to Jockusch [86, p. 25]. Let

S= '_.. By assumption, there are computable functions f and g such that )(f(e)) = (e)
and c(g(e)) = (e). Since the numbering ,'_. is based on an enumeration of the partial
computable functions, we can ensure by padding that f is injective. To modify g into an
injective function gl, it is sufficient to effectively compute from each e an infinite set Se of
indices such that p(g(i)) = c(e) for all i e Se. We proceed as follows. Let A and B be

computably inseparable c.e. sets and define computable functions k and such that, for all







e and m:

(k(em))- (e), if m B,
0, if m B
and

((e, T)) -= (e), if me A,
2", if m A

That is, we build a tree for &(k(e, m)) which exactly equals the tree for Q(e) for strings of

length s until m e B,+i, in which case no strings of length s + 1 are put into ((k(e, m)).

To build the tree for (f(e, m)), we put in all strings of length s until m e A,+1, in which

case we include only the strings of length s + 1 which are in )(e).

Now let Ce ={k(e, m) : m e A} and De { (e, m) : m A} and let Se g(Ce U De).

Then for j = g(i) e Se, it follows from the definition that b(i) = (e) and therefore
p(g(i)) = 9(g(e)). We will prove in two cases that either g(Ce) is infinite or g(De) is

infinite.

Case I: Suppose that Q(e) / 0 and suppose by way of contradiction that g(Ce)

is finite. Then S {= m : g(k(e, m)) e g(Ce)} is a computable set. Now A C g(Ce)

by definition. On the other hand, if j = g(k(e,m)) e g(Ce) where m e A, then

((j) = (k(e, m)) = ((e) / 0. But for m e B, p(g(k(e, m)) = (k(e, m)) = 0, so that
S n B = 0. This contradicts the assumption that A and B are computably inseparable.
Case II: Suppose that b(e) = 0. It follows as in Case I that g(D,) is infinite.

Thus we may assume without loss of generality that both f and g are one-to-

one. Now define a sequence {e, : n e w} and two partitions of u as follows. Let

eo = 0 and for each n, en+ is the least e such that 9p(e) / 9p(ej) for every i <, n. Let

An = {e : (e) = (en)} and B, = {e : 9p(e) = (en)}. Then w = U. AT = Un B, and each
sequence is pairwise disjoint. Furthermore, f(B,) C AT and g(A,) c B,. The remainder

of the proof follows as in the Myhill Isomorphism Theorem [86, p. 24; Also 5.8, p. 25]. O
A similar argument shows that if p is a A numbering of the II classes, then there is

a A permutation p with p = o p. It follows that each of the computable numberings








S,... Q4 are acceptable, that is, they occur in the greatest element of the semilattice

C(P). In the section 2.2.4 we will see that minimal elements exist in the semilattice that

is, injective numberings.

First, however, we provide an alternate proof of Theorem 2.2.2.

2.2.3 Equivalence of the Numberings (Alternate Proof)

At the Third International Conference of Computability and Complexity in Analysis

in Gainesville, Florida in 2006, I presented the argument for Theorem 2.2.2 as given

above; it is a modification of an argument of Jockusch for the c.e. version. Pleased with

the argument's applicability to IIo classes, I urged Robert Soare, who was present in the

audience, to keep the Jockusch argument for the c.e. version in his new and upcoming

book, Computability Theory and Applications [88]. If the alternate proof using the

recursion theorem for the c.e. version of Theorem 2.2.2 [86, p. 43] could not also be

modified to also prove Theorem 2.2.2, then he said he would. We show below that an

alternate proof is possible, modifying the recursion theorem argument.

Theorem 2.2.3. For any computable numbering o which is computably equivalent to b2,

there is a computable permutation p such that 2 = p o p.

Alternate Proof. We proceed, at first, as before. That is, there are computable func-

tions f and g such that b(f(e)) = (e) and c(g(e)) = (e). Our goal is to modify g to

obtain an injective function gl. Instead, however, we define gl differently, using an auxil-

iary computable function h obtained by the Recursion Theorem. We ensure h satisfies, for

all distinct i and j, (i) gh(e, i) / gh(e,j) and (ii) (h(e,i) = O.

We now define gl, given h satisfying (i) and (ii). Define gi(0) = gh(fg(0),0). To

define gi(k + 1), note that (i) ensures infinitely many distinct gh(e, i) for each e. Let

ao = 0 and ak+1 be the least integer i such that gh(fg(k + 1), i) > gi(k). Define

gi(k + 1) = h(fg(k + 1),ak+1). To see that for all e, '..,,e) = Q, fix e and note that

' ..,(e) = (fg(e),a) h(fg(e),a,) = fg(e) = ':. = e. To see that gi is injective, note that

the definition ensures that for all k, g (k + 1) > gi(k).







We now define h(e, *) for each e, by induction, ensuring that (i) and (ii) are satisfied.

Define h(e, 0) = e. To define h(e, k + 1), use the Recursion Theorem to obtain an n such

that

() = #e(z) if g(n) / gh(e, i) (Vi < k)
undefined otherwise
Notice that if g(n) = gh(e, i) then since i < k, by induction = h(e,i) = '..I.

Qg(T) = Q, the undefined function. Hence e = OT in all instances. Since {a : fa = ~}
is a productive [86, p. 43], let p be a corresponding productive function. Define Wr(x)

WE U {p(x)} and note that each W,(,r) is a distinct c.e. subset of A. Consequently each

gri(n) = '. ; is a distinct partial computable function. Let nk be the least i such that
r(n) / h(e,j) for all j < k. Define

h(e, k n if g(n) / gh(e, i) (Vi < k)
h(e, k+1)
rf (n) otherwise
To see that (i) holds (gh(e, i) / gh(e,j) for distinct i,j), note that '.,., = Qe /

S, gr(,) for all i > 1. Therefore if g(n) / g(e) then for all j, h(e,j) e {e} U

{r'(n)}i>o. Otherwise h(e,j) e {e} U {r'(n)} i1. In either case, for all distinct i,j,
gh(e, i) / gh(e,j).
To see that (ii) holds (/'..., = oe for all i), note that ,e = h(e, i) e {e} U

{r'(n)}i(o or 1), and r'(n) e {a :a O= ,}. So e = (h(e,i) for all i. E
2.2.4 Injective Computable Numberings

In this section, we modify Friedberg's original argument for injective computable

numbering of the c.e. sets, to provide different numbering results needed for 11 classes. In

Section 2.2.4.1, we present Friedberg's original argument. In Section 2.2.4.2, we provide

an injective computable numbering of di-iP ii 1 pairs of c.e. sets; this in needed later in

Section 2.4.1 in order to provide an injective computable numbering of the homogeneous

II classes. Finally, in Section 2.2.4.3, we construct a computable injective numbering of

the II classes in 2"; we also provide other injective numbering results for II classes.








2.2.4.1 Original c.e. sets argument

The following is the original Friedberg argument, with slight modifications in the

notation and presentation.

Theorem 2.2.4 ( [46]). There is an injective computable numbering of all c.e. sets.

Proof. Let {We}ee be the standard numbering of the c.e. sets. We will construct

a sequence of c.e. sets {Ye}ee in stages so that e v-4 Y, will be the desired injective

numbering. In the construction we will use the notion of one Y-index i following a W-

index e with the idea that in the end i will equal We. At some point, however, we may

decide that i will no longer follow e again and we will -i- that i is released. If i is never

released from following e then it is said to be a lo(,il follower and otherwise it is 1-/. ..;'/.

Once released, an index remains free and is never again the follower of any e. At any

particular stage, any Y-index that is not following any W-index is said to be free. Any

nonzero Y-index that has never followed any W-index is said to be unused.

To ensure that no c.e. set is excluded from the Y-sets, we will ensure that each We

is infinitely often given the opportunity to be followed. To do this, at stage s = (ns, eS)

all actions in the construction will be taken with respect to We,. Assume without loss of

generality that Yo = 0.

Construction: There are three possible cases at each stage s.

Case 1: If i follows e & We,s n [0, i 1] = We,,,s [0, i 1] (some e < e,), release i and

go on to stage s + 1.

Case 2: Suppose Case 1 does not occur. If Wes, = i,s-1, and either i follows some

e < e, i = 0, i is free and i < e, or i is free and i was previously displaced (see Case 3) by

e and released, then go on to stage s + 1 without taking any action.

Case 3: Suppose that Cases 1 and 2 do not occur. Now ensure that e has a follower. If it

does not, choose the lowest unused i / 0 to follow e. Now let ,, = We,,s.

For each j / i such that Yj,s_ = We,,s (in increasing order of j) put the lowest b not

yet in any W or Y into Yj and release j if it is a follower. We i e displaces j .

Verification: Given e E u, let e be the least k so that Wk = We. We will show:








(i) (we) (i) = W
(ii) i / j- Yi and Yj are not the same finite sets

(iii) i 7 j and Yj are not the same infinite sets

Verification of (i). Fix e. First note that although e can only have one follower at any

particular stage, it cannot have an infinite number of them which are each released at

some stage. For example, if s and x are sufficiently large, then for all j < e, Wj,8 n [0, x] /

W8, n [0, x]. Hence release can only occur in Case 1 a finite number of times. Furthermore,
Case 2 ensures that release in Case 3 can only occur for any s when e, < e. Therefore, by

the above, if i > x is follower of e and t > s, then Yt-1 = Wt- / We,,t. Hence i will not

be released in Case 3. Therefore release can only occur in Case 3 a finite number of times.

Now let s be a stage where e never loses a follower. If Case 3 occurs infinitely often

after stage s for e, then it has a permanent follower i so that Y = W. Therefore assume

Case 3 occurs only finitely often. Since e never loses a follower, Case 1 cannot occur.

Hence Case 2 must occur infinitely often. However there are only a finite number of

i such that the 'If' clause holds with "W,s = Ys-_1" in Case 2. (For example, {i :

i = 0, i < e, or i is displaced by e} is a finite set since only a finite number of i are

displaced due to Case 3 occurring only a finite number of times. To see also that {i :

for some s, i follows some a < e and "W,s = Y,s-1" holds} is finite, note that by the

definition of e, if a < e then Wa W4. So for sufficiently large t, if i follows a then

W,
of i such that the 'If' clause holds with "W6,s = i,s-", implies that there is a single i such

that I,-1 = W<,s for infinitely many s. Thus Y = W.

Verification of (ii). First note that at any stage s, if W, = 0 then Case 2 ensures that

Case 3 will not be reached so that ec never receives a follower. Furthermore each k / 0

is chosen at some stage s in Case 3 to follow some e and from the previous comments,

Yk,s / 0. Immediately thereafter, Yk,s is ensured to be distinct from all other Y,s (f / k).
This also continues to be true at all subsequent stages t by Case 3.







Now suppose i / j. Since we are supposing both i and Y are finite, there is some

t such that for all s > t, Y,8 = Y and Y,8 = j. As mentioned above Y,s / Y,s and

therefore i / Yj as required.

Verification of (iii). Assume both Y and Y are infinite and i / j. Now i must

eventually follow some W-index. If i is di-. il- 1, then after it is released i can acquire a

new member only when i is displaced. However i can be displaced only once by each e < i

and never by any e > i. It follows that if i is released then 1 only acquires a finite number

of elements thereafter, contradicting the fact that it is infinite. This argument shows that

both i and j are never released. We -iv that i and j are loyal followers.

Suppose now that i and j loyally follow i' and j', respectively. Then Y = Wi, and

Yj W. Now i' / j' since a single W-index cannot have more than one loyal follower.

Assume without loss of generality that i' < j. If W, = Wj,, then for all sufficiently large

s, Wi,,s n [0,j 1] = Wj,,, n [0,j 1] so that Case 1 releases j, a contradiction. Therefore

Yi = Wi, Wy = Yj as required. O

2.2.4.2 Ordered tuples of disjoint c.e. sets

In this section we modify the argument of Theorem 2.2.4 to obtain an injective

computable numbering of all ordered tuples of disjoint c.e. sets.

Theorem 2.2.5. There is an injective computable numbering of all ordered tuples of

disjoint c.e. sets.

Proof. Let {Se}ee, be the standard numbering of the c.e. sets. Then (e, i) v- (Se, Si) is a

computable numbering of all tuples of c.e. sets. Obtain a computable numbering (e, i) i-

(We, Wi) of all tuples of dli-i iiil c.e. sets as follows. At stage s + 1, if a E Se,8+1 \ Se,s then

put a into We,s+i if a g Wi,,. If b e Si,s+l \ Si,s then put b into Wi,s+l if b g We,s+1. Note

that if originally Se n Si = 0 then S, = We and Si = Wi.

Similar to Theorem 2.2.4 (in construction and terminology), we will construct in

stages a sequence of pairs of c.e. sets (Y, Y)(e,i)ew so that (e, i) (- (Y, ) will be the

desired injective numbering. So that no pair is excluded from the Y-sets, we will ensure

that each (e, i) is infinitely often given the opportunity to be followed. To do this, at stage







s = (ns, (e, is)) all actions in the construction will be taken with respect to (We, Wi).

Assume without loss of generality that (Yo, Y) = (0, 0).
Construction: There are three possible cases at each stage s.

Case 1: If (e, i) follows (e,, i) and Wa,s n [0, (e, i) 1] = We,, n [0, (e, i) 1] and

Wb,s n [0, (e, i) 1] = Wi,, n [0, (e, i) 1] for some (a, b) < (e, i,), then release (e, i) and
go on to stage s + 1.

Case 2: Suppose Case 1 does not occur. If (Y(,-1, Y,s1) = (We,,s, Wi,,s) and either (c, t)

follows some (a, b) < (e,, i), (e, t) = 0, (e, t) is free and (e, t) < (e,, is), or (e, t) is free and

(e, t) was previously displaced (see Case 3) by (es, is), then go on to stage s + 1 without
taking any action.

Case 3: Suppose Cases 1 and 2 do not occur. Now ensure that (es, is) has a follower.

If it does not, choose the least unused (e, i) / 0 to follow (es, is). Now let (Y,,, ,,s) =

(W5s, WiU s ).
For each (c, t) / (e, i) such that (Y,s_1,Yi,s-1) = (W",,s, Wis,,) (in increasing order of

(e, t)) put the lowest b not yet in any W or Y into Y, and release (c, t) if it is a follower.
We v (es, i4) displaces (, t) .

Verification: Given (e, i) e wo, let (e,i) be the least (k, ) so that (Wk, W1) =

(We, Wi). We will show:

(i) V(e,i) 3(k,<-) (Yk,Y) -= W, Wj)

(ii) [(e, i) / (c, t) & Y = Y] Y, and Y, are not the same finite sets
(iii) [(e, i) (c, t) & Y Y,] Y, and Y, are not the same infinite sets

(iv) V(e,i) Y, n 0
Verification of (i). First note that although (e, i) can only have one follower at any

particular stage, it cannot have an infinite number of them which are each released at

some stage. For example, if s and (x, y) are sufficiently large then for all (k, ) < (e, i),
either Wk,, n [0, (x, y)] / W,,, n [0, (x, y)] or W1,, n [0, (x, y)] / Wi,, n [0, (x, y)]. Hence

release can only occur in Case 1 a finite number of times. Furthermore, Case 2 ensures

that release in Case 3 can only occur for any s when (e, is) < (e,i). Therefore, by the







above, if (a, b) > (x, y) is follower of (e,i) and t > s, then Ya,t-1 = We,t-1 / W,t or

Yb,t- = Wi,t-1 / Wi,t. Hence (a, b) will not be released in Case 3. Therefore release can
only occur in Case 3 a finite number of times.
Let s be a stage where (e, i) never loses a follower. If Case 3 occurs infinitely often

after stage s for (e,i), then it has a permanent follower (e, i) so that (Ye, Y) = (W, Wi).
Therefore assume Case 3 occurs only finitely often. Since (e, i) never loses a follower, Case
1 cannot occur. Hence Case 2 must occur infinitely often.
There are only a finite number of (c, t) such that the 'If' clause holds with the

equality A(c, t, e,i, s) given by "(Y ,s-,,s-1) = (We,s, Wi,)" in Case 2. For example,

{(c, t) : (c, t) = 0, (c, t) < (e,i), or (c, t) is displaced by (e,i)} is a finite set since only a
finite number of (c, t) are displaced due to Case 3 occurring only a finite number of times.

To see also that { (, t) : for some s, (c, t) follows some (a, b) < (e,i) and A(c, t, e, i, s) holds}
is finite, note that by the definition of (e,i), if (a, b) < (e,i), then either WE / We or

Wb / Wi. So for sufficiently large t, if (c, t) follows (a, b) then (Y ,t-, ,,t- ) (W,t, Wi,t)
and so A(c, t, e, i, t) does not hold.
Now Case 2 occurring infinitely often, together with only finite number of (c, L)
such that the 'If' clause holds with A, implies that there is a single (c, t) such that

(Y,,_ 1,Y, ,_1) (We,s, Wi,8) for infinitely many s. Thus (, Y,) (We, W).

Verification of (ii). First note that at any stage s, if (We, Ws) = (0, 0) then Case 2 en-

sures that Case 3 will not be reached so that (es, is) never receives a follower. Furthermore
each (a, b) / (0, 0) is chosen at some stage s in Case 3 to follow some (e, i) and from the
previous comments, (Y,s,, Yb,) / (0, 0). Immediately thereafter, (Ys, Yb,s) is ensured to
be distinct from all other (Yk,s,Y ,s) ((a, b) / (k,f)). This also continues to be true at all
subsequent stages t by Case 3.
Now suppose (e, i) / (e, t), = Y,, and both Y, and Y, are finite. There are two
cases. Suppose first that both Yi and Y, are finite. Now there is some t such that for all

s > t, Yk,s Yk (k e {e, i, e, t}). As shown above (Y,,, Y,) / (Y,s,,Y,) and therefore

(Y, Y) / (Y, Y,). Since Y, Y, it follows that Y / Y,.







Suppose now that Yi and Y, are both infinite. Now (e, i) must eventually follow

some (e', i'). If (e, i) is ever released then it is free. Thereafter only the first coordinate

of (Y,, Y) acquires members so that Yi is finite, a contradiction. A similar argument

holds for (c, t). Therefore both (e, i) and (e, L) are never released and are Ic;,.l followers.

Suppose that they follow (e', i') and (c', t'), respectively. Then (Y, Y) = (We,, Wi) and

(Y,Y) = (We,, W,). Note that (', i') / (c', L') since a single W-index cannot have
more than one loyal follower. Assume without loss of generality that (e', i') < (e', L').

Suppose now that We = We,. By assumption, Wi, Y= =, W,, Therefore, for all

sufficiently large s, We,s n [0, (c, t) 1] = We,,s n [0, (c, t) 1] and Wi,, n [0, (c, t) 1]

W,,, n [0, (c, t) 1], so that Case 1 releases (c, t) a contradiction. Therefore We' / We so
that also Y, We/ We = Ye, as required.

Verification of (iii). Suppose that (e, i) / (e, t), Y= Y- and both Ye and Y, are infinite.

Now (e, i) must eventually follow some (e', i'). If (e, i) is di-1 ,l 1l, then after it is released

Ye can acquire a new member only when (e, i) is displaced. However (e, i) can only be

displaced once by each (c, d) < (e, i) and never by any (c, d) > (e, i). It follows that if (e, i)

is released then Ye only acquires a finite number of elements thereafter, contradicting the

fact that it is infinite. This argument shows that both (e, i) and (e, t) are never released

and are therefore loyal followers. Now apply the same argument given in the later part of

the verification of (ii) to get that Ye / Y,.

Verification of (iv). If (e, i) is a loyal follower of some (e', '), then (Ye, Y) = (We, Wi).

Therefore since (e, i) (We, Wi) is an enumeration of disjoint sets, it follows that

Ye and Yi are disjoint. Otherwise suppose that (e, i) is released at some stage s. Then

Ye,s n Y, 0 and thereafter only Ye can acquire new elements not already included in Y,s.
Therefore disjointness is preserved and Ye n = 0. O

2.2.4.3 Results for effectively closed sets

In this section, we modify the Friedberg argument of Section 2.2.4.1 to construct a

computable injective numbering of the II classes in 2". An alternative proof was sketched

by Raichev [79]. Afterwards we provide other injective numbering results for II classes.







Theorem 2.2.6. There is an injective computable numbering of all II classes in 2".

Proof. Let {W }ee be the computable enumeration of the nonempty c.e. subsets of

2". We will construct a computable numbering {Y : e E ow} in stages Ye,, of a family

of c.e. subsets of 2" so that {O(Y,)},E is an injective numbering of the E classes; our

construction and terminology will be similar to Theorem 2.2.4.

It is important to note that an injective numbering of the c.e. subsets of 2<' will not

automatically yield an injective numbering of the E classes, since each E class will equal

O(W) for many different c.e. sets W. However, if O(V) / O(W) for two c.e. sets V and

W, then there must be some interval I(a) which is included in, -, O(V) but not included

in O(W) and hence some stage s such that O(Vs) [ s / O(Ws) [ s at stage s and at any

later stage.

To ensure that no c.e. set is excluded from the Y-sets, we will ensure that each We is

infinitely often given the opportunity to be followed. To do this, at stage s = (n, es) all

actions in the construction will be taken with respect to W,. At each stage s, we initiate

at most one new Yi, so that after stage s, we have sets Yo,Y1,...,Yk, for some k, < s. Fix

Yo = 0 and Y1 {0} so that O(Yo) = 0 and O(YI) = 2", respectively.

Construction: There are three possible cases at each stage s.

Case 1: If i follows e, and there exists e < e such that O(We,s) [(i-1) = 0(W,,s) [(i-1),

then release i and go on to stage s + 1.

Case 2: Suppose that Case 1 does not occur. If O(We,,s) = (Es-i), and either i follows

some e < e, i = 0, i = 1, or i is free and either i < es or i was previously displaced (see

Case 3) by es, then go on to stage s + 1 without taking any action.

Case 3: Suppose that Cases 1 and 2 do not occur. Now ensure e has a follower. If it

does not, choose the least unused i / 0, 1 to follow e. Now let i, = We,,,.

If Yj,s_, for some j / i satisfies O(Yj,s_) = O(W,,,s), then put some cj e 2
in what follows, into Yj so that O(Yj,) / O(We,,s).

Let E = {j E c : j / i & O(Yj,_) = O(We,,s)} be the set of indices of equivalent

Y-open sets and suppose that E, = {ci < 62 < ... < IlEs}. Now define Str(k, s) = {a







2k :a O(W,,s) [ k}. Let f(s) be the least k such that IStr(k, s)| > IEs,. Then f(s) is

the least level of O(We,,s) where there is enough room to give each equivalent Yj,s an

additional string to distinguish O(Yj,_i) from O(We,s). (Notice that O(We,s) / 2" by

Case 2.) Suppose that Str(f(s), s) = {oi -< 2 ... < i Str(s(s),s) }. Now put ac into Yj

and release cj if it is a follower. We w that ce is displaced at stage s.

Verification: Given e c L, let e be the least k such that [O(Wk) = O(We)]. We will

show:

(i) (Ve)(3i) y -We;
(ii) i / j implies that O(Yi) and O(Yj) are not equal when both are clopen.
(iii) i / j implies that O(Yi) and O(Yj) are not equal when both are not clopen.

Verification of (i). Fix e. First note that although e can have different followers at

different stages, it cannot have an infinite number of di-1 i Ll followers. That is, if s and x

are sufficiently large, then by the definition of e, for all j < e, O(Wj,s) [ x / O(W,s) [ x.

Hence release can only occur in Case 1 a finite number of times. Furthermore, Case 2

ensures that release in Case 3 can only occur for any s when e < e. Therefore, by the

above, if i > x is follower of e and t > s, then O(Y,t-1) = O(W,t-) / O(W,,t). Hence i

will not be released in Case 3. Therefore release can only occur in Case 3 a finite number

of times.

Now let s be a stage after which e never loses a follower. If Case 3 occurs infinitely

often after stage s for e, then it has a permanent follower i so that O(Yi) = 0(W).

Therefore assume Case 3 occurs only finitely often. Since e never loses a follower, Case

1 cannot occur. Thus Case 2 must occur infinitely often. However there are only a finite

number of i such that the hypothesis of (ii) holds with O(W( ,s) = 0(Y,s-_). To see

this, consider the three cases. First, each e < e has only finitely many followers by the

argument above; second, there are only finitely many i < e; and third, only a finite

number of i are displaced by es, due to Case 3 occurring only a finite number of times.

This contradiction shows that e has a permanent follower, as desired.







Verification of (ii). Suppose that U = 0(Y) = O(Yj) (/ 0, 2") is clopen and let

U = O(W6). It follows from compactness, that there is some finite s such that, for all

t > s, O(Yt) [ t = O(Y,t) [t = O(Y). It follows from the verification of (i) above that

there is a stage t > s such that Case 3 applies to e. But then at least one of 0(Y), O(Yj)

must change at stage t. This contradiction verifies (ii).

Verification of (iii). Assume both O(Yi) and O(Yj) are not clopen and i $ j. It follows

that O(Yi) must change infinitely often, since of course O(Y,s) is clopen for each s, and

similarly for O(Yj). Now i must eventually follow some W-index. If i is ever released, then

it is free. Thereafter Y acquires members in Case 3 at stage s only when We,,s = Yi,s-.

This implies that Case 2 does not apply at stage s and thus e < i. But each e < i can

only displace i once, again by the hypothesis of Case 2. Thus if i is a di-1 .v I1 follower,

then in fact O(Yi) is clopen. Thus we may assume that i is a loyal follower of e and j is a

loyal follower of e'. Then O(We) = O(We') but e / e', since each e can have at most one

loyal follower. Without loss of generality suppose e < e'.

Since O(We) = O(W,), there will be a stage s large enough so that O(We) [ (i 1)

O(We,) [ (i 1). Then since i follows e < e', i will be released at stage s, contradicting the

assumption that i is a loyal follower.

This verification completes the proof. O

The problem of finding an injective enumeration of the II classes in w" remains. For

classes in 2", we have the following generalization of Theorem 2.2.6. It will be useful later.

Let C be the family of clopen subsets of 2".

Theorem 2.2.7. For any family F of II classes in 2" which has a computable numbering,

there is a 1-1 computable numbering of C U F.

Proof. Let the computable enumeration P, be given. We may assume that C c F by

simply enumerating the clopen sets as {Q2e : e < w} and letting Q2e+1 = P,. Then the

proof of Theorem 2.2.6 produces a 1-1 computable enumeration of F as desired. O

We have the following corollary from the proof of Theorem 2.2.6.







Corollary 2.2.8. There is an effective numbering of the II classes based on the total

computable functions

Proof. Modify each c.e. set in the standard numbering to enumerate an element only

as long as it is larger than any previously enumerated element. Applying Friedberg's

argument to this class of c.e. sets yields an effective injective numbering e v-4 Ce of the

computable sets [91]. Furthermore each Ce still enumerates its elements in increasing

order.

Now suppose {Xe}eew is a corresponding set of characteristic functions. One charac-

terization of a nII class P is that P = w" \ O(W) for some computable set W [20]. As a

result, e \ OCe) \ 0(({n : Xe(n) = 1}) is an alternative effective numbering

based on total computable functions (replacing noneffective Numbering 2). O

It is known, for fixed n > 0, that there is a effective injective numbering of the n-c.e.

sets [50].

Conjecture 2.2.9. For each n, there is a numbering e v-> Ne of n-c.e. sets such that there

is an injective computable numbering e v- Lw" \ O(Ne) of all closed sets of this form.

For n = 1 the conjecture is given by Theorem 2.2.6.

We next show that Theorem 2.2.6 is not obtainable by any computable procedure

that uniformly selects the minimal index of every II class.

Theorem 2.2.10. There is no computable choice function for indices of FII classes. (i.e. a

computable function f such that f(e) is an index of PF and Pi = PF = f(i) = f(e))

Proof. Suppose that f exists. Let ao, al,... be an enumeration of a noncomputable c.e.

set A. Define a computable function g and trees T,(e) so that if al = n, then


E Tg(e) eg {ao,... ,a,an}.

Then

0 if eeA

S 2" otherwise
For any a c A,e A <+- f(g(e)) = f(g(a)), making A computable. O








It is still possible, however, that some interesting proper family of II classes may be

enumerated by selecting minimal indices from the enumeration of all II classes.

2.3 String Verifiable Families of II Classes

In this section we examine families of classes which are deemed to be string ;' I-

able (e.g. decidable or homogeneous classes) or strongly string ;. ,:/7,,'.1: (e.g. strongly

decidable or strongly homogeneous classes). Any string verifiable family has an effective

numbering and any strongly string verifiable family has a computable numbering.

2.3.1 Definition and Examples

First we will define the notions of string verifiable and strongly string verifiable. We

also give some examples.

Notation 2.3.1. Let Fo, F1,... be a computable enumeration of the finite subsets of 2<,

that is, for any n, a c F, b,((ao)) = 1, where bn is the binary expression for the

natural number n. Let E denote the family of finite sequences of positive integers of even

length. Let Po = [To], P = [Ti],... be some computable enumeration of the HI classes in

2".

Definition 2.3.2 (String Verifiability). (i) A string function is a computable function

f : 2
(ii) A family '-H of trees (or, more generally, of subsets of 2<") is string verifiable if there

is a string function h : 2< -- E so that for all T, TE G-H if and only if the following

condition is satisfied for all a c T, where h(ao) = (mi, m2,..., M2n) and Di = F,m for

i = 1,..., n: There exists i < n such that D2i+l C T and T n D2i+2 = 0 that is,

[T] c S(D2i+1, D2i+2) (the family of separating sets of D2i+1 and D2i+2).

Remark 2.3.3. Note that the family of trees itself is string verifiable among the family of

all subsets of 2<", via the function h(a) =(a, b), where Fa = {r : T E a} and Fb 0.

Example 2.3.1. (a) The Homogeneous Trees. A tree T is said to be homogeneous if


(Va7, r T)[|7| = I- => (Vi)(,-i i T -^i E T)1.







Define the string verification function h as follows. Let A1, A2, A3, A4 enumerate

P({0, 1}<") and let B1,..., B21 1 enumerate the strings of length a. Let h(a)

enumerate in order the set of m2(j,k)+l and m2(j,k)+2 for 1 < j < 4 and 1 < k < 21'1,

where

F2j, = {7 i i Aj, e BkJ

and

F,n.+2 = {r : } U { i B : i Aj, re Bk-.

That is, h(a) verifies that T is homogeneous by selecting the unique set Bk {= T-

la & 7- E T} and the unique set Aj such that for 7- E Bk, -Ti T, e= i E Aj.

(b) The Extendible Trees. Recall that a closed set P is decidable if the P = [T] for some

computable tree T without dead ends. For the purpose of string verification, let us

w that a tree T is extendible if T has no dead ends. This means that that for any

a T, either a0 E T or a 1 e T. In general, Ext(T) = {a I(a) n [T] / 0}

is the set of extendible nodes of T and T is extendible if and only if T = Ext(T).

Thus we let h(a) = (mi, m2, m3, m4), such that Frn {1, 0}, F = Fm4 0 and

Fm3 = {^1}. That is, h verifies that T has no dead ends by either showing that

a-O T if F,, C T or that a-l E T if F3 C T.

Definition 2.3.4. (a) A 11 class P satisfies a finite set of relations 'i C P(2<") (i < n)

if there is a computable tree T such that P = [T] and 7H(T) for each i < n.

(b) A II class P strongly satisfies a finite set of relations Hit C P(2<") (i < n) if there is
a primitive recursive tree T such that P = [T] and 'H(T) for each i < n).

Definition 2.3.5. A family F of classes is [strongly] string verifiable (s.v.) if there is

some finite set of string verifiable relations so that: P E F if P [strongly] satisfies these

relations.

Remark 2.3.6. Note that any string verifiable family of trees contains the empty tree,

so that any string verifiable family of II classes contains the empty class. If P 0, then

P = [T] if and only if T is finite, so any tree T with P = [T] is primitive recursive. So any

strongly string verifiable family also contains the empty class.







2.3.2 Computable and Effective Numberings

We now demonstrate that strongly string verifiable and string verifiable families

possess computable and effective numberings, respectively.

Theorem 2.3.7. (a) Any strongly string-verifiable family of II classes has a com-

putable numbering.

(b) Any string-verifiable family of HI classes has an effective numbering.

Proof. Suppose F is a [strongly] string-verifiable family of II classes satisfying string

verifiable (tree) relations 'Ho, ...,'H 9, with corresponding string functions ho, hi,..., h,.

For part (a), let the standard computable enumeration of the HI classes in 2" be given

by P, = [T,], where the sequence T, is uniformly primitive recursive (for example, the

numbering ,'_. given in section 2.2.1). We will define a uniformly computable sequence

S, of trees such that the sequence Q, = [S,] enumerates exactly the family of II classes

strongly satisfying 7-o, '1, ..., K-,.

For any a e {0, 1}', we determine whether a S, as follows. First check that

a E T,. If so, for each rT E {0, 1} and each i < m, compute hi(-) = (D, D2,..., D2j)

and determine whether there exists i < j such that D2j+1 C Te and D2j+2 n T, = 0. This

process is computable since each De is a canonical finite set. If the answer is yes, for every

'- E {0, 1}', then a E Se and otherwise, a r Se. It is clear that if Te satisfies all of the

relations 7Ho, '1, ..., Hm,, then T, = S,. It follows that every II class in F occurs in the

enumeration Q = [Se]. On the other hand, if T, fails any of the relations, then S, is a

finite set and Q = 0. By assumption, Q, E F in this case as well, so that the sequence

{Q, : e < u} enumerates exactly the family F, as desired.
For part (b), let PE = (e) = T, the uniformly HI enumeration which has the

property that every computable tree occurs in the list {T, : e < u}. We need to do the

string verification in a HI fashion and in particular to check that D2j+2 n T, = 0, which

appears to be E. However, we can simply check that, if p E D2j+2 and e(p) 1, then

e(p) 0. Then the sequence S, is uniformly HI and, if T, has characteristic function ,
and satisfies the string relations, it follows that S, Te. E








We now provide some additional examples of string and strongly string verifiable

classes.

Definition 2.3.8. (i) A fIo class P is strongly decidable if there is a primitive recursive

tree T with no dead ends such that P = [T].

(ii) A TII class P is strongly homogeneous if there is a homogeneous primitive recursive

tree T with no dead ends such that P = [T].

From definition 2.3.8, we immediately have the following corollary to Theorem 2.3.7

Corollary 2.3.9. (a) The family of decidable Hi classes in 2" has an effective num-

bering and the family of strongly decidable II classes in 2" has a computable

numbering.

(b) The family of homogeneous FII classes in 2" has an effective numbering and the

family of strongly homogeneous II classes in 2" has a computable numbering.

In the following section we will provide results which demonstrate that decidable

classes in fact have in fact a 1-1 computable numbering (see Corollary 2.4.2.1). We provide

an alternate explicit proof in section 2.4.2.1. For homogeneous classes, a different approach

is needed to demonstrate that they possess a 1-1 computable numbering and we do this in

section 2.4.1.

2.3.3 Families Containing the Clopen Classes

In this section we consider string-verifiable families of classes that contain all clopen

classes. We first improve Theorem 2.3.7 to obtain a computable numbering of any string-

verifiable family which includes the clopen sets.

Theorem 2.3.10. If F is any string-verifiable family of FII classes, then there is a

computable numbering of C U F.

Proof. We modify the proof of Theorem 2.3.7 so that when the string-verifiable relations

fail, we extend all nodes rather than making them dead ends. Once again, the construc-

tion is based on the enumeration e, of the partial computable functions. The construction

is in stages, where at stage s we will have


ne,s = max{n : (Va e {0, 1}")Qe,8((a)) 1},









J,s = {a E {0, 1}ne" : e,s((a)) = I},

and

Qe,s =- J es [Se,1,.

Then Qe = n, Qe,,s [Se] will be the desired numbering. To ensure that this numbering is

computable, we will determine whether a E Se at stage |a .

For this argument, we assume that Qe(0) = 1 for all e.

Construction. At stage 0 we have ne,o = 0, Je,o = Se,o = {0} and Qe,o = 2w.

At stage s + 1, we check to see whether Qe,s,+((a)) 1 for all a E {0, 1}"'+1. If not,

then ne,s+ = ne,s, Je,s+1 Js and Se,s+1 = Se, U {ai : a S,s, i = 0, 1}. If so, then we

check to see that Qes+1 is the characteristic function of a tree on {0, 1}r"0,+1 and we verify

the string relations up to {0, l}"je+1. If this verification fails, then again ne,s,+ = ne,s and

Je,s+ Je,s. In this case, verification will also fail at all future stages, so that Qe = Qe,s is
a clopen set.

If the tree and string-verifications succeed, then ne,s + = ne,s + 1, so that Je,s+1 c

{0, 1}t"I,+1 and Qe,s+1 change as indicated above. In this case,

Se,s+ = Se,s U {a {0, 1}s+1 : a [(ne,s+l) E Je,s+l}.

If Qe is the characteristic function of the computable tree Te, and if P, = [Te] E F,

then it follows from the construction that Q = Pe, so that Qe E F and furthermore, any

II class P, E F will thereby occur in the numbering. Otherwise, the construction will

make Qe a clopen set. O

Corollary 2.3.11. For any string verifiable family F of II classes, there a 1-1 computable

numbering of C U F.

Proof. Let F be a string verifiable family. Then there is a computable numbering

of C U F by Theorem 2.3.10. It then follows from Theorem 2.2.7 that there is a 1-1

computable numbering of C U F. O








Corollary 2.3.12. There a 1-1 computable numbering of any string verifiable family of

II classes containing all clopen classes.

Corollary 2.3.13. There a 1-1 computable numbering of the decidable II classes.

2.4 Named Families of II Classes

It this section, we otain numbering results for various named families that commonly

occur in the literature. In the first two sub-sections (2.4.1, 2.4.2), we expand upon the

results from section 2.3 and we obtain positive numbering results for the homogeneous

and decidable classes. In the two sub-sections (2.4.3, 2.4.4) that follow these, we obtain

negative results for the thin, perfect thin, small, very small, and nondecidable classes.

2.4.1 Homogeneous Classes

Homogeneous II classes are a string-verifiable family of II classes. Consequently,

by Corollary 2.3.9, they possess an effective numbering. Corollary 2.3.12 falls short

of demonstrating that they possess a computable numbering, as clopen sets are not

necessarily homogeneous. We provide the necessary argument in Theorem 2.4.3 and

show, in fact, that a computable injective numbering exists. We first provide an alternate

characterization of the homogeneous classes; they may be viewed separating classes of c.e.

sets.

Definition 2.4.1. The separating class S(A, B) of two sets A, B C w is given by

S(A,B) {C C u : A C and B n C 0.

In what follows, S,(A, B) will denote the set of characteristic functions fc of C e

S(A, B).

Theorem 2.4.2. P C 2" is a nonempty homogeneous II class iff P = S,(A, B) for some

disjoint c.e. sets A and B.

Proof. (-) Suppose that P is a nonempty homogeneous class with homogeneous tree

Tp. We will show that P = S,(A,B) where A {= n : 0'0 g Tp} and B = {n : 0'1 t

Tp}.

To see that A and B are c.e., suppose that P = [T] with T computable. Then for

each i {0,1},, O' if Tp iff (3s > (n+ 1)) T n {a: lal = s and 0' i C a} =0.







To see that A and B are disjoint, suppose that n C A. Since P is not empty, there is

some x c P so that x[ n c Tp. Now 0`0 T Tp so that (x [ n) -0 Tp. Hence x(n) / 0

so that x(n) = 1. Therefore (x [ n) l E Tp. Since P is homogeneous, 0"o 1 Tp so that

n B. Hence AnB =0.

To see that S,(A, B) c P, take fc e Sc(A, B) for some C c S(A,B). Then,
nEC a ngB 0"^ leTp f (n+ 1) (fc [n)-l eTp&

ngC a ngA 0"^ 0OeTp fcr(n+1) (fcrn)- 0Tp
Since for arbitrary n, (fc [ (n + 1)) c Tp, it follows that fc E P.

To see that P C S,(A, B), suppose that x E P and let C = {n : x(n) = 1}. It is clear

that x is the characteristic function fc of C. It suffices to show that C e S(A, B) (so that

fc e Sc(A,B)). Suppose first that n e A. Then 0"0 ^ Tp so that (fc [ n)^0 i Tp.
Since x = fc C Tp, it follows that fc(n) = 1. Hence n E C and A c C. Now suppose

n E B. Then 0" 1 i Tp, so (fc [ n)l 1 Tp. As before, fc(n) = 0 so that n i C. Hence

CnB 0.

(-) Suppose A and B are disjoint c.e. sets and {A,}sE and {B,}s,, are stage

enumerations of A and B, respectively. Define P = n,[T,] where T, C 2<' is given by

a T, iff (i e A, (i) = 1) A (i B, (i) = 0). Then P is a HI class and

P = S,(A,B). Note that Tp = {a : (Vi < Ia)[(a(i) 0 A i A) V (a(i) =1 A i B)]}

is homogeneous. Furthermore, since A and B are disjoint, Sc(A, B) is nonempty. Hence

S,(A, B) is a nonempty homogeneous class. D
Theorem 2.4.3. There is an injective computable numbering of the homogeneous HI

classes.

Proof. We may obtain an injective numbering of all ordered tuples (A,, B,) of c.e. sets

by Theorem 2.2.4.2. Then by Theorem 2.4.2, P, = S(Ae, Be) is an injective numbering of

the homogeneous classes. Furthermore, S(A{, B,) = [T,] where


Ta The (Vn < a)[(n e Ae,e n a(n) = 1) & (n B,s (nc) = 0)]

This shows that the numbering is computable. O








2.4.2 Decidable Classes

In this section we provide an alternate proof of the existence of injective computable

numberings of the decidable classes (see Corolllary 2.3.13 for the first proof). We also

show that in any computable numbering Q of the decidable classes via trees, we can pro-

vide a computable numbering Q of all the trees without dead ends that occur, along with

all clopen classes. Finally, we show that some decidable class in class in the numbering Q

must necessarily have, as a tree, dead ends throughout every occurance in the numbering.

2.4.2.1 An injective computable numbering (Alternate proof)

Since decidability is string-verifiable and every clopen set is decidable, it follows from

Corollary 2.3.12 that the decidable classes have a 1-1 computable numbering.

This result could not be obtained by using the standard numbering of the FII classes

and modifying each tree as it becomes known that is has a dead end. (For example,

simply extend each such node with, wi, all ones.) This is because, as a consequence to

the following theorem, P being decidable is insufficient to ensure that the unique tree

Tp without dead ends shows up in a computable tree numbering. The following is an

alternate, explicit proof.

Theorem 2.4.4. There is a 1-1 computable numbering of all decidable classes in 2".

Proof. Let e v- We be the injective effective numbering of the computable sets as given

in the proof of Corollary 2.2.8. We will define an effective correspondence between these

sets and the nonempty computable trees without dead ends. We will do this through a

series of three one-to-one correspondences namely the correspondences between (1) the

subsets of uw and 2', (2) 2" and 3', and (3) 3' and the nonempty trees without dead ends.

In a stage construction we will then define at stage (e, s) a tree Te,s based on We,s and the

correspondences. Letting T, = nsT,, we will obtain an injective computable numbering

e v-4 [T,] of all nonempty decidable HII classes. Furthermore [T,] will correspond to We for

each e. Finally, by appending the empty class to the enumeration we obtain the desired

result. We now define the correspondences.








The one-to-one correspondence between the subsets S of w and 2" is given as follows.

Each S corresponds to xs C 2" given by xs(i) = 1 iff i E S.

The one-to-one correspondence between 2" and 3" is given as follows. Let x e 2"

and define ao 1i, al = 01, and a2 = 00. Then x corresponds to the unique sequence

(fX(i))iEW 3" where f, : 3 and x = af,(o)af,(1)af,(2) ..
The one-to-one correspondence between the set of all nonempty trees T C 2<

without dead ends and 3' is given as follows. Let T be a tree without dead ends and let

ao = 0, oa, ... be an enumeration of the elements of T in order, first by length and then

lexicographically. We define g = gr e 3" by recursion as follows. For each n, define

g(n) = 2 if an^0 and ajn are both in T, g(n) = 1 if an 0 T and aj^l E T and

g(n) = 0 if a,~^0 E T and a,^l T. For each such g we now let T, denote the unique tree

without dead ends corresponding to g.

At stage (e, s), let Te,s be the clopen tree corresponding to We,s. Then, for fixed e,

since the elements of We are enumerated into the set in increasing order, for all s we have

that Te,s+i C Te,s and each We corresponds to Te = ,sTe,s. In fact, the finite sets precisely

correspond to the clopen trees. Now, e v-4 We injectively numbers all computable sets.

Therefore {0} U {[Te]}eew is an 'injective computable numbering' of the decidable II

classes. E

2.4.2.2 Trees without dead ends: A numbering result

A class is decidable iff it is the set of infinite paths through computable trees without

dead ends. Given any computable numbering of the II classes via trees, this motivates

capturing, through some computable enumeration, those trees in the numbering that

have no dead ends. The following theorem demonstrates that this is possible injectively,

provided that all clopen trees are also included.

Theorem 2.4.5. Suppose e T-> Te is a computable numbering of computable trees in 2<.

Then there is an injective computable numbering of trees consisting precisely of all the Ti

that have no dead ends along with all clopen trees.







Proof. Assume without loss of generality that {Tje}c, contains all clopen trees. We will

construct in stages, as in the terminology of Theorem 2.2.6, a sequence of follower trees

Si. At stage i we will ensure that we have i + 1 trees So, Si,..., Si, constructed up to

level 2', following trees T(so,k ),... T(s,ki) (ki c {m, n}) which are each pairwise distinct

at level 2i. Also, at stage i, initially some of the Si will have the status of being marked

(ki = m) in which case Si will continue to follow T(s,,m) forever. If not, then Si is not

marked (ki = n) and we determine for each i, if Si should be marked. If an Si needs to be

marked then we determine a tree T(s,,m) that it shall hereafter follow. Otherwise each Si

continues to follow T(ss,T) and the stage is complete.

Construction. Stage 0. Find the first tree T, such that T, n {0, 1}20 / 0, denote this

tree as T(so,), and define So = T(so,n) n {0, 1}20

Stage j+1. So,... S have already been constructed up to level 2i and are already

following trees T(So,k)), T(sj,kj). We perform the following two actions at this stage: (1)

Construct So,... ,Sj up to level 2J+l by determining the trees T(so,k+i), .. T(si,k+l,) they

shall follow, and (2) Construct a new tree Sj+I up to level 2j+.

Action (1). Let Uj+I = {(Si, kj) : kj = n and T(s,,kj) has dead ends at level 2J+1}. All

Si such that (Si, kj) g Uj+1 keep their status as marked or unmarked, so kj = kj+l, and

continue to follow T(Si,kj+i). Those Si such that (Si, kj) E Uj+~ will hereafter be marked

and will now follow the tree T(s\,m) = { : T C a or a C r for some -r T(s,n) of length

2J}. Note that each marked Si follows a clopen tree T(Sr,m).

Action (2). Let (Sj+1, n) be the least i such that T, is distinct from all T(s,,kj+1)

(i < j) at level 2j+1 and such that T, has no dead ends. Define Sj+1 = T(s+ ,n) n {0 1} j+1
Verification. We now verify that: (i) For each i, limjT(S,,kj) = Si = T,, for some

T,, without dead ends, (ii) For all i, if Ti has no dead ends then there is a c such that

Ti = S, and (iii) i : j implies that Si / Sj.

VerY. ,,i."'. of (i). For all j, kj = n or kj = m. Fix i. By Action (2), at stage i,

(Si, ki) = (Si, n). By Action (1), ke = k+I = n for all > i if Si is never marked. If Si is
marked at stage r > i, then for all s > r, k, = k,+i = m. In either case limjo~Ikj so that







limj(Si, kj) converges to (Si, n) or (S, m). If it converges to (S, m) then Si never diverges

from following the clopen tree T(s,,m). Otherwise Si is never marked and continually

follows T(s,n). Since it is never marked it means that T(s,n) never has dead ends up to

level 2', for all r > i. So T(s,,n) is a tree without dead ends. Either way limjT(Si,kj)l= TR

for some tree T,, without dead ends. Now for all n, Si n {0, 1}<" T(S,,k,) n {0, 1} t and

T(S,k.) C T(sr,k(+,). Therefore Si limjT(s,,k) T.
Verl./ ,'I. i, of (ii). Let Ti be a tree without dead ends. There are two cases. If there

is a stage j and a c such that Ti = T(s,,m) at stage j, then by the construction Ti = Sc. If

not, let i equal the least k such that Tk Ti. Let j be large enough so that T differs from

T, at level 2i for all e < i. If at stage j there already exists a c such that T7 = T(sc,n) then

clearly Ti Sc. Otherwise, by Action (2), some tree Sc follows T7 by no later than stage

j+i.

V1er 'flr. i', of (iii). By Action (2), Si is distinct from all Sj (j < i) at level 2' and

from all Sj (j > i) at level 2j. So S, / Sj if i / j. O

2.4.2.3 Trees with dead ends: A necessity

In any computable numbering Q of Io classes via trees, some decidable class must

necessarily have, as a tree, dead ends throughout every occurance in the numbering (see

Corollary 2.4.7). Two different proofs of this fact may be obtained from Theorems 2.4.6

and 2.4.8 below.

Theorem 2.4.6. In any computable numbering of computable trees in 2<' there is a

computable tree without dead ends outside the image of the numbering.

Proof. Let {T, : e < w} be a uniformly computable sequence of trees. Now use a

diagonalization argument to construct a tree T such that for all n, T n {0, 1}"+1 /

T, n {0, 1}+1, as follows. At stage 0 let T n {0, 1}o {0}. At stage n + 1 we are given

T n {0, 1}" / 0. Therefore there are at least 2 subtrees of {0, 1}"+1 without dead ends

extending T n {0, 1}. Define T n {0, 1}"+1 to be an extension which is different from

T, n {0, 1})+1. O







Corollary 2.4.7. For any computable numbering P, = [T,] of the 10 classes in 2", there

is a decidable II class P such that P / [T,] for any Te without dead ends.

Proof. Let P = [T] where T is the computable tree without dead ends provided by

Theorem 2.4.6. Suppose that P = [T,] for some e. Since T has no dead ends, it follows

that T = Tp and if T, also had no dead ends, then T, = Tp = T. But by the construction,

T n {0, 1}e+1 / T, n {0, 1}e+1, so that T / T,. O

It follows from this corollary that in the standard numbering, {e : T, has no dead

ends} / {e : P, = [T,] is decidable}. In fact both have distinct complexities. By

Konig's Lemma, Ext(P,) = {a e 2<" : I(a) n P, / 0} is nI. So {e : T has no dead

ends} = {e : T, = Ext(P,)} is II. However, {e : P, is decidable} = {e : = [T] for some

computable T without dead ends} = {e : (3a) ~, is a characteristic function for Ext(P,)} is

E. An alternate proof of Corollary 2.4.7 is as a corollary of the following.
Theorem 2.4.8. For any acceptable numbering i of the 10 classes,

{e : i(e) is decidable} is E complete.
Proof. It suffices to prove this for the standard numbering (b2). We will make use of
the well-known [86] E completeness of {e : We is computable}. It is easy to see that

{e : Q2(e) is decidable} is ZE. For the completeness, define the uniformly computable trees
Tf(e) so that
(i) 0" e Tf(e) for all n;

(ii) 0'1" E Tf () a n We,,.

It follows that 0O1 e Ext(Tf(e)) U n We, so that if /f(e) is decidable, then We

is computable. On the other hand, Ext(Tf(e)) ={0" : n e } U {O0') : s e t, n i We}, so

that if We is computable, then (f (e)) is decidable. Thus We is computable if and only if

b(f (e)) is decidable. O
Note that in [23], a II class P = [Te] in the standard numbering is said to be

decidable if T, has no dead ends, which we now see is probably not the right approach.







2.4.3 Thin and Perfect Thin Classes

In the literature, a Martin-Pour El theory is a consistent c.e. propositional theory

with additional 'thinness' conditions. The conditions imposed have varied depending

upon the context and motivation of the authors, but include: (1) few c.e. extensions,

(2) essentially undecidable, and (3) well-generated. Some authors have chosen to only

impose (1) [21], while others (1) and (2) [19], [24], and finally others (1), (2), and (3)

[34], [40], [29]. The complete consistent extensions of these theories correspond to thin,
perfect thin (or equivalently, special thin [19]), and homogenous thin classes, respectively.

This section is devoted towards demonstrating the nonexistence of computable numbering

of the first two cases by modifying the classical Martin-Pour El construction of a perfect

thin class. Recently Solomon [89] also modified this theorem to construct a homogeneous

thin class and therefore we conjecture that no computable numberings exist for these

classes.

2.4.3.1 The Martin-Pour El Construction

Recall that a II class P is thin if for every II subclass Q C P, there is a clopen set U

such that Q = U P. It is perfect iff it has no isolated points.

A perfect class may be defined by a function g : 2<" 2< such that for all a, -, a K

'r implies g(a) C -r; let us -iv that g is extension preserving. Let G(x) = U g(x [n). Then

G(2") is a perfect class. If g is defined in uniformly computable, extension-preserving

stages g8 (with corresponding G : 2" 2'), so that gs(a) C g+,1(a), then we have

G(2") = neG(2"), so that G(2") is a nI class.

Theorem 2.4.9 (\! ,itin-Pour-El). For any computable extension-preserving function

g : 2<' 2<', there exists a perfect thin II class P C G(2").

Proof. Let {P, = [T,] : e E w} be the standard numbering of the II classes and

{(1 : e E w} be the standard numbering {0, 1}-valued partial computable functions.
We will construct a computable tree S, corresponding II class P = [S], and a surjective

homeomorphism F : 2" P. F will be constructed by means of an extension-preserving

map f : 2<" S, with corresponding map F : 2" 2" defined by F(x) = UJ f(x [ n).







We will define f in stages to obtain uniformly computable, extension-preserving functions

f, so that f = limsf. To ensure that P is thin, we will meet the following requirement for

each e:

Thin(e): (Va E {0, 1}e+l)(V-) [(f(a) E Te A a C r) f(r) E Te]

To see that Thin(e) makes P thin, let U {I(f (a)) : 1a = e + 1 & f(a) e T,} and observe

that if P, C P, then P, Pn U.

Construction. Let fo = g. At stage t + 1, we define ft+l as follows. Look for

e < t + 1, r {0, l}e+1, and Tr a with r
If no such e, a, and r exist, then ft+i = ft. Otherwise take the least such e and the

lexicographically least a and r for that e. For all p E 2'<, let ft+l(a p) = ft(>'p); for

p C a (with p / a) or p incomparable with a, let ft+i(p) ft(p).

Verification. It is easy to see by induction on a that for each a, f,(a) converges

to a limit f(a). Then by induction on e, each requirement Thin(e) is satisfied. To see

that f is injective, suppose towards a contradiction that f(a) = f(r) for a / T. By

the construction, a and r must be comparable. Assume, without loss of generality, that

r = a p (p / 0). By induction it is clear that for all t, ft(a) / ft(-a p) = ft(-). Let

F,(X) = Unf,(r [n), so that P = nF,(2). Since fo = it follows that P C G(2"). O

2.4.3.2 Non-existence of computable numberings

We now modify the Martin-Pour El construction to obtain, as a corollary, non-

existence of computable numbering for thin and perfect thin classes.

Theorem 2.4.10. Any computable numbering of FIl classes in 2" of Lebesgue measure

zero omits some perfect thin class from its image.

Proof. Let P, = [T,], where {T, : e E u} is uniformly computable. We will construct

a computable extension-preserving function g 2<' 2<' such that for all e and

all a e {0, 1}e+l, g(a) T,. Then letting G(x) = U,g(x [ n) we will ensure that
G(2") n P, = 0. Replacing fo by g in Theorem 2.4.9, we obtain a perfect thin class P such

that P n P, 0 (and hence certainly P / P,), for all e.








We define g : 2<" 2< recursively, as follows. Define g(0) = 0. Then for each

a E {0, 1}t, compute the shortest and lexicographically least extension r of g(a) such that

7T Tg. Since [T,] has measure zero, it is nowhere dense and thus such a T alv7--v exists.

Then let g(a^i)= r^i for i E {0, 1}. O

Corollary 2.4.11. There is no computable numbering of all thin or of all perfect thin II

classes.

Proof. All thin classes have Lebesgue measure zero [85]. Therefore if e v-> P, were a

numbering of (perfect) thin classes then Theorem 2.4.10 would provide a (perfect) thin

class P such that P / P, for all e, a contradiction. O

2.4.4 Small, Very Small, and Nondecidable Classes

Binns defined in [11] the notions of small and very small classes as a means of

guaranteeing incompleteness in the lattice of the Medvedev and Muchnik degrees of

subsets of uw. A nonempty II class P is small if there is no computable function I) such

that for all n, ITp n 2(")l n. Let I(n) be the least k such that ITp n ok'| > n. A

nonempty II class P is very small if the function T dominates every computable function

g; that is, I(x) > g(x) for all but finitely many x. Let A be a coinfinite c.e. set, i

A = {ao < al < ...}. Recall that A is i,;' ":',,il. if there is no computable function f

such that f(n) > aT for all n and it is dense simple if n v-> an dominates every computable

function. In this section we will use these sets to show that no effective numbering exists

for the small, very small, or decidable classes.

2.4.4.1 Numberings and high/noncomputable sets

En route to demonstrating our theorem, we now proceed show that there are no

effective numberings of the high or of the noncomputable sets. As we shall eventually

characterize small an very small classes in terms the degrees of these sets, these results will

be crucial to our argument.

First, we modify Shoenfield's Thickness Lemma [86, p. 131]. Some definitions are

needed. For B C w, let B[y] {(y, z) B : z e w} and that B is piecewise computable








if B[] is computable for all y. For B C A C u, we iv that B is a thick subset of A if for

all y, Bly \ AM is finite.

Lemma 2.4.12 (Thickness Lemma). For any uniformly c.e. sequence {Wi : i E u} of

noncomputable c.e. sets and any piecewise computable c.e. set B, there is a thick c.e.

subset A of B so that Wn %r A for all n.

Proof. The proof as in [86] is modified to ensure that the length and restraint functions

and the requirements incorporate the pair (i, k) in place of the single argument i to make

the argument go through with each Wi in conjuction with each functional Tk. E

We obtain the following corollary.

Corollary 2.4.13. For any uniformly c.e. sequence {W,1 n E u} of noncomputable c.e.

sets, there is a high c.e. set A such that for all i, Wi ,T A.

Proof. This follows from the modified thickness lemma above by the same argument

found in [86, p. 133]. O

Corollary 2.4.14. (a) There is no uniformly c.e. numbering of all high c.e. sets.

(a) There is no uniformly c.e. numbering of all noncomputable c.e. sets.

In fact, it follows that there is no uniformly c.e. numbering of the high or noncom-

putable c.e. degrees.

2.4.4.2 Non-existence of effective numberings

We now proceed characterize the small and very small classes in term of the noncom-

putable degrees and the high degrees, respectively. The degree of a II class P is defined to

be the degree of Tp and is thus alv--iv- a c.e. degree (since Tp is a co-c.e. set).

We will use the following two classic results.

(1) \! irtin] Any high degree contains a maximal (and hence dense simple) set [86,

pp. 211-217].

(2) [Dekker] Any noncomputable c.e. degree contains a hypersimple set [86, p. 81].

Proposition 2.4.15. A c.e. degree is high if and only if it contains a very small II class

PC 2".







Proof. (-) Suppose a is high, and let A E a be a maximal set, and let p be the

principal function of u A, so that p dominates every computable function. Now let

PA = {0} U {0(t10 : n A}. Then PA is a io class and for each n, the least k such that

ITp n {0, 1}' > k is precisely p(n) + 1 for n > 0 and hence dominates every computable

function.

(-) Let a be a c.e. degree and suppose that Tp Ea for some very small P. Then the

function f(n) = (least k)[ITp n {0, 1}k > n], which dominates every computable function,

is computable from Tp. It follows from Martin's Theorem [86, p. 208] that Tp is high.


Proposition 2.4.16. A c.e. degree is noncomputable if and only if it contains an infinite,

small II P C 2".

Proof. (-) Suppose a is a noncomputable c.e. degree, let A E a be hypersimple, and p

be the principal function of u A, so that p is not dominated by any computable function.

Then the ni class PA as defined in the proof of Proposition 2.4.15 will have degree a and

will be small.

(<--) Suppose that P is an infinite II0 class and Tp is computable. Then the function

g(n) = (least k)[ITp n {0, 1} > n] is computable and it follows that P is not small. O
Theorem 2.4.17. There is no effective (i.e. ni ) numbering of all nondecidable, of all
infinite small, or of all very small n0 classes in 2".

Proof. Suppose, towards a contradiction, that {Q, = [T,] : n E u} is an effective

numbering of HI classes such that each Q, is nondecidable. Then W,I {(a) : a

Ext(T,)} is a uniformly c.e. numbering of noncomputable c.e. sets. By Corollary 2.4.13,

there is a high c.e. set A such that for all n, Wn, T A. Therefore A is a high degree that

contains a (very) small class not amongst the Qi, a contradiction. O







CHAPTER 3
RANDOM CLOSED SETS

The following chapter is joint work with George Barpalias, Douglas Cenzer, Seyyed

Dashti, and Rebecca Weber and appears in the Journal for Logic and Computation (no.

17, 2007, pages 1041-1062) as an article entitled Algorithmic Randomness of Closed

Sets [7]. A preliminary version of this research was originally presented at the Com-

putability in Europe Conference in Swansea, Wales in 2006 by D. Cenzer. This prelimi-

nary work was published in the referred conference proceedings as Random Closed Sets

(P. Brodhead, D. Cenzer, and S. Dashti) in Proceedings of CIE 2006: Logical Approaches

to Computational Barriers, (A. Beckmann, U. Berger, B. Loewe, and J. Tucker, eds.),

Springer Lecture Notes in Computer Science, Vol. 3988 (2006), pages 55-64 [14].

Portions of this work were also presented by P. Brodhead at the Greater Boston

Logic Conference (\! ly 2006, Boston, MA), the 2006 SACNAS Conference (October

2006, Tampa, FL), the AMS Fall 2006 Eastern Sectional Meeting (October 2006, Storrs,

CT), the Conference on Logic, Computability, and Randomness (January 2007, Buenos

Aires, Argentina), and the Workshop on Computability and Randomness (December 2007,

Auckland, New Zealand).

Portions of this work were also presented by D. Cenzer at a randomness workshop

at the American Institute of Mathematics (August 2006, Palo, Alto, CA). R. Weber also

presented portions of this material at a randomness workshop at the University of Chicago

(September 2007, Chicago, IL).

3.1 Overview

The literature abounds with results in algorithmic randomess as pertaining to reals

over a finite alphabet, especially within the last few years. Little is known, or even

developed, however, with respect to randomness for closed sets of binary reals. This

chapter is a first approach in this direction.

In this chapter, we consider a notion of effective (i.e. algorithmic) randomness on

the space C of nonempty closed subsets P of 2N; to accomplish this task, we will need use

the definition and machinery of effective randomness for reals, since, through appropriate








coding of closed sets, we will define a closed set to be random iff its code, as a real, is

random. (In fact, later in Chapter 4, we will approach a definition of randomness for

continuous functions in a similar fashion.) Consequently we begin this chapter with an

introduction to algorithmic randomness, including a brief historical background.

More specifically, this chapter is organized as follows. In Section 3.2, we provide an

introduction to algorithmic randomness for reals over a finite alphabet. In Section 3.3,

we give a probability measure and define a version of the Martin-L6f Test for closed sets,

leading to a definition of randomness for closed sets. In Section 3.4, we tackle the question

of which types of elements necessarily belong, or do not b. 1.in. to random closed sets. For

instance, every random closed set contains random and non-random elements, but no n-

c.e. elements. In Section 3.5, we show that random closed sets have measure zero and box

dimension log2 4. In Sections 3.6- 3.7, we explore alternate notions for randomness, such as

the problem of compressibility of trees. Finally, in Section 3.8, we consider the problem of

when a randomly chosen closed set meets a closed Q; this is the study of capacities.

3.2 Effective Randomness of Reals

In this section, we present a basic introduction, including a brief historical back-

ground, for randomness of reals over a finite alphabet.

3.2.1 Introduction

The study of algorithmic randomness has been of great interest in recent years. The

basic problem is to quantify the randomness of a single real number. Early in the last

century, von Mises [94] -1-i::. -1. I that a random real should obey reasonable statistical

tests, such as having a roughly equal number of zeroes and ones of the first n bits, in the

limit. Thus a random real would be stochastic in modern parlance. If one considers only

computable tests, then there are countably many and one can construct a real satisfying all

tests.

An early approach to randomness was through betting. Effective betting on a random

sequence should not allow one's capital to grow unboundedly. The betting strategies used








are constructive martingales, introduced by Ville [93] and implicit in the work of Levy [65],

which represent fair double-or-nothing gambling.

Martin-L6f [69] observed that stochastic properties could be viewed as special kinds

of measure zero sets and defined a random real as one which avoids certain effectively

presented measure zero sets; see Section 3.2.4. At the same time Kolmogorov [55] defined

a notion of randomness for finite strings based on the concept of i...iii., ..i.;././ A

stronger notion of prefix-free complexity was developed by Levin [64], Gacs [48] and

C'I 1 i [27] and extended to infinite words.

In the following sections, we formalize the notions of constructive martingale ran-

domness, Martin-L6f randomness, and prefix-free randomness. After their entry into the

literature, Schnorr later proved [83] that all of these notions are equivlant; this is a funda-

mental result in the theory of algorithmic randomness. While these definitions and results

are usually given for binary strings and sequences, they carry over to k-ary strings and

sequences as well. See, for example, Calude [17, 18], or Section 3.2.4 below, where we do

this for the Martin-L6f definition of randomness.

3.2.2 Constructive Martingale Randomness

The betting approach to randomness is formalized as follows.

Definition 3.2.1 (Ville [93]). (i) A martingale is a function m : k< [0, oo) such that

for all a E k<',
k-1
Mr(7) = m(17i).
i=0
(ii) A martingale m succeeds on X E kN if


lim sup d(X [n) = oc.
n-Too

That is, the betting strategy results in an unbounded amount of money made on the

k-ary infinite sequence X.

(iii) The success set of m is the set S~ [m] of all sequences on which m succeeds.

That is, a martingale on 2<" is the capital function of a fair double-or-nothing betting

strategy. When working on 3<" the strategy is triple-or-nothing.








Definition 3.2.2. A martingale m is constructive (or effective, or c.e.) if it is lower

semi-computable; that is, if there is a computable function m : k x N Q such that

(i) for all a and t, m(ao, t) < mat(a, t + 1) < m(a), and

(ii) for all o, limtoo rh(o, t) = m(a).

In other words, m(w) is approximated from below by rationals uniformly in w.

Definition 3.2.3 (Constructive Martingale Randomness). A sequence in kV is construc-

tive martingale random if no constructive martingale succeeds on it.

Comment 3.2.4 (Nonmonotonic Martingales). Some flexibility may be gained by

also considering nonmonotonic martingales; i.e., martingales which bet on the bits of a

sequence out of order. While for a monotonic martingale only the amount of the next

bet is determined from the bits seen previously, for a nonmonotonic martingale both the

amount and the location of the next bet are determined from the bits seen previously

(the next bit may precede them, follow them, or lie in the middle). These martingales

must obey two rules: the standard fair-betting rule that monotonic martingales obey, and

the rule that they never bet on the same bit twice. We refer the reader to Downey and

Hirschfeldt [37] for the formal definition.

Although a priori allowing nonmonotonic martingales strengthens the notion of

randomness, since more strategies must be defeated, in fact in the c.e. case they are

equivalent. Muchnik, Semenov, and Uspensky [71] (Theorem 8.9) show that ML-random

sequences defeat all computable nonmonotonic martingales (in fact they show this with

respect to general measures, not just the coin-toss measure). The proof does not depend

on the computability of the martingale, however; the martingale is used to define a

Martin-Lof test which may be enumerated equally well alongside the enumeration of the

martingale. Therefore, as defeating all c.e. nonmonotonic martingales is clearly sufficient

to be ML-random, the two are equivalent.

3.2.3 Prefix-free Randomness

Prefix-free randomness for reals is defined as follows. A Turing machine M which

takes inputs from A*, where A is a finite alphabet, is called prefix-free if it has prefix-free







domain dom(M); that is, if a 1 r are strings in dom(M), then a must equal r. For any

finite string r, the prefix-free ,-.. i/l. iH.;, of 'r with respect to M is

KM() = min{ 7, oO : M(a) = r}.

There is a universal prefix-free function U such that, for any prefix-free M, there is a

constant c such that for all r

Ku(7) KM(r) + c.

We let K(r) = Ku(7) and call it the prefix-free .i'.i- ili, of r.

Definition 3.2.5 (Prefix-free Randomness). x E {0, 1}" is called prefix-free random if

there is a constant c such that K(x [n) > n c for all n.

This latter inequality means that the initial segments of x are not compressible.

3.2.4 Martin-Lif (n-)randomness

According to the Martin-Lof definition of randomness, a random real must avoid

certain effectively presented measure zero sets. Inherent in the definition, therefore, is

the chosen measure being used. Fix an alphabet k= {0, ,..., k 1}. We present the

definition of a general probability measure on k", as well as different named measure

types. Typically, however, we will make use of the alphabets {0, 1} or {0, 1, 2}.

Definition 3.2.6 (General Probability Measures). Let f : {0, 1, 2}* -- [0, 1] be a function

such that f(0) = 1 and f(a) = i=0,1,2 f(a^ i) for all a. The f-probability measure vf is

defined so that the vf-measure of the interval [a] is such that vf([a]) = f(a).

Definition 3.2.7 (Different Measures Types). Let f and vf be as in Definition 3.2.6.

(i) vf is computable if f is computable.
(ii) vf is nonatomic, or continuous, if for all x E 3, vf({x}) = 0.

(iii) f and vf are bounded if (3b, c E (0, l))(Va)(Vi)[b f(a) < f(a i) < c f(a)].

[Note that any bounded measure is also continuous.]

(iv) vf is regular if (3bo, bi, b2 (0, 1))(Va) f(a^i) = be f(a)

Now fix a probability measure p. In the literature, p is most typically the Lebesgue

measure.







Definition 3.2.8. A real x E k" is Martin-Lif random if for every effective sequence

S, S2,... of c.e. open sets with p(S,) < 2-", x i n, S,.
The latter condition is equivalent to the condition we get if we replace 2-" with q,

for a computable sequence (qi) of rationals such that limit qi = 0. We can also consider an

extended definition of Martin-Lif randomness, in terms of E sets.
Definition 3.2.9. (i) A E test is a computable collection {V, : n E 2N} of E classes

such that p(Vk) < 2-k;
(ii) A real a is E random or n-random if and only if it passes all E tests (i.e., if

{V,: n E 2N} is a computable collection of Z classes such that p(Vk) < 2-k, then
a nn>oVl).
Thus 1-random reals are just Martin-Lbf random reals. See [36] for details on random

and n-random reals. Kurtz [59] and Kautz [54] proved the following result. Let .I' denote

the n-th jump of 0.
Theorem 3.2.10. Let q be a rational number.

(i) For each E class S we can uniformly compute from q and a E index for S, the
index of a EY' 1) class U D S such that U is an open E class and p(U) P(S) < q.

(ii) For each 11 class T, we can uniformly compute from q and a II index for T, the

index of a "II(') class V D T such that V is a closed 1 class and p(V) p(T) < q.

(iii) For each EO class S, we can uniformly compute from q, and a EY index for S and an

oracle for .I ', the index of a I1n_ class V C S such that V is a closed nI1_ class

and p(S) p(V) < q. Moreover, if p(S) is a real computable from .i -1) then the

index for V can be found computably from -1)

(iv) For each 11 class T, we can uniformly compute from q and 110 index for T and an

oracle for .I ', the index of a YZ_1 class U C T such that U is an open ZE_1 class

and p(T) p(U) < q. Moreover, if p(S) is a real computable from -1), then the

index for U can be found computably from -1

Comment 3.2.11. It follows that a real is n + 1-random if and only if it is 1-random

relative to .







Theorem 3.2.12 (van Lambalgen [92]). The following are equivalent.

1. A B is n-random.

2. A is n-random and B is n-A-random.

3. B is n-random and A is n-B-random.

4. A is n-B-random and B is n-A-random.

3.3 Martin-LMf Randomness of Closed Sets

In this section we define a measure on the space C of nonempty closed subsets of 2"

and use this to define the notion of randomness for closed sets. We then obtain several

properties of random closed sets.

3.3.1 The Hit-or-Miss Topology on C

The standard (hit-or-miss) topology on C has as a sub-basis, the following two types

of sets, where Q is any closed set: V(Q) = {K : K n Q / 0}; W(Q) = {K : K C Q}. A

basis for the hit-or-miss topology, then, is formed by taking finite intersections of these.

We now consider a refinement of the sub-basis sets and obtain a basis for the Borel

sets. We will use the following notation. T" denotes the set T n {0, 1}", and T4" denotes

the set Tn {0, 1}i

Definition 3.3.1. For any tree S and any n, define


C,(S) = {Q C : S = TQ}

That is, C,(S) is the set of closed sets Q E C that agree with S up to level n.

Claim 3.3.2. A basis for the Borel sets is given by the agreement set A:


A = {C,(S) : S is a tree, n E uc}.


To see this, first note that any closed set [T] is the decreasing intersection of clopen

sets

[T'] : UJ{[] : Ee T}.

Therefore we may rewrite sub-basis elements V([T]) and W([T]) as

V([T]) nV([T"]) (by definition) and







W([T])= nW([T"]) (by compactness)
But then,

V([T"]) U{n(S) : Sn T" / 0} and

W([T]) = U{ C(s) : s~ c T}
To see the first equality, for example, note that Q n [T"] / 0 if and only if Q E C,(S) for
some S with S n T" / 0. The latter equality holds similarly.
3.3.2 Toward a Measure

To define p(V(Q)) and p(W(Q)), for some fixed measure p and any closed set Q, it
suffices to define p(V(Q,)) and p(W(Q,)) for clopen sets Q, where Q = nQ,. We would

simply define p(V(Q)) =lim,,(V(Q,)) and p(W(Q)) =lim,,(W(Q,)). However, for any
clopen Q, W(Q) is the complement V(2N \ Q). Hence it furthermore suffices to define

p(V(Q)) for clopen sets to get a measure on C.
From the justication of Claim 3.3.2, the latter task may be accomplished by defining a
measure on all Borel basis elements, namely the agreement sets C,(S). To accomplish this,
in the following section we will encode all closed sets Q E C with a canonical code xQ e C.

Then using the Lebesgue measure on 3N, we will define a measure on the sets C,(S) which,
in fact, defines a measure on all closed sets.

3.3.3 Canonical Coding and Measure

The Canonical Coding. An effective one-to-one correspondence between the space
C and the space 3" is defined as follows. Let a closed set Q be given and let T = TQ be the

tree without dead ends such that Q = [T]. Define the canonical code x = xQ E {0, 1, 2}"
for Q as follows. Let A = 0, ol, 2,... enumerate the elements of T in order, first by

length and then lexicographically. We now define x = xQ = XT by recursion as follows. For

each n, x(n) = 2 if a'O and a'l are both in T, x(n) = 1 if ajO T and al E T and
x(n) = 0 if a0 E T and a1- i T. For example, if Q = {0,1}", then xQ = (2,2,...) and
if Q = {y}, then XQ = y. Let Qx denote the unique closed set Q such that xQ = x.
Definition 3.3.3 (The Measure). Define the measure p* on C by


p*(X) = ({xQ Q: Q X}).







where p is the Lebesque measure (i.e. the regular measure Pd with bo = bl = b3 = (see

Definition 3.2.7) on 3.

Informally this means that given a E TQ, there is probability 1 that both a^0 E TQ

and a-tl TQ and, for i = 0, 1, there is probability that only aji E TQ. In particular,

this means that Q n I(a) / 0 implies that for i = 0, 1, Q n I(a i) / 0 with probability .

Comment 3.3.4. At this stage, we have fixed the uniform measure (i.e. all bi = )

towards defining randomness of closed sets. This allows us to more easily demonstrate the

validity of many results. Later, in Section 3.7.1, we will show that the results hold with

any regular measure. Proposition 3.3.5, however, demonstrates that the defined measure

on C, above, holds for any generalized probability measure Pd (see Definition 3.2.6).

Justification for the Coding. Let us also comment briefly on why some other

natural representations were rejected. Suppose first that we simply enumerate all strings

in {0, 1}* as ao, a ,... and then represent T by its characteristic function so that XT(n)

1 <= a,n T. Then in general a code x might not represent a tree. That is, once we

have (01) T we cannot later decide that (011) E T. Suppose then that we allow the

empty closed set by using codes x E {0, 1, 2, 3}* and modify our original definition as

follows. Let x(n) = i have the same definition as above for i < 2 but let x(n) = 3 mean

that neither aO 0 nor a l is in T. Informally, this would mean that for i = 0, 1, a E T

implies that a'i E T with probability '. The advantage here is that we can now represent

all trees. But this is also a disadvantage, since for a given closed set P, there are many

different trees T with P = [T]. The second problem with this approach is that we would

have [T] = 0 with positive probability. We briefly return to this subject in Section 3.7.2.

Now recall the definition of a general probability measure on 3N(Definition 3.2.6). Let

d : 10, 1,2}* [0,1] be a function such that d(0) = 1 and d(a) = E 0,1,2 d(a i) for all

a. Then pd([a]) is defined to be d(a). We may now define, for any such d, pi exactly as in

Definition 3.3.3. Furthermore p* we be deemed computable if d is computable.







Proposition 3.3.5. For any d, the measure p* is defined on all Borel sets in the hit-or-

miss topology on C. Furthermore, if d is computable, then p* is computable on the family
of clopen sets.

Proof. As discussed in section 3.3.2, it suffices to show that p*(C,(S)) is defined for all

C,(S) E A. Fix a tree S and suppose that {a C : r E S'} is ordered ao < ... < Jk(s,n),
first by length and then lexicographically. Then it is easy to see that

Q G C,(S) I ,[.,n] (k(S, n) + ) C ExQ

Consequently C,(S) = ncs V([al) is clopen in C. Furthermore,

p (C,(S))= ([x[s ] (k(S, n) + )]).

This also demonstrates the computability of p*. O

Definition 3.3.6 (Random Closed Sets). A closed set Q E C is (\! irtin-Lbf) random iff
its canonical code XQ is Martin-L6f random.
This definition clearly relativizes to any oracle in accordance with the definitions of
relative randomness in the Cantor space. Since random reals exist, it follows that random

closed sets exist. Furthermore, there are Ao random reals, so we have the following.
Theorem 3.3.7. There exists a random closed set Q such that TQ is A. O
Note that if TQ is A, then Q must contain A elements (in particular the leftmost

path). Since there exist strong HII classes with no Ao elements, there are strong HII classes
Q such that TQ is not A. The following lemma will be needed throughout.
Lemma 3.3.8. For any Q C 2" which is either closed or open,


p*({P: P C Q}) < P(Q).

Proof. Let Pc(Q) denote {P : P C Q}. We first prove the result for nonempty
clopen sets U in place of Q by the following induction. Suppose U = U 1es I(a), where
S C {0, 1}. For n = 1, either p(U) =1 p*(Pc(U)) or p(U) = and p*(Pc(Q)) = .
For the induction step, let Si {a : ica E S}, let Ui = Ues, I(a), let ui = p(Ui) and let







= p*(Pc(Ui)), for i = 0, 1. Then considering the three cases in which S includes both

initial branches or just one, we calculate that


P*(Pc(U))


1
+ V + I XI).
3


Thus by induction we have


1
*(P c(U)) t (uo + u1 + uoui).
3


Now


2u0oul U2 + U U< + 1,


and therefore
1 1
P*(Pc(U)) < t(uo + uI + Uou1) < ( o + ui) p(U).
3 2
For a closed set Q, let Q = n U,, where U, is clopen and U,,+ c U, for all n. Then

P C Q if and only if P C U, for all n. Thus




so that


(Pc(Q)) =

Finally, for an open set Q, let Q

sets U,. Then, by compactness,


lim *(p'(UW,)) < lim i(Un) = i(Q).
n-*oo n-*oo

= UT Un be the union of an increasing sequence of clopen



'Pc(Q)= UJPc(Un),


so that


^(Pc(Q)) = lim P*(Pc(U.)) lim /1(U) P (Q).

This completes the proof of the lemma.

3.3.4 Ghost Coding

We wish now to introduce a second method of coding, the ghost coding. A ghost

code of Q is an infinite ternary string whose terms correspond to all nodes of 2< in







lexicographical order. The terms corresponding to the nodes of Q's tree (the "canonical

nodes") agree with the corresponding terms in the canonical code; the remaining ;!I -I

nodes" may hold any values. Ghost codes are non-unique, and every closed set has a

non-random ghost code (if the closed set itself is random take the code with ghost nodes

all equal to zero, v). This method of coding is more convenient for some purposes;

for example, we will use it to show that if Qo, Qi are closed sets and Q = {^x : x E

Qo} U { lx : x E Q1}, Q is random if and only if the Qi are random relative to each other.
3.3.5 Coding Equivalance

The utility of the ghost codes rests on the following correspondence. Recall van

Lambalgen's theorem (Theorem 3.2.12).

Theorem 3.3.9. The canonical code of a closed set Q C 2" is random if and only if Q has

some random ghost code. Furthermore, for any y, the canonical code r is y-random if and

only if Q has a ghost code which is y-random.

Proof. (-) Suppose the canonical code of Q is nonrandom. Then there is a c.e. mar-

tingale m that succeeds on it. From any initial segment a of a ghost code g for Q, the

subsequence a of exactly the canonical nodes of a is computable. Therefore it is com-

putable whether the bit of g after a is canonical or ghost. From m, define the martingale

m' which bets as follows:


m'(a i) ={ r(ai) next bit is a canonical node
m'(a) next bit is a ghost node.

That is, m' holds its money on ghost nodes and bets identically to m on canonical nodes.

It is clear that m' succeeds on the ghost code g and thus g is nonrandom.

(-) Now suppose the canonical code r for Q is random, and let q be an infinite ternary

string that is random relative to r (and so by Theorem 3.2.12 r E q is random). We claim

the ghost code g obtained by using the bits of r as the canonical nodes and the bits of q in

their original order as the ghost nodes is random. It is clear that g is a ghost code for Q.

Suppose m is a c.e. martingale that bets on g. From m it is straightforward to define

a nonmonotonic martingale m' which mimics m's bets exactly but performs them on r D q,








succeeding whenever m succeeds. As r and q were chosen to be relatively random, this will

show g is random.

As discussed previously, from g n it is computable whether g(n) will be a ghost node

or a canonical node, and which position in g or r it occupies in either case. Therefore,

assuming the bits seen so far may be assembled into an initial segment a of g, m' takes

the values m(a i), i < 3, as its bets on the corresponding bit of r or g, whichever is

appropriate. Having seen that bit, then, it can assemble a (1a1 + 1)-length initial segment

of g and repeat the process. As m' makes identical bets to m and has identical outcomes,

since it cannot succeed on r E g, m cannot succeed on g and g is random.

To relativize (-), suppose that r is y-random, so that r E y is random by Van

Lambalgen's Theorem 3.2.12. Then in the proof simply choose q to be random relative

to r E y, and then g will be random relative to y. The other direction relativizes in a

straightforward way. O

3.3.6 Coding and Joins of Closed Sets

The primary purpose of the ghost codes is to remove the dependence on the particular

closed set under discussion when interpreting bits of the code as nodes of the tree. This is

especially useful when subdividing the tree, as in the following definition.

Definition 3.3.10. The tree join of closed sets Po and P1 is the closed set


Q {0x : x e Po} U {'x: x e Pi}.


Given ghost codes ro, rl for the Pi, their tree join ro B rl is the code for Q with the

corresponding ghost node values.

The standard recursion-theoretic join is defined by


ro ri (ro(0),ri(0),ro(l),ri(1),...).


We wish to relate the recursion-theoretic join and the tree join.

Lemma 3.3.11. Given two ghost codes ro, rl, the tree join ro B rl is random if and only if

the recursion theoretic join ro E ri is random.








Proof. It is clear that there is a computable permutation 7 which uniformly maps any

tree join ro E] r to the recursion-theoretic join ro D rl. That is, in ro D rl, the entries of

ro and rl alternate, whereas ro E rl starts with a 2, followed by blocks from ro and ri,

as follows. First ro(0), ri(0), then ro(l), ro(2), ri(1), ri(2), and continuing with pairs of

blocks of size 4, 8 and so on. The result now follows from the Von-Mises-C '!i. !I-Wald

Computable Selection Theorem [94]; the theorem states that, for any random sequence x

and any computable 1-1 function g, the sequence z(n) = x(g(n)) is random. E

We now obtain the following corollary of Theorems 3.2.12 and 3.3.9 and Lemma

3.3.11.

Corollary 3.3.12. Suppose Pi, i = 0, 1, are closed sets with canonical codes ri and let P

be the tree join of Po, P1. Then P is random if and only if ro E rl is random.

Proof. (-) Suppose that ro E ri is random. Then by Theorem 3.2.12, ro and rl are

mutually relatively random. By Theorem 3.3.9, Po has a ghost code go which is random

relative to ri, and so also vice-versa, and then P1 has a ghost code gl which is random

relative to go. Again by 3.2.12, the recursion-theoretic join go g gi is random, so by

Theorem 3.3.11 the tree join go ] gi is also random, and hence P possesses a random ghost

code and is random.

(-) Suppose now that P is random, and therefore possesses a random ghost code g. The

code g may be thought of as a tree join go ] gi, which is therefore random, and so by

Theorem 3.3.11, go gi is random. By Theorem 3.2.12, the individual codes go, gi are

therefore mutually relatively random. Now by the relative version of Theorem 3.3.9, ro is

random relative to gl. But ri is computable from gl and hence ro is random relative to ri

as well. Similarly, ri is ro-random and thus, again by 3.2.12, ro rl is random. E

3.4 Members of Random Closed Sets

In this section we tackle the question of which types of elements necessarily belong,

or do not belong, to random closed sets. The former is addressed in Section 3.4.1 and the

latter in Section 3.4.2.








3.4.1 Positive Results

We shall see, as a consequence of Theorem 3.4.19, that every closed set is perfect and

contains continuum ,rn ir: elements. In this section, we demonstrate other positive results.

For example, every random closed set contains random and nonrandom elements. Other

examples abound. We begin with the first example.

Theorem 3.4.1. Every random closed set contains a random element.

Proof. Suppose that a closed set Q has no random element and consider the following

Martin-L6f test on the space C:


U = P IP C and PCV}


where (Vi) is a universal Martin-L6f test on the Cantor space. By Lemma 3.3.8, p*(Ui) <

/ip() < 2-' so that (Ui) is a Martin-L6f test on C. But Q E nU, so Q is not random. O

As a converse to Theorem 3.4.1 we have the following.

Theorem 3.4.2. For any random r E 2", there exists a random closed set containing r as

a path.

The proof of this theorem was originally given by Joe Miller and Antonio Montalbdn

and has been subsequently improved thanks to the anonymous referee.

Proof. Let r be a random real and let x be the canonical code of an r-random closed set.

We alter x to the code x' of a closed set guaranteed to contain r but changed as little as

possible to achieve that.

To determine x'(n), assume x' [ n has been defined. If x(n) = 2 or x(n) corresponds

to a node not along r, set x'(n) = x(n). If x(n) c {0, 1} corresponds to r(k), set

x'(n) = r(k).

The closed set defined by x' will clearly contain r. For a contradiction, assume x' is

nonrandom and let m' be a c.e. martingale that succeeds on it. We build a nonmonotonic

martingale m to bet on x E r. On bits of x, m will be a triple-or-nothing martingale; on r,

it will be double-or-nothing.








First note that from initial segments of x and r we may reconstruct an initial segment

of x' computably, and we ah--bi- know from an initial segment of x' whether the next bit

is along r or not, and which bit of r it is. We will construct m so that after every stage of

betting (which will be one bet by m' and one or two bets by m), the value of m is equal to

the value of m'. At every stage it will be clear we have revealed enough bits of x and r to

reconstruct x' to the needed length.

Suppose inductively m and m' hold equal capital after the stage of betting on the

last node of a E x'. If the bit x'(n) following a is not on r, m bets identically to m';

i.e., m(x(n) = i) = m'(aui) for i < 3. In that case x(n) = x'(n) so our inductive

hypothesis holds. If x'(n) is on r, set m(x(n) = 2) = m'(au2) and for i = 0, 1, set

m(x(n) = i) = [m '(a0) + m'(a 1)]. If x'(n) = 2, then the capital for both m

and m' is m'(a'2), so the inductive hypothesis holds and we proceed to the next stage.

Otherwise m bets on r(k) for the appropriate k, setting m(r(k) i) = m'(a i) for

i = 0, 1. On r(k), the sum of m's capital on each of the two outcomes must average to

the previous capital; as the previous capital was 1 [m'(a.'0) + m'(a' 1)] this clearly holds.

By construction r(k) = x'(n) = i, so both m and m' now have capital m'(a i) and the

inductive hypothesis holds. As m' is c.e., m will also be.

As the values of m' along x' are a subsequence of the values of m along x E r, if m'

succeeds so does m, contradicting our assumption on x E r. Therefore x' is the code of a

random closed set containing the given random path r. E

The previous results might si--,-' -1 that every element of a random closed set is a

random real. However, it turns out that every random closed set contains a non-random

real.

We need the following classic result of Chernoff [28] (a version of Bernoulli's Weak

Law of LIr,,/. Numbers) here and also for another theorem to follow. See [67] for an

exposition.

Lemma 3.4.3 (C('! i i. .11). Let E be an event which we will refer to as 'success'. If E

occurs with probability p, then for any natural numbers n and any E with 0 < E < 1, the








probability that out of n mutually independent trials, the number of successes differs from

pn by > Epn is < 2-e2pn/3

Theorem 3.4.4. Every random closed set contains a non-random real; in particular, the

leftmost and rightmost paths in a random closed set are not random reals.

Proof. We will show that, for a random closed set Q, the leftmost path is not stochas-

tically random, that is, the ..i-mptotic frequency of 0's is 2. Since an effectively random

real in 2" must have ..ii-',iiiil1 ic frequence of 1 for 0's and l's, this will suffice to prove

that the leftmost path is not random. We define a Martin-L6f test as follows. Fix a ratio-

nal E such that 0 < E < 1. For each n, let S, be the family of closed sets (that is, codes

for closed sets) such that the first n bits of the leftmost path have either < 2(1 E)n, or
> (1 + E)n occurrences of 0. By the definition of our probability measure, we have


(n 2\ m 1 n-m
3 E*nj ( ) )


It now follows from C'!I. Ii1 all's Lemma 3.4.3 that


p*(Sn) < 2e-E22n/9.

Thus the measures of the test sets S, have effective limit zero. It is easy to see that the

sequence {S,} is computably enumerable. For each n, S, is a clopen set and in fact the

union of the finite family of intervals I(c) in C such that a codes a tree up to level n in

which the leftmost path has either < ( )n, or > (1 + )n occurrences of 0.

Furthermore, S' J= U>~ S, is also a Martin-L6f test. It follows that for any random

closed set Q, and any E > 0, there is an n such that for all m > n, the frequency of 0's in

the first m bits of the leftmost path is alv--,v- within E of j. Thus the leftmost path is not

effectively random. D

Recall that the leftmost and rightmost elements of any strong A closed set are

A. Given Theorems 3.4.1 and 3.4.4, we ask: Does a A random closed set contain a A

random path?

Theorem 3.4.5. Every random strong A closed set contains a random A real.








Proof. Let Q be a random strong A0 class. By Theorem 3.4.1, Q contains a random real

x. Let P be a II class in the Cantor space which contains only randoms and contains x

(this exists since the class of random reals is an effective union of I H classes). Note that

P n Q is a non-empty strong A0 class and it follows that the leftmost path of P n Q is a

A real which must be random since it belongs to P. E

The above theorem does not combine with the low basis theorem to establish the

existence of a low random real in any random strong A class. We can use the low basis

theorem, however, to demonstrate the existence of a low random real in any random closed

set with low canonical code.

Theorem 3.4.6. Every random closed set with low canonical code contains a low random

element.

Proof (Kjos-Hanssen). Let Q be a random closed set with low canonical code. By

Theorem 3.4.1, Q contains a random element x. Therefore x E 2" \ U, for some n and

some open U, from the universal Martin-L6f test. So, in particular, Q n 2" \ U, is non-

empty. Now Q n 2" \ U, is ni relative to TQ. By the low basis theorem, Q n 2" \ U, is I0

has a member y such that y'
Furthermore, since TQ is low, y is also low. O

It is open whether every random closed set with a A canonical code has a low

random element; we conjecture that the answer is no. In the following section, we will

show that there is a random closed set not containing any A0 path.

Our next result, Theorem 3.4.8, uses a method which was used in [56] to show that

every random real is effectively bi-immune. We first define this latter notion.

Definition 3.4.7. (i) A set A is effectively immune if it is infinite and there is a

computable function g(x) such that if W1 c A, |W,| < g(x).

(ii) A is effectively bi-immune if A and A are both effectively immune.

Note, in particular, that an effectively immune set cannot contain an infinite c.e. set.

Theorem 3.4.8. If P is a random closed set then all elements of P are effectively bi-

immune.







Proof. Suppose that P is a random closed set and A E P. Let (Ui)ijN be a Martin-Lif

test (in the space 3N) such that there is a computable function f with the property that

if (Ve)jEN is the e-th Martin-L6f test (under some effective enumeration of all Martin-Lif

tests) then for all e, i, V (,i) C U, (a standard construction of a universal test gives one

with this property). Since P is random, there is some k such that (the canonical code of)

P is not in Uk; let U = Uk. It suffices to find a computable function g such that


[A P and We CA] == W,| < g(x) (3-1)

for all sets A and all x (the proof that A is effectively immune is entirely similar). Let B,,

be 0 if IWe I n, and otherwise the class of (canonical codes of) trees which contain a path

containing the first n + 1 elements in the standard enumeration of Wx. Then (Bx,,) is a

uniform double sequence of ZE classes and (by the definition of the probability measure on

3N),

pt*(B1) (2)

So for each x, (Bx,2,) is a Martin-L6f test in the space 3" and from x we can calculate

the index of it. Then by using the computable function f mentioned above we get a

computable function g such that for all x, Bx,g(9) C U. This means g satisfies (3-1). O

It is well known that effectively immune sets can compute a fixed point free function,

so we have the following.

Corollary 3.4.9. The paths through a random tree are of fixed point free degree. That is,

each of them computes some fixed point free function.

It is known that every real can be computed by some random real. It is not known,

however, whether any real can be computed by all the paths of some random closed set.

The next theorem, an observation of Ted Slaman at a randomness workshop in Chicago in

2007, is a step in that direction. First, we need the following definitions, which are, in fact,

equivalent notions.

Definition 3.4.10. (i) A real x is K-trivial if K(x[n) < K(n) + c for some c.








(ii) A real x is a base for 1-randomness if there is some y >r x such that y is 1-x-

random.

Theorem 3.4.11. Any set which is computable from all paths of a random closed set is

K-trivial.

Proof. If A is computable from all paths, then A is computable from the leftmost and

rightmost paths. Note that each of these latter paths are each computable from the two

halves of the tree. Furthermore these two halves are relatively random to each other.

Hence each is random relative to anything the other computes. So half 1, for example,

computes A and hence half 2 is random relative to A. On the other hand, half 2 also

computes A. Therefore this is an example of something in the cone above A that is

A-random. So A is a base for 1-randomness. O

Much interesting work has been done on the K-trivial reals. C'!i il ii showed that if A

is K-trivial, then A r 0'. Solovay constructed a noncomputable K-trivial real. Downey,

Hirschfeldt, Nies and Stephan [39] showed that no K-trivial real is c.e. complete. The

notion of a K-trivial closed set was introduced in [9]. It was shown in particular that every

K-trivial class contains a K-trivial member, but there exist K-trivial II0 classes with no

computable members.

3.4.2 Negative Results

Random closed sets can never contain n-c.e., isolated, or 1-generic paths, or paths of

incomplete c.e. degree. We build to these facts and prove others along the way.

Theorem 3.4.12. Random closed sets contain no computable elements.

Proof. For any finite string a of length n, the probability that a closed set Q meets

I(a) is ( )". For a computable real y, the sqeuence {Q : Q n I(y[n) / 0} thus forms a

Martin-L6f test in the space C of closed sets, which shows that y does not belong to any

Martin-L6f random closed set. That is, for each n, {x : Qx n I(y[n) / 0} is a c.e. open set

and has measure (2)" in {0, 1, 2}1, where Qx is the closed set with code x. O

We prove an even stronger result in Theorem 3.4.17. First, however, recall that a

HI class P is decidable if Tp is decidable. It follows that a nonempty decidable HI class







must contain a computable element (for example, the leftmost path). By Theorem 3.4.12,

it follows that no decidable II class can be random. As every random class contains a

random element (Theorem 3.4.1) and has, as we shall show, measure zero (Theorem 3.5.1),

the following proposition demonstrates that this extends to arbitrary II classes.

Proposition 3.4.13. If P is a II class of measure 0, then P has no random elements.

Proof. Let T be a computable tree such that P [T], and for each n, let P,= U{I():

a E T n {0, 1}"}. Then {Pn},EN is an effective sequence of clopen sets with P = P,

and lim, (P) = p(P) = 0. Furthermore, p(P,) = 2- T n {0, 1}'"; this is a computable

sequence. Thus {P,},}N is a Martin-L6f test and P has no random elements. O

Alternatively, we can show that no II class is random through the following stronger

result, combined with an appeal to Theorem 3.5.1, from the next section.

Theorem 3.4.14. Let Q be a II class with measure 0. Then no subset of Q is random.

Proof. Let T be a computable tree (possibly with dead ends) and let Q = [T]. Then

Q = n ,, where U, = [T,]. Since p(Q) = 0, it follows from Lemma 3.3.8 that
lim, p*(Pc(Un)) = 0. But Pc(U,) is a computable sequence of clopen sets in C and

p*(Pc(U,)) is a computable sequence of rationals with limit 0. Thus Pc(U,) is a Martin-
Lof test, so that for any random closed set, there exists n such that P Pc(U,) and hence

P is not a subset of U,. O

Corollary 3.4.15. No II class can be random. O

We now provide an even stronger version of Theorem 3.4.12; we need the following
definition.

Definition 3.4.16 (f-c.e. reals). For any computable, non-decreasing function f, we -v

that a real 3 E {0, 1}" is f-c.e. if there exists a computable approximating function Q such

that, for all i E N,

(i) Q(i,0) 0
(ii) lim Q0(i, s) = 3(i);

(iii) {s : Q(i, s + 1) / Q(i, s)} has cardinality < f(i).







The reals which are f-c.e. for some computable function f are part of the well-known

Ershov hierarchy [43, 86].

Theorem 3.4.17. Suppose that f is computable and bounded by a polynomial. Then no

random closed set has any f-c.e. paths.

Proof. Let f be as above, 3 an f-c.e. real and P a closed set containing 3. Let

be the f-approximating function for f. Also let .i C {0, 1}" be the set of different

Q-approximations to f[n during the stages.

A priori, '.1 is exponential. However, for a fixed n, 3 [n can change at most

Ei<, f(i) times, so 'V is also bounded by a polynomial, i.e. there is k E N such
that for almost all n, '1 < nk. Now let


Sn, U {PI P P C& J(7) /0}. (3-2)
o-fMn

Then (S,) is a uniformly c.e. sequence of open sets in the space C of closed sets of 2" and

for all n, P E S,. Also for almost all n,


*(S,) 4 p ({P | P t C & Pn IJ(o) / 0}) = .-V | [ Gk *
(TcEM

Since lim, [nk ()"] 0 there is a computable subsequence of (S,) which is a Martin-Lof

test and so P is not random. E

For any K-trivial real A and any unbounded nondecreasing computable function h,

A is h-c.e. (Nies [75]). Thus it follows from Theorem 3.4.17 that a random closed set can

have no K-trivial paths. We observe that Theorem 3.4.17 cannot be extended to w-c.e. in

general, because there are left-c.e. (and hence w-c.e.) random reals, and by Theorem 3.4.2

each of these belongs to a random closed set.

Also, recall from Corollary 3.4.9 that that paths through a random tree are of fixed

point free degree. It is known that fixed point free degrees cannot be 1-generic (see [42] for

a proof) or incomplete c.e., and that if they are A0 they compute a promptly simple set

and no pair of them forms a minimal pair (see [57]). So we have the following.

Corollary 3.4.18. The following hold:







No path of a random tree is 1-generic.

No pair of A paths of a random tree can be a minimal pair.

Every A path of a random tree computes a promptly simple set.

No path of a random tree can have incomplete c.e. degree.

Theorem 3.4.19. If Q is a random closed set, then Q has no isolated elements.

Proof. Let Q = [T] and suppose by way of contradiction that Q contains an isolated

path x. Then there is some node a E T such that Q n I(a) = {x}. For each n, let

S', = {P C : {re {0,1}" : P n I(a^ ) O0}| = 1}.


That is, P E S, if and only if the tree Tp has exactly one extension of a of length n + 1|.

It follows that

Pn I(a)| = 1 (Vn)P e S,

Now for each n, S,, is a clopen set in C and again by induction, S, has measure () ". Thus

the sequence So, Si,... is a Martin-L6f test. It follows that for some n, Q S,. Thus

there are at least two extensions in TQ of a of length n + lal, contradicting the assumption

that x was the unique element of Q n I(a). O

As mentioned previously, it follows that every random closed set is perfect and hence

contains continuum rn ii: elements.

Next we want to find a random closed set which does not contain a A path. Now

it is easy [20, 24] to construct a strong 1 class P of positive measure which contains no

A elements; of course P must contain a random real since it has measure 1. The difficult

problem is to construct a random strong 1 class with no A elements. We have the

following result in this direction, which yields a random strong A closed set with no A

elements.

Theorem 3.4.20. For any set A there is an A-random closed set Q such that TQ
but Q has no elements
Proof. It is enough if we prove the claim for A = 0 because the argument relativises to

any oracle A in a straightforward way. For A = 0 we use a finite injury construction over







0' to construct Q with the above properties. In the construction we will 0'-approximate

the canonical code of a tree T which has no A paths. To make sure that the tree T is

random we fix a 1 class P of positive measure in the space 3" (where the code for T lies)

which contains only randoms, and we make sure that at every stage our approximation

(as a finite ternary string) to T's canonical code can be extended to a path in P. Then by
compactness the canonical code of our tree will be in P and so the tree will be random.

The changes in the approximations are motivated by the requirements:

Re : if 4' is total then the real it defines is not in [T].

Let a, be a finite string approximation of the canonical code a we are building. We will

have |as = s. Strategy Re will come into power after stage e and will restrain a up to

some re > e (the default value is re [0] e). Also it might request some changes in a after

the e-th bit. We start with ao = 0 and at stage s + 1, assuming inductively that as I and

[ac] n P / 0 we ask for the least i < s such that Ri requires attention. This happens if
(i) The longest defined initial segment T of 4+0' is larger than ever before;

(ii) there exists a e {0, 1,2}* such that a, [(maxj
and T is not consistent with the finite tree with code a.

If there is no such i then we extend as by one bit such that [ac8+] n P / 0. Otherwise

we let a8s+ = a and ri[s + 1] = s + 1. The construction proceeds in a straightforward way

and we can prove inductively that for every e, Re is satisfied, stops requiring attention and

re reaches a limit. Then the limit a = lim8 a, exists and we also have that a is random

by compactness. The satisfaction of the requirements comes from a measure-theoretic

fact. Consider Re and inductively assume that after stage se no Ri with i < e requires

attention. Then r = maxi
[a [maxi
p([a Fr] n P) > 0

and on the other hand, if 3 = KL' we have seen that


pu{7 | 7 3' and 7 is the canonical code of a tree which has 3 as a path} = 0.







This means that if at stage s, the requirement Re is not yet satisfied, it will receive

attention at a later stage and get satisfied permanently. E

3.5 Measure and Dimension

In this section we show that random closed set have measure zero (Theorem 3.5.1)

and box dimension log2 1 (Theorem 3.5.2).

3.5.1 Measure

Theorem 3.5.1. If Q is a random closed set, then p(Q) = 0.

Proof. We will show that in the space C of closed sets, the p*-probability that a closed

set P has Lebesgue measure 0, is 1. This is proved by showing that for each m, I(P) >

2-m with p*-probability 0. For each m, let


S,= {P : p(P) > 2-}.


We claim that for each m, It*(Sm) = 0. The proof is by induction on m.

For m = 0, we have pI(P) > 1 if and only if P = 2', which is if and only if

xp = (2, 2,...), so that So is a singleton and thus has measure 0.

Now assume by induction that Sm has measure 0. Then the probability that a closed

set P = [T] has measure > 2-"-1 can be calculated in two parts.

(i) If T does not branch at the first level, -i To {(0)} without loss of generality.

Now consider the closed set Po {y : O-y E P}. Then p(P) > 2-m-1 if and only if

p(Po) > 2-m, which has probability 0 by induction, so we can discount this case.

(ii) If T does branch at the first level, let Pi {y : i^y e P} for i = 0, 1. Then

p(P) (p(Po) +(Pi)), so that p(P) > 2-m-1 implies that at least one of p(P,) > 2- m-

(Note that the reverse implication is not alvb- true.) Let p = p*(Sm+1). The observations

above imply that
1 2 1
p < (1 (1- p)2) p 2,
3 3 3
and therefore p = 0.







To see that a random closed set Q must have measure 0, fix m and let S = S,. Then

S is the intersection of an effective sequence of clopen sets V, where for P = [T],

P Ve p([T,]) > 2-.

Since these sets are uniformly clopen, the sequence me = p*(V) is computable. Since

lime m =- 0, it follows that this is a Martin-Lof test and therefore no random set Q

belongs to [n V. Then in general, no random set can have measure > 2-' for any m. E

3.5.2 Dimension

Surprisingly, we can compute the (Kolmogorov) box dimension of a random closed

set, and in fact it turns out that all random closed sets have the same dimension. The

intuition for this comes from the following lemma. For any function F mapping the space

C of closed sets into Z, the expected value of F on C is the integral f F(P) with respect to

the probability measure p*.

Lemma 3.5.2. In the space C of closed sets, the expected cardinality of {a E {0, 1})

QIn(a) / 0} is exactly ( )" for every n, where Q is chosen uniformly at random according

to /t*.

Proof. Let S,= {-a {0, 1}" : Q n I(a) / 0}, for a randomly chosen Q from C.

The proof is by induction on n. For n 1, we have two cases. With probability 2,

card(S1) = 1 and with probability card(S) = 2. Thus the expected value is exactly

. For n + 1, there are again two cases. With probability 2, card(S1) = 1, so that the

expected card(S,+1) equals the expected card(S,), which is (4)" by induction. With

probability card(SI) =2, in which case the expected card(S,+1) is twice the expected

card(S,), that is, 2(4)". Thus we have the expected value

2 /4" 1 /4 4Y
card(S, +) = ()+ 2 () (4) +
3 3 3 3
El








The box dimension of a closed set in the Cantor space, if it exists, is given by the

following limit:
F ) log2(card(TQ n {0, 1}))
dimB F(Q) = lim
n-oo 7

(See [6] for this formulation of the box dimension in {0, 1}".) Now by Lemma 3.5.2, the

expected value of card(TQ n {0, 1}") for a random closed set Q is (4)", which s,-. -I that

the box dimension of Q should be log2 3.

Lemma 3.5.3. Let Q be a random closed set. Then for any E > 0, there exists a m e N

such that, for all n > m, ( )"(1 ))n < card(TQ n {0, 1}) < ( )"(1 + E).

Proof. For each n, let c,(Q), or just c,, denote card(TQ n {0, 1}"). We will use three

applications of ('!,. il l's Lemma 3.4.3. First we show that there exists m such that for

all n > m, C6n > n. Since the tree TQ n {0, 1}6" -1 has at least 6n nodes, it follows from

('!. i i .l'"s Lemma that the number of branching nodes is less than n with probability

S2-/6 Thus c6n < n with probability < 2-n/6. Then the probability that C6n < K for any

n > m is less than
00 2-m/6
E 2-n/6
1 2-1/6"
n=mn
This provides a computable sequence of clopen sets with measures bounded by a com-

putable sequence with limit zero and hence a Martin-L6f test. It follows that for any

random closed set Q, there exists mo such that c6n > n0 for all n > mo. Now for n > o0,

there are at least 6n2 nodes in TQ n {0, 1}412n-1 {0, 1}6n-1, so that again by Chernoff's

Lemma, the probability that < n2 of these are branching nodes is < 2-n /6. It follows as

above that there exists mi > 3 such that cl2n > n 2 for all n > mi. Now suppose that

m 12mi and that 12n < m < 12(n + 1) < 16n. Then n > mi, so that


cm > c12n > n2 > (m/16)2

Again by ('!, i Ii ll"'s Lemma, the probability that the number of branching nodes from

TQ n {0, 1}" differs from c by >
that e+i differs from 4cn by > -c_ c4. For n > mi, we know that c, > (4) so that

A and c4 and hence 2- /9 < 2-n/144. Thus the probability p, that c,+i







differs from c by more than is < 2 n/144. Then the probability that for any n > ml,

c+I differs from c, by more than 3rc, is bounded by

n n1 2-m/144
S 2-/144 1 2-144'
n=m n=m
This again provides a Martin-L6f test which shows that for any random closed set Q, there

exists m2 so that for n > m2,


(*)

Now given E, choose m > m2

Then for any k,

Cm(f4)2k(_
3


4 1 ) 4 1
3 ) 3 and i <

so that (1 + 1)2 < 1 + and 1 F < (1 -


c)k < Cm( 2k( 2k < Cm+2k
3 c2

< )( 2k(1 + )2k < Cn( )2k( + )k
3 \//n 3


Now let k be large enough so that

(1 E)m+k C ()~1( + E)m+k


Then the desired inequality


( )"(1


)" < c < (4 )"(1 + E)


will hold for even n > m + 2k. For odd n, this inequality will hold by the inequality (*)

above.

Theorem 3.5.4. For any random closed set Q, the box dimension of Q is log2 .*

Proof. Given E > 0, let m be given by Lemma 3.5.3. Then for n > m, we have


4
n log2 nlog2(1
0


so that
4
log2 + log(1-
and therefore dim ) li
and therefore dimB(Q) = linm


4
E) < log2(card(TQ n {0,1}") <, nlog2 + nlog2(1 + ),
3


log2(card(TQ n {0, 1})) 4
E < 1092 10n-2- + 109g E),

, 1 1. !. n of ,j )) 4g


1)2.
V7'-







3.6 Prefix-Free Complexity of Closed Sets

In this section, we consider randomness for closed sets in terms of incompressibility

of trees. Of course, Schnorr's theorem tells us that P is random if and only if the code

xp E {0, 1, 2}1 for P is prefix-free random, that is, K3((xp n) > n 0(1). (Schnorr's

theorem for arbitrary finite alphabets is shown in [18].) Here we write K3 to indicate that

we would be using a universal prefix-free function U : {0, 1, 2}* {0,, 12}*. However,

many properties of trees and closed sets depend on the levels Tn = T n {0, 1}" of the tree.

For example, if [T,] = U{I(a) : a c T}, then [T] = n, [T,] and p([T]) = lim", p([T,]).

So we want to consider the compressibility of a tree in terms of K(T,). Now there is

a natural representation of Tn as a string of length 2". That is, list {0, 1}" in lexicographic

order as a1,..., a2. and represent Tn by the string el,..., e2~ where ei = 1 if ai E T and

ei = 0 otherwise. Henceforth we identify T, with this natural representation.

It is interesting to note that the code for Tn will have a shorter length than the

natural representation. For example, if [T] = {y} is a singleton, then x = y and for each

n, the code for Tn is x[n. If x is the code for the full tree {0, 1}*, then x= (2, 2,...) and

the code for Tn is a string of (2" 1) 2's, those labels attached to nodes of length < n. For

the remainder of this section, we will use T, to mean the natural representation and x, to

mean the code.

Question. Is there is a formulation of randomness for closed sets in terms of the

incompressibility of T,?

It seems plausible that P = [T] is random if and only if there is a constant c such

that K(T,) > 2" c for all n. However, we will see that this is not possible for any tree.

On the one hand, in Section 3.6.1 we achieve a lower bound for incompressibility. That

is, we show that if P = [T] is random then there is a constant c such that K(T,) >

(0)" c for all n. On the other hand, in Section 3.6.2, we see that the 2" is too high of

an incompressibility bound since there is some c and some random closed set such that

K(T,) < 2" c for all n.







In a larger sense, we seek a formulation of randomness, in terms of the incompressibil-

ity of T,, fo thr th objects such as II classes or II classes. In the following two sections

we consider these questions and achieve some lower and upper bounds for these classes of

objects.

3.6.1 Lower Complexity Bounds

First we give a lower bound for the prefix-free complexity of a random tree.

Theorem 3.6.1. If P is a random closed set and T = Tp, then there is a constant c such

that K(T,) > ()" c for all n.

Proof. Let P = [T] be a random closed set. Let m be given by Lemma 3.5.3, for E =

so that for n > m,

card(TT,)> ( .

It follows that the code x, for T, has length > (7). Since x is random, we know that, for

n K m,
K3(X ) > (i a,

for some constant a. Now we can compute x, from T,, so that


K (T,) 3 K (x) b,

for some constant b. The result now follows.

That is, let U (mapping {0, 1}* to {0, 1}*) be a universal prefix-free Turing machine

and let K(T,) = min{|(a : U(a) = T,}. Let M be a prefix-free machine M (mapping

{0, 1}* to {0, 1,2}*) such that M(T,) = x,. Then define V by

V(a) = (U(a)).

Then Kv(x n) < K(T,), so that for some constant e, K3(x,) < K(T,) + e and hence

K(T.) > K3x)- > (. -e.

E







The standard example of a random real, C' iil ,ii's Q [27], is a c.e. real and therefore

A. Thus there exists a A random tree T and by Theorem 3.6.1, K(T ) > ( c for

some c.

We have a more modest result for II classes. That is, there is an effectively closed set

with not too much compressibility, in the following sense.

Theorem 3.6.2. There is a 1 class P [T] such that K(T,) > n for all n.

Proof. Recall the universal prefix-free machine U and let S {-a E Dom(U) : |U(a)|l

21' }. Then S is a c.e. set and can be enumerated as a1, a2,.... The tree T = QT T8 where

T" is defined at stage s. Initially we have To {0, 1}*. We -i that at requires attention

at stage s > t when T = U(Ut) T, for some n (so that r = 2") and n > lat Action is

taken by selecting some path pt E T, of length n and defining T"+l to contain all nodes of

T" which do not extend pt. Then T / T,+1 and furthermore T / T, for any r > s + 1 since

future action will only remove more nodes from T,.

At stage s + 1, look for the least t < s + 1 such that at requires action and take the

action described if there is such a t. Otherwise, let T"+ = T".

Let A be the set of t such that action is ever taken on Ut. Recall from the Kraft

Inequality that t 2- < 1. Since pt > Ut, it follows that EteA 2-Pt < 1 as well. Now

/I([T]) =1 >t 2-1IPt > 0 and therefore [T] is nonempty.
It follows from the construction that for each t, action is taken for at at most once.

Now suppose by way of contradiction that U(a) = T, for some at with la < n. There

must be some stage r > t such that for all s > r, T, = T, and such that action is never

taken on any t' < t after stage r. Then at will require action at stage r + 1 which makes

T+1 / TA, a contradiction. E

There is a stronger result for closed 1 classes. Namely, there is a closed 1 class with

the following stronger incompressibility property.

Theorem 3.6.3. There is a nII class P [T] such that K(T4) > 2 for all .







Proof. We will construct a tree T such that T,2 can not be computed from fewer than 2"

bits. We will assume that U(0) T to take care of the case n = 0. At stage s, we will define

the (nonempty) level T,2 of T, using an oracle for 0'.

We begin with To {0}*.

At stage s > 0, we consider


D, {a e Dom(U) : al < 2"}.

Since U is prefix-free, card(D,) < 22". Now there are at least 222 1 trees of height s2

which extend T(s_1)2 and we can use the oracle to choose some finite extension T' = T,2 of

T(s_1)2 such that, for any a E Ds, U(a) / T' and furthermore, U(a) / Tr for any possible

extension T, with s2 < r. That is, since there are < 22s finite trees which equal U(a) for

some a E Ds, there is some extension T' of T(s_1)2 which differs from all of these at level

s2. We observe that the oracle for 0' is used to determine the set Ds.

At stage s, we have ensured that for any extension T C {0, 1}* of T,2, any a with

lal < 22 and any n > s2, U(a) / Tn. It is immediate that K(T,) > 2 '. E

3.6.2 Upper Complexity Bounds

In Theorem 3.6.1 we achieved a lower bound of (')" for the prefix-free complexity of

a tree Tp of a random closed set P. It seems plausible that we might be able to achieve

a higher bound of 2". If true this would actually provide and immediate characterization

of randomness of closed sets in terms of prefix-free complexity of trees. That is, a closed

would be random iff K(T,) > 2' c for some constant c. (To see this, note that from

x [2" we can compute T, uniformly so that K3(x [2) > K(T,) b for some b.) However

the following theorem provides an upper complexity bound less than 2", refuting such a

possibility.

Theorem 3.6.4. For any tree T C {0, 1}*, there are constants k > 0 and c such that

K(T)) < 2' 2-k + c for all f.

Proof. For the full tree {0, 1}*, this is clear so suppose that a T for some a E {0, 1}'.

Then for any level f > m, there are 2-'m possible nodes for T which extend a and Te may








be uniformly computed from a and from the characteristic function of T. restricted to the

remaining set of nodes. That is, fix a of length m and define a prefix-free computer M as

follows. The domain of M is strings of the form 01-T where I = 2 m 2-'m. M outputs

the standard representation of a tree Tf such that no extension of a is in Tf and such that

7 tells us whether strings not extending a are in T. It is clear that M is prefix-free and

we have KM (T+) = + 1 + 2' 2-'-. Thus K(T ) < + + 1 + 2' 2-'- + c for some constant

c. Now + 1 < 2-'/-1 for sufficiently large f and thus by adjusting the constant c, we can

obtain c' so that

K(T<) < 2Y 2-'m-1 + c'.



The following theorem also refutes the possibility that K(T{) > 2e-c is a charac-

terization of random closed sets in terms of prefix-free randomness. It shows that closed

sets with small measure, such as random closed sets which have measure zero (see Theo-

rem 3.5.1), are more compressible.

Theorem 3.6.5. If p([T]) < 2-k, then there exists c such that, for all ,


K(T) < 2-k+1 + c.

Proof. Suppose that p([T]) < 2-k. Then for some level n, T, has < 2 -k nodes

al,...,at. Now for any f > n, T. can be computed from the fixed list a,,..., at and the

list of nodes of Te taken from the at most 2 -k extensions of al,..., o-t. It follows as in the

proof of Theorem 3.6.4 above that for some constant c and all ,


K (T) < 2- + f + 1 + c.

Thus for large enough so that f + 1 < 2 -k, we have


K(TT) < 2-k+1 + c,

as desired. O








We conjecture that a bound of (4)" would characterize random closed sets in terms of

prefex-free complexity. It would suffice, then, to show that (4)" is a lower bound and that

this bound implies randomness.

We also still seek upper bounds for II or closed II classes towards establishing

prefix-free complexity characterizations of these classes. It seems plausible that II classes

are more compressible (i.e. necessarily have smaller lower bounds) than random closed sets

and we would like to explore this notion further.

3.7 Other Notions of Randomness for Closed Sets

Other notions of randomness that depend on different probability measures, or the

inclusion of trees with dead ends in the encoding, might also be considered.

3.7.1 Randomness with Regular Probability Measures

For any regular measure v, we can define the notion of a v-Martin-L6f test and the

resulting notion of a v-Martin-L6f -random (or just v-random) real. It is easy to see that

v-random reals exist for any v and hence v-random closed sets exist. The results on ghost

codes and joins will hold for any regular measure. The corresponding version of Lemma

3.3.8 will hold if v is regular with bo and bl < 1. The proofs of Theorem 3.4.14 and

Corollary 3.4.15, that no subset of a measure-zero II class is random, also go through

under this assumption.

Some of the results in this chapter may also be obtained for vf where f(a i) <

for i = 0, 1. For example, with respect to vf, a random closed set will have no isolated

elements and it will alv--x contain a random element. For any regular measure, either the

leftmost or the rightmost path will be nonrandom, since either b0 + b2 > or bl + b2 The

proof of Theorem 3.4.19 that every random closed set has measure 0 seems to require, for

vf-randomness, that f(a 2) < 1 for all a.

3.7.2 Randomness with the Inclusion of Trees with Deads Ends

Returning to the notion of randomness which allows trees with dead ends, let b3 now

be the probability that a given node has no extensions and let the probability be regular

as above. Then a simple recursion shows the probability p of a given closed set being







empty satisfies the equation


p = b3 + (bo + b)p + b22.

Solving for p, we obtain

(p 1)(b2p b3) 0.

Thus either p = 1 or p = It follows that if b2 < b3, then p 1, that is, almost every

closed set is empty. Suppose now that b3 < b2 and let p, be the probability that a given

tree T has no paths of length n. Then it can be seen by induction that pi < b for all n.

That is, p b3 < b and then


b2

Hence in this case, the probability that a given closed set is empty is b3 < 1. In this case,

one could presumably develop a notion of a random tree and a random closed set and

explore the properties of random closed sets.

3.8 Random Closed Sets and Effective Capacity

In this section we will consider, given a closed set Q, the probability that a randomly

chosen closed set meets Q. This probability is given by pi(V(Q)), where V(Q) is a sub-

basis set for the hit-or-miss topology on C (as given in section 3.3.1) and pi is a given

probability measure. If we define Td(Q) to be precisely p*(V(Q)), it turns out that Id is

a computablee) capacity, in the sense defined below; furthermore, the converse also also

holds. That is, if T is a computablee) capacity, then there is some probability measure ld

for which T = 7d. We then explore the capacities of random and effectively closed sets,

under the uniform measure (i.e. bi = for all i in Definition 3.2.6). This is joint work with

Douglas Cenzer.

3.8.1 Computable Capacities

Definition 3.8.1. A capacity on C is a function T: C -- [0, 1] with T(0) = 0 such that

(i) T is monotone increasing, that is,


Qi C Q2 (Q1) <(Q2)







(ii) For n > 2 and any Qi,... Q E C


T7( Q) < i{(- i)'+7T(U Q):0 o$/ I c ,2,..., n}}.
i=1 itI
This is the alternating of infinite order property.

(iii) If Q = nQ, and Q,+1 c Q, for all n, then T(Q) = lim,,,T(Q,).

We will assume, unless otherwise specified, that T(2N) = 1 for a given capacity T.
Definition 3.8.2 (Computable Capacities). A capacity T is computable if it is com-

putable on the family of clopen sets.

It follows that the capacity of any II class is upper semi-computable. Finally, the

following notational definition will be used throughout.

Definition 3.8.3 (Td(Q), for Q e C). Suppose Q E C. Define Td(Q) := p*(V(Q)), where
V(Q) is a sub-basis set for the hit-or-miss topology on C (as given in section 3.3.1) and pl

is a given probability measure.

That is, Td(Q) is the probability that a randomly chosen closed set meets Q.

We now show, in the following two theorems, that a computable capacity is alv--,

obtainable from, or a consequence of, a computable probability measure pi for some d. The

following theorem, in particular, is well-known. For details on capacity and random set

variables, see [73].
Theorem 3.8.4. If p* is a computablee) probability measure on C, then 7d is a (com-

putable) capacity.

Proof. This is easily verified. Certainly T(0) = 0. The alternating property follows

by basic probability. For (iii), suppose that Q = nQ is a decreasing intersection.
Then by compactness, Q n K / 0 if and only if QT n K / 0 for all n. Furthermore,

V(Q,~+) C V(Q,) for all n. Thus


T(Q) -= (V(Q)) -= (nV(Q,)) = lim,1(V(Q,)) = lim,mT(Q).







The computability of T is easily verified. That is, for any clopen set I(al) U .. U I(ak)

where each ai E {0, 1}", we compute the probability distribution for all trees of height n

and add the probabilities of those trees which contain one of the ci. O

This result has a converse, due to Choquet. See [73] for the general result.

Theorem 3.8.5. If T is a computable capacity, then there is a computable measure p on

the space of closed sets such that T = d.

Proof. Given the values T(U) for all clopen sets I(al) U ... U I(ok) where each ai E

{0, 1}", there is in fact a unique probability measure Pd on these clopen sets such that

T = d and this can be computed as follows.
Suppose first that T(I(i)) = ai for i < 2 and note that each a, < 1 and ao + a1 > 1

by the alternating property. If T = d, then we must have d((0)) + d((2)) = ao and

d((1)) + d((2)) = ai and also d((0)) + d((1)) + d((2)) = 1, so that d((2)) = ao + al 1,

d((0)) = 1 al and d((1)) = 1 ao. This will imply that T(r) = d(r) when rl = 1. Now
suppose that we have defined d(r) and that r is the code for a finite tree with elements

jo,... Un= a and thus d(r i) is giving the probability that a will have one or both

immediate successors. We proceed as above. Let T(I(ai)) = ai T(I(a)) for i < 2. Then

as above d(T-r2) = d(r) (ao + al 1) and d(- i) = d(r) (1 ai) for each i. O

3.8.2 Regular Measures and Capacities of Closed Sets

In this section all results are with respect to pj fixed as the uniform measure (i.e.

the regular measure with bo b b2 = see Definition 3.2.6). With this measure, we

will consider the capacities of random closed sets and effectively closed sets. We -iv that

Q E C is pI -random if XQ is (\! ,rtin-Lof ) random with respect to the measure Pd.

Theorem 3.8.6. For the regular measure Pd with bi = if R is a p/-random closed set,

then Td(R) = 0.

Proof. Fix d as described above so that d(a i) d() and let p~* = We will

compute the probability, given two closed sets Q and K, that Q n K is nonempty. Let


Q. = JI(a7): a CIO, 1} & Q n J(a) / 0}







and similarly for K,. Then Q n K / 0 if and only if Q, n K, / 0 for all n. Let p, be

the probability that Q, n K, / 0 for two arbitrary closed sets K and Q, relative to our

measure i*. It is immediate that pi = since Q n K1 = 0 only when Qi I(i) and

K1 = 1(1 i). Next we will determine the quadratic function f such that p,+ = f(pn).

There are 9 possible cases for Qi and K1, which break down into 4 distinct cases.

Case I: There are two chances that Qi n K1 = 0.

Case II: There are two chances that Q = K1 = I(i), so that Q,+1 n K,+I / 0 with

probability p,.

Case III: There are four chances where Qi = 2" and K = I(i) or vice versa, so that

once again Q,+1 n K,+I / 0 with probability pn.

Case IV: There is one chance that Qi = K1 = 2N, in which case Q,+1 n Ki / 0

with probability 1 (1 p,)2 = 2p, p2. This is because Q,+1 n K,+ = 0 if and only if

both Q,+1 n I(i) n K,, = 0 for both i = 0 and i =1.

Adding these cases together, we see that

6 1 8 12
Pn+1 = Pn + (2p p2) Pn 9P
9 9 9 9

It follows that the sequence (pn)E, is computable and we will see that the limit is zero.
Let f(p) = _p _- p2. Elementary calculus shows that f has fixed points at p -1 and

p = 0 and that for 0 < p < 1, 0 < f(p) < p. Since po = it follows that the sequence

(p }) is monotonic decreasing. Thus the limit exists and limnp = 0 (since it must be a
fixed point of f).

Thus the probability that Q n K / 0 equals limp, = 0. Next we will obtain a

Martin-L6f test to prove our result.

For each m, n E u, let


Am,n = {Q : p*({K : Km n Qm / 0}) > 2-n}.

Let Cm be the number of trees of height m without dead ends.







For each Q E Am,, there are 2-"Cm possible choices for Km such that Km n Qm / 0

and thus at least Dm,n = ()2-"lp*(Am,nT)C2 choices for (K,Q) E C x C such that

Km n Qm / 0 with Q E Am,, (since each pair might be counted twice).

Now define a computable sequence (m,),E,, so that pm, < 2-2n-1. Then


Dr ,,n 2-(n+l) p*(Am,n)
C2
It follows that


p*(Am ,n) < 2n+lP p < 2n+1-2n-1 2-".

Letting


Sn Ur>nAm,r

it follows that p*(S,) < 2-" as well.

Now let R be a random closed set. The sequence (Sn,),, is a computable sequence

of c.e. open sets with measure < 2-", so that there is some n such that R S,. Thus for

r > n, p*({K : Km, n Rm,, 0}) < 2-r and it follows that



*({K : K nR / 0}) limp*({K : K n Rm, / 0}) = 0.

Thus Td(R) = 0, as desired. D

This result seems to depend on the measure. For different regular measures, the

capacity of a random closed set can have different values.

Theorem 3.8.7. For the regular measure Pd with bi = there is a measure zero IIo class

Q such that Td(Q) > 0.

Proof. First let us compute the capacity of X, = {x : x(n) = 0}. For n = 0, we have

Td(Qo) = Now the probability Td(X,+1) that an arbitrary closed set K meets X,+I may
be calculated in two distinct cases. Let K& be as in the proof of Theorem 3.8.6.

Case I If Ko = 2N, then Td(X,+1) =1 (1 d(X,))2.







Case II If Ko = I((i)) for some i < 2, then Td(Xn+) = T(X,).

It follows that d(Xn+l) = 7(X,) + (2d(X,) (7(X,))2) (X,)) ((X,))2.
Now the function f(p) = p lp2 has the property that f(p) > p for 0 < p < 1 and

f(l) = 1. Since Td(X,+1) = f(l(X,)), it follows that lim,Td(X,) = 1 and is the limit of a
computable sequence.
For any a = (no, n1,... n ), with no < ni < .. < nk, similarly define X, { x : (Vi <
k)x(ni) = 0}. A similar argument to that above shows that lim'Td(X,-)/Td(X,) = 1.
Now consider the decreasing sequence ck 2= k+2 ith limit 1. C'!-- i = no such
that Td(X,) > co and for each k, choose n = nk+1 such that Td(X(no,...,nk,n)) > Ck+1.
This can be done since Ck+1 < Ck. Finally, let Q n= X(no,..., ). Then Td(Q)
limkTTd(X(o,...,nk)) >- limkCk .
This result can easily be extended to any bounded measure.







CHAPTER 4
RANDOM CONTINUOUS FUNCTIONS

The following chapter is joint work with George Barpalias, Douglas Cenzer, Jeffrey

B. Remmel, and Rebecca Weber and will appear in the Archive for Mathematical Logic as

an article entitled Al,..., I///.:: Randomness of Continuous Functions [8]. A preliminary

version of this research was originally presented at the Third International Conference

of Computability and Complexity in Analysis in Gainesville, Florida in 2006 by J. B.

Remmel. This preliminary work was published in the referred conference proceedings

as Random Continuous Functions (P. Brodhead, D. Cenzer, and J. B. Remmel) in

Proceedings of CCA 2006 (D. Cenzer, R. Dillhage, T. Grubb and Klaus Weihrauch, eds.),

Information Berichte, FernUniversitit (2006), pages 76-89 and in Springer Electronic

Notes in Theoretical Computer Science, Elsevier Science 167 (2007) [15].

Portions of this work were also presented by P. Brodhead at the AMS Fall 2006

Eastern Sectional Meeting (October 2006, Storrs, CT) and the Conference on Logic,

Computability, and Randomness (January 2007, Buenos Aires, Argentina).

4.1 Overview

In C!I Ipter 3, we considered a notion of randomness for closed sets. We do the same

for continuous functions here. An introduction to randomness for reals is provided in

Section 3.2.

This chapter is organized as follows. In Section 4.2, we provide a definition of

randomness for continuous functions and show that it is sound. In Section 4.3, we prove

various results for (images of) random continuous functions perfectness, non-injectivity,

and instances of non-surjectivity; we also study images of computable elements. In

Section 4.4, we tie random closed sets to random closed functions through images: inverse

images of 0" are random closed sets, but images, in general, are not. Continuing on the

theme of inverse images of 0, in Section 4.5 we consider pseudo-distance functions. In

Section 4.6, we briefly consider how the results of Chapters 3 and 4 can be relativized for

n-randomness. Finally, in Section 4.7, we describe some directions for future research.







4.2 Definining Randomness for Continuous Functions

A function F : 2" 2" is continuous iff it has a closed graph. It seems reasonable,

then, to define continuous function f to be random, iff the its graph Gr(F) = {x y

y F(x)} is random. However if [T] is the graph of a function and a E T has even

length, then we must have a^0 E T and a^ l T. This means that the family of closed

sets which are the graphs of functions has measure 0 in the space of closed sets and hence

a random closed set will not be the graph of a function. We need, therefore, a different

method to define randomness for continuous functions. We do this below.

4.2.1 Representing Functions

Given a continuous function F : 2" 2N, we are interested in representing it in such

a way so as to be able to consider a notion of algorithmic randomness. We show below

that any such function F may be represented by infinitely many representing functions of

the form f : {0, 1}* {0, 1, 2*. This will allow us, in the following section, to be able to

represent continuous function as elements of 3', so-called representing sequences, and to

consider a such a function as random if it possesses a random representing sequence.

Information output, the key to representation. For any continuous function

F on 2" and any a {0, 1}*, there is a natural number n and binary string T of length

n such that for all u E I(a), F(u) [n = T. In particular, F(u)(n) = r(n) for every such

u. In general, the length of a may be much larger than n, so we may have to extend a by

several bits to get uniformity of F(u) [(n + 1) within the interval around a's extension.

Representing functions. Taking the above into consideration, we may recursively

represent any continuous function F : 2" 2" by some function f : {0, 1}* -- {0, 1, 2}*

as follows. Suppose F is given. Let f(0) = 0. For al = m + 1, having defined f(a [i) = ec

for all i < m, let p = (n,..., nk) be the result of deleting all 2s from (el,..., e,). If for all

u I(a), F(u) [k pj, j E {0, 1}, we may let em+l j. If not we must have em+l = 2;

even if so we allow em+l = 2.







The canonical represention. Notice, from the above, that any continuous F has

infinitely many representing functions f : {0, 1}* -- {0, 1, 2}*. The representation which

uses as few 2s as possible we shall call the canonical representation.

4.2.2 Representing Sequences

We want to code the representing function as an element of 3" to discuss its algo-

rithmic randomness. To do so, first enumerate {0, 1}* = {0} as ao, oa,..., ordered first

by length and then lexicographically. Thus ao = (0), a1 = (1), 2 = (00), etc. We define

representing sequences below.

Definition 4.2.1. (i) (INF, Rem2) Let INF equal the set of y {0, 1, 2}" such that

{n : y(n) / 2} is infinite and, for y E INF, let Rem2(y) be the result of removing
from x all occurrences of 2.

(ii) (Representing functions) A function f : {0, 1}* {0, 1, 2} represents a function
F : 2N 2N if for all x E 2", the sequence y, defined by y(n) = f(x [n) belongs to

INF and Rem2() F(x).

(iii) (Representing sequences) A sequence r E {0, 1, 2} represents the continuous function

F (written F = F,) if the function fr : {0, 1}* {0, 1,2}, defined by f,(a,) = r(n),

represents F.

(iv) (Labelled 2"-trees) Given a representing sequence r, the function f, gives rise to a
labelled 2"-tree. We attach, or associate, the value of f,(a) with each node a.

Example 4.2.1 (The 2"-tree for the Identity, A Geometric Intrepretation). The identity

function can be represented by placing an e on any node a which ends in e. This can also

be pictured geometrically as representing the graph of F as the intersection of a decreasing

sequence of clopen subsets of the unit square. Initially the choice of f((0)) and f((1))

selects from the 4 quadrants. That is, for example, f((O)) = (0) = f((1)) implies that the

graph of F is included in the bottom half of the square and f((0)) = 0 and f((1)) = (1)

implies that the graph excludes the lower right hand quadrant. Successive values of f

continue to restrict the graph of F in a similar fashion.







4.2.3 A Sound Definition

In this section we define a measure /** on the space of functions F : 2" 2" that

allows us to define the notion of randomness for functions on 2N. In short, a function

is random if it possesses a random representing function, or equivalently, a random

representing sequence. We will show that no canonical representing function can be

random, so that necessarily, the definition of randomness for functions is in this existential

format. It may be, however, that no continuous function has a random representing

function. We show that this is not so. In fact, we show that every random representing

function is continuous. Clearly then, random continuous functions exist and, in fact, A

random continuous functions exist. Therefore the definition is sound and the structure

begins to manifest itself as rich.

The Measure for Randomness. The measure which is used to define random-

ness for continuous functions is the Lebesgue measure on the space 3" of representing

sequences. Thus for each new bit of input, there is equal probability 1 that f, gives a new

output of 0 for Fr, gives a new output of 1 for Fr, or gives no new output for Fr. This

measure now induces a measure, p** v, on the space F of continuous functions.

Definition 4.2.2. A function F : 2 2" is random if there is a representing sequence

r E 3 for F that is random with respect to the measure **.

We first show that every random representing function is continuous. The following

lemma is needed.

Lemma 4.2.3. Let E be a finite set and let Q C E" be a II class of measure 0. Then no

element of Q is Martin-L6f random.

Proof. Let E {0, 1, 2} without loss of generality. Let Q = [T] where T C {0, 1, 2}* is a

computable tree (possibly with dead ends). For each n, let T = T n {0, 1, 2}" and let


Q. = UJ{I(a ) a c T }.

Let g(n) = p(Q,) = -T. Then g(n) is a computable sequence and


lims.g(n) = p(Q) = 0.







This Martin-L6f test shows that Q has no random elements. (As observed by Solovay, it

is sufficient to have a computable sequence approaching zero rather than the stricter test

with a sequence of measures g(n) < 2-".) O

Theorem 4.2.4. (i) The set of representing functions for total functions has measure

one.

(ii) Every random function is continuous.

Proof. (i) Let r E 3 and suppose that f, does not represent a total function. Then

there is some x E 2" and some r E {0, 1}* such that f,(x n) = r for almost all n.

Without loss of generality we may assume that 7 = 0. Let A be the set of functions

f : {0, 1}* {0, 1}* such that f(a) = 0 for arbitrarily long strings a and let p = p**(A).

Then certainly p < 9, since if r(0) and r(1) are both in {0, 1}, then f, A. Considering

the 9 cases for the initial choices of f((0)) and f((1)), we see that

4 1
S= p + [1 (1 -p)2],
9 9

so that p2 + p 0, which implies that p 0. (That is, there are 4 cases in which

If((i)) = 1 for i = 0,1 so that immediately f A, there are 4 cases in which only one

of f((i)) = 0, in which case the remaining function g, defined by g(a) = f(i a) must be
in A, and there is one case in which f((i)) = 0 for i = 0, 1, in which case at least one of

the remaining functions must be in A.) Consequently, the set of representing functions for

total functions has measure one.

(ii) Observe that A is a II class, since fr E A if and only if (Vn)(3a e {0, }t")f,(a) =

0. It follows from Lemma 4.2.3 that no representing function on 2* for a random function

on 2" can be in A. As all functions representing partial functions on 2" occur in A, it

follows that every random function is total. Since the graph of a total function is a closed

set, it follows that random functions are continuous. O

Now the set of Martin-L6f random elements of {0, 1, 2}" has measure one and there

exists a A0 Martin-L6f real. Hence we have the following.

Theorem 4.2.5. There exists a random continuous function which is A computable.








We also first observe that any continuous function will have a representation which is

not random. In fact, the canonical representation itself can never be random.

Proposition 4.2.6. For any continuous function F, the canonical representation is not

random.

Proof. The idea is that whenever the canonical representation labels a node a with 2,

then the two labels on the successor nodes a-0 and a l cannot be both 0, or both 1.

Thus we have the following Martin-L6f test. Assume by way of contradiction that r is

random and canonical. Let S, be the set of r E 3" such that r has at least e occurrences

of 2 and such that, for the first e occurrences of 2 in r, the corresponding successor values

are not both 0 or both 1. Since r is random, it must have infinitely many occurrences of 2

and since r is canonical, it must belong to every S,. But each S, is a c.e. open set and has

measure < (7), so that no random sequence can belong to every Se. O

The theorem, in fact, demonstrates the need for the existential part of definition of

random functions. In the following sections we will obtain some additional properties of

random continuous functions.

4.3 Random Continous Functions and Images

4.3.1 Perfect Images, in every instance

In this section we show that all random continuous functions alv-i-i have perfect

images. This is a consequence of the following theorem.

Theorem 4.3.1. If F is a random continuous function, then the image F[2N] has no

isolated elements.

Proof. Let f be the random representing function for F and let Q = F[2]. Suppose by

way of contradiction that Q contains an isolated path y. Then there is some finite -r C y

such that y is the unique element of I(r) n Q. Fix a such that f(a) = -.

For each n, let S, be the set of all g e F such that for all pi, p2 E {0, 1},

1. g(a pl) is compatible with g(a-p2),

2. T z g(a pi), and

3. T E g(a^p2)







Then for any each m < n and each p E {0, 1}", we are restricted to at most 7 of the 9

possible choices for f(p 0) and f(p l). This same scenario applies for all p E {0, 1}-1,

so that in general, p(S,) < (7)2" 1

Now for each n, S, is a clopen set in F and thus the sequence So, Si,... is a Martin-

Lof test. It follows that for some n, F S,. Thus there are two extensions of a of length

n which have incompatible images, contradicting the assumption that y was the unique

element of Q n I((). E

It follows that the image of a random continuous function is perfect and has contin-

uum many elements.

4.3.2 Non-injective Images, in every instance

In this section we show that no random continuous function is injective. We will use

the following theorem en route.

Theorem 4.3.2. For any a E {0, 1}*, the probability that the image of a continuous

function F meets I(a) is alv--v- > .

Proof. The proof is by induction on lal. Without loss of generality, we assume that
a = 0". For each n > 0, let q, be the probability that F[2N] meets I((0")). Let f be the

representing function for F. For n = 1, there are 9 equally probable choices for the pair

f((0)) and f((1)), breaking down into 4 distinct cases.

Case 1. If f((0)) = (1) = f((1)), then F[2"] does not meet I((0)). This occurs just

once.

Case 2. If f((0)) = (0) or f((I)) (0), then F[2"] meets I((0)). This occurs in 5 of

the 9 choices.

Case 3. If f((i)) 0 and f((1 i)) (1), then F[2"] meets I((0)) if and only if

F() [2"] meets I((0)). This occurs in 2 of the 9 choices, with probability qi.

Case 4. If f((0)) = 0 = f((1)), then F[2"] meets I((0)) if at least one of F() [2"]

meets I((0)). This occurs in 1 of the choices, with probability 1 (1 q,)2. That is, F[2N]

fails to meet I((0)) if both F(o)[2"] and F(o)[2"] fail to meet I((0)).







Putting these cases together, we see that


5 2 1
qi + 9qi+ (2qi q),
99 9

so that qi satisfies the quadratic equation

x2 + 5x 5 = 0.

Thus qi is the unique solution in [0,1] of this equation, that is,

v'45 5
5-5
2

which is indeed > .75.

Now let q, = q and let q,+1 = p. Once again we consider the 9 initial choices, now

breaking down into 6 distinct cases.

Case 1. If f((0)) = (1) f((I)), then F[2"] does not meet I((0"+1)). This occurs

just once.

Case 2. If f((0)) = (0) f((1)), then F[2"] meets I((0"+)) if and only if at

least one of F(o) and F(1) meets I((0")). This occurs just once, and with probability

1-(1- q)2 2q-q2.

Case 3. If f((i)) (0) and f((1 i)) (1), then F[2"] meets I((0"n+)) if and only if

F(4)[2"] meets I((0O)). This occurs in 2 of the 9 choices, with probability q.

Case 4. If f((i)) 0 and f((1 i)) (1), then F[2"] meets I((0"+')) if and only if

F() [2"] meets I((0+'1)). This occurs in 2 of the 9 choices, with probability p.

Case 5. If f((0)) = 0 = f((1)), then F[2"] meets I((0"+')) if at least one of F(i)[2"]

meets I((0"+1)). This occurs just once, with probability 1 (1 p)2.

Case 6. If f((i)) 0 and f((1 i)) (0), then F[2"] meets I((0"+')) if at least one

of the following two things happens. Either F(i)[2"] meets I((0+'1)), or F(li)[2N] meets

I((0")). This occurs in 2 of the 9 choices, with probability 1 (1 p)(l q).








Putting these cases together, we see that


2 12 2 2 12
p p- p pq + q q,
3 9 9 3 9

so that p = qn,+ satisfies the equation


p2 + 3p + 2pq 6q + q2 = 0.

We note that for p = q, the solutions are p = q = 0 and p = q = This explains the value

Sin the statement of theorem.

Now assume by induction that q > |. Suppose by way of contradiction that p < |. It

follows that
9 9 3
S+ + -q 6q+ q2 > 0.
16 4 2
Simplifying, this implies that 16q2 72q + 45 > 0. But this factors into (4q 3)(4q 15)

and is only > 0 when either q < 4 or q > -. Since the latter is impossible, we obtain the

desired contradiction that q < |. D

Theorem 4.3.3. No random continuous function is injective.

Proof. Let p be the probability that an arbitary continuous function F is injective. It

follows from Theorem 4.3.2 that there is a chance that F has a zero in I(0) and also

in I(1), so that p < i7. Now in general, if F is injective, then it must be injective when

restricted to I(0) and when restricted to I(1). It follows that p < p2, which happens only

for p = 0 and p = 1, given that 0 < p < 1. Since p ,< it follows that p = 0, as

desired. This can be reformulated as a Martin-L6f test as follows. First we observe that

F is injective if and only if, the images of each pair of ldi-i ;il 1 intervals I(a) and I(r) are

disjoint. Let

D(, 7) = {F : F[I(a)] n F[I(r)] =0.

Then D(a, 7) is uniformly c.e. since F[I(a)] n F[I(r)] 0 if and only if there exists n such

that for all extensions a' of a and r' of 7 of length n, f(a') and f(7') are incompatible.

Now let

Sm, {F: (Va, 7 {0,1}m)(a7 / T 7 F E D(a, T)}.








It follows from the observation above that F is injective if and only if F E Pm S,. The

argument above shows that p**(SI) < T6 and that p**(S,m+) < l**(S,)2 and hence

7


It follows that {Sm : m e u} is a Martin-Lof test and therefore no random continuous

function may belong to every S, and hence no random continuous function can be

injective. E

4.3.3 Non-surjective Images, in instances

In this section we show that random continuous functions are not necessarily onto.

Definition 4.3.4 (F,, the restriction of F to I(a)). For any function F on 2" and any

a E {0, 1}*, define the restriction F, of F to I(a) by


F,(x) = F(a^-x).

Clearly any such restriction of a random continuous function will be random, but

more can be said. Recall van Lambalgen's theorem, Theorem 3.2.12.

Proposition 4.3.5. F is a random continuous function if and only if the functions F(o)

and F(1) are relatively random.

Proof. Let r represent F. Suppose first that F is random. It follows, as in Corol-

lary 3.3.12, that F(o) E F(1) is random and hence F(o) and F(1) are relatively random by van

Lambalgen's theorem.

Next suppose that F(o) and F(1) are relatively random and let ri represent F(i) for

i = 0, 1. Let d be any martingale, which we think of as betting on r. Then for i = 0, 1,

we can define a martingale di with oracle rl-i as follows. We will give the definition for do

and leave dl for the reader. Given a = ro(0),..., ro(2 + q 2) where 0 < q < 2P, use rl

to compute T = r(0),..., r(2"+1 + q 2) and then define di to bet in the same proportion

as d. That is, di(a j'j)/di(a) = d(>- j)/d(r) for j < 3. Thus for any node on the left side

of the labelled tree for F, do is making the same bet on the next label that d would have

made, and similarly for di and the right side.








Since the F(i) are relatively random for i = 0, 1, it follows that di does not succeed

and hence there exist upper bounds Bi for {di(ri [n)},E. But it follows from the above

definitions of di that for any p,


d(r [2P 2) do(ro 2P 1) d(ri [2P 1).


This is because the martingale d alternates using do and dl and the result can be viewed

in each alternation as multiplying the capital by some factor. Then in general, for 0 < q <

2P,

d(r [2+ + q 2) do(ro [2P + q 1) di(r [2 1)

and

d(r [2P+ + 2 + q 2) = do(ro [2" 1) d(r, 2P + q 1).

It follows that Bo B1 is an upper bound for {d(r[k) : k E N}, so that d does not succeed

on r. E

Corollary 4.3.6. A random continuous function is not necessarily onto.

Proof. It follows from Proposition 4.3.5 that, for any r e {0, 1}*, there is a random

continuous function with image C I(r). Thus a random continuous function is not

necessarily onto. O

4.3.4 Images of computable elements

In this section we will show that the image of computable element under a random

continuous function is necessarily non-computable. In fact, it is random. We need the

following proposition.

Proposition 4.3.7. Suppose A C B are two finite sets of symbols. Given X E BN,

let X E A" be the sequence obtained by deleting all symbols in B A from X. If X is

1-random, then X is 1-random.

Proof. Given X, X as in the proposition, suppose X is not random and let d be a

constructive martingale on A" that succeeds on X. We will construct a martingale d on

B" that succeeds on X. Essentially, d will keep its capital constant on symbols in B A;

it will bet according to d, repeating its bets after bits which hold symbols from B A.







Define d(A) = d(A), and for a E B* and a the corresponding string of A*,


d(a; x) d(c7d ) xcA
d(a) x B A

The function d is clearly constructive, since d is. To show d is a martingale, consider

the sum

d(,7-x) dC r-x) d(,7) + Y d(17)
d(axx)d (
xEB -x A B-A

Sd() Sd( ( + d(a)lB Al d(a)[AI + IB AI].
xECA
It remains to show that d succeeds on X. However, that is clear, as on bits which

are in X but not X, d keeps its capital constant, and on bits from X, it acts exactly as d

would. Therefore since d succeeds on X, d succeeds on X and X is nonrandom. O

It is easy to see that, for any random continuous function F and any computable real

x, F(x) is not computable. This follows from our next result.

Theorem 4.3.8. If F is a random continuous function, then, for any computable real x,

F(x) is not computable.

Proof. Suppose that F is random and let x and y be computable reals. For each n, let


S, = {G: G(x[n) y[n}.

Then So, Si,... is an effective sequence of c.e. open sets in F, and an easy induction

shows that p**(S,) = (2/3)". This is a Martin-Lof test and it follows that F S, for some

n, so that F(x) / y. O

We now strengthen this result to show that the image of a computable element is

random.

Theorem 4.3.9. If F is a random continuous function, then, for any computable real x,

F(x) is a random real.







Proof. Suppose that F is random with representing function fr, let x be a computable

real and let y = F(x). Define the computable function g so that, for each n,


7g(,) X [rn.

By the Von-Mises(-C'lnl !- Wald Computable Selection Theorem, the subsequence

z(n) = r(g(n)) is random in {0, 1,2}". Now y = F(x) may be computed from z by

removing the 2's. Thus F(x) is random by Proposition 4.3.7. E

We note that Fouche [45] has used a different approach to randomness for continuous

functions connected with Brownian motion, first presented by Asarin and Prokovskiy [5],

and has shown that, under this approach, it is also true that for any random continuous

function F, F(x) is not computable for any computable input x.

It follows that a random function F can never be computably continuous and hence

the graph of F is not a IIt class.

4.4 Random Closed Sets arising from random continuous functions

4.4.1 A Positive Result: Inverse Images of 0'

In this section we prove that for any random continuous function F, the set Z(F)

{x : F(x) = 0} is a random closed set. For any subset S of C, let Zs = {F F : Z(F) e
S}.

Lemma 4.4.1. For any open set S, p**(Zs) < i*(S).

Proof. It suffices to prove the result for intervals S = I(a). We will show by induction

on a,| that p**(ZI(,)) ()1I4, whereas of course p*(I(a)) = ()1'1. Recall from Corollary

4.4.3 that 0 E F[2"] with probability exactly 3. For a = 1, there are two distinct cases.

Case I Suppose first that a = (i), where i E {0, 1}. Then F E Zs if and only if F has

a zero in I((i)) and has no zero in I((1 i)). Now F has a zero in I((i)) if f((i)) E {0, 2}

and if the restricted function has a zero, which gives probability 2 3= Thus the

combined probability that F E Zs is .

Case II Suppose next that a = (2). Then F E Zs if and only if F has zeroes in both

I((0)) and I((1)). It follows from the argument in Case I that p* (Zs) = .







Notice that Z{} {F : F has no zeroes} has positive measure 1 but p*({0}) = 0.

Now suppose a|| = n and let r = a i; suppose by induction that p**(Ziy,)) <

p*(JI()). Interpret T as the code for a (finite) binary tree and let p E {0, 1}* be the
terminal node of that tree such that i indicates the branching of p. Again there are two

cases.

Case I Suppose first that i c {0, 1}. Then F E ZI(T) if and only if F E ZI(,) and

furthermore F has a zero in I((p i)) and has no zero in I((p 1 i)). It follows as above

that p**(ZI(T)) (**(Z( ) ( )"1.

Case II Suppose next that i = 2. Then F E ZI(T) if and only if F has zeroes in both

I(p 0) and I(p^1). It follows as above that p**(ZI(T)) p**(Z(,)) (n)"+1

An arbitrary open set is a dl-i-P iil union of intervals and thus the desired inequality

can be extended to open sets. E
Theorem 4.4.2. For any random continuous function G : 2" 2", the set of zeroes of G

is either empty or is a random closed set.

Proof. Suppose that G is a random continuous function which has at least one zero, and

let So, Si,... be a Martin-L6f test in C. Then there is a computable function Q such that

Si = UI(a7(i,,)). We may assume without loss of generality p*(Si) < 2-i-2 and that each

Si is not clopen and that, for each i, the intervals I(ar(i,)) are pairwise disjoint. We will

define a Martin-L6f test So, S',... in the space F and use the fact that G must satisfy

{Si}iE to show that Z(G) satisfies {Si}iE.
Fix an interval I(a) in C and let C, = ZI(). Observe that there is a clopen set

B, C 2N and a corresponding finite set To, ... Tk-1 of strings such that B, = Uj
associated with a such that, for any Q E C with code r, rE I(a) if and only if Q C B,

and Q n I(rj) / 0 for all j < k. It follows that C is a difference of IIo classes. That is,

F E C if and only if the following two conditions hold.

(i) For each j, F has a zero in I(r7); by compactness, this is equivalent to -wicing that

for any there is an extension -T {0, 1} of T such that f(r,) E {0, 2}171, where f is the

function on strings representing F.







(ii) F has no zeroes outside of B. Let 2" B = UTAI(T). By compactness, F has no
zeroes outside of B if and only if


(E)(VT e A)(VT' > T) [|T'| f > (E(m)(f(T'[m) 1)]. (4-1)

Note that the measure of Co may be computed uniformly from a given the calculation
from Corollary 4.4.3 that whenever f(a) c {0, 2}1'1, then the probability that F has a
zero in I(a) is exactly |. For each o, we will uniformly compute a c.e. open set S, CF
such that C, C B, and such that p**(B,) < 2 p. **(C,). There are two stages in the
construction of B,.
Stage I: Let U be the set of codes a' for partial functions f' such that 4-1 holds with
f' in place of f, and such that furthermore for every j and f such that f' is defined on all
length-f extensions 7 of T, there is such a T with f'(p) E {0, 2} Vp _- '. It is clear that for
any F E Co, there exists a' E U with F E I(W') and hence

C' C U {I(J') : e U}.

As usual, we may then uniformly compute from U a set U' such that the intervals I(c') for
a' E U' are pairwise dl -ii.il in F and

U{I(uT) : U 7 } {U- JI(t') :t' E U'T}.

For each o' E U', let Q(c') C I(a) be the II class in F consisting of those extensions of a'
which actually have zeroes in each I(-j). Then in fact we have

C = U Q(-') : 7 u'}.

As noted above, we can actually compute the measure p**(Q(c')) uniformly from a'
by expressing Q(c') as an effective decreasing intersection of clopen sets. Thus for
each a', we can compute a clopen set B(a') such that Q(a') C B(a') C I(a') and







p**(B(a') 2- **(Q(a')). Let


B,= U B(a') : U'}.

Then we have C, C B, and p**(B,) < p**(C,).

Finally, for each i, let

S( = U i,n)

Then by Proposition 4.3.7, p**(S ) < 2 p*.(Si) < 2- -1 and therefore there exists some

i such that G Sj, since F is random. But this means that Z(G) Si and hence Z(F)

meets the Martin-Lof test. Thus Z(F) is random, as desired. O

4.4.2 A Negative Result: Images, in general

In general, the image of a random continuous function need not be a random closed

set. To see this, recall the statement of Theorem 4.3.2. That is, given a {0, 1}*, the

probability that the image of a continuous function F meets I(a) is .1'. i-. > 3. We

obtain the following corollary.

Corollary 4.4.3. For any y e 2",

(a) p**({F : yeF[2"]}) ;
(b) there exists a random continuous function F with y e F[2N].

Proof. (a) Let p be the probability that y e F[2N]. It follows that for each a {0, 1}",

the probability that y E F[I(a)], given that f(a) is consistent with y, also equals p. It

follows from the proof of Theorem 4.3.2 that p = .

(b) Since the random continuous functions have measure 1 in C(2"), it follows that

some random continuous function has y in the image. O

This allows us to demonstrate our result.

Theorem 4.4.4. The image of a random continuous function need not be a random

closed set.

Proof. It was shown in Theorem 3.4.12 that a random closed set has no computable

members. Let F be a random continuous function with 0 in the image, as given by

Corollary 4.4.3. Then F[2N] is not a random closed set. O







4.5 Pseudo-Distance Functions

In section 4.4.1 we showed that if A is a random continuous function, then A-'(0")

is a random closed set, if it is nonempty. This motivates the study of pseudo-distance

functions.

Definition 4.5.1. A : 2" 2" is a pseudo-distance function for Q C 2" if A is continuous

and A-1(0w) = Q.

Comment 4.5.2 (Background). The name comes from a modification of the distance

function distQ : 2" [0, 1] for a closed set Q. For x E 2", distQ(x) is defined to be

min{d(x, y) : y Q} where d is a metric on 2" given by

0 if x = y;
d(x,y) =
2-" if n is the least such that x(n) / y(n).

This may be viewed as a computable mapping from 2" x 2" into 2" by representing 0 as

0" and 2-" as 0"'10. From this viewpoint, we may view distQ similarly:


0" if x e Q;
distQ(x) =
0"'10 otherwise, where n is the least such that x n TQ.

Every closed set has a characterization in terms of pseudo-distance functions, as

follows.

Theorem 4.5.3. Q C 2" is closed iff there is pseudo-distance function for Q.

Proof. First suppose that Q is closed and Q = [T]. Define a map A : 2< 2<" by

recursion as follows, with initial mapping 0 v-> 0.

A((a)'0 if a^i E T;

(a) 1 otherwise.

Letting A : 2" 2 be defined so that A(X) is the unique Y E n,[A(X [ n)], we obtain

the required pseudo-distance function.








Now suppose that A : 2" 2" is a pseudo-distance function for Q. This means that

there is some A : 2< 2<" such that A(X) is the unique Y c n[A(X [ n)]. Define the

required tree T (so that Q = [T]) as follows. Put a E T iff A(a) E 0". O

For effectively closed sets, such a characterization in terms of pseudo-distance

functions might not seem as immediate, as an effectively closed set may possess a noncom-

putable distance function. Nonetheless, the we have the following theorem.

Theorem 4.5.4. Q C 2" is effectively closed iff there is computable pseudo-distance

function for Q.

Proof. The follows as in the proof of Theorem 4.5.3, except that A, in both directions, is

now computable. O

It seems plausible that there is a pseudo-distance function characterization for random

closed sets. Looking first to distance functions, every random closed set possess the non-

random distance function distQ. (To see why distQ is non-random, note that if a TQ,

then d is constant on the interval I(a).) If a characterization exists, as in the case for

effectively closed sets, it appears to be not so immediate. On the other hand, we know, by

Theorem 4.4.2, that if Q is a closed set with a random pseudo-distance function, then Q is

random. This leads us to the following conjecture.

Conjecture 4.5.5. Q C 2" is random closed iff there is a random pseudo-distance

function for Q.

4.6 n-Randomness

Recall that a real is n + 1-random if and only if it is 1-random relative to .I (see

Remark 3.2.11). Now the analogue of Theorem 3.2.10 also holds for {0, 1, 2} for our

measures p* or /**. Therefore our approach also allows us to define the notion of n-

random continuous functions (or n-random closed sets), as follows. A continuous function

F : 2N 2N (or closed set) is defined to be n-random if and only if it is Martin Lof

random relative to I' '. One can then easily relativize the results of previous sections to

obtain similar results for n-random continuous functions.








4.7 Future Work

We close this chapter noting that random Brownian motions as studied by Fouche [45]

are a special case of random continuous functions on the real line. This is another area

of interest for further research. That is, we would like to extend the notion of a random

continuous function to functions on the real unit interval [0, 1] and the real line R by

representing functions again in terms of the images of subintervals. We conjecture that

a random continuous real function cannot be left or right computable and in fact, not

weakly computable. We also conjecture that a random continuous function is nowhere

differentiable.







CHAPTER 5
CONTINUITY OF CAPPING IN CBT

The following chapter is joint work with Angsheng Li and Weilin Li and will appear

in the Annals of Pure and Applied Logic as an article entitled C 'ii<.;,.;/ of Capping in

CbT [16]. For this project, P. Brodhead acknowledges support from the National Science
Foundation (under grant number 0714151 as the principal investigator) to conduct this

joint work in Beijing during the summer of 2007 as part of the East Asia and Pacific

Summer Institutes (EAPSI).

This work was presented by P. Brodhead at the First Joint AMS-NZMS Meeting

(December 2007, Wellington, New Zealand).

5.1 Introduction

Given sets A, B C w, we i- that A is Turing reducible to B, if there is an oracle

Turing machine -i'*-, such that A 4= B (denoted by A T B). Furthermore, if the bits

of oracle queries are bounded by a computable function, then using recent nomenclature

from Soare [88] we ;i- that A is bounded Turing reducible to B, written A
literature often refers to this as the weak truth table reducibility, written
and a bounded Turing (or bT, for short) degree is the equivalence class of a set under

the Turing reductions and the bounded Turing reductions respectively. A degree is called

computably enumerable (c.e.), if it contains a c.e. set. Let C and CbT be the structures

of the c.e. degrees under the Turing reductions and the bounded Turing reductions

respectively.

During the past decades, the studies of the structures C, CbT focused on that of the

algebraic properties, leading to n1 iP r achievements such as the decidability results of

the Zi-theory of C, and the E2-theory of CbT (Ambos-Spies, P. F iP r, S. Lempp and M.

Lerman [3]), and the undecidability results of the E3-theory of C (Lempp, Nies, and

Slaman [63]), and of the E4-theory of CbT (Lempp and Nies [62]). This progress brings the

decidability problems of the E2-theory of C, and the E3-theory of CbT into sharper focus,

for which new ingredients are welcome.








In the recent years, the study of the computably enumerable degrees has focused on

Turing definability in the structure C. For instance, Slaman asked in 1985 if there are any

c.e. degrees that are incomplete and nonzero which are definable in the c.e. degrees C.

This question of Slaman is still open. A natural approach to this problem is to find some

definable substructures of C that have nontrivial minimal/maximal and/or least/greatest

members. As a result, topics such at the continuity of the c.e. degrees, started by Lachlan

in 1967, have renewed interest.

In this chapter, we demonstrate the continuity of capping in CbT. This refutes the

existence of a maximal non-bounding degree. It also brings the question of the E3-theory

of CbT into sharper focus, as the statement is one of Es-complexity.

To motivate these ideas further, we begin with a brief history of relevant continuity

results in Section 5.2. This motivates our main result and method of proof, described in

Section 5.2.2. The main substance of the proof involves demonstrating that Theorem 5.2.3

holds, that local noncappability holds in CbT. Sections 5.3-5.6 are devoted to proving this

theorem.

5.2 Continuity Results

The Turing degrees form an upper-semilattice; that is, each pair of elements a, b

has a least upper bound (or join) a V b. A greatest lower bound a A b may or may not

exist. Given a (bounded) Turing degree a, we -iv- that a is capable if a (bounded-)

Turing degree b / 0 exists such that a A b = 0. We -iv- that a is cuppable if there is a

degree b / 0' such that a V b = 0'. The study of continuity properties the (bounded)

Turing degrees is with respect to meets and joins, and related notions such as capping and

cupping.

5.2.1 Continuity Results in C

In 1979, Lachlan [60] proved the existence of a non-bounding c.e. degree namely, a

non-computable c.e. degree with no minimal pair below it; Cooper demonstrated in 1974,

that no high c.e. degree can be non-bounding [30]. Returning to the Slaman question,

Downey, Lempp, and Shore demonstrated in 1993 that Cooper's non-bounding degree








could be made high2 [41], leading to the possibility of a maximal non-bounding c.e. degree.

Such a degree could be used to show the existence of a discontinuity, which could be

used to prove the definibility of a c.e. singleton. (To see the former, note that if b > 0

is nonbounding, then for any c.e. a > b, there is some minimal pair r A s below a. Then

either b A r 0 or b As 0, but neither a A r nor a A s equals 0.) However, Seetapun

refuted this possibility (albeit earlier in 1991), demonstrating the non-existence of a

maximal non-bounding degree [84]. Welch proved a complimentary result in 1981: there is

no maximal bounding degree, in the sense that for all a / 0', there is are b, c such that

bAc 0 and b,c % a [95].

Continuing with capping results, Harrington and Soare [51] proved in 1989, the

nonexistence of maximal minimal pairs that is, for any non-trivial minimal pair (a, b) of

c.e. Turing degrees a, b, there exists a c.e. Turing degree c > a such that (c, b) is still a

minimal pair. Seetapun [84] showed an even stronger result, the continuity of capping: for

any c.e. Turing degree b / 0, 0', there exists a c.e. Turing degree a > b such that for any

c.e. Turing degree x, if x < a, then a A x = 0 if and only if b A x = 0.

Ambos-Spies, Lachlan, and Soare [4] proved the dual case of the Harrington and

Soare's result: for any non-trivial splitting x, y of 0', there exists a c.e. degree a < x such

that a V y = 0'. Cooper and Li [31] showed the dual of the Seetapun theorem, that for any

c.e. Turing degree b / 0, 0', there exists a c.e. Turing degree a < b such that for any c.e.

Turing degree x, x V a = 0' if and only if b V x = 0', answering Lachlan's in i, i subdegree

problem.

5.2.2 Continuity Results in CbT and Main Result

There are few results in the topic of definability in the c.e. bT-degrees, CbT. For

instance, we know nothing about the Slaman problem as described in Section 5.1 or the

characterization of definable ideals in CbT. As a matter of fact, little is known of the

continuity properties of the c.e. bT-degrees. An interesting partial result was given by

Stob [90]: in both C and CbT, there are c.e. degrees ao, al > 0, with ao being the unique

complement of al in the interval [0, ao V all, such that if b < ao V al is capable with








any x < ao (i.e. b A x = 0), then x < a. This can be interpreted as a continuity or

discontinuity result in both C and CbT.

The main result of this chapter is the following theorem, an analogue of Seetapun

continuity result.

Theorem 5.2.1 (Continuity of Capping in CbT). For any c.e. bT-degree b : 0, O', there is

a c.e. bT-degree a > b such that for any c.e. bT-degree x, b A x 0 a A x 0.

For this, it suffices to prove Theorem 5.2.3 below, an analog of the Seetapun local

noncappability theorem for the c.e. Turing degrees [84].

Definition 5.2.2 (Local N. i1 '11' ,1ility). A degree b : 0 is locally non-cappable if there

is some a > b such that for all x < a, if x A b = 0 then x = 0.

Theorem 5.2.3 (Local Noncappability in CbT). For any c.e. bT-degree b, if b / 0, 0',

then there is a c.e. bT-degree a > b such that if x < a is noncomputable, then x A b / 0.

Proof of Theorem 5.2.1. Assuming Theorem 5.2.3, we can see Theorem 5.2.1. Given

b, let a be the degree in Theorem 5.2.3. For a fixed x e CbT, by Theorem 5.2.3, we con-

sider only the case where x % a. Clearly if x A a = 0, then x A b = 0. Assume a A x : 0.

We can choose a c.e. bT-degree y such that y / 0 and y < a, x. Therefore 0 < y < a and

by Theorem 5.2.3, we have y A b / 0, so that x A b / 0. Theorem 5.2.1 follows. O

As a consequence of Theorem 5.2.1, no maximal non-bounding degrees exist in the

c.e. bT-degrees. To see this, suppose b : 0 is a degree which bT-bounds no minimal

pairs in CbT, and let a > b be the degree in Theorem 5.2.1. We claim that there are no

bT-minimal pairs below a. Suppose to the contrary that x, y is a minimal pair below a.

Since a Ax = x : 0 and a A y =y 0, we can choose nonzero xl to be below both x

and b, and nonzero yi below both y and b. Then (xi,yi) is a minimal pair below b, a

contradiction. An alternative approach is to use the fact that a maximal non-bounding c.e.

degree is equivalent to a non-bounding degree which is not locally non-cappable [49, 84].

Consequently, by Theorem 5.2.3, no maximal non-bounding degree can exist.

Our approach to the proof of theorem 5.2.3 is similar to Seetapun's approach for the

c.e. Turing degrees, but it is non-obvious due to the computable bounds of oracle query








bits in both the conditions and conclusions of requirements. That is, bT-reductions are

stronger than Turing reductions. So when we require the reductions being built to be

bT-reductions, we must satisfy stronger conditions and, in this sense, the problem becomes

harder to solve. For example, A. Li, W. Li, Y. Pan, and L. Tang [66] have shown that the

solution to the i i" sub-degree problem in CbT (i.e. the dual to the continuity problem)

is completely different from the result for C (see Cooper and Li [31]). They show that the

statement of the solution in C fails badly in CbT: there exist c.e. bT-degrees a, b such that

0 < a < 0', and for any c.e. bT degree x, b V x = if and only if x > a.

Our approach might not be the only one. Klaus Ambos-Spies proved that for any c.e.

set, its Turing degree is capable in the Turing degrees iff its bT-degree is capable in the

bT-degrees [2]; we thank an anonymous referee for pointing this out. Therefore, another

possible approach might be to prove, if possible, that for any two c.e. sets, their Turing

degrees form a minimal pair in the Turing degrees iff their bT-degrees form a minimal pair

in the bounded Turing degrees. As consequence, our continuity result would immediately

follow from Seetapun's continuity result. We comment that although our continuity proof

might be non-obvious, from the above perspective, oftentimes bT-degrees can be handled

much more easily than Turing degrees [1]. Ambos-Spies provides various examples [1]. For

example, density of the c.e. bT-degrees can be proved by a finite injury priority argument,

whereas the same result requires an infinite injury argument for the c.e. Turing degrees.

The rest of this chapter is devoted to proving Theorem 5.2.3, the main result. In

section 5.3, we formulate the conditions of the theorem by requirements; in section 5.4, we

arrange all strategies to satisfy the requirements on the nodes of a tree, or more precisely,

the 1,<.:' i.:/; tree T. In section 5.5, we use the priority tree to describe a stage-by-stage

construction of the objects we need. Finally, in section 5.6 we verify that the construction

in section 5.5 satisfies all of the requirements, finishing the proof of the theorem.

Our notation and terminology are standard and generally follow Soare [86]. During

the course of a construction, notations such as A, KP are used to denote the current

approximations to these objects, and if we want to specify the values immediately at







the end of stage s, then we denote them by As, I[s] etc. For a partial computable

(p.c., or for simplicity, also a Turing) functional, K -iv, the use function is denoted by

the corresponding lower case letter Q. The value of the use function of a converging

computation is the greatest number which is actually used in the computation. For a

Turing functional, if a computation is not defined, then we define its use function = -1.

During the course of a construction, whenever we define a parameter, p -iv, as fresh, we

mean that p is defined to be the least natural number which is greater than any number

mentioned so far. In particular, if p is defined afresh at stage s, then p > s.

5.3 Requirements and Strategies

In this section we provide the requirements and strategies for proving Theorem 5.2.3.

We restate it here for convenience.

Theorem 5.2.3 (Local Noncappability in CbT). For any c.e. bT-degree b, if b / 0, 0',

then there is a c.e. bT-degree a > b such that if x < a is noncomputable, then x A b / 0.

5.3.1 The requirements

Given a c.e. set B, we will build a c.e. set A to satisfy the following requirements:

e: A (B) V K bT B

Re : Xe= ,e(A,B) (3 c.e. De)[De
Se,i: De / Ai X V T VB
where e, i E u, {(4e, e, Xe) : e E W} is an effective enumeration of all triples (+, ', X) of

all bounded Turing (bT, for short) reductions (, T, and of all c.e. sets X; {A I i E w} is

an effective enumeration of all partial computation functions A; and K is a fixed creative

set. De for all e, are c.e. sets built by us.

Let a, b, x, d be the bT-degrees of A D B, B, X, D, respectively. By the P-

requirements, a > b (unless b was already the degree of 0'), and by the R-requirements,

if x < a there is a d below both x and b such that d / 0 unless either x = 0 or b = 0.

Therefore the requirements are sufficient to prove the theorem.







Before describing the strategies, we introduce some conventions of the bounded

Turing reductions. We will assume that for any given bounded Turing reduction KP or T,

the use functions 0 and i will be increasing in arguments.

5.3.2 A P-strategy

A P-strategy will try to satisfy a P-requirement, P i- (we drop the index in the

following discussion). We use a node on a tree, 7y i-, to denote a P-strategy. It aims

to ensure that if A = T(B), then there is a bounded Turing reduction A such that

A(B) = K. Therefore the P-strategy 7 will try to build a bounded Turing reduction A.
A will be built by an cu-sequence of ;/. /' k. Each cycle k of 7 will be responsible for

defining A(B; k) as follows: first 7 chooses a fresh witness a(k) and waits for a stage, v

-, at which we have I(B; a(k)) =0 = A(a(k)). When this occurs, we define A(B; k)

to be K(k) with use function 6(k) = (a(k)). Since i is partial computable, so is the use

function 6 of A. We will ah--i,-x assume that whenever B changes below the 6-use, 6(k)

-v, the corresponding computation A(B; k) becomes undefined simultaneously. Suppose

that at a later stage s > v, k is enumerated into K, and B has not changed since A(B; k)

was last created, then A(B; k) / K(k). In this case, we enumerate a(k) into A so that an

inequality '(B; a(k)) / A(a(k)) is created. The key point is that, if A(B; k) / K(k) is a

permanent inequality, so is '(B; a(k)) 0 / 1= A(a(k)).

The P-strategy 7 will start cycles k in increasing order of k. Cycle k acts only if the

following conditions occur:

1. For all k' < k, A(B; k') = K(k').

2. Either a(k) T, or A(B; k) T and I(B; a(k)) = A(a(k)), or A(B; k) 1= 0 / 1= K(k).

As we have seen in the above analysis, if there is a permanent inequality between

A(B) and K, there is a corresponding permanent inequality between T(B) and A. Since

A is a bounded Turing reduction (with use bound 6), we have that if A is built infinitely

many times, then A(B) is total, and A(B) = K. Hence K is bT-reducible to B. Suppose

this never occurs, then the P-strategy 7 acts only finitely many times, which will be

denoted by 1. Therefore a P-strategy 7 has only one possible outcome 1, unless K






5.3.3 An R-strategy

Before describing the R-strategy, we introduce a convention of the bounded Tur-

ing reduction 4. We assume that for any x and any s, if x enters X at stage s, then

S(A,B;x)[s] = 1.

Given an R-requirement, R ,i-, we define the length function of agreement as usual.

That is to -,-: At stage s, the length function of agreement f between 4(A, B) and X is

defined as the largest x such that N(A, B) and X agree on all values y < x:

= (X, (A, B))[s] maxx : (Vy < x)[(A, B;y)[s]] = X(y)[s]}

Stage s is said to be R-expansionary if the length function of agreement increases; that

is, if for all v < s, [s] > [v]. At R-expansionary stages, an R-strategy builds bounded

Turing reductions H(X), _(B) (with use functions 0, (, respectively) so that for the least

undefined x < f:


6(X, x) I= D(x) with 0(x) = x and E(B, x) = D(x) with (x) = 0(x) (5-1)


where 0 is the use of 4(A, B), and D is a c.e. set, whose elements are enumerated into it

by lower priority S-strategies associated with R, to satisfy the R-requirement.

Suppose that a is an R-strategy. As above, the use functions 0 and of the bounded

Turing reductions 6 and E built by a are the identity function and 0 respectively, so that

both 6 and E are bounded Turing reductions. (Notice that 4(A, B) is a bounded Turing

reduction, so that the use 0 is a partial computable function.)

To satisfy O(X) = D and E(B) = D, the R-strategy a will impose the following

constraints on all S-strategies with the same global index as the R-strategy a:

For any s, and any d, d is allowed to be enumerated into D, at stage s only if both

O,(X; d) and 2E,(B; d) are undefined during stage s.

We assume that for the bounded Turing reductions 6 and E, any computation will

automatically become undefined, whenever the oracle changes below the corresponding

use.







By the building of 0, and E,, and by the constraints of a, we have that if 0, and oa

are built infinitely many times, then both O,(X) and ES(B) are total, and both equal Da.

Hence R is satisfied.

Therefore the key point towards the satisfaction of R is that if there are infinitely

many R-expansionary stages, then both O, and Eo are built infinitely many times.

We thus define the possible outcomes of the R-strategy a by


0
to denote infinite and finite expansionary stages respectively.

5.3.4 An S-strategy

Suppose that we want to satisfy an S-requirement, Se,i v-. For simplicity, we use

R and S to denote 7R and Se,i respectively. Suppose that a and 3 are the 7- and S-

strategies respectively. Let a^(0) C 3.

3 attempts to find some d such that A(d) 1= 0 with an expectation of enumerating d

into D to create an inequality A(d) = 0 / 1 = D(d). However 3 can enumerate a number

d into Da at a stage, s ,-, only if both O,(X; d) and E (B; d) are undefined during

stage s as required by the R-strategy a. Therefore, 3 will prepare a sequence of possible

candidates c's such that

A(c) 1 0 =D(c),

Oa(X; c) and

(B; c) 1.
For the largest c, we build a partial computable f3 as follows:

for every y < ~a(c), if f3(y) is undefined, then define fy(y) B(y).

define d(3) = c, and set c to be undefined, which allows us to define a larger c.

Suppose that there is an error between f3 and B, in the sense that there is a y such

that f3(y) 1= 0 / 1 = B(y) occurs at a stage, v ~-,-. Then we open an A-gap:

build a partial computable function g to simulate Xa as follows: for every x < d(3),

if g(x) is undefined, define g3(x) = Xa(x),







set f3 to be totally undefined (the fa proves wrong, so it is cancelled),

drop the A-restraint by defining rA(3) = -1, and

create a link (a, /3).

[Notice that at stage v, _,(B; d(3)) is undefined due to the B-change in the domain

of fa. We regard this as a B-permission for the enumeration of d(o3) into D,. This B-

permission will be kept until the current link (a, 3) is either travelled or cancelled so that,

in either case, the link is removed.]

Suppose that 3 creates a link (a, /3) at stage v. Then the link (a, /3) will be travelled

at the next a-expansionary stage s > v. Now we consider two cases:

Case 1. There is an error between gp and X,.

In this case, there is an x <, d(3) which has entered X, since stage v. Therefore

Oc(X; d(/3)) is currently undefined. Together with the condition that E,(B; d(3)) 1,

found at the stage we created the current link (a, /3), 3 is qualified to enumerate d(3) into

Do. S is satisfied by A(d(3)) =0 / 1= D,(d(3)).

Case 2. Otherwise, we know that gp is correct during the gap. Therefore we preserve

g9 on its domain until 3 opens another A-gap. For this, we implement:

for every y < Q(d(3)), if f3(y) is undefined, then define f(y) = B(y), and

define the A-restraint rA(/) of 3 to be Q(d(3)).

[Notice that although we have A-restraint at this stage, X, may change due to a

B-change below Q(d(3)). The definition of fp at this stage allows us to immediately open

an A-gap once such a B-change occurs, in which case, we do not increase the domain of g3

but resume with the old candidate d(3).]

Therefore the S-strategy 3 is a gap/cogap strategy. It will build partial computable

functions ft and g3 and will proceed as follows:

1. Define a possible candidate c(/3) as fresh.

2. (Building fa) Wait for a stage v at which

A(c(3)) 1= 0= D(c(3)),
O(X,; c(/3)) 1= 0 D (c(3)), and







E,(B; c(/)) 1 0 = D,(c(3)).

Then for c= c(/),

for every y 4< (c), if fy3(y) T, then define f3(y) = B(y),

define d(3) = c,

set c(3) to be undefined, and go back to step 1.

3. (Creating a link (c,/3)) Let s be the current stage. Suppose that there is a b in the

domain of f, that enters B at stage s. Notice that the domain of f, is precisely

everything 4< Q(d(/3)) = 0(d(/3)), so that E,(B; d(3)) becomes undefined at stage s.

Then:

for every x < d(3) = O0(d(3)), if ga(x) is undefined, define it to be X,(x),

define the A-restraint of 3 by rA(/) -1,

set f3 to be totally undefined, and

create a link (ca, /).

4. (Travelling the link (c, /)) We travel the link (c, /) at the next c-expansionary stage

t > s. There are two cases:

Case 4a. (Successful closure) There is an x such that g9(x) 1= 0 / 1 = Xa(x).

(This x must enter X, since the current link (c, /) was created.) Then:

enumerate d(f) into Da, and stop.

Case 4b (Unsuccessful Closure) Otherwise, then

for every y < Q(d(/)), if fs(y) is undefined, then define fs(y) = B(y),

define an A-restraint of 3 by rA(/) (d(/3)).

The Possible Outcomes

We consider the following cases.

Case 1. Case 4a occurs at some stage t.

In this case, lim d(/) [s] 1= d(/) < u, and A(d(/)) = 0 / 1 = D (d(/)) is created. S

is satisfied.

Case 2. Otherwise, and Case 4b occurs infinitely many times.







Notice that g3 is never set to be totally undefined, and that for a fixed number d,

((d) is a fixed number, so that B changes below Q(d) only finitely many times, and so that
Step 3 occurs with the same d only finitely many times. Since Case 4b occurs infinitely

many times, we have that d(j3)[s] will be unbounded over the course of the construction,

and that whenever Step 3 occurs, we build ga on the initial segment of the current d(3).
Therefore g, is built as a computable function.

For an arbitrarily given x, we prove ga(x) I= X,(x). Let s be the stage at which

ga(x) is created. Suppose that si are all stages s' > s at which Step 3 of 3 occurs, and
that for each si, t, E (si, si+l) is the stage at which the link (a, P) created at stage si is

travelled through Case 4b.

By the choice of si, so = s. Since Case 4b occurs at stage to, and to is c-expansionary,

we have that

(i) g3(x) = Xj[so](x) (a will never be visited at stage so).
(ii) For any s E [so,to], g3(x) = X,[s](x).

(iii) g9(x) = 4(A, B; x)[to] = X,[to (x).

By the A-restraint rA(3)[to], and the convention of 4, we have that for any t E [to, Si),

(iv) g(x) = ,(A,B; x)[t] X,[t](x).
Suppose by induction that for n, we have that

(A) For any s E [s,,t,,], gp(x) = X[s](x).

(B) g(x) = X (x)[t1] = 4 (A, B; x) [t,],
(C) For any t E [tn, s,+l), g3(x) = (A, B; x)[t] = X,[t](x), and

(D) gp(x) = X,[sn+l](x).
By (C), (D) for n, and by the choice of tn+l, (A) holds for (n + 1). By (A) for (n + 1),
and the choice of t,+l, we have (B) for (n + 1). By (B) for (n + 1), by the A-restraint at

stage t,1+, and by the convention of 4, (C) holds for (n + 1). (D) for (n + 1) follows from

(C) for (n + 1) and the assumption that a is not visited at stage s,+l.

[Remark. We will arrange the construction so that an A-gap can be opened only at

odd stages, and that no R-strategies can be visited at these stages. This means that no








X, for any R-strategy a can receive elements at odd stages, since we assume that X, is

enumerated only at stages at which a is visited.]

Therefore for any s > so, either ga(x) = X,(x)[s] or ga(x) = (4(x)[s]. Since

) (A, B; x) equals Xa(x), we have that ga(x) = Xa(x).

This is a global win for the requirement R, so that we don't consider any other

S-requirements with the same global index with the requirement R.

Case 3. Otherwise, and Step 2 occurs infinitely many times.

Since step 3 occurs only finitely many times, f, is set to be totally undefined only

finitely many times. Let f be the final version of fp. By the assumption that step 2 occurs

infinitely many times, f is built as a computable function.

By the choice of f, for any x, once f(x) is created, we have that f(x) = B(x). B is

computable, contradicting the assumption of the theorem. So we assume that this case

will never occur.

Case 4. Otherwise, then by the strategy, we have that lim, c($3)[s] 1= c(3) < w exists,

c(3) g D,, and A(c(3)) = 0 never occurs. Therefore A(c(3)) / 0 D (c(3)). S is satisfied

again.

We use d, g, and w to denote the possible outcomes of 3 corresponding to case 1,

case 2, and case 4 respectively. To guarantee that the true outcome will be the one on the

leftmost visited path, we define the priority ordering of the possible outcomes as follows:



d
With this ordering, we notice that case 3, in case it happens, will be an outcome

between g and w.

5.4 The Priority Tree

In this section, we will build a priority tree of strategies T C A< with A

{0, 1, d, g, w}. Note that there are infinitely many copies of a fixed computable func-

tion Ai in {Ai : i E w}. Therefore, to satisfy a fixed 7R requirement, it suffices to satisfy all








Se,i with i > e. Let P < R denote that the priority ranking of P is higher than that of R.

Also let
Definition 5.4.1. (i) Define a priority ranking of the requirements so that, Ve e w:

'Pe < R, < So,e < Sl,e < < Se,e < P'+1

(ii) The possible outcome of a P-strategy is only 1.

(iii) The possible outcomes of an R-strategy are 0
(iv) The possible outcomes of an S-strategy are d
Definition 5.4.2. Given a node we i- that:

(i) P, is satisfied at ( if there is a 'P,-strategy 7 C c

(ii) 7R is satisfied at ( if either

there is some R,-strategy a such that a (l1) C _, or

there is some S,i-strategy f3 (for some i) such that 3^ (g) C

(iii) R, is active at if R, is not satisfied at and there there is an R,-strategy a with

a (0) C i, such that there is no S,,i',-strategy f3' with a c(0) C P3' c f'^ (g) C for any

Ce < e.

In this case, a is unique, and we i- that R, is active at via a.

(iv) Se, is satisfied at if 7R is satisfied at or 7R is active at via a, i-, and there is

an S,,i-strategy f3 such that a (0) C 3 C 3 .

We now define the priority tree T inductively as follows.

Definition 5.4.3 (Priority Tree). (i) Define the root node 0 to be a Po-strategy.

(ii) The immediate successors of a node are the possible outcomes of the corresponding

strategy.

(iii) A node -i-, will work on the highest priority ranking requirement which is not

satisfied and not active at .

Definition 5.4.4. The index I(i) of a node is the index of the requirement on which

the node acts. For example, if is an Re- or P,-strategy, we define I(i) = e, and if is an

S,i-strategy, we define I() = (e, i).

As usual, the priority tree T will have the following properties.







Proposition 5.4.5. Let f be an infinite path through T. Then for any requirement X,
there is a node 0o C f such that either (i) or (ii) below holds,

(i) X is satisfied at ( for any ( with o c ( C f.
(ii) X is active at f for any with Co c C f.
Proof. Let {Xi : i c w} be the priority ranking of the requirements so that Xi < Xi+1 for
all i. We prove this by induction. Assume that the proposition holds for all Xi with i < n.
Let X = X,+1. Fix (' C f so that the proposition holds for all Xi with i < n. Let 1o be an
X-strategy such that (' c $o c f. Note that 1o must exist by the priority ranking of the
requirements unless X already satisfies Proposition 5.4.5 with the given '. There are three
cases.
Case 1. o is a P-strategy.
The only outcome of a P-strategy is 1. Therefore, necessarily o^(1) C f E [T]. So the
proposition holds with t = -o^(l).
Case 2. io is a Re-strategy for some e.
If o^(1) C f, then the proposition holds with = -o^(1). Otherwise o = o^(0) C f.
Now if Ri (i < e) is satisfied at all ( with 0o C ( C f, then by the priority ranking of the
requirements, there is no Si,e-strategy f3 D o^(0). Otherwise, by assumption Ri is active at
all ( with 0o C ( C f. So the outcome of Sie for each such Ri must necessarily be d or w.
Hence 1 = (o^(0)^A C f where A is the concatenated outcomes of all Si,-strategies (i < e)
such that Ri is active at all ( with 0o c ( C f.
Now l^(k) C f for some k e {d, w, g}. If k = g then the proposition holds with

( = (^(g). If i = (I^(i) C f (i {d, w}) then by the same reasoning above concerning
the Rj (j < e), Re is active at as long as i C ( C f.
Case 3. io is an S,i-strategy.
By the priority ranking of the requirements and by the assumption on the Xi (i < n),
it follows the Re is active at all with 0o C ( C f. Therefore either ,w = o^(w) C f
or d = 0o^(d) C f. By the assumption on Re, the proposition holds with j = k, with

appropriate choice of k so that k C f.








Definition 5.4.6. If 3 is an Se,i-strategy for some e, i, then define top(3) to be the

longest R,-strategy a such that a^(0) C 3.

5.5 The Construction

Our construction will perform different actions at even and odd stages. At even

stages, strategies on the tree will act to satisfy the requirements.

Suppose that B is enumerated at odd stages only, and that at every odd stage, there

is exactly one element that enters B. Given a B-permission in the construction, we want

to open A-gaps for as many S-strategies as possible. This allows us to specify a P-strategy

so that we enumerate its witness into A. So we will ensure that A-restraints drop at odd

stages, and also that A is only enumerated at odd stages.

During the course of the construction, we may initialize a node, ( ,-iv, which means

that all the actions taken by ( previously, are cancelled, or set to be totally undefined.

Precisely, if an R-strategy a is initialized, then both 0, and E are set to be totally

undefined, D, is set to be the empty set 0, and all links associated with a are cancelled.

If an S-strategy 3 is initialized, then both ga and fa are set to be totally undefined,

parameters d(3) and c(3) are both set to be undefined, and any link associated with 3 is

cancelled. If a P-strategy 7 is initialized, then A, is set to be totally undefined, and all

witnesses of 7 are cancelled.

Notice that an S-strategy 3 opens its A-gap, exactly at stages at which an error

between fa and B occurs, which gives a B-permission for its current candidate d(3). Our

problem is to make sure that there are infinitely many stages at which all the S-strategies

on the true path (or the current approximation of the true path) drop their A-restraints

simultaneously.

Given a node suppose that 31 c /2 C .. C P are all S-strategies 3 with P^(g) C (.

To guarantee that if is a P-strategy, then there are infinitely many stages at which for

all i = 1, 2, n, the A-restraints rA (i) of i3 drop to -1 infinitely often, proceed as

follows. Let fi be f, for all i = 1, 2, n.

We will arrange the building of various fa, such that: for any s,







1. for any i, if fi[s] is empty, then the current A-restraint rA(3i) is -1, and

2. for any i < j, if both fi[s] and fj[s] are not empty, then dom(fi[s]) 3 dom(fj[s]),

where dom(f) is the domain of f.

Our construction will ensure that for any s, at the end of stage s, the two properties

above hold for all nodes .

Using these properties, we know that whenever we find an error between f, and B,

the same error occurs for fi for all i < n, allowing us to open A-gaps for the S-strategies

3i simultaneously, except for those /3's which are already in their A-gaps. Therefore,

if f3, opens an A-gap at stage s, then the current stage s is in the A-gap of 3i for all

iE {1,2, ,n}.
We are ready to describe the stage-by-stage construction.

Definition 5.5.1. (The Construction) The construction will be defined as follows.

Stage s = 0. Initialize every node and set A = 0.

Stage s = 2n + 1. Let b be the number that enters B at stage s.

Run the following procedure:

1. Let 3 be the <-minimal and c-maximal S-strategy, if it exists, such that f3(b) 1.

Suppose that 31 C /02 C '.. C f n-1 are all nodes 3' such that 3'^(g) C 3. Let 3 = 3,.

2. Initialize all nodes with 3,^(g)
3. In increasing order of i, for 3i, and ci = top(Qi), if f, / 0, then:

for every x < d(Ao), if g3(x) is undefined, then define g(x) = X,(x),

create a link (a~i, /),

set rA(3,i) --1, and

set fa to be totally undefined.
We -w that a P-strategy 6 requires attention at stage s if:

There is some x such that A6(B; x) J/ K(x) and a (x) g A;

For all 3 with 3P(g) C 6, rA(0) = -1 hold during stage s.

4. If there is a P-strategy which requires attention at stage s, then:

let 7 be the <-least such P-strategy,








let k be the least x such that A (B; x) J/ K(x) and a (x) g A,

enumerate a((k) into A, and

initialize all nodes with > 7, and go to stage s + 1.

5. Otherwise, then go to stage s + 1.

Stage s = 2n + 2. We first specify the root node to be eligible to act at substage

t = 0. At each substage t, we allow the strategy which is eligible to act at this substage to

take action, and then either close the current stage or specify a new node to be eligible to

act at the next substage of stage s.

Substage t. Let ( be the node which is eligible to act at substage t of stage s. If '

has length s, then initialize all nodes % and close the current stage. Otherwise, there

are three cases corresponding to different types of strategy .

Case 1. = 7 is a P-strategy. Then run the following:

Program 7: 7 will build a bounded Turing reduction A,, and define witnesses a,(k).

For simplicity, we drop the subscription 7 in the description of the program.

1. If there is an n such that a(n) is defined, and 1(WT,(B), A) / a(n), then let 7^(1) be

eligible to act next (i.e. at substage t + 1 of stage s).

2. Otherwise, let k be the least x such that A(B; x) is undefined. Then:

if a(k) 1, then define A(B; k) = K(k) with 6(k) = Q(a(k)),

otherwise, then define a(k) to be fresh, and

initialize all nodes ( > 7, and go to stage s + 1.

Case 2. ( = a is an Re-strategy for some e. Run the following

Program a:

1. If s is not c-expansionary, let a^(1) be eligible to act next.

2. Otherwise, and there is a link (c, 3) which was created and has never been cancelled

or travelled. Let /o be the <-least such 3, and let /o be eligible to act at the next

substage.

3. Otherwise, then,

let x be the least y such that either O,(X,; y) or E,(B; y) is undefined,







if 0,(X,; x) T, define e,(X,; x) = D (x) with 0(x) = x,

if ,(B; x) T, then define E,(B; x)[s] : D,(x)[s] with use ((x) := (x), where

is the use of N(A, B), and

let a^(0) be eligible to act next.

Case 3. = 3 is an Si-strategy for some e, i. Let a = top(/3). We perform the

following,

Program 3:

1. If 3 has already been satisfied, as defined in 2a below, then 3^(d) is eligible to act.

2. (Travel a link (a, 3)) If a link (a, 3) was created and it has never been cancelled or

travelled since it was created, then travel the link (a, 3) by cases.

Case 2a. (Successful closure) (Ex) g (x) X/ Xa(x). (Notice that g was correct

at the stage we created the current link (a, 3), so this error must occur during the

A-gap of the S-strategy 3.) Then:

enumerate d(3), the largest confirmed candidate of 3, into Da,

we that 3 is -.-,I.:f/ .1 at stage s,

initialize all nodes > 3, and go to stage s + 1.

Case 2b. (Unsuccessful closure) Otherwise. Then:

define u =max{((y) : g(y) 1} = (d(3)),

set f r[ (O(d(3)) + 1) -B ( (d(3))+ 1),

set rA(0) u+1, and

initialize all nodes to the right of 3^(g), and go to stage s + 1.

In either case, the link (a, 3) is removed.

[Remark. We have that if step 2 of program 3 occurs at stage s, then a is visited at

stage s.]

3. (Building f) If c(3) 1= c, AX(c) I= D,(c) = 0, 0,(X,; c) 1, and E,(B; c) 1 then:

Case 3a. (/3') 3'^(g) C 3 and dom(f3,) C [0, (c)]. Then:

initialize all nodes > P3(w), and go to stage s + 1.

Case 3b. Otherwise, then:







for any x < Qa(c), if fa(x) T, then define fa(x) I= B(x),

set d(/) = c(3); d is said to be confirmed,

cancel c(3), so that c(/) T, and

let ^3(g) be eligible to act next.

4. If c(/) T, then define c(/) as fresh, initialize nodes > P^(w), and go to stage s + 1.

5. Otherwise, let P^(w) be eligible to act at the next substage.

This completes the description of the construction.

5.6 The Verification

In this section, we verify the satisfaction of the requirements. First we investigate

some global properties that hold at the end of an arbitrary stage. These properties ensure

that the construction is implemented properly.

Proposition 5.6.1. Let s be a stage.

(i) There is at most one link that is travelled during stage s.

(ii) If a link (ca, ) is travelled at stage s, then before we travel the link, a is visited and

step 2 of program a occurs at stage s.

(iii) There are no a,i, 2, /1, and /a such that a, C a2 C 01 C /2 and both links (ai, P1)

and (a2, 02) exist at the end of stage s.

Proof. It is easy to see that both (i) and (ii) hold by observing the construction.

For (iii), suppose to the contrary that s is the least stage such that there are

al, a2, 1, and 32 with ac C a2 C 13 C /2, and such that both links (al, P/) and

(a2, 02) exist at the end of stage s. By the minimality of s, exactly one of the two links

(cai, 1), and (a2, 02) is created during stage s. We consider two cases.
Case 1. The link (ac, 3i) is created at stage s.

By the construction, s = 2n + 1 for some n, and ft, is set to be totally undefined at

stage s. We analyze the location of 2. If l^(w) C /32, then by the construction at stage s,

/2 is initialized during stage s, so the link (a2, 32) is removed during stage s, contradicting
the choice of 32. If f3 (g) C i2, then by the definition of the priority tree T, top(/32) I a2,

so that there is no link (c2, 2) which can be created in the construction. If fl^(d) C 0/2,







then the current d(32) must be defined after /3 created its inequality at argument d(/31),

after which no link (ci, 31) can be created, since 31 has satisfied its requirement through

A1, (d(31)) = 0 / 1 D(d(31)). So case 1 does not happen.
Case 2. The link (a2, 32) is created at stage s.

Let si < s be the stage at which the current link (al, 31) was created. By the proof in

case 1, we only need to consider the case of 1 ^(w) C /32. By the construction at stage Si,

32 was initialized during stage si. So f3, is totally undefined at the end of stage si. And
in fact, all nodes > 3i^(w) were initialized at stage il. Therefore /32 cannot be visited at

any stage > si unless the link (ca, 0/3)[si] has been removed. This contradicts the choice of

s.

(iii) holds.

The Proposition follows. E

Proposition 5.6.2. (i) Let 3 be an S-strategy. Then for any s, t, if f3 is totally

undefined at substage t of stage s, then rA3() = -1 holds at the end of substage t of

stage s.

(ii) Let f be an S-strategy, and s be a stage. Let s- be the greatest stage t < s such

that 3 was initialized. Then for any s- < sI < 82 < s, if both f3[sl] and f[s2] are

not empty, then


dom(f/p[si]) C dom(f3[s2])

(iii) Let /3, be S-strategies with 3'^(g) C 3. For any s, if both f3 and fy, are non-

empty at the end of stage s, then dom(f3[s]) C dom(f3,[s]).

Proof. Both (i) and (ii) are easy facts by observing the construction.

For (iii), we prove the proposition by induction on the stages. Suppose that (iii) holds

for all s' < s. Consider a stage s at which f3 is built. By program 3 in the construction,

there are two cases.

Case 1. A link (ca,/3) for a = top(/) is travelled unsuccessfully at stage s. Let Si be

the stage at which the current link (ca,/3) was created. Then dom(f3[s]) = dom(f3[si 1)].







Let b = dom(f/[s]), and let so be the first stage at which f3 was defined on [0, b].

By program /3, case 3b of program 3 occurred at stage so. By the construction, there

was no link that was travelled by the substage at which 3 was visited during stage so.

Therefore, 3' was visited at stage so, and case 3b of program 3' occurred at stage so. By

the assumption in case 3b of program /3, we have that both fy and f, are non-empty

at the end of stage So, and dom(f/,[so]) D dom(f/[so]). By the choice of s, 3' has not

been initialized during stages [so, s], by (ii) if f/,[s] is not empty, then dom(f/,[s]) D

dom(f, [so]), (iii) follows in case 1.

Case 2. Case 3b of program 3 occurs at stage s. As the same as that in case 1, by

observing the construction, we have that for any C /3, is visited at stage s, so that

/3' is visited at stage s, and furthermore, case 3b of program 3' occurs at stage s. By the

assumption of case 3b of program /3, the domain of fo is larger than that of f, at the end

of stage s. (iii) follows in case 2.

The proposition follows. O

Definition 5.6.3. Suppose that K (bT B and B %T 0.

(i) Let 6, be the last node which is eligible to act at stage s.

(ii) Define the true path TP e [T] of the construction by TP =liminf,68.

Hereafter whenever we consider a node on the true path, we will use the notation

I E TP rather than I C TP.

Proposition 5.6.4. (Existence of the true path) Suppose ( e TP. Then there is some

a such that ^ (a) is visited infinitely often and initialized only finitely many times. Hence

^(a) e TP.
Proof. We prove by induction on the length of Suppose by induction that the

proposition holds for all c and e TP. Let so be minimal after which will never be

initialized. By the inductive hypothesis, will be visited infinitely often.

We prove the proposition for by cases.

Case 1. = 6 is a P,-strategy for some e.

By program 6, there are two subcases to consider.







Subcase la. There are infinitely many 6-expansionary stages.

By the construction, step 2 of program 6 occurs infinitely many times, so that A6 is

built infinitely many times, and that A6(B) is built as a total function. Since K bT B,

there is some m such that As(B; m) J/ K(m) holds permanently. Let n be the least such

m, and let A6(B; n) be created at stage v > so. By the choice of n, Ta(B; as(n))[v] = 0

and it will hold permanently. Let u > v be the stage at which n enters K.

Suppose that Pf c 02 C ... C f3 are all S-strategies f with P^(g) C 6.

By the choice of so, fQ will never be set to be totally undefined after stage so by

initialization for any j. Therefore, for every j E {1, 2, ... ,}, fa is set to be totally

undefined after stage so only if an error occurs between f, and B.

By inductive hypothesis, case 3b of program /3 occurs infinitely many times, so that

fa3 will be built infinitely many times for all j E {1, 2, ,1}.
In particular, f3, will be built infinitely many times. By the assumption of B %T 0,

we can choose a stage s, > u at which there is a number b such that f/3(b) 1= 0 / 1 =

B(b) occurs.

By Proposition 5.6.2 (iii), for any j E {1, 2, }, if f, is not empty at the

beginning of stage si, then there is an error between f3 and B that occurs exactly at

stage si. We have that for every j E {1, 2, .. ,1}, if fa / 0 at the beginning of stage si,

then 3j opens its A-gap during stage sl.

By Proposition 5.6.2 (i), for any j E {1, 2, ,1}, if f, is empty at the beginning of

stage si, rA(/j) = -1 holds at both the beginning and the end of stage si.

By program 6, we have that 6 requires attention at stage si, and we let 6 receive

attention by enumerating its witness as(n) into A.

By the choice of n, for any s > si, we have that Ta(B; as(n)) = 0 / 1 = A(a6(n))

holds during stage s, contrary to there being infinitely many S-expansionary stages. This

case is impossible.

Subcase lb. Otherwise.








In this case A6 is built only finitely many times. Let si > so be minimal after which

A6 will never be built.

By the choice of si, 6^(1) will never be initialized after stage sl, and by program 6, for

any s > si, if J is visited at stage s, so is 6^(1).

The proposition follows in case 1.

Case 2. =- a is an R-strategy.

Observing program a, we consider two subcases.

Subcase 2a. Step 3 of program a occurs infinitely many times.

Then a^(0) E TP. By choice of so, a^(0) will never be initialized after stage so.

Furthermore a^(0) is visited infinitely often. Therefore a^(0) e TP and the proposition

holds in this case.

Subcase 2b. Otherwise.

Suppose that Step 3 of program a occurs at most finitely many times so that a^(0)

is visited at most a finite number of times. We will show that a^(1) e TP. To show that

a^(1) is initialized finitely often, first note that by the choice of so, only nodes ^ D ac(0)

can initialize a^(1).

Let si > so be minimal after which step 3 of program a will never occur. Then for

any s > si, if a node 3' D a^(0) is visited at stage s, then there is an R-strategy a' C a,

and a link (a', P') which is travelled at stage s.

Suppose that a C a2 C ... C ac~- are all J--strategies a' with a'^(0) C a. Let

O = a.

We prove by induction that for each i < n, there is a stage after which there will be

no links (aj, 3j) that can be either created or travelled for all j > i and for all pj D ac(0).

For i = n. Define



bF = max{( (x) [t] | t < si, a (x)[t] [}

For every s > si, define









p.[s] = max{y I fa(y)[s] 1, top(3) = a,}


By the construction, we have that for every s > si,

if a link (ac, P') is travelled for some i < n and some f3' D a,^(0), then there is no f

such that f3 is built during stage s for any f with top(3) = a,. In this case, we have

p,[s] < p,[s I], and
if a link (a~, P3) is travelled at stage s, then p[s] < b,.

Therefore in any case we have that for all s > si, p,[s] < b,. By the construction

at odd stages, a link (a,, ,,) can be created at a stage s > si only if there is an element

b < bn that enters B at stage s. Since bn is a fixed number, B changes below bn only
finitely many times. Therefore there are only finitely many stages at which we create links

(ac, /,). Since once a link is travelled, it is removed immediately, there are only finitely
many stages at which a link (ca, P) is either created or travelled.

Let v, > si be minimal after which there will be no link (c,, 3,) that can be either

created or travelled.

Suppose by induction that [ i is a minimal stage after which there will be no link

(ac, 3j) which is either created or travelled for all j E {i + 1, i + 2, n}, and all

4, D c'^(0).
Define


bi = max{, (x)[t] I B(x) 1, t < I }

For any s > i ,I, define


pi[s] = max{y | f(y)[s] 1, 3 aD a(0), top(3) = a}

By the construction, it is easy to see from an inductive argument that for all s > ,,

if a link (acj, s) is travelled for some j < i, and some 3j D ca^(0), then pi[s] <

i [s 1]; and







if a link (a 03) for some f3i a^ (0) is travelled at stage s, then pi[s] < bi.

This shows that for all s > I ,I pi[s] < bi. By the construction, if a link (a /3) for

some 3,i D ca(0) is created at a stage s > I, t, then there is a number b < bi which enters

B at stage s. Since bi is a fixed number, the creation of links (a ,A) for f D a^ (0) occurs

only finitely many times, so that there is a stage > ., I i-, after which there will be no

link of the form (a 03) for any f3i a (0) which can be either created or travelled.

Therefore there is a stage vi -',- after which no link (cai, /) can be created or travelled

for any i < n and any 3 D ca^(0). So there are only finitely many stages at which some

node ( D a^ (0) is visited.

Thus ca^(1) is initialized only finitely many times.

By the proof above, there are only finitely many stages at which either Step 2 or Step

3 of program a occurs. a^(1) is visited at almost every stage at which a is visited.

Therefore in Subcase 2b, we have that a^(1) is initialized only finitely many times and

visited infinitely often.

Case 3. = 3 is an S-strategy.

Let a = top(3).

Subcase 3a. a link (c, 3) is travelled once and successfully closed.

Then 3^(d) is visited infinitely often and only initialized finitely many times. So

^ (d) e TP.
Subcase 3b. (c, P) is travelled infinitely often and unsuccessfully closed p's A-gap.

As in Subcase 3a, P^{(g) TP.

Subcase 3c. Otherwise.

By the assumption of this case, fa is built only finitely many times, since if this is not

true, then ft is built as a computable function, and ft = B, contradicting the hypothesis

B T 0.

By program 3, lim, c() [s] 1= c() < uc. By the choice of c(3), c(3) g D,. Let si be

the stage after which neither of the step 1, 2, 3, or 4 of program P occurs, therefore, for

any s > si, if f is visited at stage s, so is f^(w).







Therefore f^(w) is initialized only finitely many times, and visited infinitely often,

0^(w) E TP.
The proposition follows in Case 3. D

Since the true path exists only if both K %bT B, and B %T 0 hold as proved in

Proposition 5.6.4, we alv--- assume the two conditions from now on.

Proposition 5.6.5 (Possible outcomes along TP). Given ( e TP:

(i) If 6 is a Pe-strategy for some e, then 6^(1) e TP and A / Te(B).

(ii) If (= a is an R-strategy, then

(a) if a^(0) e TP, then D, O(X) = _E(B);

(b) if ac(1) e TP, then ko is partial or 4)(A,B) / X,.

(iii) If = p3 is an S-strategy, then for a =top(3), we have:

(a) if PT(d) e TP, then limd(3)[s] = d(3) < u and A (d(3)) = 0 / 1 D,(d(3)).

(b) if P/(g) e TP, then g3 is a computable function and g3 = Xa.

(c) if /^(w) E TP, then limc(3)[s] 1= c(3) < u and A (c(3)) / 0 = D(c(3)).

Proof. By Proposition 5.6.4, we can choose so minimal after which ( e TP will never be

initialized.

For (i). By Proposition 5.6.4, 6^(1) E TP. Suppose to the contrary that A = e,(B).

By program 6, step 2 of program 6 occurs infinitely many times. Therefore, A6(B) is

total. Since K %bT B, we can choose the least n such that a permanent inequality

A6(B; n) = 0 / 1 = K(n) appears. Let v > so be the stage at which the computation

A6(B; n) was created. Notice that Te(B; a6(n))[v] = 0 and B will never change below
'. (a6(n)) after stage v.

By the proof in case 1 of Proposition 5.6.4, there is a stage Si at which we can

enumerate a6(n) into A.

By the choice of n and si, 4'(B; a6(n)) = 0 / 1 = A(a6(n)) will be preserved forever.

A contradiction.

Therefore we have that A / e(B).

For (ii). Let a^(0) e TP.







By Proposition 5.6.3, both 0, and E, are built infinitely many times.

By Step 3 of program a, it suffices to prove that the following constraints of a are

satisfied during the course of the construction:

For any d, and any s > so, if d is enumerated into D, at stage s, then both O,(X; d)

and E,(B; d) are undefined during stage s.

By Proposition 5.6.1 (i), there is at most one link which is travelled during stage s.

Let d be enumerated into D, at stage s. Then there is an S-strategy f such that

top(3) = a and case 2a of program 3 occurs at stage s, and d = d(3)[s].

By Proposition 5.6.1 (ii), a is visited at stage s, and there is a link (a, 3) which

was created at a stage s-(> so) < s and travelled at stage s successfully. By the

construction at stage s-, there was an error between f3 and B occurred at stage s-, since

d = d(3)[s] d(3)[s-] dom(fg[s-]), we have that E,(B; d(3)) becomes undefined during

stage s-. By the link (a, 3) [s-], step 3 of program a has never occurred during stages

[s-, s]. Therefore, E,(B; d) is undefined during and after stage s.

By the assumption of case 2a of program 3 at stage s, there has been an error

between g3 and X, during stages [s-, s], therefore, 0,(X,; d) has become undefined

during stage s.

Therefore, for any number d chosen after stage so, the enumeration of d into D, does

alv--i- respect the constraints imposed by a. Both @,(Xa) = D, and E,(B) = D, are

satisfied.

Let a^(1) E TP. We prove that Xe / ~e(A, B). Suppose to the contrary that

Xe = L(A, B). By the assumption of this case, step 3 of program a occurs only finitely

many times, and that there are infinitely many a-expansionary stages. Therefore, there

is a stage Si > so after which step 3 of program a will never occur. However there are

infinitely many stages at which we travel a link (a, 3) for some 3.

By the choice of sl, E, is a finite set, let s2 be the stage > sl after which B will never

change below max{(~(x)[Sl] I ,(B; x)[si] 1}.







By the S-strategies, for any 3 with top(3) = a, if f3 is created after stage s2, then

there will be no link (a, 3) which can be created by using the difference between B and fp.

Therefore we can choose a stage s3 > s2 after which there is no link (a, 3) which

can be created for any 3. By the construction, once we travel a link (a, 3), it is removed.

There is a stage s4 > s3 after which there is no link from a to any S-strategy 3 which can

be either created or travelled. This contradicts the assumption that step 2 of program a

occurs infinitely many times.

We have that Xe / (A, B).

For (iii)(a). If 3^(d) E TP then there is some stage s where Case 2a of program 3

enumerates d(3) = lim, d(3)[s] into D,. Then at all stages t > s, program 3 enacts Case

1. Furthermore since d(3) is only defined when A(d(3)) = 0, we have that Ap(d(3)) 0 /

1 = D (d(3)).

For (iii)(b). By the choice of so, and by the assumption of this case, 0^(g) is never

initialized after stage so.

By program 3, case 3b of program 3 occurs infinitely many times. By the choice of

so, f3 becomes totally undefined at any stage s > so only if a link (a, P) is created at stage
s, and this link will certainly be travelled unsuccessfully, instead of being initialized.

For a fixed number d, 0,(d) is a fined number, so that B changes below 0,(d) only

finitely many times. Therefor d(3)[s] will be unbounded in the construction. By the

construction at odd stages, if a link (a, 3) is created at stage s, then d(/3)[s] is defined, and

gp is built on the initial segment d(3) [s]. Therefor g, is built as a computable function.
Notice that g, will never be set totally undefined after stage so.

We prove that for any x, if ga(x) is created at a stage v > so, then for any s > v,

g (X) = X [s](x).
Given an x, let v > so be the stage at which ga(x) is created and defined as 0 (if it is

1, then ga(x) = Xa(x) takes already the permanent value).

Suppose that i,, < vl < v2 < are all stages > v at which a link (a, 3) is created,

and let ti be the stage at which the link (a,/3P), ] is travelled. Then vo = v.







By the choice of to, the link (c,/) [,,,] is unsuccessfully travelled at stage to, this

means that for any t E [vo, to], g3(x) = X[t](x). Since to is c-expansionary, we have

g9(x) = X,[to](x) = (A, B; x) [to]. By the A-restraint rA(03)[to], the definition of fa [to],
and by the convention of 4,, we have that for any s E [to, vi), g(x) = 4(A, B; x)[s] =

Xa[s](x) is preserved.
Suppose by inductive hypothesis we have:
1. for any s E [va,tn], g9(x) = X[s](x).
2. gp(x) = X,[t](x) = t.(A,B; x)[t,].

3. for any s E [t., v,u+), g/(x) = I)(A, B; x)[s] = X[s](x).

4. g/(y) = Xa[v,+l].
Since the link (a, P)3[vT+l] is unsuccessfully travelled at stage t1+l, (1) holds for n + 1,

and since t+l is c-expansionary, (2) for (n + 1) holds, and furthermore, by the A-restraint

at stage tn+l, (3) for (n + 1) holds, (4) holds since a will never be visited at odd stages, so
there are no elements which enter X, at odd stages.
This proves that g9(x) = Xa(x).
Since x is arbitrarily given, we have that for almost every x, g9(x) = Xa(x), X, is

computable.

(iii)(b) follows.
For (iii)(c). By the assumption in this case, g9 is built only finitely many times. If

fp is built infinitely many times, then the final version of fp, denoted by f, is built as a
computable function, and f = B. A contradiction. Therefore f3 is built only finitely many

times. Let Si > so be such that f3 will never be built at any stage s > si. Furthermore,

we can choose a stage s2 > si after which none of the steps 1, 2, 3, or 4 of program 3 will
occur. Therefore lim, c(/)[s] = c(3) must be chosen before stage s2. Clearly c(3) g D,.

Since 3 is visited infinitely many times, the only reason that case 3b will never occur after

stage s2 is that AX(c(3)) / 0. (iii)(c) follows. O
Proposition 5.6.6 (P-satisfaction Proposition). For each e, P, is satisfied.







Proof. Given e, let 6 be the P,-strategy E TP. By Proposition 5.6.5 (i), necessarily

A / ',(B) so that P, is satisfied. The proposition follows. E
Proposition 5.6.7. For each ee c, if X, 4), (A, B) and X, and B are not computable,

then they do not form a minimal pair.

Proof. By Proposition 5.4.5, let a be the longest Re-strategy on the true path TP. By

Proposition 5.6.5 (ii), a^(0) E TP and D, = O,(X,) = -E(B). By Proposition 5.6.5 (iii),

for any S-strategy 3, if top(3) = a and /3 TP then either / (w) e TP or ^ (d) e TP.

In either case A3 / D, so that X, and B do not form a minimal pair. The proposition

follows. O







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BIOGRAPHICAL SKETCH

Paul Steven Brodhead was born in Oak Park, IL in 1980. He moved to Richland

Center, WI in 1989 and continued his schooling there until he graduated from Richland

Center High School in 1997. As a Chancellor's scholar, Paul attended the University of

Wisconsin-Madison from 1997 until 2000, when he earned a B.S. in mathematics. During

the summer of 2000, Paul participated in the NSF-funded undergraduate mathematics

research experience SIMU, at the University of Puerto Rico- Humacao. In 2003 Paul

started graduate school at the University of Florida and earned a master's degree in

mathematics in 2005. During the summer of 2006, Paul was a graduate assistant for an

NSF SEAGEP-funded undergraduate mathematics research experience at the University

of the Virgin Islands. He went to the Chinese A, i1I in'v of Sciences, Institute of Software

during the summer of 2007 as an NSF fellow, participating in the East Asia and Pacific

Summer Institutes. As an NSF SEAGEP fellow during the fall of 2007, Paul went to

Victoria University of Wellington (Wellington, New Zealand). As an invited visitor and

lecturer, Paul went to the University of Hawaii at Manoa in the spring of 2008. Paul

earned his Ph.D. in mathematics from the University of Florida in 2008.





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ACKNOWLEDGMENTSIgivethanks,rstandforemost,tomyPh.D.advisorDouglasCenzer,whomustbenotedinthisgenerationasamongthemostpatientandirrevocablykindofmen.Hissenseofhumoralsotrumpsthecloudiestofdays,betheyeventhecloudsofhurricanes{which,infact,occurredasmystudiesprogressedinFlorida.Thesecharacteristics,combinedwithmydetermination,helpedmegainanunderstanding,broadandin-depth,ofcomputabilitythatallowedmetosuccessfullyundertakeanddeveloptheresearchpresentedhere.Theroadtothispointbeganlongago,butreacheditspinnacleatFloridaunderhisdirection.TheroadbeganattheUniversityofWisconsin,inmyrstsemester;Iwasrstin-spiredtopursuemathematicswhiletakingAlgebraandTrigonometry.ProfessorArnoldJohnsonmadethecourseenjoyableandchallenging,makingitunlikeanymathematicscourseIevertook.HimIthank,forashaveIalwayshadhighgoals,hisinuencewascriticalasitwasatthispointinlifethatIdecidedthatIwastopursueaPh.D.inmath-ematics.Later,inmylastsemesteratWisconsin,hehelpedmedevelopamathematicalmaturitybeyondexpectations,asheledmeinanundergraduateresearchprojectinhisarea,Algebra.Yet,betweentheseexperiences,manyothersmustbethanked.Duringmyundergraduateyears,manyinspiringteacherspushedthemathematicalenvelope,deliveringtheendeavoroftheages,theenjoyablementalfrayofthebreakingofmathematicalbarriers.Ithank:SteenLempp,ReedSolomon,H.JeromeKeisler,ArnoldMiller,MauryBramson,SteveBauman,SimonHellerstein,DanielRider,DanielShea,andJenniferZiebarth.Inparticular,IthankSteenLemppforintroducingmetocomputabilitytheorywhileIwasinhislinearalgebraclass,asitwasatthismomentthatIdeterminedmyPh.D.specialization.Mymindsetwaseverafterwardsdirectedthisway.Forexample,IthankArnoldMillerforhispatienceinlisteningtomynumerousattemptsinusingcomputabilitytheoryinhisyear-longabstractalgebraclass.IthankH.JeromeKeislerforlisteningtomeinhislogiccourse.IalsothankReedSolomonforhelpingmeevenfurtherinhisgraduatelogiccourseandhissubsequentguidanceinmyrstundergraduateresearchproject,thestudyofHilbert'sTenthProblem.IthankforDaniel 4

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SheaalsotheundergraduateadvisorforacceptingmydestinedcourseinmathematicsasIsatinhisabstractanalysiscourse,vigorouslydrivenunderadepartmentmathematicsscholarship,whenyearspriorIenteredhisocewhilststillinAlgebraandTrigonometry,demandinghelpinsettingupmyundergraduatecurriculumforaneventualPh.D.inmathematics.UpontheclosingofmyWisconsinexperience,excellenceinmathematics,ahigherlevelstill,evolvedattheUniversityofPuertoRico-Humacaoduringtheimmediatesummer.IparticipatedintheNSF-fundedundergraduateresearchexperience,theSummerInstituteofMathematicsforUndergraduates.Ithank:ReinhardLaubenbacher,AbdulJarrah,RebeccaGarcia,MalarieCummings,CoraSeidler,IvelisseRubio,andHerbertA.Medina.Herbert,Ivelisse,andReinhard,inparticular,helpedmetoseetheurgencyinthetacklingoftheproblemsathand.Iamgreatlyappreciativeofthishelptothisday.MyresearchrigorandcoremathematicaltalentbloomedingraduateschoolatFlorida.Ithank:DouglasCenzer,BillMitchell,JindrichZapletal,JeanLarson,AlexandreTurull,PaulRobinson,JamesKeesling,JonathanKing,ScottMcCullough,StephenSummers,andShariMoscow.IespeciallythankAlexandreTurull.Ihadtheprivilegeofhavingsixsemesterswithhim;hehelpedmetoseeaproblem,itsvalue,anditscomplexities.Ithankmydissertationcommittee:DouglasCenzer,RickSmith,BillMitchell,MuraliRao,andBeverlySanders.IalsothankBernardMair,RickSmith,MuraliRao,andJuanLiu;myexperienceswiththemconvincedmetomoveinvaluabledirectionsthattookmearoundtheglobeandbeyond.Mycollaborators,nearandabroad,youIthank:DouglasCenzer,AngshengLi,WeilinLi,GeorgeBarmpalias,SeyyedDashti,RebeccaWeber,JereyRemmel,RodDowney,NoamGreenberg,KengMengNg,andBjrnKjos-Hanssen.ManymanymoredoIalsothank{Ihavecollaboratedwithmanypeople,ifevenforashorttime,andtheircontributionsareappreciated.Ithank:PeterHinman,DenisHirschfeldt,CarlJockusch,RussellMiller,JohannaFranklin,BakhadyrKhoussainov,JanRiemann,TedSlaman, 5

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AndreNies,RobertSoare,SteveSimpson,AntonioMontalban,BarbaraCsima,JoeMiller,VeronicaBecher,andWolfgangMerkle.Ithankthepeopleandinstitutionswhocontibutedtheirsupportthroughfunding:theSoutheastAllianceforGraduateEducationandtheProfessoriate,theNationalScienceFoundation,theUniversityofFlorida,theChineseAcademyofSciences,theAssociationforSymbolicLogic,andothers.Manypeopledeservecreditfarbeyondthebrief,orevenomitted,acknowledgementhere;Iamextremelygratefultoyou.Ofspecialconsiderationismyadvisor.Themag-nitudeofhissupportisaboveanymeasurablescale.Hissupportinallmyendeavors,professionallyandbeyond,willsurelycontinuetohelpshapeme.Thisdissertationandtheworktherein,andtheexperiencesassociatedwiththeattainmentthereofareawitnesstothis.Tothemanypeoplewhohelpedmealongthisjourney,whichisactuallyjustabeginning,youIthank. 6

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 ABSTRACT ........................................ 10 1INTRODUCTION .................................. 12 1.1GeneralOverview ................................ 12 1.2ClassicalComputability ............................ 14 1.3ClosedSetsinComputability ......................... 15 2EFFECTIVELYCLOSEDSETSANDENUMERATIONS ............ 18 2.1Introduction ................................... 18 2.2TheFamilyof01Classes ............................ 20 2.2.1NumberingsintheLiterature ...................... 20 2.2.2EquivalenceoftheNumberings ..................... 22 2.2.3EquivalenceoftheNumberingsAlternateProof .......... 25 2.2.4InjectiveComputableNumberings ................... 26 2.2.4.1Originalc.e.setsargument .................. 27 2.2.4.2Orderedtuplesofdisjointc.e.sets .............. 29 2.2.4.3Resultsforeectivelyclosedsets .............. 32 2.3StringVeriableFamiliesof01Classes .................... 37 2.3.1DenitionandExamples ........................ 37 2.3.2ComputableandEectiveNumberings ................ 39 2.3.3FamiliesContainingtheClopenClasses ................ 40 2.4NamedFamiliesof01Classes ......................... 42 2.4.1HomogeneousClasses .......................... 42 2.4.2DecidableClasses ............................ 44 2.4.2.1AninjectivecomputablenumberingAlternateproof ... 44 2.4.2.2Treeswithoutdeadends:Anumberingresult ....... 45 2.4.2.3Treeswithdeadends:Anecessity .............. 47 2.4.3ThinandPerfectThinClasses ..................... 49 2.4.3.1TheMartin{PourElConstruction .............. 49 2.4.3.2Non-existenceofcomputablenumberings .......... 50 2.4.4Small,VerySmall,andNondecidableClasses ............. 51 2.4.4.1Numberingsandhigh/noncomputablesets ......... 51 2.4.4.2Non-existenceofeectivenumberings ............ 52 3RANDOMCLOSEDSETS ............................. 54 3.1Overview .................................... 54 3.2EectiveRandomnessofReals ......................... 55 3.2.1Introduction ............................... 55 3.2.2ConstructiveMartingaleRandomness ................. 56 3.2.3Prex-freeRandomness ......................... 57 3.2.4Martin-Lofn-randomness ....................... 58 7

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3.3Martin-LofRandomnessofClosedSets .................... 60 3.3.1TheHit-or-MissTopologyonC .................... 60 3.3.2TowardaMeasure ............................ 61 3.3.3CanonicalCodingandMeasure .................... 61 3.3.4GhostCoding .............................. 64 3.3.5CodingEquivalance ........................... 65 3.3.6CodingandJoinsofClosedSets .................... 66 3.4MembersofRandomClosedSets ....................... 67 3.4.1PositiveResults ............................. 68 3.4.2NegativeResults ............................. 73 3.5MeasureandDimension ............................ 78 3.5.1Measure ................................. 78 3.5.2Dimension ................................ 79 3.6Prex-FreeComplexityofClosedSets ..................... 82 3.6.1LowerComplexityBounds ....................... 83 3.6.2UpperComplexityBounds ....................... 85 3.7OtherNotionsofRandomnessforClosedSets ................ 87 3.7.1RandomnesswithRegularProbabilityMeasures ........... 87 3.7.2RandomnesswiththeInclusionofTreeswithDeadsEnds ...... 87 3.8RandomClosedSetsandEectiveCapacity ................. 88 3.8.1ComputableCapacities ......................... 88 3.8.2RegularMeasuresandCapacitiesofClosedSets ........... 90 4RANDOMCONTINUOUSFUNCTIONS ..................... 94 4.1Overview .................................... 94 4.2DeniningRandomnessforContinuousFunctions .............. 95 4.2.1RepresentingFunctions ......................... 95 4.2.2RepresentingSequences ......................... 96 4.2.3ASoundDenition ........................... 97 4.3RandomContinousFunctionsandImages .................. 99 4.3.1PerfectImages,ineveryinstance .................... 99 4.3.2Non-injectiveImages,ineveryinstance ................ 100 4.3.3Non-surjectiveImages,ininstances .................. 103 4.3.4Imagesofcomputableelements ..................... 104 4.4RandomClosedSetsarisingfromrandomcontinuousfunctions ....... 106 4.4.1APositiveResult:InverseImagesof0! ................ 106 4.4.2ANegativeResult:Images,ingeneral ................. 109 4.5Pseudo-DistanceFunctions ........................... 110 4.6n-Randomness .................................. 111 4.7FutureWork ................................... 112 5CONTINUITYOFCAPPINGINCBT ....................... 113 5.1Introduction ................................... 113 5.2ContinuityResults ............................... 114 5.2.1ContinuityResultsinC ......................... 114 5.2.2ContinuityResultsinCbTandMainResult .............. 115 8

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5.3RequirementsandStrategies .......................... 118 5.3.1Therequirements ............................ 118 5.3.2AP-strategy ............................... 119 5.3.3AnR-strategy .............................. 120 5.3.4AnS-strategy .............................. 121 5.4ThePriorityTree ................................ 125 5.5TheConstruction ................................ 128 5.6TheVerication ................................. 132 REFERENCES ....................................... 144 BIOGRAPHICALSKETCH ................................ 150 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCOMPUTABLEASPECTSOFCLOSEDSETSByPaulBrodheadMay2008Chair:DouglasCenzerMajor:MathematicsAclosedsetinf0;1gNmaybeviewedthesetofinnitepathsthroughatree;asetAiscomputableifthereisacomputerprogramwhichhaltsandgivesthecorrectansweroneveryquerytothemembershippredicateforA.Anumbering,orenumeration,isamapfromNontoacountablecollectionofobjects;ifthereisacomputablenumberingontoasetANthenwesaythatAiscomputablyenumerableorc.e..Thesetofinnitepathsthroughacomputable,orequivalentlyaco-c.e.,treeiscalledaneectivelyclosedset.Inthiswork,weinvestigate:numberingsfordierentfamiliesofeectivelyclosedsets,notionsofrandomnessfornonemptyclosedsubsetsof2N,notionsofrandomnessforcontinuousfunctionsfrom2Nto2N,andcontinuitypropertiesofCbT,thec.e.degreesundertheTuringreduction6bTthatrequiresthateachusebeboundedbyacomputablefunction.Numberingsandeectivelyclosedsets.Weshowthatcertainfamiliesorclassesoffamiliesofeectivelyclosedsets{suchasthedecidable,homogeneous,thin,smallortheentirefamilyofeectivelyclosedsets,orstringveriablefamilies{possess,ordonotpossess,injectivecomputableoreectivenumberings.ThisworksbuildsupontheseminalworkbyFriedberg[ 46 ],whoconstructedaninjectivenumberingofthec.e.sets.Randomnessofclosedsets.Inthespaceofclosedsets,wegiveaprobabilitymeasureanddeneaversionoftheMartin-LofTestforrandomness.Weshowthatrandomclosedsetsarenevereectivelyclosed,butare,ontheotherhand,alwaysperfect,havemeasurezero,andhaveboxdimensionlog24 3.Everyrandomclosedsetcontainsrandomandnon-randomelements,butnon-c.e.elements.Wealsoexplorealternate 10

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notionsforrandomness,suchastheproblemofcompressibilityoftrees.Finally,weconsidertheproblemofwhenarandomlychosenclosedsetmeetsaclosedQ;thisisthestudyofcapacities.Randomnessofcontinuousfunctions.Asin,wegiveaprobabilitymeasureanddeneaversionoftheMartin-LofTestforrandomness.Weshowthattheimageofarandomcontinuousfunctionisalwaysnon-injectiveandperfect,butnotnecessarilysurjective.Furthermore,computableelementsmaptorandomelements.Also,randomclosedsetsariseasinverseimagesof0!,butnot,ingeneral,asimages.Theformermotivatesastudyofpseudo-distancefunctions.Finally,weconsiderourresultsinthecontextofn-randomness.ContinuitypropertiesinCbT.WeshowinCbTthatforanyb6=0;00,thereisana>bsuchthatforanyx,b^x=0ia^x=0.WeprovethisbyrstshowingthatthattheSeetapunlocalnoncappabilitytheoreminthec.e.Turingdegrees[ 84 ]alsoholdsinCbT.Thistheoremdemonstratesthateveryb6=0;00isnoncappablewithanynontrivialdegreebelowsomea>bi.e.ifx
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CHAPTER1INTRODUCTIONThisthesisiaanaccumulationofmyworkasagraduatestudentattheUniversityofFlorida.Muchofthisworkisjointandpublished,ortobepublished.Thecitationsarelistedatthebeginningoftheappropriatechapters.Inthischapterweintroducevariousnotionsandexpoundupontheseinlaterchapters.EachofChapters 2 { 5 containsadistincttopicfromcomputabilitytheory.1.1GeneralOverviewComputabilitytheoryisaeldofmathematicallogic;thesubjectcapturestheprecisenotionofanalgorithmicprocesstowardsthestudyofdecidable/undecidableproblemsinmathematicsandnature.ItsmostnotablehistoricalcontributiontomathematicsisthedisruptionofHilbert'sProgrambyGodel'sIncompletenessTheorem.Morerecentworkhasshownthatothermathematicalproblemsareunsolvable,inthesensethatnocomputeralgorithmcansolveeveryinstantiationoftheproblem.ExamplesincludeHilbert'sTenthProblemtodecidewhetheragivenDiophantineequationhassolutions,thewordproblemforgroupstodecidewhetheragivenproductofgeneratorsandtheirinversesistheidentityelementofagroupdenedbyanitesetofequationsbetweensuchproducts,andthehomeomorphyproblemtodecidewhetherthetopologicalspacesdenedbyagivenpairofsimplicialcomplexesarehomeomorphic[ 32 ].Maturationofcomputability,however,throughapplicationsandtechniques,hasbroadeneditsinteractionswithotherelds,mostnotablycomputerscience.EachofChapters 2 { 5 isawitnesstothisbroadening.InChapter 2 ,wefocusonthestudyofeectivelyclosedsetsofbinaryreals.Thoughtofasasetrepresentingthesolutionstosomeproblem,eectivelyclosedsetscharacterizemanystructuresinmathematicsandcomputerscience.Inalgebra,forexample,theyrepresenttheprimeidealsofacomputableenumerableBooleanalgebraoracommutativeringwithidentity[ 47 ].Ingraphtheory,theyrepresentthesetofsolutionstomanyproblemswithcomputablegraphs,suchasHamiltoniancircuitsorvertexpartitions[ 20 ].Incomputerscience,eectivelyclosedsetsariseinthestudyofnon-monotoniclogicsand 12

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!-languages[ 25 68 70 ].Giventhewidevarietyofapplications,theworkinChapter 2 focusesonmethodsofenumeratingvariousfamiliesofeectivelyclosedsets.Theideaistoprovide,basedondesiredproperties,completelistingsoftheobjects{inthiscase,eectivelyclosedsets{representingthesetofsolutionstoproblemsofacertaintype.Weshowthatthereisaninjectivecomputableenumerationoftheentireclassofsets,ofcertainfamiliesofstringveriableclasses,andofthedecidableandhomogeneousclasses.Wealsoshowthatnocomputableenumerationexistsforthin,perfectthin,small,verysmall,ornondecidableclasses.InChapters 3 and 4 ,weextendnotionsrelatedtoeectiverandomnessforbinaryreals,toclosedsetsChapter 3 andcontinuousfunctionsChapter 4 ;variousglobalpropertiesareobtained.Abinaryrealiseectivelyrandomifitisimpossibleforacomputertondregularityorpatternsinit.Foraclosedset,representingsomesetofsolutions,thismeansthatitisdicultforacomputertopreciselyobtainorlocallydescribethissetofsolutions,giventhelackofpattern.Inbothchapters,weobtainso-calledbasisandantibasistheorems.Forinstance,everyrandomclosedsetcontainsrandomandnon-randomelements,butomitsvariouselementsofcomputability-theoreticinterest,suchasthepropertiesofbeingf-c.e.fapolynomial,1-generic,orofincompletec.e.degree.Wealsoshowthatconceptsfrombothchaptersareclosedrelated;randomclosedsetsariseasinverseimagesofrandomcontinuousfunctionsmappingto0!,butnot,ingeneral,asimages.Methodsemployedinallofthisworkrangefromtechniquesincomputabilityandeectiverandomness,totechniquesrelatedtothestudyofeectiveHausdordimensionandclassicalprobability.Finally,inconnectionwiththerootsofcomputabilitytheory,inChapter 5 wefocusontheclassicationofinformationcontentbymeansofTuringdegreetheory.Mathemati-calstructuresorobjectsareoftenencodedassetsofnaturalnumbers.Reductionmethods,suchastheTuringreducibility,allowtheinformationcontentofthesesetstobeclassiedintoequivalenceclasses,calleddegrees.InChapter 5 ,thefocusisontheboundedTuringreducibility[ 16 ];weshowthatcapping,theoperationwhichtakesthemeetofagiven 13

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noncomputableincompletec.e.degreewithanothernoncomputableincompletec.e.degreesuchthattheresultingmeetisthedegree0ofthecomputablesets,iscontinuous.Thatis,foranyb6=0;00,thereisana>bsuchthatforanyx,b^x=0ia^x=0.Asthisisathree-quantierstatementinthec.e.bT-degrees,thisresultgivesinsightintothethreequaniertheoryofthesame,whosedecidability/undecidabilityiscurrentlyunknown.Asanaside,inrecentworkbySoare,theboundedTuringreducibilitywiththeidentityusehasledtoapplicationsofdegreetheorytodierentialgeometry[ 72 87 ].Therestofthischapterisdevotedtointroducingtobasicdenitions,terminology,andnotationsthatwillbeusedthroughoutthisentirework.Section 1.2 coversbasicno-tionsandnotationsfortopicsinclassicalcomputabilitye.g.computablesets,computablyenumerablesets,partialcomputablefunctions,Turingdegrees.Section 1.3 coverstheba-sicsofclosedsetsinNN.AsalgorithmicrandomnessisatopiccoveredonlyinChapters 3 and 4 ,wepostponeageneralintroductionofthistopicandprovideitinSection 3.2 .1.2ClassicalComputabilityWegenerallyfollowthenotationofSoare[ 86 ]fornotionsthatarisefromclassicalcomputability.Foranin-depthtreatmentofthebasicfoundationsofthesubject,wereferthereaderthere.AsetAiscomputableifthereisacomputerprogram,orequivalentlyacomputablefunction,whichhaltsandgivesthecorrectansweroneveryquerytothemembershippredicateforA.Itiscomputablyenumerablec.e.ifthecomputerprogramisrequiredtohaltonlyonqueriesforelementsinA;thisgivesrisetothestandardexampleofac.e.set,namelythehaltingproblem{thesetofTuringmachines,ociallyencodedasasetofnaturalnumbers,thathaltwhengiventheirownbinaryinput.Computablyenumerablesetsarethedomains,therefore,ofso-calledpartialcomputablefunctions;weindexthepartialcomputablef0;1g{valuedfunctionsasfege2!.Partialcomputablefunctionalsthattakenaturalnumbermandrealxinputsareindexedase;wewillwritexemfortheresultofapplyingetomandx. 14

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Otherrelatednotationsarestandard:e;sdenotesthatportionedenedbystages,andex#meansthateisdenedonxand"meansundened.Wealsoindextheprimitiverecursivefunctions,asmallerclassoftotalfunctions,asfege2!.h;i:!2!!istypicallyacomputablebijectionsuchthath0;0i=0. AandPAdenotethecomplementandpowersetofA,respectively.z=xyisthecodingtogetheroftworealsxandy,sothatzn=xnandzn+1=ynforalln.Incomputability,areductionisabinaryrelationonsubsetsofNthatcapturesarelationshipbetweentheinformationcontentoftwosets.TheTuringreduction6Tisthemainreductionusedincomputabilitytheory.AisTuringreducibletoB,writtenA6TB,ifmembershipinAcanbedeterminedbyacomputeralgorithmthathasfullaccesstothemembershippredicateforB.Intuitively,theinformationcontentofAisviewedascomputable,orrecoverable,fromB.FurthermoreBisviewedasanoracle,intermsofinformationcontent,fordeterminingmembershipinA.Variousrestrictionsonhowmuchoracleinformationisallowedtobeusedindeter-miningthemembershipofasingleelementhavegivenrisetodierentkindsofreductions.Forexample,theboundedTuringreduction6bTrequiresthatquerytoanoracleBfordeterminingmembershipofxinAuseatmostfxamountofB,wheref,calledtheuse,isboundedbyacomputablefunction.TheidentityboundedTuringreductionsinsiststhattheusefbeboundedbytheidentityfunction.Reductionsoftengiverisetoequivalenceclasses,calleddegrees,wheretwosetsareequivalentiftheyaremutuallyreducibile.TheTuringdegreesthatcontainc.e.setsarecalledc.e.Turingdegrees.WedenoteCandCbTasthestructuresofthec.e.degreesundertheTuringreductionsandtheboundedTuringreductionsrespectively.ThestudyoftheTuringdegreeshasbeenoneofthemajorthemesincomputabilityresearch;thesedegreescapturethestructureoftheundecidableproblemsinarithmeticandnature.1.3ClosedSetsinComputabilityWegenerallyfollowthenotationofCenzer[ 19 ]forclosedsets:Foranitestring2!n,letjj=n.Welet;denotetheemptystring,whichhaslength0.Awordaof 15

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length1ismaybeidentiedwiththesymbola.Fortwostrings;,saythatextendsandwritevifjj6jjandi=ifori
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thatSn=2N)]TJ/F24 11.955 Tf 11.961 0 Td[(Pfnforalln.Foranyc.e.setW,wedenethec.e.opensetgeneratedbyWtobeOW=[fI:hi2Wg:AlsoletOWn=fxn:x2OWand8j
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CHAPTER2EFFECTIVELYCLOSEDSETSANDENUMERATIONSThefollowingchapterisjointworkwithDouglasCenzerandhasbeensubmittedasanarticleentitledEectivelyClosedSetsandEnumerations[ 13 ].ApreliminaryversionofthisresearchwasoriginallypresentedattheThirdInternationalConferenceofComputabilityandComplexityinAnalysisinGainesville,Floridain2006byP.Brodhead.ThispreliminaryworkwaspublishedinthereferredconferenceproceedingsasEnumerationsof01Classes:AcceptabilityandDecidableClassesP.BrodheadinhardcopyandinSpringerElectronicNotesinTheoreticalComputerScience,ElsevierScience16707,289-301[ 12 ].PortionsofthisworkwerealsopresentedbyP.Brodheadatthe2005SACNASConferenceOctober2005,Denver,CO,7thAnnualGraduateStudentConferencesinLogicApril2006,Madison,WI,the2007AssociationforSymbolicLogicAnnualMeetingMarch2007,Gainesville,FL,the2ndNewYorkGraduateStudentConferenceinLogicMarch2007,NewYork,NY,andthe8thAnnualGraduateStudentConferenceinLogicApril2007,Chicago,IL.Duetoinclimateweather,R.MillerpresentedinplaceofP.BrodheadattheNewYorkconference.2.1IntroductionThegeneraltheoryofnumberingswasinitiatedinthemid-1950sbyKolmogorov,andcontinuedunderthedirectionofMal'tsevandErshov[ 44 ].Anumbering,orenumeration,ofacollectionCofobjectsisasurjectivemapF:!!C.Inoneoftheearliestresults,Friedbergconstuctedaninjectivecomputablenumberingofthe01orcomputablyenumerablec.e.setssuchthattherelationn2e"isitself01.Moregenerally,wewillsaythatanumberingofcollectionofobjectswithcomplexityCsuchasn-c.e.,0n;or0niseectiveiftherelationx2e"hascomplexityC.Wewillalsoconsiderenumerationswheretherelationx2e"hasadierentcomplexitythanC.Forexample,thereisac.e.,butnotcomputable,numberingofthecomputablesets.Anumberingisacceptablewithrespecttoanumbering,denoted6,ithereisatotalcomputablefunctionfsuchthat=f.Ifisacceptablewithrespect 18

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toalleectivenumberings,thenissaidtobeacceptable.Theordering6givesrisetoanequivalencerelation,andtwonumberingsinthesameequivalenceclassarecalledequivalent.Furthermore,thestructureLCofallnumberingsofCmoduloformsanuppersemilatticeunder6.Hereinjectivenumberingsoccuronlyintheminimalelementsandacceptablenumberingsoccuronlyinthegreatestelement.Inthischapter,westudyeectivenumberingsoffamiliesofeectivelyclosedsetsi.e.01classes.Enumerationsof01classesweregivenbyLempp[ 61 ]andCenzerandRemmel[ 23 24 ],whereindexsetsforvariousfamiliesof01classeswereanalyzed.Foragivenenumeratione=Peofthe01classesandapropertyRofsets,fe:RPegissaidtobeanindexset.Forexample,fe:Pehasacomputablemembergisa03completeset.See[ 20 ]formanymoreresultsonindexsets.Certaintypesof01classesareofparticularinterest.LetPbea01class.WewillsaythatPisthinifforevery01subclassQofP,thereisclopensetUsuchthatQ=UP.WesaythatPishomogenousif,givendistinct;2TPofthesamelength,_i2TP_i2TP:ForPf0;1g!,PishomogeneousifandonlyifPistheclassofseparatingsetsSA;Bfortwodisjointc.e.setsA;B,thatis,SA;B=fC!:ACandBC=;g:Pissmallifthereisnocomputablefunctionsuchthat,foralln,cardTP!n>n.LetPnbetheleastksuchthatcardTP!k>n;thenPisverysmallifthefunctionPdominateseverycomputablefunctiong{thatis,Pn>gnforallbutnitelymanyn.Anumberinge7![Te]of01classesiscalledatreenumberingandwrittene7!Te.Numberingsbasedonprimitiverecursivetreesandon01treesarebothstudiedintheliteraturesee[ 23 24 20 ].Ifthesetfe;:2Tegiscomputable,thenthenumberinge=[Te]issaidtobeacomputablenumbering. 19

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Webeginourstudywiththefamilyof01classesinSection 2.2 .InSections 2.2.1 { 2.2.3 ,severalcommonlyusednumberingsarestudiedandshowntobeequivalentviaacomputablepermutation.InSection 2.2.4 ,wegivegiveaFriedbergnumberingofthe01classes;thismotivatesastudyofageneralclassoffamiliesof01classes,calledstringveri-ablefamiliesinSection 2.3 .InSection 2.4 ,weconsidernamedfamiliesof01classes.WeobtainpositiveresultsforhomogeneouscandecidableclassesinSections 2.4.1 and 2.4.2 .Weobtainnegativeresultsforthin,perfectthin,small,verysmall,andnondecidableclassesinSections 2.4.3 and 2.4.4 .2.2TheFamilyof01Classes2.2.1NumberingsintheLiteratureInthissection,wepresentseveraldierentcomputablenumberingsof01classesthathaveappearedintheliterature.Wealsopresentaneective,butnotcomputable,numbering.Ineachcasewedemonstratethateachprovidesacompletenumberingofthe01classes.Numbering1:PrimitiveRecursiveFunctions[ 23 ]Foreache,letebetheethprimitiverecursivefunctionfrom!to!andlet2Ue8vehi=1:ThenUeisauniformlyprimitiverecursivetreeforalleandiff:hi=1gisanyprimitiverecursivetree,thenUeisthattree.ThereforethesequenceU0;U1;:::containsallprimitiverecursivetreesandhencethemapping1e=[Ue]isacomputablenumberingofthe01classes.Numbering2:ComputablyEnumerableSets[ 20 ]Let2e=!!)-222(OWe:Thisisaneectivenumberingsincetherelationx22e"is01.Thiscanactuallybeimprovedtoacomputablenumbering,asfollows. 20

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Foreache,recallthatWe;sisthesetofelementsenumeratedintotheethc.e.setWebystagesandlet2Se8vhi=2We;jj:ThenSeisauniformlyprimitiverecursivetreeforalle.LetP=[T]bea01class,whereTisacomputabletree.Itfollowsthatforsomee,2Thi=2We:ThenP=[Se].Itfollowsthatthesequence[S0];[S1];:::containsall01classesandhencethemappinge=[Se]isacomputablenumberingofthe01classes.Itiseasytoseethatinfact[Se]=2e.Numbering3:Universal01Relation[ 52 ,p.73]Thereisauniversal01relationU!2!suchthatifQxisa01class,thenthereisane2!suchthatQ=fx:Ue;xg.UisdenedintermsoftheKleenT-predicate,sothatessentiallyUe;xxe":Dene3e=fx:Ue;xgtoobtainaneectivenumbering.Toseethatthisisacomputablenumbering,let2Ree":sothat3e=[Re]andthetreesReareuniformlyprimitiverecursive.Numbering4:TheHaltingProblem[ 53 ]Considerthemappinggivenby4e=fx:xeee"g:Thisisacomputablenumbering,since4e=[Te],where2Teee": 21

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ForanycomputabletreeT,chooseasothatanconvergesifandonlyif2T.Then2Taa#;sothat[T]=4a.Numbering5:TotalComputableFunctionsHerewewillconsideraneective,butnotcomputablenumberingbasedoncomputabletrees.ThisnumberingwillbeusedinconnectionwithstringveriablefamiliesofclassesinSection 2.3 .Let5e=[Te],where2Te8[ehi#)167(!ehi=1]:Thisenumerationisuniformly01,butisnotcomputable,sincetherelationem#isc.e.non-computable.Clearlyeach5eisa01class.IfeistotalandTisatreesuchthat,forall,wehave2Tehi=1,thenTe=Tandisa01class.Hencethisenumerationhasthecrucialpropertythat,foreverycomputabletreeT,thereexistsesuchthatT=Te.2.2.2EquivalenceoftheNumberingsInthissectionweshowthatthecomputablenumberingsofsection 2.2.1 aremutuallyequivalentviaacomputablepermutation.Eachoftheseisequivalenttotheeectiveenumerationofsection 2.2.1 viaa03-permutation.Weneedthefollowingproposition. Proposition2.2.1. a Foreachpairi;jwith16i65and16j64,thereisacomputablefunctionfsuchthatj=if. b Foreachj65,thereisa03functionfsuchthat5=if.Proof.162:UsetheS)]TJ/F24 11.955 Tf 11.955 0 Td[(m)]TJ/F24 11.955 Tf 11.955 0 Td[(nTheoremtodenefsothatWfe=fn:en6=1g:Then2Ue2Sfe. 22

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263:Denefsothat,forallm,xfem=leastnhxni2We:Then2e=3fe.364:Denefsothatxan=xforalln.Thenx23ex24fe.461:Recallthat4e=[f:ee"g].Denetheprimitiverecursivefunctiongsothatforeache,fehi=8>><>>:1;ifee";0;otherwise.Then1e=4ge.165:Denetheprimitiverecursivefunctionfsuchthat,foreache,fehi=8>><>>:1;if8vehi=1;0;otherwise.Then1e=5fe.Therestoftheprooffollowsbycomposition. Theorem2.2.2. Foranycomputablenumbering'whichiscomputablyequivalentto2,thereisacomputablepermutationpsuchthat2='p.Proof.TheproofisamodicationofanargumentduetoJockusch[ 86 ,p.25].Let=2.Byassumption,therearecomputablefunctionsfandgsuchthatfe='eand'ge=e.Sincethenumbering2isbasedonanenumerationofthepartialcomputablefunctions,wecanensurebypaddingthatfisinjective.Tomodifygintoaninjectivefunctiong1,itissucienttoeectivelycomputefromeacheaninnitesetSeofindicessuchthat'gi='eforalli2Se.Weproceedasfollows.LetAandBbecomputablyinseparablec.e.setsanddenecomputablefunctionskand`suchthat,forall 23

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eandm:ke;m=8>><>>:e;ifm=2B;;;ifm2Band`e;m=8>><>>:e;ifm2A;2!;ifm=2AThatis,webuildatreeforke;mwhichexactlyequalsthetreeforeforstringsoflengthsuntilm2Bs+1,inwhichcasenostringsoflengths+1areputintoke;m.Tobuildthetreefor`e;m,weputinallstringsoflengthsuntilm2As+1,inwhichcaseweincludeonlythestringsoflengths+1whichareine.NowletCe=fke;m:m2AgandDe=f`e;m:m2AgandletSe=gCe[De.Thenforj=gi2Se,itfollowsfromthedenitionthati=eandtherefore'gi='ge.WewillproveintwocasesthateithergCeisinniteorgDeisinnite.CaseI:Supposethate6=;andsupposebywayofcontradictionthatgCeisnite.ThenS=fm:gke;m2gCegisacomputableset.NowAgCebydenition.Ontheotherhand,ifj=gke;m2gCewherem2A,then'j=ke;m=e6=;.Butform2B,'gke;m=ke;m=;,sothatSB=;.ThiscontradictstheassumptionthatAandBarecomputablyinseparable.CaseII:Supposethate=;.ItfollowsasinCaseIthatgDeisinnite.Thuswemayassumewithoutlossofgeneralitythatbothfandgareone-to-one.Nowdeneasequencefen:n2!gandtwopartitionsof!asfollows.Lete0=0andforeachn,en+1istheleastesuchthat'e6='eiforeveryi6n.LetAn=fe:e=engandBn=fe:'e=eng.Then!=SnAn=SnBnandeachsequenceispairwisedisjoint.Furthermore,fBnAnandgAnBn.TheremainderoftheprooffollowsasintheMyhillIsomorphismTheorem[ 86 ,p.24;Also5.8,p.25]. Asimilarargumentshowsthatif'isa03numberingofthe01classes,thenthereisa03permutationpwith'=2p.Itfollowsthateachofthecomputablenumberings 24

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1;:::;4areacceptable,thatis,theyoccurinthegreatestelementofthesemilatticeLP.Inthesection 2.2.4 wewillseethatminimalelementsexistinthesemilattice{thatis,injectivenumberings.First,however,weprovideanalternateproofofTheorem 2.2.2 .2.2.3EquivalenceoftheNumberingsAlternateProofAttheThirdInternationalConferenceofComputabilityandComplexityinAnalysisinGainesville,Floridain2006,IpresentedtheargumentforTheorem 2.2.2 asgivenabove;itisamodicationofanargumentofJockuschforthec.e.version.Pleasedwiththeargument'sapplicabilityto01classes,IurgedRobertSoare,whowaspresentintheaudience,tokeeptheJockuschargumentforthec.e.versioninhisnewandupcomingbook,ComputabilityTheoryandApplications[ 88 ].Ifthealternateproofusingtherecursiontheoremforthec.e.versionofTheorem 2.2.2 [ 86 ,p.43]couldnotalsobemodiedtoalsoproveTheorem 2.2.2 ,thenhesaidhewould.Weshowbelowthatanalternateproofispossible,modifyingtherecursiontheoremargument. Theorem2.2.3. Foranycomputablenumbering'whichiscomputablyequivalentto2,thereisacomputablepermutationpsuchthat2='p.AlternateProof.Weproceed,atrst,asbefore.Thatis,therearecomputablefunc-tionsfandgsuchthatfe='eand'ge=e.Ourgoalistomodifygtoobtainaninjectivefunctiong1.Instead,however,wedeneg1dierently,usinganauxil-iarycomputablefunctionhobtainedbytheRecursionTheorem.Weensurehsatises,foralldistinctiandj,ighe;i6=ghe;jandiihe;i=e.Wenowdeneg1,givenhsatisfyingiandii.Deneg1=ghfg;0.Todeneg1k+1,notethatiensuresinnitelymanydistinctghe;iforeache.Leta0=0andak+1betheleastintegerisuchthatghfgk+1;i>g1k.Deneg1k+1=hfgk+1;ak+1.Toseethatforalle,g1e=e,xeandnotethatg1e=ghfge;ae=hfge;ae=fge=ge=e.Toseethatg1isinjective,notethatthedenitionensuresthatforallk,g1k+1>g1k. 25

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Wenowdenehe;foreache,byinduction,ensuringthatiandiiaresatised.Denehe;0=e.Todenehe;k+1,usetheRecursionTheoremtoobtainannsuchthatnz=8><>:ezifgn6=ghe;i8i6kundenedotherwiseNoticethatifgn=ghe;ithensincei6k,byinductione=he;i=ghe;i=gn=n,theundenedfunction.Hencee=ninallinstances.Sincefa:a=ngisaproductive[ 86 ,p.43],letpbeacorrespondingproductivefunction.DeneWrx=Wx[fpxgandnotethateachWrinisadistinctc.e.subsetofA.Consequentlyeachgrin=rinisadistinctpartialcomputablefunction.Letnkbetheleastisuchthatrin6=he;jforallj6k.Denehe;k+1=8><>:nifgn6=ghe;i8i6krnknotherwiseToseethatiholdsghe;i6=ghe;jfordistincti;j,notethatge=e6=rin=grinforalli>1.Thereforeifgn6=gethenforallj,he;j2feg[fringi>0.Otherwisehe;j2feg[fringi>1.Ineithercase,foralldistincti;j,ghe;i6=ghe;j.Toseethatiiholdshe;i=eforalli,notethate=n,he;i2feg[fringi>or1,andrin2fa:a=ng.Soe=he;iforalli. 2.2.4InjectiveComputableNumberingsInthissection,wemodifyFriedberg'soriginalargumentforinjectivecomputablenumberingofthec.e.sets,toprovidedierentnumberingresultsneededfor01classes.InSection 2.2.4.1 ,wepresentFriedberg'soriginalargument.InSection 2.2.4.2 ,weprovideaninjectivecomputablenumberingofdisjointpairsofc.e.sets;thisinneededlaterinSection 2.4.1 inordertoprovideaninjectivecomputablenumberingofthehomogeneous01classes.Finally,inSection 2.2.4.3 ,weconstructacomputableinjectivenumberingofthe01classesin2!;wealsoprovideotherinjectivenumberingresultsfor01classes. 26

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2.2.4.1Originalc.e.setsargumentThefollowingistheoriginalFriedbergargument,withslightmodicationsinthenotationandpresentation. Theorem2.2.4[ 46 ]. Thereisaninjectivecomputablenumberingofallc.e.sets.Proof.LetfWege2!bethestandardnumberingofthec.e.sets.Wewillconstructasequenceofc.e.setsfYege2!instagessothate7!Yewillbethedesiredinjectivenumbering.IntheconstuctionwewillusethenotionofoneY-indexifollowingaW-indexewiththeideathatintheendYiwillequalWe.Atsomepoint,however,wemaydecidethatiwillnolongerfolloweagainandwewillsaythatiisreleased.Ifiisneverreleasedfromfollowingethenitissaidtobealoyalfollowerandotherwiseitisdisloyal.Oncereleased,anindexremainsfreeandisneveragainthefollowerofanye.Atanyparticularstage,anyY-indexthatisnotfollowinganyW-indexissaidtobefree.AnynonzeroY-indexthathasneverfollowedanyW-indexissaidtobeunused.Toensurethatnoc.e.setisexcludedfromtheY-sets,wewillensurethateachWeisinnitelyoftengiventheopportunitytobefollowed.Todothis,atstages=hns;esiallactionsintheconstructionwillbetakenwithrespecttoWes.AssumewithoutlossofgeneralitythatY0=;.Construction:Therearethreepossiblecasesateachstages.Case1:Ififollowses&We;s[0;i)]TJ/F15 11.955 Tf 12.044 0 Td[(1]=Wes;s[0;i)]TJ/F15 11.955 Tf 12.044 0 Td[(1]somee
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i 8e9iYi=W^e ii i6=j)167(!YiandYjarenotthesamenitesets iii i6=j)167(!YiandYjarenotthesameinnitesetsVericationofi.Fixe.Firstnotethatalthough^ecanonlyhaveonefolloweratanyparticularstage,itcannothaveaninnitenumberofthemwhichareeachreleasedatsomestage.Forexample,ifsandxaresucientlylarge,thenforallj<^e,Wj;s[0;x]6=W^e;s[0;x].HencereleasecanonlyoccurinCase1anitenumberoftimes.Furthermore,Case2ensuresthatreleaseinCase3canonlyoccurforanyswhenes<^e.Therefore,bytheabove,ifi>xisfollowerof^eandt>s,thenYi;t)]TJ/F23 7.97 Tf 6.586 0 Td[(1=W^e;t)]TJ/F23 7.97 Tf 6.586 0 Td[(16=Wes;t.HenceiwillnotbereleasedinCase3.ThereforereleasecanonlyoccurinCase3anitenumberoftimes.Nowletsbeastagewhere^eneverlosesafollower.IfCase3occursinnitelyoftenafterstagesfor^e,thenithasapermanentfollowerisothatYi=W^e.ThereforeassumeCase3occursonlynitelyoften.Since^eneverlosesafollower,Case1cannotoccur.HenceCase2mustoccurinnitelyoften.Howeverthereareonlyanitenumberofisuchthatthe`If'clauseholdswithW^e;s=Yi;s)]TJ/F23 7.97 Tf 6.586 0 Td[(1"inCase2.Forexample,fi:i=0;i6^e;oriisdisplacedby^egisanitesetsinceonlyanitenumberofiaredisplacedduetoCase3occuringonlyanitenumberoftimes.Toseealsothatfi:forsomes;ifollowssomea<^eandW^e;s=Yi;s)]TJ/F23 7.97 Tf 6.587 0 Td[(1"holdsgisnite,notethatbythedenitionof^e,ifa<^ethenWa6=W^e.Soforsucientlylarget,ififollowsathenW^e;t6=Yi;t)]TJ/F23 7.97 Tf 6.586 0 Td[(1.ThereforeCase2occuringinnitelyoften,togetherwithonlynitenumberofisuchthatthe`If'clauseholdswithW^e;s=Yi;s)]TJ/F23 7.97 Tf 6.586 0 Td[(1",impliesthatthereisasingleisuchthatYi;s)]TJ/F23 7.97 Tf 6.587 0 Td[(1=W^e;sforinnitelymanys.ThusYi=W^e.Vericationofii.Firstnotethatatanystages,ifWes=;thenCase2ensuresthatCase3willnotbereachedsothatesneverreceivesafollower.Furthermoreeachk6=0ischosenatsomestagesinCase3tofollowsomeeandfromthepreviouscomments,Yk;s6=;.Immediatelythereafter,Yk;sisensuredtobedistinctfromallotherY`;s`6=k.ThisalsocontinuestobetrueatallsubsequentstagestbyCase3. 28

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Nowsupposei6=j.SincewearesupposingbothYiandYjarenite,thereissometsuchthatforalls>t,Yi;s=YiandYj;s=Yj.AsmentionedaboveYi;s6=Yj;sandthereforeYi6=Yjasrequired.Vericationofiii.AssumebothYiandYjareinniteandi6=j.NowimusteventuallyfollowsomeW-index.Ifiisdisloyal,thenafteritisreleasedYicanacquireanewmemberonlywheniisdisplaced.Howevericanbedisplacedonlyoncebyeachei.ItfollowsthatifiisreleasedthenYionlyacquiresanitenumberofelementsthereafter,contradictingthefactthatitisinnite.Thisargumentshowsthatbothiandjareneverreleased.Wesaythatiandjareloyalfollowers.Supposenowthatiandjloyallyfollowi0andj0,respectively.ThenYi=Wi0andYj=Wj0.Nowi06=j0sinceasingleW-indexcannothavemorethanoneloyalfollower.Assumewithoutlossofgeneralitythati0
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s=hns;hes;isiiallactionsintheconstructionwillbetakenwithrespecttoWes;Wis.AssumewithoutlossofgeneralitythatY0;Y0=;;;.Construction:Therearethreepossiblecasesateachstages.Case1:Ifhe;iifollowshes;isiandWa;s[0;he;ii)]TJ/F15 11.955 Tf 21.148 0 Td[(1]=Wes;s[0;he;ii)]TJ/F15 11.955 Tf 21.148 0 Td[(1]andWb;s[0;he;ii)]TJ/F15 11.955 Tf 19.707 0 Td[(1]=Wis;s[0;he;ii)]TJ/F15 11.955 Tf 19.707 0 Td[(1]forsomeha;bi
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above,ifha;bi>hx;yiisfollowerofhe;iiandt>s,thenYa;t)]TJ/F23 7.97 Tf 6.587 0 Td[(1=We;t)]TJ/F23 7.97 Tf 6.586 0 Td[(16=Wes;torYb;t)]TJ/F23 7.97 Tf 6.586 0 Td[(1=Wi;t)]TJ/F23 7.97 Tf 6.586 0 Td[(16=Wis;t.Henceha;biwillnotbereleasedinCase3.ThereforereleasecanonlyoccurinCase3anitenumberoftimes.Letsbeastagewherehe;iineverlosesafollower.IfCase3occursinnitelyoftenafterstagesforhe;ii,thenithasapermanentfollowerhe;iisothatYe;Yi=We;Wi.ThereforeassumeCase3occursonlynitelyoften.Sincehe;iineverlosesafollower,Case1cannotoccur.HenceCase2mustoccurinnitelyoften.Thereareonlyanitenumberofh;isuchthatthe`If'clauseholdswiththeequality;;e;i;sgivenbyY;s)]TJ/F23 7.97 Tf 6.587 0 Td[(1;Y;s)]TJ/F23 7.97 Tf 6.587 0 Td[(1=We;s;Wi;s"inCase2.Forexample,fh;i:h;i=0;h;i6he;ii;orh;iisdisplacedbyhe;iigisanitesetsinceonlyanitenumberofh;iaredisplacedduetoCase3occuringonlyanitenumberoftimes.Toseealsothatfh;i:forsomes;h;ifollowssomeha;bit,Yk;s=Ykk2fe;i;;g.AsshownaboveYe;s;Yi;s6=Y;s;Y;sandthereforeYe;Yi6=Y;Y.SinceYi=Y,itfollowsthatYe6=Y. 31

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SupposenowthatYiandYarebothinnite.Nowhe;iimusteventuallyfollowsomehe0;i0i.Ifhe;iiiseverreleasedthenitisfree.ThereafteronlytherstcoordinateofYe;YiacquiresmemberssothatYiisnite,acontradiction.Asimilarargumentholdsforh;i.Thereforebothhe;iiandh;iareneverreleasedandareloyalfollowers.Supposethattheyfollowhe0;i0iandh0;0i,respectively.ThenYe;Yi=We0;Wi0andY;Y=W0;W0.Notethathe0;i0i6=h0;0isinceasingleW-indexcannothavemorethanoneloyalfollower.Assumewithoutlossofgeneralitythathe0;i0ihe;ii.Itfollowsthatifhe;iiisreleasedthenYeonlyacquiresanitenumberofelementsthereafter,contradictingthefactthatitisinnite.Thisargumentshowsthatbothhe;iiandh;iareneverreleasedandarethereforeloyalfollowers.NowapplythesameargumentgiveninthelaterpartofthevericationofiitogetthatYe6=Y.Vericationofiv.Ifhe;iiisaloyalfollowerofsomehe0;i0i,thenYe;Yi=We0;Wi0.Thereforesincehe;ii7!We;Wiisanenumerationofdisjointsets,itfollowsthatYeandYiaredisjoint.Otherwisesupposethathe;iiisreleasedatsomestages.ThenYe;sYi;s=;andthereafteronlyYecanacquirenewelementsnotalreadyincludedinYi;s.ThereforedisjointnessispreservedandYeYi=;. 2.2.4.3ResultsforeectivelyclosedsetsInthissection,wemodifytheFriedbergargumentofSection 2.2.4.1 toconstructacomputableinjectivenumberingofthe01classesin2!.AnalternativeproofwassketchedbyRaichev[ 79 ].Afterwardsweprovideotherinjectivenumberingresultsfor01classes. 32

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Theorem2.2.6. Thereisaninjectivecomputablenumberingofall01classesin2!.Proof.LetfWege2!bethecomputableenumerationofthenonemptyc.e.subsetsof2
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2k:62OWes;skg.Let`sbetheleastksuchthatjStrk;sj>jEsj.Then`sistheleastlevelofOWes;swherethereisenoughroomtogiveeachequivalentYj;s)]TJ/F23 7.97 Tf 6.587 0 Td[(1anadditionalstringtodistinguishOYj;s)]TJ/F23 7.97 Tf 6.586 0 Td[(1fromOWes;s.NoticethatOWes;s6=2!byCase2.SupposethatStr`s;s=f12:::jStr`s;sjg.NowputjintoYjandreleasejifitisafollower.Wesaythatjisdisplacedatstages.Verication:Givene2!,let^ebetheleastksuchthat[OWk=OWe].Wewillshow: i 8e9iYi=W^e; ii i6=jimpliesthatOYiandOYjarenotequalwhenbothareclopen. iii i6=jimpliesthatOYiandOYjarenotequalwhenbotharenotclopen.Vericationofi.Fixe.Firstnotethatalthough^ecanhavedierentfollowersatdierentstages,itcannothaveaninnitenumberofdisloyalfollowers.Thatis,ifsandxaresucientlylarge,thenbythedenitionof^e,forallj<^e,OWj;sx6=OW^e;sx.HencereleasecanonlyoccurinCase1anitenumberoftimes.Furthermore,Case2ensuresthatreleaseinCase3canonlyoccurforanyswhenes<^e.Therefore,bytheabove,ifi>xisfollowerof^eandt>s,thenOYi;t)]TJ/F23 7.97 Tf 6.587 0 Td[(1=OW^e;t)]TJ/F23 7.97 Tf 6.587 0 Td[(16=OWes;t.HenceiwillnotbereleasedinCase3.ThereforereleasecanonlyoccurinCase3anitenumberoftimes.Nowletsbeastageafterwhich^eneverlosesafollower.IfCase3occursinnitelyoftenafterstagesfor^e,thenithasapermanentfollowerisothatOYi=OW^e.ThereforeassumeCase3occursonlynitelyoften.Since^eneverlosesafollower,Case1cannotoccur.ThusCase2mustoccurinnitelyoften.HoweverthereareonlyanitenumberofisuchthatthehypothesisofiiholdswithOW^e;s=OYi;s)]TJ/F23 7.97 Tf 6.587 0 Td[(1.Toseethis,considerthethreecases.First,eache
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Vericationofii.SupposethatU=OYi=OYj6=;;2!isclopenandletU=OW^e.Itfollowsfromcompactness,thatthereissomenitessuchthat,forallt>s,OYi;tt=OYj;tt=OYi.Itfollowsfromthevericationofiabovethatthereisastaget>ssuchthatCase3appliesto^e.ButthenatleastoneofOYi;OYjmustchangeatstaget.Thiscontradictionveriesii.Vericationofiii.AssumebothOYiandOYjarenotclopenandi6=j.ItfollowsthatOYimustchangeinnitelyoften,sinceofcourseOYi;sisclopenforeachs,andsimilarlyforOYj.NowimusteventuallyfollowsomeW-index.Ifiiseverreleased,thenitisfree.ThereafterYiacquiresmembersinCase3atstagesonlywhenWes;s=Yi;s)]TJ/F23 7.97 Tf 6.586 0 Td[(1.ThisimpliesthatCase2doesnotapplyatstagesandthuses
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Corollary2.2.8. Thereisaneectivenumberingofthe01classesbasedonthetotalcomputablefunctionsProof.Modifyeachc.e.setinthestandardnumberingtoenumerateanelementonlyaslongasitislargerthananypreviouslyenumeratedelement.ApplyingFriedberg'sargumenttothisclassofc.e.setsyieldsaneectiveinjectivenumberinge7!Ceofthecomputablesets[ 91 ].FurthermoreeachCestillenumeratesitselementsinincreasingorder.Nowsupposefege2!isacorrespondingsetofcharacteristicfunctions.Onecharac-terizationofa01classPisthatP=!!nOWforsomecomputablesetW[ 20 ].Asaresult,e7!!!nOCe=!!nOfn:en=1gisanalternativeeectivenumberingbasedontotalcomputablefunctionsreplacingnoneectiveNumbering2. Itisknown,forxedn>0,thatthereisaeectiveinjectivenumberingofthen-c.e.sets[ 50 ]. Conjecture2.2.9. Foreachn,thereisanumberinge7!Neofn-c.e.setssuchthatthereisaninjectivecomputablenumberinge7!!!nONeofallclosedsetsofthisform.Forn=1theconjectureisgivenbyTheorem 2.2.6 .WenextshowthatTheorem 2.2.6 isnotobtainablebyanycomputableprocedurethatuniformlyselectstheminimalindexofevery01class. Theorem2.2.10. Thereisnocomputablechoicefunctionforindicesof01classes.i.e.acomputablefunctionfsuchthatfeisanindexofPeandPi=Pefi=feProof.Supposethatfexists.Leta0;a1;:::beanenumerationofanoncomputablec.e.setA.DeneacomputablefunctiongandtreesTgesothatifjj=n,then2Tge$e62fa0;:::;ang:ThenPge=8><>:;ife2A2!otherwiseForanya2A;e2A$fge=fga,makingAcomputable. 36

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Itisstillpossible,however,thatsomeinterestingproperfamilyof01classesmaybeenumeratedbyselectingminimalindicesfromtheenumerationofall01classes.2.3StringVeriableFamiliesof01ClassesInthissectionweexaminefamiliesofclasseswhicharedeemedtobestringver-ablee.g.decidableorhomogeneousclassesorstronglystringveriablee.g.stronglydecidableorstronglyhomogeneousclasses.Anystringveriablefamilyhasaneectivenumberingandanystronglystringveriablefamilyhasacomputablenumbering.2.3.1DenitionandExamplesFirstwewilldenethenotionsofstringveriableandstronglystringveriable.Wealsogivesomeexamples. Notation2.3.1. LetF0;F1;:::beacomputableenumerationofthenitesubsetsof2
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Denethestringvericationfunctionhasfollows.LetA1;A2;A3;A4enumeratePf0;1g
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2.3.2ComputableandEectiveNumberingsWenowdemonstratethatstronglystringveriableandstringveriablefamiliespossesscomputableandeectivenumberings,respectively. Theorem2.3.7. a Anystronglystring-veriablefamilyof01classeshasacom-putablenumbering. b Anystring-veriablefamilyof01classeshasaneectivenumbering.Proof.SupposeFisa[strongly]string-veriablefamilyof01classessatisfyingstringveriabletreerelationsH0;H1;:::;Hm,withcorrespondingstringfunctionsh0;h1;:::;hm.Forparta,letthestandardcomputableenumerationofthe01classesin2!begivenbyPe=[Te],wherethesequenceTeisuniformlyprimitiverecursiveforexample,thenumbering2giveninsection 2.2.1 .WewilldeneauniformlycomputablesequenceSeoftreessuchthatthesequenceQe=[Se]enumeratesexactlythefamilyof01classesstronglysatisfyingH0;H1;:::;Hm.Forany2f0;1gn,wedeterminewhether2Seasfollows.Firstcheckthat2Te.Ifso,foreach2f0;1gnandeachi6m,computehi=D1;D2;:::;D2janddeterminewhetherthereexistsi
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Wenowprovidesomeadditionalexamplesofstringandstronglystringveriableclasses. Denition2.3.8. i A01classPisstronglydecidableifthereisaprimitiverecursivetreeTwithnodeadendssuchthatP=[T]. ii A01classPisstronglyhomogeneousifthereisahomogeneousprimitiverecursivetreeTwithnodeadendssuchthatP=[T].Fromdenition 2.3.8 ,weimmediatelyhavethefollowingcorollarytoTheorem 2.3.7 Corollary2.3.9. a Thefamilyofdecidable01classesin2!hasaneectivenum-beringandthefamilyofstronglydecidable01classesin2!hasacomputablenumbering. b Thefamilyofhomogeneous01classesin2!hasaneectivenumberingandthefamilyofstronglyhomogeneous01classesin2!hasacomputablenumbering.Inthefollowingsectionwewillprovideresultswhichdemonstratethatdecidableclassesinfacthaveinfacta1-1computablenumberingseeCorollary 2.4.2.1 .Weprovideanalternateexplicitproofinsection 2.4.2.1 .Forhomogeneousclasses,adierentapproachisneededtodemonstratethattheypossessa1-1computablenumberingandwedothisinsection 2.4.1 .2.3.3FamiliesContainingtheClopenClassesInthissectionweconsiderstring-veriablefamiliesofclassesthatcontainallclopenclasses.WerstimproveTheorem 2.3.7 toobtainacomputablenumberingofanystring-veriablefamilywhichincludestheclopensets. Theorem2.3.10. IfFisanystring-veriablefamilyof01classes,thenthereisacomputablenumberingofC[F.Proof.WemodifytheproofofTheorem 2.3.7 sothatwhenthestring-veriablerelationsfail,weextendallnodesratherthanmakingthemdeadends.Onceagain,theconstruc-tionisbasedontheenumerationeofthepartialcomputablefunctions.Theconstructionisinstages,whereatstageswewillhavene;s=maxfn:82f0;1gne;shi#g; 40

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Je;s=f2f0;1gne;s:e;shi=1g;andQe;s=[Je;s=[Se;s]:ThenQe=TsQe;s=[Se]willbethedesirednumbering.Toensurethatthisnumberingiscomputable,wewilldeterminewhether2Seatstagejj.Forthisargument,weassumethate=1foralle.Construction.Atstage0wehavene;0=0,Je;0=Se;0=f;gandQe;0=2!.Atstages+1,wechecktoseewhethere;s+1hi#forall2f0;1gns+1.Ifnot,thenne;s+1=ne;s,Je;s+1=JsandSe;s+1=Se;s[f_i:2Se;s;i=0;1g.Ifso,thenwechecktoseethate;s+1isthecharacteristicfunctionofatreeonf0;1gne;s+1andweverifythestringrelationsuptof0;1gne;s+1.Ifthisvericationfails,thenagainne;s+1=ne;sandJe;s+1=Je;s.Inthiscase,vericationwillalsofailatallfuturestages,sothatQe=Qe;sisaclopenset.Ifthetreeandstring-vericationssucceed,thenne;s+1=ne;s+1,sothatJe;s+1f0;1gne;s+1andQe;s+1changeasindicatedabove.Inthiscase,Se;s+1=Se;s[f2f0;1gs+1:ne;s+12Je;s+1g:IfeisthecharacteristicfunctionofthecomputabletreeTe,andifPe=[Te]2F,thenitfollowsfromtheconstructionthatQe=Pe,sothatQe2Fandfurthermore,any01classPe2Fwilltherebyoccurinthenumbering.Otherwise,theconstructionwillmakeQeaclopenset. Corollary2.3.11. ForanystringveriablefamilyFof01classes,therea1-1computablenumberingofC[F.Proof.LetFbeastringveriablefamily.ThenthereisacomputablenumberingofC[FbyTheorem 2.3.10 .ItthenfollowsfromTheorem 2.2.7 thatthereisa1-1computablenumberingofC[F. 41

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Corollary2.3.12. Therea1-1computablenumberingofanystringveriablefamilyof01classescontainingallclopenclasses. Corollary2.3.13. Therea1-1computablenumberingofthedecidable01classes.2.4NamedFamiliesof01ClassesItthissection,weotainnumberingresultsforvariousnamedfamiliesthatcommonlyoccurintheliterature.Inthersttwosub-sections 2.4.1 2.4.2 ,weexpandupontheresultsfromsection 2.3 andweobtainpositivenumberingresultsforthehomogeneousanddecidableclasses.Inthetwosub-sections 2.4.3 2.4.4 thatfollowthese,weobtainnegativeresultsforthethin,perfectthin,small,verysmall,andnondecidableclasses.2.4.1HomogeneousClassesHomogeneous01classesareastring-veriablefamilyof01classes.Consequently,byCorollary 2.3.9 ,theypossessaneectivenumbering.Corollary 2.3.12 fallsshortofdemonstratingthattheypossessacomputablenumbering,asclopensetsarenotnecessarilyhomogeneous.WeprovidethenecessaryargumentinTheorem 2.4.3 andshow,infact,thatacomputableinjectivenumberingexists.Werstprovideanalternatecharacterizationofthehomogeneousclasses;theymaybeviewedseparatingclassesofc.e.sets. Denition2.4.1. TheseparatingclassSA;BoftwosetsA;B!isgivenbySA;B=fC!:ACandBC=;g.Inwhatfollows,ScA;BwilldenotethesetofcharacteristicfunctionsfCofC2SA;B. Theorem2.4.2. P2!isanonemptyhomogeneous01classiP=ScA;Bforsomedisjointc.e.setsAandB.Proof.!SupposethatPisanonemptyhomogeneousclasswithhomogeneoustreeTP.WewillshowthatP=ScA;BwhereA=fn:0n_062TPgandB=fn:0n_162TPg.ToseethatAandBarec.e.,supposethatP=[T]withTcomputable.Thenforeachi2f0;1g,0n_i62TPi9s>n+1Tf:jj=sand0n_ivg=;. 42

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ToseethatAandBaredisjoint,supposethatn2A.SincePisnotempty,thereissomex2Psothatxn2TP.Now0n_062TPsothatxn_062TP.Hencexn6=0sothatxn=1.Thereforexn_12TP.SincePishomogeneous,0n_12TPsothatn62B.HenceAB=;.ToseethatScA;BP,takefC2ScA;BforsomeC2SA;B.Then,n2C!n62B!0n_12TP!fCn+1=fCn_12TP&n62C!n62A!0n_02TP!fCn+1=fCn_02TPSinceforarbitraryn,fCn+12TP,itfollowsthatfC2P.ToseethatPScA;B,supposethatx2PandletC=fn:xn=1g.ItisclearthatxisthecharacteristicfunctionfCofC.ItsucestoshowthatC2SA;BsothatfC2SCA;B.Supposerstthatn2A.Then0n_062TPsothatfCn_062TP.Sincex=fC2TP,itfollowsthatfCn=1.Hencen2CandAC.Nowsupposen2B.Then0n_162TP,sofCn_162TP.Asbefore,fCn=0sothatn62C.HenceCB=;.SupposeAandBaredisjointc.e.setsandfAsgs2!andfBsgs2!arestageenumerationsofAandB,respectively.DeneP=s[Ts]whereTs2
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2.4.2DecidableClassesInthissectionweprovideanalternateproofoftheexistenceofinjectivecomputablenumberingsofthedecidableclassesseeCorolllary 2.3.13 fortherstproof.Wealsoshowthatinanycomputablenumberingofthedecidableclassesviatrees,wecanpro-videacomputablenumbering^ofallthetreeswithoutdeadendsthatoccur,alongwithallclopenclasses.Finally,weshowthatsomedecidableclassinclassinthenumberingmustnecessarilyhave,asatree,deadendsthroughouteveryoccuranceinthenumbering.2.4.2.1AninjectivecomputablenumberingAlternateproofSincedecidabilityisstring-veriableandeveryclopensetisdecidable,itfollowsfromCorollary 2.3.12 thatthedecidableclasseshavea1-1computablenumbering.Thisresultcouldnotbeobtainedbyusingthestandardnumberingofthe01classesandmodifyingeachtreeasitbecomesknownthatishasadeadend.Forexample,simplyextendeachsuchnodewith,say,allones.Thisisbecause,asaconsequencetothefollowingtheorem,PbeingdecidableisinsucienttoensurethattheuniquetreeTPwithoutdeadendsshowsupinacomputabletreenumbering.Thefollowingisanalternate,explicitproof. Theorem2.4.4. Thereisa1-1computablenumberingofalldecidableclassesin2!.Proof.Lete7!WebetheinjectiveeectivenumberingofthecomputablesetsasgivenintheproofofCorollary 2.2.8 .Wewilldeneaneectivecorrespondencebetweenthesesetsandthenonemptycomputabletreeswithoutdeadends.Wewilldothisthroughaseriesofthreeone-to-onecorrespondences{namelythecorrespondencesbetweenthesubsetsof!and2!,2!and3!,and3!andthenonemptytreeswithoutdeadends.Inastageconstructionwewillthendeneatstagee,satreeTe;sbasedonWe;sandthecorrespondences.LettingTe=sTe;s,wewillobtainaninjectivecomputablenumberinge7![Te]ofallnonemptydecidable01classes.Furthermore[Te]willcorrespondtoWeforeache.Finally,byappendingtheemptyclasstotheenumerationweobtainthedesiredresult.Wenowdenethecorrespondences. 44

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Theone-to-onecorrespondencebetweenthesubsetsSof!and2!isgivenasfollows.EachScorrespondstoxS22!givenbyxSi=1ii2S.Theone-to-onecorrespondencebetween2!and3!isgivenasfollows.Letx22!anddenea0=1;a1=01,anda2=00.Thenxcorrespondstotheuniquesequencefxii2!23!wherefx:!!3andx=afxafxafx:::Theone-to-onecorrespondencebetweenthesetofallnonemptytreesT2
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Proof.AssumewithoutlossofgeneralitythatfTege2!containsallclopentrees.Wewillconstructinstages,asintheterminologyofTheorem 2.2.6 ,asequenceoffollowertreesSi.Atstageiwewillensurethatwehavei+1treesS0;S1;:::;Si,constructeduptolevel2i,followingtreesTS0;ki;:::;TSi;kiki2fm;ngwhichareeachpairwisedistinctatlevel2i.Also,atstagei,initiallysomeoftheSiwillhavethestatusofbeingmarkedki=minwhichcaseSiwillcontinuetofollowTSi;mforever.Ifnot,thenSiisnotmarkedki=nandwedetermineforeachi,ifSishouldbemarked.IfanSineedstobemarkedthenwedetermineatreeTSi;mthatitshallhereafterfollow.OtherwiseeachSicontinuestofollowTSi;nandthestageiscomplete.Construction.Stage0.FindthersttreeTisuchthatTif0;1g206=;,denotethistreeasTS0;n,anddeneS0=TS0;nf0;1g620.Stagej+1.S0;:::;Sjhavealreadybeenconstructeduptolevel2jandarealreadyfollowingtreesTS0;kj;:::;TSj;kj.Weperformthefollowingtwoactionsatthisstage:ConstructS0;:::;Sjuptolevel2j+1bydeterminingthetreesTS0;kj+1,:::,TSj;kj+1theyshallfollow,andConstructanewtreeSj+1uptolevel2j+1.Action.LetUj+1=fSi;kj:kj=nandTSi;kjhasdeadendsatlevel2j+1g.AllSisuchthatSi;kj62Uj+1keeptheirstatusasmarkedorunmarked,sokj=kj+1,andcontinuetofollowTSi;kj+1.ThoseSisuchthatSi;kj2Uj+1willhereafterbemarkedandwillnowfollowthetreeTSi;m=f:vorvforsome2TSi;noflength2jg.NotethateachmarkedSifollowsaclopentreeTSi;m.Action.LetSj+1;nbetheleastisuchthatTiisdistinctfromallTSi;kj+1i6jatlevel2j+1andsuchthatTihasnodeadends.DeneSj+1=TSj+1;nf0;1g6j+1.Verication.Wenowverifythat:iForeachi,limjTSi;kj#=Si=TniforsomeTniwithoutdeadends,iiForalli,ifTihasnodeadendsthenthereisacsuchthatTi=Sc,andiiii6=jimpliesthatSi6=Sj.Vericationofi.Forallj,kj=norkj=m.Fixi.ByAction,atstagei,Si;ki=Si;n.ByAction1,k`=k`+1=nforall`>iifSiisnevermarked.IfSiismarkedatstager>i,thenforalls>r,ks=ks+1=m.Ineithercaselimj>ikj#sothat 46

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limjSi;kjconvergestoSi;norSi;m.IfitconvergestoSi;mthenSineverdivergesfromfollowingtheclopentreeTSi;m.OtherwiseSiisnevermarkedandcontinuallyfollowsTSi;n.SinceitisnevermarkeditmeansthatTSi;nneverhasdeadendsuptolevel2r,forallr>i.SoTSi;nisatreewithoutdeadends.EitherwaylimjTSi;kj#=TniforsometreeTniwithoutdeadends.Nowforalln,Sif0;1g6n=TSi;knf0;1g6nandTSi;knTSi;kn+1.ThereforeSi=limjTSi;kj=Tni.Vericationofii.LetTibeatreewithoutdeadends.Therearetwocases.IfthereisastagejandacsuchthatTi=TSc;matstagej,thenbytheconstrctionTi=Sc.Ifnot,letbiequaltheleastksuchthatTk=Ti.LetjbelargeenoughsothatTbidiersfromTeatlevel2jforalleiatlevel2j.SoSi6=Sjifi6=j. 2.4.2.3Treeswithdeadends:AnecessityInanycomputablenumberingof01classesviatrees,somedecidableclassmustnecessarilyhave,asatree,deadendsthroughouteveryoccuranceinthenumberingseeCorollary 2.4.7 .TwodierentproofsofthisfactmaybeobtainedfromTheorems 2.4.6 and 2.4.8 below. Theorem2.4.6. Inanycomputablenumberingofcomputabletreesin2
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Corollary2.4.7. ForanycomputablenumberingPe=[Te]ofthe01classesin2!,thereisadecidable01classPsuchthatP6=[Te]foranyTewithoutdeadends.Proof.LetP=[T]whereTisthecomputabletreewithoutdeadendsprovidedbyTheorem 2.4.6 .SupposethatP=[Te]forsomee.SinceThasnodeadends,itfollowsthatT=TPandifTealsohadnodeadends,thenTe=TP=T.Butbytheconstruction,Tf0;1ge+16=Tef0;1ge+1,sothatT6=Te. Itfollowsfromthiscorollarythatinthestandardnumbering,fe:Tehasnodeadendsg6=fe:Pe=[Te]isdecidableg.Infactbothhavedistinctcomplexities.ByKonig'sLemma,ExtPe=f22
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2.4.3ThinandPerfectThinClassesIntheliterature,aMartin-PourEltheoryisaconsistentc.e.propositionaltheorywithadditional`thinness'conditions.Theconditionsimposedhavevarieddependinguponthecontextandmotivationoftheauthors,butinclude:fewc.e.extensions,essentiallyundecidable,andwell-generated.Someauthorshavechosentoonlyimpose1[ 21 ],whileothersand2[ 19 ],[ 24 ],andnallyothers,2,and[ 34 ],[ 40 ],[ 29 ].Thecompleteconsistentextensionsofthesetheoriescorrespondtothin,perfectthinorequivalently,specialthin[ 19 ],andhomogenousthinclasses,respectively.ThissectionisdevotedtowardsdemonstratingthenonexistenceofcomputablenumberingsofthersttwocasesbymodifyingtheclassicalMartin-PourElconstructionofaperfectthinclass.RecentlySolomon[ 89 ]alsomodiedthistheoremtoconstructahomogeneousthinclassandthereforeweconjecturethatnocomputablenumberingsexistfortheseclasses.2.4.3.1TheMartin{PourElConstructionRecallthata01classPisthinifforevery01subclassQP,thereisaclopensetUsuchthatQ=UP.Itisperfectiithasnoisolatedpoints.Aperfectclassmaybedenedbyafunctiong:2
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Wewilldenefinstagestoobtainuniformlycomputable,extension-preservingfunctionsfssothatf=limsfs.ToensurethatPisthin,wewillmeetthefollowingrequirementforeache:Thine:82f0;1ge+18[f2Te^v!f2Te]ToseethatThinemakesPthin,letU=fIf:jj=e+1&f2TegandobservethatifPeP,thenPe=PU.Construction.Letf0=g.Atstaget+1,wedeneft+1asfollows.Lookfore
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Wedeneg:2n.LetnbetheleastksuchthatjTP!kj>n.Anonempty01classPisverysmallifthefunctiondominateseverycomputablefunctiong;thatis,x>gxforallbutnitelymanyx.LetAbeacoinnitec.e.set,say A=fa0anforallnanditisdensesimpleifn7!andominateseverycomputablefunction.Inthissectionwewillusethesesetstoshowthatnoeectivenumberingexistsforthesmall,verysmall,ordecidableclasses.2.4.4.1Numberingsandhigh/noncomputablesetsEnroutetodemonsratingourtheorem,wenowproceedshowthattherearenoeectivenumberingsofthehighorofthenoncomputablesets.Asweshalleventuallycharacterizesmallanverysmallclassesintermsthedegreesofthesesets,theseresultswillbecrucialtoourargument.First,wemodifyShoeneld'sThicknessLemma[ 86 ,p.131].Somedenitionsareneeded.ForB!,letB[y]=fhy;zi2B:z2!gandsaythatBispiecewisecomputable 51

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ifB[y]iscomputableforally.ForBA!,wesaythatBisathicksubsetofAifforally,B[y]nA[y]isnite. Lemma2.4.12ThicknessLemma. Foranyuniformlyc.e.sequencefWi:i2!gofnoncomputablec.e.setsandanypiecewisecomputablec.e.setB,thereisathickc.e.subsetAofBsothatWn66TAforalln.Proof.Theproofasin[ 86 ]ismodiedtoensurethatthelengthandrestraintfunctionsandtherequirementsincorporatethepairhi;kiinplaceofthesingleargumentitomaketheargumentgothroughwitheachWiinconjuctionwitheachfunctionalk. Weobtainthefollowingcorollary. Corollary2.4.13. Foranyuniformlyc.e.sequencefWn:n2!gofnoncomputablec.e.sets,thereisahighc.e.setAsuchthatforalli,Wi66TA.Proof.Thisfollowsfromthemodiedthicknesslemmaabovebythesameargumentfoundin[ 86 ,p.133]. Corollary2.4.14. a Thereisnouniformlyc.e.numberingofallhighc.e.sets. a Thereisnouniformlyc.e.numberingofallnoncomputablec.e.sets.Infact,itfollowsthatthereisnouniformlyc.e.numberingofthehighornoncom-putablec.e.degrees.2.4.4.2Non-existenceofeectivenumberingsWenowproceedcharacterizethesmallandverysmallclassesintermofthenoncom-putabledegreesandthehighdegrees,respectively.Thedegreeofa01classPisdenedtobethedegreeofTPandisthusalwaysac.e.degreesinceTPisaco-c.e.set.Wewillusethefollowingtwoclassicresults.[Martin]Anyhighdegreecontainsamaximalandhencedensesimpleset[ 86 ,pp.211{217].[Dekker]Anynoncomputablec.e.degreecontainsahypersimpleset[ 86 ,p.81]. Proposition2.4.15. Ac.e.degreeishighifandonlyifitcontainsaverysmall01classP2!. 52

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Proof.!Supposeaishigh,andletA2abeamaximalset,andletpbetheprincipalfunctionof!)]TJ/F24 11.955 Tf 12.821 0 Td[(A,sothatpdominateseverycomputablefunction.NowletPA=f0!g[f0n10!:n=2Ag.ThenPAisa01classandforeachn,theleastksuchthatjTPf0;1gnj>kispreciselypn+1forn>0andhencedominateseverycomputablefunction.Letabeac.e.degreeandsupposethatTP2aforsomeverysmallP.Thenthefunctionfn=leastk[jTPf0;1gkj>n],whichdominateseverycomputablefunction,iscomputablefromTP.ItfollowsfromMartin'sTheorem[ 86 ,p.208]thatTPishigh. Proposition2.4.16. Ac.e.degreeisnoncomputableifandonlyifitcontainsaninnite,small01P2!.Proof.!Supposeaisanoncomputablec.e.degree,letA2abehypersimple,andpbetheprincipalfunctionof!)]TJ/F24 11.955 Tf 11.489 0 Td[(A,sothatpisnotdominatedbyanycomputablefunction.Thenthe01classPAasdenedintheproofofProposition 2.4.15 willhavedegreeaandwillbesmall.SupposethatPisaninnite01classandTPiscomputable.Thenthefunctiongn=leastk[jTPf0;1gkj>n]iscomputableanditfollowsthatPisnotsmall. Theorem2.4.17. Thereisnoeectivei.e.01numberingofallnondecidable,ofallinnitesmall,orofallverysmall01classesin2!.Proof.Suppose,towardsacontradiction,thatfQn=[Tn]:n2!gisaneectivenumberingof01classessuchthateachQnisnondecidable.ThenWn=fhi:=2ExtTngisauniformlyc.e.numberingofnoncomputablec.e.sets.ByCorollary 2.4.13 ,thereisahighc.e.setAsuchthatforalln,Wn66TA.ThereforeAisahighdegreethatcontainsaverysmallclassnotamongsttheQi,acontradiction. 53

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CHAPTER3RANDOMCLOSEDSETSThefollowingchapterisjointworkwithGeorgeBarpalias,DouglasCenzer,SeyyedDashti,andRebeccaWeberandappearsintheJournalforLogicandComputationno.17,2007,pages1041{1062asanarticleentitledAlgorithmicRandomnessofClosedSets[ 7 ].ApreliminaryversionofthisresearchwasoriginallypresentedattheCom-putabilityinEuropeConferenceinSwansea,Walesin2006byD.Cenzer.Thisprelimi-naryworkwaspublishedinthereferredconferenceproceedingsasRandomClosedSetsP.Brodhead,D.Cenzer,andS.DashtiinProceedingsofCIE2006:LogicalApproachestoComputationalBarriers,A.Beckmann,U.Berger,B.Loewe,andJ.Tucker,eds.,SpringerLectureNotesinComputerScience,Vol.39882006,pages55-64[ 14 ].PortionsofthisworkwerealsopresentedbyP.BrodheadattheGreaterBostonLogicConferenceMay2006,Boston,MA,the2006SACNASConferenceOctober2006,Tampa,FL,theAMSFall2006EasternSectionalMeetingOctober2006,Storrs,CT,theConferenceonLogic,Computability,andRandomnessJanuary2007,BuenosAires,Argentina,andtheWorkshoponComputabilityandRandomnessDecember2007,Auckland,NewZealand.PortionsofthisworkwerealsopresentedbyD.CenzeratarandomnessworkshopattheAmericanInstituteofMathematicsAugust2006,Palo,Alto,CA.R.WeberalsopresentedportionsofthismaterialatarandomnessworkshopattheUniversityofChicagoSeptember2007,Chicago,IL.3.1OverviewTheliteratureaboundswithresultsinalgorithmicrandomessaspertainingtorealsoveranitealphabet,especiallywithinthelastfewyears.Littleisknown,orevendeveloped,however,withrespecttorandomnessforclosedsetsofbinaryreals.Thischapterisarstapproachinthisdirection.Inthischapter,weconsideranotionofeectivei.e.algorithmicrandomnessonthespaceCofnonemptyclosedsubsetsPof2N;toaccomplishthistask,wewillneedusethedenitionandmachineryofeectiverandomnessforreals,since,throughappropriate 54

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codingofclosedsets,wewilldeneaclosedsettoberandomiitscode,asareal,israndom.Infact,laterinChapter 4 ,wewillapproachadenitionofrandomnessforcontinousfunctionsinasimilarfashion.Consequentlywebeginthischapterwithanintroductiontoalgorithmicrandomness,includingabriefhistoricalbackground.Morespecically,thischapterisorganizedasfollows.InSection 3.2 ,weprovideanintroductiontoalgorithmicrandomnessforrealsoveranitealphabet.InSection 3.3 ,wegiveaprobabilitymeasureanddeneaversionoftheMartin-LofTestforclosedsets,leadingtoadenitionofrandomnessforclosedsets.InSection 3.4 ,wetacklethequestionofwhichtypesofelementsnecessarilybelong,ordonotbelong,torandomclosedsets.Forinstance,everyrandomclosedsetcontainsrandomandnon-randomelements,butnon-c.e.elements.InSection 3.5 ,weshowthatrandomclosedsetshavemeasurezeroandboxdimensionlog24 3.InSections 3.6 3.7 ,weexplorealternatenotionsforrandomness,suchastheproblemofcompressibilityoftrees.Finally,inSection 3.8 ,weconsidertheproblemofwhenarandomlychosenclosedsetmeetsaclosedQ;thisisthestudyofcapacities.3.2EectiveRandomnessofRealsInthissection,wepresentabasicintroduction,includingabriefhistoricalback-ground,forrandomnessofrealsoveranitealphabet.3.2.1IntroductionThestudyofalgorithmicrandomnesshasbeenofgreatinterestinrecentyears.Thebasicproblemistoquantifytherandomnessofasinglerealnumber.Earlyinthelastcentury,vonMises[ 94 ]suggestedthatarandomrealshouldobeyreasonablestatisticaltests,suchashavingaroughlyequalnumberofzeroesandonesoftherstnbits,inthelimit.Thusarandomrealwouldbestochasticinmodernparlance.Ifoneconsidersonlycomputabletests,thentherearecountablymanyandonecanconstructarealsatisfyingalltests.Anearlyapproachtorandomnesswasthroughbetting.Eectivebettingonarandomsequenceshouldnotallowone'scapitaltogrowunboundedly.Thebettingstrategiesused 55

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areconstructivemartingales,introducedbyVille[ 93 ]andimplicitintheworkofLevy[ 65 ],whichrepresentfairdouble-or-nothinggambling.Martin-Lof[ 69 ]observedthatstochasticpropertiescouldbeviewedasspecialkindsofmeasurezerosetsanddenedarandomrealasonewhichavoidscertaineectivelypresentedmeasurezerosets;seeSection 3.2.4 .AtthesametimeKolmogorov[ 55 ]denedanotionofrandomnessfornitestringsbasedontheconceptofincompressibility.Astrongernotionofprex-freecomplexitywasdevelopedbyLevin[ 64 ],Gacs[ 48 ]andChaitin[ 27 ]andextendedtoinnitewords.Inthefollowingsections,weformalizethenotionsofconstructivemartingaleran-domness,Martin-Lofrandomness,andprex-freerandomness.Aftertheirentryintotheliterature,Schnorrlaterproved[ 83 ]thatallofthesenotionsareequivlant;thisisafunda-mentalresultinthetheoryofalgorithmicrandomness.Whilethesedenitionsandresultsareusuallygivenforbinarystringsandsequences,theycarryovertok-arystringsandsequencesaswell.See,forexample,Calude[ 17 18 ],orSection 3.2.4 below,wherewedothisfortheMartin-Lofdenitionofrandomness.3.2.2ConstructiveMartingaleRandomnessThebettingapproachtorandomnessisformalizedasfollows. Denition3.2.1Ville[ 93 ]. i Amartingaleisafunctionm:k
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Denition3.2.2. Amartingalemisconstructiveoreective,orc.e.ifitislowersemi-computable;thatis,ifthereisacomputablefunction^m:k
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domaindomM;thatis,ifvarestringsindomM,thenmustequal.Foranynitestring,theprex-freecomplexityofwithrespecttoMisKM=minfjj;1:M=g:Thereisauniversalprex-freefunctionUsuchthat,foranyprex-freeM,thereisaconstantcsuchthatforallKU6KM+c:WeletK=KUandcallittheprex-freecomplexityof. Denition3.2.5Prex-freeRandomness. x2f0;1g!iscalledprex-freerandomifthereisaconstantcsuchthatKxdn>n)]TJ/F24 11.955 Tf 11.955 0 Td[(cforalln.Thislatterinequalitymeansthattheinitialsegmentsofxarenotcompressible.3.2.4Martin-Lofn-randomnessAccordingtotheMartin-Lofdenitionofrandomness,arandomrealmustavoidcertaineectivelypresentedmeasurezerosets.Inherentinthedenition,therefore,isthechosenmeasurebeingused.Fixanalphabetk=f0;1;:::;k)]TJ/F15 11.955 Tf 12.58 0 Td[(1g.Wepresentthedenitionofageneralprobabilitymeasureonk!,aswellasdierentnamedmeasuretypes.Typically,however,wewillmakeuseofthealphabetsf0;1gorf0;1;2g. Denition3.2.6GeneralProbabilityMeasures. Letf:f0;1;2g![0;1]beafunctionsuchthatf;=1andf=Pi=0;1;2f_iforall.Thef-probabilitymeasurefisdenedsothatthef-measureoftheinterval[]issuchthatf[]=f. Denition3.2.7DierentMeasuresTypes. LetfandfbeasinDenition 3.2.6 i fiscomputableiffiscomputable. ii fisnonatomic,orcontinuous,ifforallx23N,ffxg=0. iii fandfareboundedif9b;c2;188i[bf
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Denition3.2.8. Arealx2k!isMartin-LofrandomifforeveryeectivesequenceS1;S2;:::ofc.e.opensetswithSn62)]TJ/F25 7.97 Tf 6.586 0 Td[(n,x=2TnSn.Thelatterconditionisequivalenttotheconditionwegetifwereplace2)]TJ/F25 7.97 Tf 6.587 0 Td[(nwithqnforacomputablesequenceqiofrationalssuchthatlimiqi=0.WecanalsoconsideranextendeddenitionofMartin-Lofrandomness,intermsof0nsets. Denition3.2.9. i A0ntestisacomputablecollectionfVn:n22Ngof0nclassessuchthatVk62)]TJ/F25 7.97 Tf 6.587 0 Td[(k; ii Arealis0nrandomorn-randomifandonlyifitpassesall0ntestsi.e.,iffVn:n22Ngisacomputablecollectionof0nclassessuchthatVk62)]TJ/F25 7.97 Tf 6.587 0 Td[(k,then=2n>0Vn.Thus1-randomrealsarejustMartin-Lofrandomreals.See[ 36 ]fordetailsonrandomandn-randomreals.Kurtz[ 59 ]andKautz[ 54 ]provedthefollowingresult.Let;ndenotethen-thjumpof;. Theorem3.2.10. Letqbearationalnumber. i Foreach0nclassS,wecanuniformlycomputefromqanda0nindexforS,theindexofa;n)]TJ/F18 5.978 Tf 5.756 0 Td[(11classUSsuchthatUisanopen0nclassandU)]TJ/F24 11.955 Tf 11.955 0 Td[(S
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Theorem3.2.12vanLambalgen[ 92 ]. Thefollowingareequivalent. 1. ABisn-random. 2. Aisn-randomandBisn-A-random. 3. Bisn-randomandAisn-B-random. 4. Aisn-B-randomandBisn-A-random.3.3Martin-LofRandomnessofClosedSetsInthissectionwedeneameasureonthespaceCofnonemptyclosedsubsetsof2Nandusethistodenethenotionofrandomnessforclosedsets.Wethenobtainseveralpropertiesofrandomclosedsets.3.3.1TheHit-or-MissTopologyonCThestandardhit-or-misstopologyonChasasasub-basis,thefollowingtwotypesofsets,whereQisanyclosedset:VQ=fK:KQ6=;g;WQ=fK:KQg.Abasisforthehit-or-misstopology,then,isformedbytakingniteintersectionsofthese.Wenowconsiderarenementofthesub-basissetsandobtainabasisfortheBorelsets.Wewillusethefollowingnotation.TndenotesthesetTf0;1gn,andT6ndenotesthesetTf0;1g6n Denition3.3.1. ForanytreeSandanyn,deneCnS=fQ2C:S6n=T6nQgThatis,CnSisthesetofclosedsetsQ2CthatagreewithSuptoleveln. Claim3.3.2. AbasisfortheBorelsetsisgivenbytheagreementsetA:A=fCnS:Sisatree;n2!g:Toseethis,rstnotethatanyclosedset[T]isthedecreasingintersectionofclopensets[Tn]:=[f[]:2Tng:Thereforewemayrewritesub-basiselementsV[T]andW[T]as V[T]=nV[Tn]bydenitionand 60

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W[T]=nW[Tn]bycompactnessButthen, V[Tn]=SfCnS:STn6=;gand W[Tn]=SfCnS:SnTgToseetherstequality,forexample,notethatQ[Tn]6=;ifandonlyifQ2CnSforsomeSwithSTn6=;.Thelatterequalityholdssimilarly.3.3.2TowardaMeasureTodeneVQandWQ,forsomexedmeasureandanyclosedsetQ,itsucestodeneVQnandWQnforclopensetsQnwhereQ=nQn.WewouldsimplydeneVQ=limnVQnandWQ=limnWQn.However,foranyclopenQ,WQisthecomplementVNnQ.HenceitfurthermoresucestodeneVQforclopensetstogetameasureonC.FromthejusticationofClaim 3.3.2 ,thelattertaskmaybeaccomplishedbydeningameasureonallBorelbasiselements,namelytheagreementsetsCnS.Toaccomplishthis,inthefollowingsectionwewillencodeallclosedsetsQ2CwithacanonicalcodexQ2C.ThenusingtheLebesguemeasureon3N,wewilldeneameasureonthesetsCnSwhich,infact,denesameasureonallclosedsets.3.3.3CanonicalCodingandMeasureTheCanonicalCoding.Aneectiveone-to-onecorrespondencebetweenthespaceCandthespace3Nisdenedasfollows.LetaclosedsetQbegivenandletT=TQbethetreewithoutdeadendssuchthatQ=[T].Denethecanonicalcodex=xQ2f0;1;2gNforQasfollows.Let=0;1;2;:::enumeratetheelementsofTinorder,rstbylengthandthenlexicographically.Wenowdenex=xQ=xTbyrecursionasfollows.Foreachn,xn=2if_n0and_n1arebothinT,xn=1if_n0=2Tand_n12Tandxn=0if_n02Tand_n1=2T.Forexample,ifQ=f0;1gN,thenxQ=;2;:::andifQ=fyg,thenxQ=y.LetQxdenotetheuniqueclosedsetQsuchthatxQ=x. Denition3.3.3TheMeasure. DenethemeasureonCbyX=fxQ:Q2Xg: 61

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whereistheLebesquemeasurei.e.theregularmeasuredwithb0=b1=b3=1 3seeDenition 3.2.7 on3N.Informallythismeansthatgiven2TQ,thereisprobability1 3thatboth_02TQand_12TQand,fori=0;1,thereisprobability1 3thatonly_i2TQ.Inparticular,thismeansthatQI6=;impliesthatfori=0;1,QI_i6=;withprobability2 3. Comment3.3.4. Atthisstage,wehavexedtheuniformmeasurei.e.allbi=1 3towardsdeningrandomnessofclosedsets.Thisallowsustomoreeasilydemonstratethevalidityofmanyresults.Later,inSection 3.7.1 ,wewillshowthattheresultsholdwithanyregularmeasure.Proposition 3.3.5 ,however,demonstratesthatthedenedmeasureonC,above,holdsforanygeneralizedprobabilitymeasuredseeDenition 3.2.6 .JusticationfortheCoding.Letusalsocommentbrieyonwhysomeothernaturalrepresentationswererejected.Supposerstthatwesimplyenumerateallstringsinf0;1gas0;1;:::andthenrepresentTbyitscharacteristicfunctionsothatxTn=1n2T.Theningeneralacodexmightnotrepresentatree.Thatis,oncewehave01=2Twecannotlaterdecidethat12T.Supposethenthatweallowtheemptyclosedsetbyusingcodesx2f0;1;2;3gandmodifyouroriginaldenitionasfollows.Letxn=ihavethesamedenitionasabovefori62butletxn=3meanthatneither_n0nor_1isinT.Informally,thiswouldmeanthatfori=0;1,2Timpliesthat_i2Twithprobability1 2.Theadvantagehereisthatwecannowrepresentalltrees.Butthisisalsoadisadvantage,sinceforagivenclosedsetP,therearemanydierenttreesTwithP=[T].Thesecondproblemwiththisapproachisthatwewouldhave[T]=;withpositiveprobability.WebrieyreturntothissubjectinSection 3.7.2 .Nowrecallthedenitionofageneralprobabilitymeasureon3NDenition 3.2.6 .Letd:f0;1;2g![0;1]beafunctionsuchthatd;=1andd=Pi=0;1;2d_iforall.Thend[]isdenedtobed.Wemaynowdene,foranysuchd,dexactlyasinDenition 3.3.3 .Furthermoredwebedeemedcomputableifdiscomputable. 62

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Proposition3.3.5. Foranyd,themeasuredisdenedonallBorelsetsinthehit-or-misstopologyonC.Furthermore,ifdiscomputable,thendiscomputableonthefamilyofclopensets.Proof.Asdiscussedinsection 3.3.2 ,itsucestoshowthatdCnSisdenedforallCnS2A.FixatreeSandsupposethatfv:2Sngisordered0<:::
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vi=PCUi,fori=0;1.ThenconsideringthethreecasesinwhichSincludesbothinitialbranchesorjustone,wecalculatethatPCU=1 3v0+v1+v0v1:ThusbyinductionwehavePCU61 3u0+u1+u0u1:Now2u0u16u20+u216u0+u1;andthereforePCU61 3u0+u1+u0u161 2u0+u1=U:ForaclosedsetQ,letQ=TnUn,whereUnisclopenandUn+1Unforalln.ThenPQifandonlyifPUnforalln.ThusPCQ=nPCUn;sothatPCQ=limn!1PCUn6limn!1Un=Q:Finally,foranopensetQ,letQ=SnUnbetheunionofanincreasingsequenceofclopensetsUn.Then,bycompactness,PCQ=[nPCUn;sothatPCQ=limn!1PCUn6limn!1Un=Q:Thiscompletestheproofofthelemma. 3.3.4GhostCodingWewishnowtointroduceasecondmethodofcoding,theghostcoding.AghostcodeofQisaninniteternarystringwhosetermscorrespondtoallnodesof2
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lexicographicalorder.ThetermscorrespondingtothenodesofQ'streethecanonicalnodes"agreewiththecorrespondingtermsinthecanonicalcode;theremainingghostnodes"mayholdanyvalues.Ghostcodesarenon-unique,andeveryclosedsethasanon-randomghostcodeiftheclosedsetitselfisrandomtakethecodewithghostnodesallequaltozero,say.Thismethodofcodingismoreconvenientforsomepurposes;forexample,wewilluseittoshowthatifQ0;Q1areclosedsetsandQ=f0_x:x2Q0g[f1_x:x2Q1g,QisrandomifandonlyiftheQiarerandomrelativetoeachother.3.3.5CodingEquivalanceTheutilityoftheghostcodesrestsonthefollowingcorrespondence.RecallvanLambalgen'stheoremTheorem 3.2.12 Theorem3.3.9. ThecanonicalcodeofaclosedsetQ2NisrandomifandonlyifQhassomerandomghostcode.Furthermore,foranyy,thecanonicalcoderisy-randomifandonlyifQhasaghostcodewhichisy-random.Proof.SupposethecanonicalcodeofQisnonrandom.Thenthereisac.e.mar-tingalemthatsucceedsonit.FromanyinitialsegmentofaghostcodegforQ,thesubsequence^ofexactlythecanonicalnodesofiscomputable.Thereforeitiscom-putablewhetherthebitofgafteriscanonicalorghost.Fromm,denethemartingalem0whichbetsasfollows:m0_i=8><>:m^_inextbitisacanonicalnodem0nextbitisaghostnode.Thatis,m0holdsitsmoneyonghostnodesandbetsidenticallytomoncanonicalnodes.Itisclearthatm0succeedsontheghostcodegandthusgisnonrandom.!NowsupposethecanonicalcoderforQisrandom,andletqbeaninniteternarystringthatisrandomrelativetorandsobyTheorem 3.2.12 rqisrandom.Weclaimtheghostcodegobtainedbyusingthebitsofrasthecanonicalnodesandthebitsofqintheiroriginalorderastheghostnodesisrandom.ItisclearthatgisaghostcodeforQ.Supposemisac.e.martingalethatbetsong.Frommitisstraightforwardtodeneanonmonotonicmartingalem0whichmimicsm'sbetsexactlybutperformsthemonrq, 65

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succeedingwhenevermsucceeds.Asrandqwerechosentoberelativelyrandom,thiswillshowgisrandom.Asdiscussedpreviously,fromgdnitiscomputablewhethergnwillbeaghostnodeoracanonicalnode,andwhichpositioningorritoccupiesineithercase.Therefore,assumingthebitsseensofarmaybeassembledintoaninitialsegmentofg,m0takesthevaluesm_i,i<3,asitsbetsonthecorrespondingbitofrorg,whicheverisappropriate.Havingseenthatbit,then,itcanassembleajj+1-lengthinitialsegmentofgandrepeattheprocess.Asm0makesidenticalbetstomandhasidenticaloutcomes,sinceitcannotsucceedonrg,mcannotsucceedongandgisrandom.Torelativize!,supposethatrisy-random,sothatryisrandombyVanLambalgen'sTheorem 3.2.12 .Thenintheproofsimplychooseqtoberandomrelativetory,andthengwillberandomrelativetoy.Theotherdirectionrelativizesinastraightforwardway. 3.3.6CodingandJoinsofClosedSetsTheprimarypurposeoftheghostcodesistoremovethedependenceontheparticularclosedsetunderdiscussionwheninterpretingbitsofthecodeasnodesofthetree.Thisisespeciallyusefulwhensubdividingthetree,asinthefollowingdenition. Denition3.3.10. ThetreejoinofclosedsetsP0andP1istheclosedsetQ=f0_x:x2P0g[f1_x:x2P1g:Givenghostcodesr0;r1forthePi,theirtreejoinr0r1isthecodeforQwiththecorrespondingghostnodevalues.Thestandardrecursion-theoreticjoinisdenedbyr0r1=r0;r1;r0;r1;::::Wewishtorelatetherecursion-theoreticjoinandthetreejoin. Lemma3.3.11. Giventwoghostcodesr0;r1,thetreejoinr0r1israndomifandonlyiftherecursiontheoreticjoinr0r1israndom. 66

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Proof.Itisclearthatthereisacomputablepermutationwhichuniformlymapsanytreejoinr0r1totherecursion-theoreticjoinr0r1.Thatis,inr0r1,theentriesofr0andr1alternate,whereasr0r1startswitha2,followedbyblocksfromr0andr1,asfollows.Firstr0,r1,thenr0,r0,r1,r1,andcontinuingwithpairsofblocksofsize4,8andsoon.TheresultnowfollowsfromtheVon-Mises{Church{WaldComputableSelectionTheorem[ 94 ];thetheoremstatesthat,foranyrandomsequencexandanycomputable1-1functiong,thesequencezn=xgnisrandom. WenowobtainthefollowingcorollaryofTheorems 3.2.12 and 3.3.9 andLemma 3.3.11 Corollary3.3.12. SupposePi,i=0;1,areclosedsetswithcanonicalcodesriandletPbethetreejoinofP0;P1.ThenPisrandomifandonlyifr0r1israndom.Proof.Supposethatr0r1israndom.ThenbyTheorem 3.2.12 ,r0andr1aremutuallyrelativelyrandom.ByTheorem 3.3.9 ,P0hasaghostcodeg0whichisrandomrelativetor1,andsoalsovice-versa,andthenP1hasaghostcodeg1whichisrandomrelativetog0.Againby 3.2.12 ,therecursion-theoreticjoing0g1israndom,sobyTheorem 3.3.11 thetreejoing0g1isalsorandom,andhencePpossessesarandomghostcodeandisrandom.!SupposenowthatPisrandom,andthereforepossessesarandomghostcodeg.Thecodegmaybethoughtofasatreejoing0g1,whichisthereforerandom,andsobyTheorem 3.3.11 ,g0g1israndom.ByTheorem 3.2.12 ,theindividualcodesg0;g1arethereforemutuallyrelativelyrandom.NowbytherelativedversionofTheorem 3.3.9 ,r0israndomrelativetog1.Butr1iscomputablefromg1andhencer0israndomrelativetor1aswell.Similarly,r1isr0-randomandthus,againby 3.2.12 ,r0r1israndom. 3.4MembersofRandomClosedSetsInthissectionwetacklethequestionofwhichtypesofelementsnecessarilybelong,ordonotbelong,torandomclosedsets.TheformerisaddressedinSection 3.4.1 andthelatterinSection 3.4.2 67

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3.4.1PositiveResultsWeshallsee,asaconsequenceofTheorem 3.4.19 ,thateveryclosedsetisperfectandcontainscontinuummanyelements.Inthissection,wedemonstrateotherpositiveresults.Forexample,everyrandomclosedsetcontainsrandomandnonrandomelements.Otherexamplesabound.Webeginwiththerstexample. Theorem3.4.1. Everyrandomclosedsetcontainsarandomelement.Proof.SupposethataclosedsetQhasnorandomelementandconsiderthefollowingMartin-LoftestonthespaceC:Ui=fPjP2CandPVigwhereViisauniversalMartin-LoftestontheCantorspace.ByLemma 3.3.8 ,Ui6Vi62)]TJ/F25 7.97 Tf 6.587 0 Td[(isothatUiisaMartin-LoftestonC.ButQ2iUi,soQisnotrandom. AsaconversetoTheorem 3.4.1 wehavethefollowing. Theorem3.4.2. Foranyrandomr22N,thereexistsarandomclosedsetcontainingrasapath.TheproofofthistheoremwasoriginallygivenbyJoeMillerandAntonioMontalbanandhasbeensubsequentlyimprovedthankstotheanonymousreferee.Proof.Letrbearandomrealandletxbethecanonicalcodeofanr-randomclosedset.Wealterxtothecodex0ofaclosedsetguaranteedtocontainrbutchangedaslittleaspossibletoachievethat.Todeterminex0n,assumex0nhasbeendened.Ifxn=2orxncorrespondstoanodenotalongr,setx0n=xn.Ifxn2f0;1gcorrespondstork,setx0n=rk.Theclosedsetdenedbyx0willclearlycontainr.Foracontradiction,assumex0isnonrandomandletm0beac.e.martingalethatsucceedsonit.Webuildanonmonotonicmartingalemtobetonxr.Onbitsofx,mwillbeatriple-or-nothingmartingale;onr,itwillbedouble-or-nothing. 68

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Firstnotethatfrominitialsegmentsofxandrwemayreconstructaninitialsegmentofx0computably,andwealwaysknowfromaninitialsegmentofx0whetherthenextbitisalongrornot,andwhichbitofritis.Wewillconstructmsothataftereverystageofbettingwhichwillbeonebetbym0andoneortwobetsbym,thevalueofmisequaltothevalueofm0.Ateverystageitwillbeclearwehaverevealedenoughbitsofxandrtoreconstructx0totheneededlength.Supposeinductivelymandm0holdequalcapitalafterthestageofbettingonthelastnodeof@x0.Ifthebitx0nfollowingisnotonr,mbetsidenticallytom0;i.e.,mxn=i=m0_ifori<3.Inthatcasexn=x0nsoourinductivehypothesisholds.Ifx0nisonr,setmxn=2=m0_2andfori=0;1,setmxn=i=1 2[m0_0+m0_1].Ifx0n=2,thenthecapitalforbothmandm0ism0_2,sotheinductivehypothesisholdsandweproceedtothenextstage.Otherwisembetsonrkfortheappropriatek,settingmrk=i=m0_ifori=0;1.Onrk,thesumofm'scapitaloneachofthetwooutcomesmustaveragetothepreviouscapital;asthepreviouscapitalwas1 2[m0_0+m0_1]thisclearlyholds.Byconstructionrk=x0n=i,sobothmandm0nowhavecapitalm0_iandtheinductivehypothesisholds.Asm0isc.e.,mwillalsobe.Asthevaluesofm0alongx0areasubsequenceofthevaluesofmalongxr,ifm0succeedssodoesm,contradictingourassumptiononxr.Thereforex0isthecodeofarandomclosedsetcontainingthegivenrandompathr. Thepreviousresultsmightsuggestthateveryelementofarandomclosedsetisarandomreal.However,itturnsoutthateveryrandomclosedsetcontainsanon-randomreal.WeneedthefollowingclassicresultofCherno[ 28 ]aversionofBernoulli'sWeakLawofLargeNumbershereandalsoforanothertheoremtofollow.See[ 67 ]foranexposition. Lemma3.4.3Cherno. LetEbeaneventwhichwewillrefertoas`success'.IfEoccurswithprobabilityp,thenforanynaturalnumbersnandany"with06"61,the 69

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probabilitythatoutofnmutuallyindependenttrials,thenumberofsuccessesdiersfrompnby>"pnis62)]TJ/F25 7.97 Tf 6.586 0 Td[("2pn=3. Theorem3.4.4. Everyrandomclosedsetcontainsanon-randomreal;inparticular,theleftmostandrightmostpathsinarandomclosedsetarenotrandomreals.Proof.Wewillshowthat,forarandomclosedsetQ,theleftmostpathisnotstochas-ticallyrandom,thatis,theasymptoticfrequencyof0'sis2 3.Sinceaneectivelyrandomrealin2Nmusthaveasymptoticfrequenceof1 2for0'sand1's,thiswillsucetoprovethattheleftmostpathisnotrandom.WedeneaMartin-Loftestasfollows.Fixaratio-nal"suchthat0<"<1.Foreachn,letSnbethefamilyofclosedsetsthatis,codesforclosedsetssuchthattherstnbitsoftheleftmostpathhaveeither<2 3)]TJ/F24 11.955 Tf 12.295 0 Td[("n,or>2 3+"noccurrencesof0.Bythedenitionofourprobabilitymeasure,wehaveSn=Xjm)]TJ/F18 5.978 Tf 7.782 3.259 Td[(2 3nj>2 3"n0B@nm1CA2 3m1 3n)]TJ/F25 7.97 Tf 6.587 0 Td[(m:ItnowfollowsfromCherno'sLemma 3.4.3 thatSn62e)]TJ/F25 7.97 Tf 6.586 0 Td[("22n=9:ThusthemeasuresofthetestsetsSnhaveeectivelimitzero.ItiseasytoseethatthesequencefSngiscomputablyenumerable.Foreachn,SnisaclopensetandinfacttheunionofthenitefamilyofintervalsIinCsuchthatcodesatreeuptolevelninwhichtheleftmostpathhaseither<2 3)]TJ/F24 11.955 Tf 11.955 0 Td[("n,or>2 3+"noccurrencesof0.Furthermore,S0n=Sp>nSpisalsoaMartin-Loftest.ItfollowsthatforanyrandomclosedsetQ,andany">0,thereisannsuchthatforallm>n,thefrequencyof0'sintherstmbitsoftheleftmostpathisalwayswithin"of2 3.Thustheleftmostpathisnoteectivelyrandom. Recallthattheleftmostandrightmostelementsofanystrong02closedsetare02.GivenTheorems 3.4.1 and 3.4.4 ,weask:Doesa02randomclosedsetcontaina02randompath? Theorem3.4.5. Everyrandomstrong02closedsetcontainsarandom02real. 70

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Proof.LetQbearandomstrong02class.ByTheorem 3.4.1 ,Qcontainsarandomrealx.LetPbea01classintheCantorspacewhichcontainsonlyrandomsandcontainsxthisexistssincetheclassofrandomrealsisaneectiveunionof01classes.NotethatPQisanon-emptystrong02classanditfollowsthattheleftmostpathofPQisa02realwhichmustberandomsinceitbelongstoP. Theabovetheoremdoesnotcombinewiththelowbasistheoremtoestablishtheexistenceofalowrandomrealinanyrandomstrong02class.Wecanusethelowbasistheorem,however,todemonstratetheexistenceofalowrandomrealinanyrandomclosedsetwithlowcanonicalcode. Theorem3.4.6. Everyrandomclosedsetwithlowcanonicalcodecontainsalowrandomelement.ProofKjos-Hanssen.LetQbearandomclosedsetwithlowcanonicalcode.ByTheorem 3.4.1 ,Qcontainsarandomelementx.Thereforex22!nUnforsomenandsomeopenUnfromtheuniversalMartin-Loftest.So,inparticular,Q2!nUnisnon-empty.NowQ2!nUnis01relativetoTQ.Bythelowbasistheorem,Q2!nUnis01hasamemberysuchthaty06TT0Q.Inparticular,sincey2Q2!nUn,itisrandom.Furthermore,sinceTQislow,yisalsolow. Itisopenwhethereveryrandomclosedsetwitha02canonicalcodehasalowrandomelement;weconjecturethattheanswerisno.Inthefollowingsection,wewillshowthatthereisarandomclosedsetnotcontainingany02path.Ournextresult,Theorem 3.4.8 ,usesamethodwhichwasusedin[ 56 ]toshowthateveryrandomrealiseectivelybi-immune.Werstdenethislatternotion. Denition3.4.7. i AsetAiseectivelyimmuneifitisinniteandthereisacomputablefunctiongxsuchthatifWxA,jWxj6gx. ii Aiseectivelybi-immuneifAand Aarebotheectivelyimmune.Note,inparticular,thataneectivelyimmunesetcannotcontainaninnitec.e.set. Theorem3.4.8. IfPisarandomclosedsetthenallelementsofPareeectivelybi-immune. 71

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Proof.SupposethatPisarandomclosedsetandA2P.LetUii2NbeaMartin-Loftestinthespace3NsuchthatthereisacomputablefunctionfwiththepropertythatifVeii2Nisthee-thMartin-LoftestundersomeeectiveenumerationofallMartin-Lofteststhenforalle;i,Vefe;iUiastandardconstructionofauniversaltestgivesonewiththisproperty.SincePisrandom,thereissomeksuchthatthecanonicalcodeofPisnotinUk;letU=Uk.Itsucestondacomputablefunctiongsuchthat[A2PandWxA]jWxj6gx3{1forallsetsAandallxtheproofthat Aiseectivelyimmuneisentirelysimilar.LetBx;nbe;ifjWxj6n,andotherwisetheclassofcanonicalcodesoftreeswhichcontainapathcontainingtherstn+1elementsinthestandardenumerationofWx.ThenBx;nisauniformdoublesequenceof01classesandbythedenitionoftheprobabilitymeasureon3N,Bx;n62 3n+1:Soforeachx,Bx;2nisaMartin-Loftestinthespace3Nandfromxwecancalculatetheindexofit.Thenbyusingthecomputablefunctionfmentionedabovewegetacomputablefunctiongsuchthatforallx,Bx;gxU.Thismeansgsatises 3{1 Itiswellknownthateectivelyimmunesetscancomputeaxedpointfreefunction,sowehavethefollowing. Corollary3.4.9. Thepathsthrougharandomtreeareofxedpointfreedegree.Thatis,eachofthemcomputessomexedpointfreefunction.Itisknownthateveryrealcanbecomputedbysomerandomreal.Itisnotknown,however,whetheranyrealcanbecomputedbyallthepathsofsomerandomclosedset.Thenexttheorem,anobservationofTedSlamanatarandomnessworkshopinChicagoin2007,isastepinthatdirection.First,weneedthefollowingdenitions,whichare,infact,equivalentnotions. Denition3.4.10. i ArealxisK-trivialifKxdn6Kn+cforsomec. 72

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ii Arealxisabasefor1-randomnessifthereissomey>Txsuchthatyis1-x-random. Theorem3.4.11. AnysetwhichiscomputablefromallpathsofarandomclosedsetisK-trivial.Proof.IfAiscomputablefromallpaths,thenAiscomputablefromtheleftmostandrightmostpaths.Notethateachoftheselatterpathsareeachcomputablefromthetwohalvesofthetree.Furthermorethesetwohalvesarerelativelyrandomtoeachother.Henceeachisrandomrelativetoanythingtheothercomputes.Sohalf1,forexample,computesAandhencehalf2israndomrelativetoA.Ontheotherhand,half2alsocomputesA.ThereforethisisanexampleofsomethingintheconeaboveAthatisA-random.SoAisabasefor1-randomness. MuchinterestingworkhasbeendoneontheK-trivialreals.ChaitinshowedthatifAisK-trivial,thenA6T00.SolovayconstructedanoncomputableK-trivialreal.Downey,Hirschfeldt,NiesandStephan[ 39 ]showedthatnoK-trivialrealisc.e.complete.ThenotionofaK-trivialclosedsetwasintroducedin[ 9 ].ItwasshowninparticularthateveryK-trivialclasscontainsaK-trivialmember,butthereexistK-trivial01classeswithnocomputablemembers.3.4.2NegativeResultsRandomclosedsetscannevercontainn-c.e.,isolated,or1-genericpaths,orpathsofincompletec.e.degree.Webuildtothesefactsandproveothersalongtheway. Theorem3.4.12. Randomclosedsetscontainnocomputableelements.Proof.Foranynitestringoflengthn,theprobabilitythataclosedsetQmeetsIis2 3n.Foracomputablerealy,thesqeuencefQ:QIydn6=;gthusformsaMartin-LoftestinthespaceCofclosedsets,whichshowsthatydoesnotbelongtoanyMartin-Lofrandomclosedset.Thatis,foreachn,fx:QxIydn6=;gisac.e.opensetandhasmeasure2 3ninf0;1;2gN,whereQxistheclosedsetwithcodex. WeproveanevenstrongerresultinTheorem 3.4.17 .First,however,recallthata01classPisdecidableifTPisdecidable.Itfollowsthatanonemptydecidable01class 73

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mustcontainacomputableelementforexample,theleftmostpath.ByTheorem 3.4.12 ,itfollowsthatnodecidable01classcanberandom.AseveryrandomclasscontainsarandomelementTheorem 3.4.1 andhas,asweshallshow,measurezeroTheorem 3.5.1 ,thefollowingpropositiondemonstratesthatthisextendstoarbitrary01classes. Proposition3.4.13. IfPisa01classofmeasure0,thenPhasnorandomelements.Proof.LetTbeacomputabletreesuchthatP=[T],andforeachn,letPn=SfI:2Tf0;1gng.ThenfPngn2NisaneectivesequenceofclopensetswithP=TnPnandlimnPn=P=0.Furthermore,Pn=2)]TJ/F25 7.97 Tf 6.586 0 Td[(njTf0;1gnj;thisisacomputablesequence.ThusfPngn2NisaMartin-LoftestandPhasnorandomelements. Alternatively,wecanshowthatno01classisrandomthroughthefollowingstrongerresult,combinedwithanappealtoTheorem 3.5.1 ,fromthenextsection. Theorem3.4.14. LetQbea01classwithmeasure0.ThennosubsetofQisrandom.Proof.LetTbeacomputabletreepossiblywithdeadendsandletQ=[T].ThenQ=TnUn,whereUn=[Tn].SinceQ=0,itfollowsfromLemma 3.3.8 thatlimnPCUn=0.ButPCUnisacomputablesequenceofclopensetsinCandPCUnisacomputablesequenceofrationalswithlimit0.ThusPCUnisaMartin-Loftest,sothatforanyrandomclosedset,thereexistsnsuchthatP=2PCUnandhencePisnotasubsetofUn. Corollary3.4.15. No01classcanberandom. WenowprovideanevenstrongerversionofTheorem 3.4.12 ;weneedthefollowingdenition. Denition3.4.16f-c.e.reals. Foranycomputable,non-decreasingfunctionf,wesaythatareal2f0;1gNisf-c.e.ifthereexistsacomputableapproximatingfunctionsuchthat,foralli2N, i i;0=0 ii limsi;s=i; iii fs:i;s+16=i;sghascardinality6fi. 74

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Therealswhicharef-c.e.forsomecomputablefunctionfarepartofthewell-knownErshovhierarchy[ 43 86 ]. Theorem3.4.17. Supposethatfiscomputableandboundedbyapolynomial.Thennorandomclosedsethasanyf-c.e.paths.Proof.Letfbeasabove,anf-c.e.realandPaclosedsetcontaining.Letbethef-approximatingfunctionfor.AlsoletMnf0;1gnbethesetofdierent-approximationstodnduringthestages.Apriori,jMnjisexponential.However,foraxedn,dncanchangeatmostPi
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Nopathofarandomtreeis1-generic. Nopairof02pathsofarandomtreecanbeaminimalpair. Every02pathofarandomtreecomputesapromptlysimpleset. Nopathofarandomtreecanhaveincompletec.e.degree. Theorem3.4.19. IfQisarandomclosedset,thenQhasnoisolatedelements.Proof.LetQ=[T]andsupposebywayofcontradictionthatQcontainsanisolatedpathx.Thenthereissomenode2TsuchthatQI=fxg.Foreachn,letSn=fP2C:jf2f0;1gn:PI_6=;gj=1g:Thatis,P2SnifandonlyifthetreeTPhasexactlyoneextensionofoflengthn+jj.ItfollowsthatjPIj=18nP2SnNowforeachn,SnisaclopensetinCandagainbyinduction,Snhasmeasure2 3n.ThusthesequenceS0;S1;:::isaMartin-Loftest.Itfollowsthatforsomen,Q=2Sn.ThusthereareatleasttwoextensionsinTQofoflengthn+jj,contradictingtheassumptionthatxwastheuniqueelementofQI. Asmentionedpreviously,itfollowsthateveryrandomclosedsetisperfectandhencecontainscontinuummanyelements.Nextwewanttondarandomclosedsetwhichdoesnotcontaina02path.Nowitiseasy[ 20 24 ]toconstructastrong02classPofpositivemeasurewhichcontainsno02elements;ofcoursePmustcontainarandomrealsinceithasmeasure1.Thedicultproblemistoconstructarandomstrong02classwithno02elements.Wehavethefollowingresultinthisdirection,whichyieldsarandomstrong03closedsetwithno02elements. Theorem3.4.20. ForanysetAthereisanA-randomclosedsetQsuchthatTQ6TA00butQhasnoelements6TA0.Proof.ItisenoughifweprovetheclaimforA=;becausetheargumentrelativisestoanyoracleAinastraightforwardway.ForA=;weuseaniteinjuryconstructionover 76

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;0toconstructQwiththeaboveproperties.Intheconstructionwewill;0-approximatethecanonicalcodeofatreeTwhichhasno02paths.TomakesurethatthetreeTisrandomwexa01classPofpositivemeasureinthespace3NwherethecodeforTlieswhichcontainsonlyrandoms,andwemakesurethatateverystageourapproximationasaniteternarystringtoT'scanonicalcodecanbeextendedtoapathinP.ThenbycompactnessthecanonicalcodeofourtreewillbeinPandsothetreewillberandom.Thechangesintheapproximationsaremotivatedbytherequirements:Re:if;0eistotalthentherealitdenesisnotin[T]:Letsbeanitestringapproximationofthecanonicalcodewearebuilding.Wewillhavejsj=s.StrategyRewillcomeintopowerafterstageeandwillrestrainuptosomere>ethedefaultvalueisre[0]=e.Alsoitmightrequestsomechangesinafterthee-thbit.Westartwith0=;andatstages+1,assuminginductivelythats#and[s]P6=;weaskfortheleasti0andontheotherhand,if=;0ewehaveseenthatfj23Nandisthecanonicalcodeofatreewhichhasasapathg=0: 77

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ThismeansthatifatstagesetherequirementReisnotyetsatised,itwillreceiveattentionatalaterstageandgetsatisedpermanently. 3.5MeasureandDimensionInthissectionweshowthatrandomclosedsethavemeasurezeroTheorem 3.5.1 andboxdimensionlog24 3Theorem 3.5.2 .3.5.1Measure Theorem3.5.1. IfQisarandomclosedset,thenQ=0.Proof.WewillshowthatinthespaceCofclosedsets,the-probabilitythataclosedsetPhasLebesguemeasure0,is1.Thisisprovedbyshowingthatforeachm,P>2)]TJ/F25 7.97 Tf 6.587 0 Td[(mwith-probability0.Foreachm,letSm=fP:P>2)]TJ/F25 7.97 Tf 6.586 0 Td[(mg:Weclaimthatforeachm,Sm=0.Theproofisbyinductiononm.Form=0,wehaveP>1ifandonlyifP=2N,whichisifandonlyifxP=;2;:::,sothatS0isasingletonandthushasmeasure0.NowassumebyinductionthatSmhasmeasure0.ThentheprobabilitythataclosedsetP=[T]hasmeasure>2)]TJ/F25 7.97 Tf 6.587 0 Td[(m)]TJ/F23 7.97 Tf 6.587 0 Td[(1canbecalculatedintwoparts.iIfTdoesnotbranchattherstlevel,sayT0=fgwithoutlossofgenerality.NowconsidertheclosedsetP0=fy:0_y2Pg.ThenP>2)]TJ/F25 7.97 Tf 6.587 0 Td[(m)]TJ/F23 7.97 Tf 6.587 0 Td[(1ifandonlyifP0>2)]TJ/F25 7.97 Tf 6.587 0 Td[(m,whichhasprobability0byinduction,sowecandiscountthiscase.iiIfTdoesbranchattherstlevel,letPi=fy:i_y2Pgfori=0;1.ThenP=1 2P0+P1,sothatP>2)]TJ/F25 7.97 Tf 6.587 0 Td[(m)]TJ/F23 7.97 Tf 6.587 0 Td[(1impliesthatatleastoneofPi>2)]TJ/F25 7.97 Tf 6.587 0 Td[(m)]TJ/F23 7.97 Tf 6.587 0 Td[(1.Notethatthereverseimplicationisnotalwaystrue.Letp=Sm+1.Theobservationsaboveimplythatp61 3)]TJ/F15 11.955 Tf 11.955 0 Td[()]TJ/F24 11.955 Tf 11.956 0 Td[(p2=2 3p)]TJ/F15 11.955 Tf 13.15 8.088 Td[(1 3p2;andthereforep=0. 78

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ToseethatarandomclosedsetQmusthavemeasure0,xmandletS=Sm.ThenSistheintersectionofaneectivesequenceofclopensetsV`,whereforP=[T],P2V`[T`]>2)]TJ/F25 7.97 Tf 6.586 0 Td[(m:Sincethesesetsareuniformlyclopen,thesequencem`=V`iscomputable.Sincelim`m`=0,itfollowsthatthisisaMartin-LoftestandthereforenorandomsetQbelongstoT`V`.Theningeneral,norandomsetcanhavemeasure>2)]TJ/F25 7.97 Tf 6.587 0 Td[(mforanym. 3.5.2DimensionSurprisingly,wecancomputetheKolmogorovboxdimensionofarandomclosedset,andinfactitturnsoutthatallrandomclosedsetshavethesamedimension.Theintuitionforthiscomesfromthefollowinglemma.ForanyfunctionFmappingthespaceCofclosedsetsinto<,theexpectedvalueofFonCistheintegralRFPwithrespecttotheprobabilitymeasure. Lemma3.5.2. InthespaceCofclosedsets,theexpectedcardinalityoff2f0;1gn:QI6=;gisexactly4 3nforeveryn,whereQischosenuniformlyatrandomaccordingto.Proof.LetSn=f2f0;1gn:QI6=;g,forarandomlychosenQfromC.Theproofisbyinductiononn.Forn=1,wehavetwocases.Withprobability2 3,cardS1=1andwithprobability1 3,cardS1=2.Thustheexpectedvalueisexactly4 3.Forn+1,thereareagaintwocases.Withprobability2 3,cardS1=1,sothattheexpectedcardSn+1equalstheexpectedcardSn,whichis4 3nbyinduction.Withprobability1 3,cardS1=2,inwhichcasetheexpectedcardSn+1istwicetheexpectedcardSn,thatis,24 3n.ThuswehavetheexpectedvaluecardSn+1=2 34 3n+1 324 3n=4 3n+1: 79

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TheboxdimensionofaclosedsetintheCantorspace,ifitexists,isgivenbythefollowinglimit:dimBFQ=limn!1log2cardTQf0;1gn n:See[ 6 ]forthisformulationoftheboxdimensioninf0;1gN.NowbyLemma 3.5.2 ,theexpectedvalueofcardTQf0;1gnforarandomclosedsetQis4 3n,whichsuggeststhattheboxdimensionofQshouldbelog24 3. Lemma3.5.3. LetQbearandomclosedset.Thenforany">0,thereexistsam2Nsuchthat,foralln>m,4 3n)]TJ/F24 11.955 Tf 11.955 0 Td[("nm,c6n>n.SincethetreeTQf0;1g66n)]TJ/F23 7.97 Tf 6.586 0 Td[(1hasatleast6nnodes,itfollowsfromCherno'sLemmathatthenumberofbranchingnodesislessthannwithprobability62)]TJ/F25 7.97 Tf 6.587 0 Td[(n=6.Thusc6nmislessthan1Xn=m2)]TJ/F25 7.97 Tf 6.586 0 Td[(n=6=2)]TJ/F25 7.97 Tf 6.587 0 Td[(m=6 1)]TJ/F15 11.955 Tf 11.956 0 Td[(2)]TJ/F23 7.97 Tf 6.586 0 Td[(1=6:Thisprovidesacomputablesequenceofclopensetswithmeasuresboundedbyacom-putablesequencewithlimitzeroandhenceaMartin-Loftest.ItfollowsthatforanyrandomclosedsetQ,thereexistsm0suchthatc6n>nforalln>m0.Nowforn>m0,thereareatleast6n2nodesinTQf0;1g612n)]TJ/F23 7.97 Tf 6.586 0 Td[(1)-229(f0;1g66n)]TJ/F23 7.97 Tf 6.587 0 Td[(1,sothatagainbyCherno'sLemma,theprobabilitythat3suchthatc12n>n2foralln>m1.Nowsupposethatm>12m1andthat12n6m<12n+1<16n.Thenn>m1,sothatcm>c12n>n2>m=162:AgainbyCherno'sLemma,theprobabilitythatthenumberofbranchingnodesfromTQf0;1gndiersfrom1 3cnby>1 3c)]TJ/F18 5.978 Tf 7.782 3.259 Td[(1 4ncnis<2)]TJ 6.586 5.248 Td[(p cn=9.Butthisisexactlytheprobabilitythatcn+1diersfrom4 3cnby>1 3c)]TJ/F18 5.978 Tf 7.782 3.259 Td[(1 4ncn.Forn>m1,weknowthatcn>)]TJ/F25 7.97 Tf 8.34 -4.977 Td[(n 162,sothatp cn>n 16andc)]TJ/F18 5.978 Tf 7.782 3.258 Td[(1 4n64 p nandhence2)]TJ 6.587 5.248 Td[(p cn=962)]TJ/F25 7.97 Tf 6.586 0 Td[(n=144.Thustheprobabilitypnthatcn+1 80

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diersfrom4 3cnbymorethancn 9p nis<2)]TJ/F25 7.97 Tf 6.587 0 Td[(n=144.Thentheprobabilitythatforanyn>m1,cn+1diersfrom4 3cnbymorethan4 3p ncnisboundedby1Xn=mpn=1Xn=m2)]TJ/F25 7.97 Tf 6.587 0 Td[(n=144=2)]TJ/F25 7.97 Tf 6.586 0 Td[(m=144 1)]TJ/F15 11.955 Tf 11.956 0 Td[(2)]TJ/F23 7.97 Tf 6.586 0 Td[(144:ThisagainprovidesaMartin-LoftestwhichshowsthatforanyrandomclosedsetQ,thereexistsm2sothatforn>m2,4 31)]TJ/F15 11.955 Tf 18.699 8.088 Td[(1 p ncn6cn+164 31+1 p ncn:Nowgiven",choosem>m2sothat+1 p m2<1+"and1)]TJ/F24 11.955 Tf 11.955 0 Td[("<)]TJ/F23 7.97 Tf 18.307 4.707 Td[(1 p m2.Thenforanyk,cm4 32k)]TJ/F24 11.955 Tf 11.955 0 Td[(km+2k.Foroddn,thisinequalitywillholdbytheinequalityabove. Theorem3.5.4. ForanyrandomclosedsetQ,theboxdimensionofQislog24 3.Proof.Given">0,letmbegivenbyLemma 3.5.3 .Thenforn>m,wehavenlog24 3+nlog2)]TJ/F24 11.955 Tf 11.955 0 Td[("6log2cardTQf0;1gn6nlog24 3+nlog2+";sothatlog24 3+log1)]TJ/F24 11.955 Tf 11.955 0 Td[("6log2cardTQf0;1gn n6log24 3+log2+";andthereforedimBQ=limnlog2cardTQf0;1gn n=log24 3. 81

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3.6Prex-FreeComplexityofClosedSetsInthissection,weconsiderrandomnessforclosedsetsintermsofincompressibilityoftrees.Ofcourse,Schnorr'stheoremtellsusthatPisrandomifandonlyifthecodexP2f0;1;2gNforPisprex-freerandom,thatis,K3xPdn>n)]TJ/F24 11.955 Tf 12.658 0 Td[(O.Schnorr'stheoremforarbitrarynitealphabetsisshownin[ 18 ].HerewewriteK3toindicatethatwewouldbeusingauniversalprex-freefunctionU:f0;1;2g!f0;1;2g.However,manypropertiesoftreesandclosedsetsdependonthelevelsTn=Tf0;1gnofthetree.Forexample,if[Tn]=[fI:2Tng,then[T]=Tn[Tn]and[T]=limn!1[Tn].SowewanttoconsiderthecompressibilityofatreeintermsofKTn.NowthereisanaturalrepresentationofTnasastringoflength2n.Thatis,listf0;1gninlexicographicorderas1;:::;2nandrepresentTnbythestringe1;:::;e2nwhereei=1ifi2Tandei=0otherwise.HenceforthweidentifyTnwiththisnaturalrepresentation.ItisinterestingtonotethatthecodeforTnwillhaveashorterlengththanthenaturalrepresentation.Forexample,if[T]=fygisasingleton,thenx=yandforeachn,thecodeforTnisxdn.Ifxisthecodeforthefulltreef0;1g,thenx=;2;:::andthecodeforTnisastringofn)]TJ/F15 11.955 Tf 11.957 0 Td[(12's,thoselabelsattachedtonodesoflength2n)]TJ/F24 11.955 Tf 12.273 0 Td[(cforalln.However,wewillseethatthisisnotpossibleforanytree.Ontheonehand,inSection 3.6.1 weachievealowerboundforincompressibility.Thatis,weshowthatifP=[T]israndomthenthereisaconstantcsuchthatKTn>)]TJ/F23 7.97 Tf 6.675 -4.977 Td[(7 6n)]TJ/F24 11.955 Tf 12.446 0 Td[(cforalln.Ontheotherhand,inSection 3.6.2 ,weseethatthe2nistoohighofanincompressibilityboundsincethereissomecandsomerandomclosedsetsuchthatKTn62n)]TJ/F24 11.955 Tf 11.955 0 Td[(cforalln. 82

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Inalargersense,weseekaformulationofrandomness,intermsoftheincompressibil-ityofTn,forotherobjectssuchas01classesor02classes.Inthefollowingtwosectionsweconsiderthesequestionsandachievesomelowerandupperboundsfortheseclassesofobjects.3.6.1LowerComplexityBoundsFirstwegivealowerboundfortheprex-freecomplexityofarandomtree. Theorem3.6.1. IfPisarandomclosedsetandT=TP,thenthereisaconstantcsuchthatKTn>)]TJ/F23 7.97 Tf 6.675 -4.977 Td[(7 6n)]TJ/F24 11.955 Tf 11.955 0 Td[(cforalln.Proof.LetP=[T]bearandomclosedset.LetmbegivenbyLemma 3.5.3 ,for"=7 6,sothatforn>m,cardTn>7 6n:ItfollowsthatthecodexnforTnhaslength>)]TJ/F23 7.97 Tf 6.675 -4.977 Td[(7 6n.Sincexisrandom,weknowthat,forn>m,K3xn>7 6n)]TJ/F24 11.955 Tf 11.955 0 Td[(a;forsomeconstanta.NowwecancomputexnfromTn,sothatKTn>K3xn)]TJ/F24 11.955 Tf 11.956 0 Td[(b;forsomeconstantb.Theresultnowfollows.Thatis,letUmappingf0;1gtof0;1gbeauniversalprex-freeTuringmachineandletKTn=minfjj:U=Tng.LetMbeaprex-freemachineMmappingf0;1gtof0;1;2gsuchthatMTn=xn.ThendeneVbyV=MU:ThenKVxdn6KTn,sothatforsomeconstante,K3xn6KTn+eandhenceKTn>K3xn)]TJ/F24 11.955 Tf 11.955 0 Td[(e>7 6n)]TJ/F24 11.955 Tf 11.955 0 Td[(b)]TJ/F24 11.955 Tf 11.955 0 Td[(e: 83

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Thestandardexampleofarandomreal,Chaitin's[ 27 ],isac.e.realandtherefore02.Thusthereexistsa02randomtreeTandbyTheorem 3.6.1 ,KT`>)]TJ/F23 7.97 Tf 6.675 -4.976 Td[(7 6n)]TJ/F24 11.955 Tf 12.409 0 Td[(cforsomec.Wehaveamoremodestresultfor01classes.Thatis,thereisaneectivelyclosedsetwithnottoomuchcompressibility,inthefollowingsense. Theorem3.6.2. Thereisa01classP=[T]suchthatKTn>nforalln.Proof.Recalltheuniversalprex-freemachineUandletS=f2DomU:jUj>2jjg.ThenSisac.e.setandcanbeenumeratedas1;2;:::.ThetreeT=TsTswhereTsisdenedatstages.InitiallywehaveT0=f0;1g.Wesaythattrequiresattentionatstages>twhen=Ut=Tsnforsomensothatjj=2nandn>jtj.Actionistakenbyselectingsomepatht2TsoflengthnanddeningTs+1tocontainallnodesofTswhichdonotextendt.Then6=Ts+1nandfurthermore6=Trnforanyr>s+1sincefutureactionwillonlyremovemorenodesfromTn.Atstages+1,lookfortheleastt6s+1suchthattrequiresactionandtaketheactiondescribedifthereissuchat.Otherwise,letTs+1=Ts.LetAbethesetoftsuchthatactionisevertakenont.RecallfromtheKraftInequalitythatPt2jtj<1.Sincejtj>jtj,itfollowsthatPt2A2jtj<1aswell.Now[T]=1)]TJ/F29 11.955 Tf 11.955 8.967 Td[(Pt2jtj>0andtherefore[T]isnonempty.Itfollowsfromtheconstructionthatforeacht,actionistakenfortatmostonce.NowsupposebywayofcontradictionthatU=Tnforsometwithjj6n.Theremustbesomestager>tsuchthatforalls>r,Tsn=Tnandsuchthatactionisnevertakenonanyt02p `forall`. 84

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Proof.WewillconstructatreeTsuchthatTn2cannotbecomputedfromfewerthan2nbits.WewillassumethatU;"totakecareofthecasen=0.Atstages,wewilldenethenonemptylevelTs2ofT,usinganoraclefor00.WebeginwithT0=f;g.Atstages>0,weconsiderDs=f2DomU:jj<2sg:SinceUisprex-free,cardDs<22s.Nowthereareatleast222s)]TJ/F18 5.978 Tf 5.756 0 Td[(1treesofheights2whichextendTs)]TJ/F23 7.97 Tf 6.586 0 Td[(12andwecanusetheoracletochoosesomeniteextensionT0=Ts2ofTs)]TJ/F23 7.97 Tf 6.587 0 Td[(12suchthat,forany2Ds,U6=T0andfurthermore,U6=TrforanypossibleextensionTrwiths26r.Thatis,sincethereare<22snitetreeswhichequalUforsome2Ds,thereissomeextensionT0ofTs)]TJ/F23 7.97 Tf 6.587 0 Td[(12whichdiersfromalloftheseatlevels2.Weobservethattheoraclefor00isusedtodeterminethesetDs.Atstages,wehaveensuredthatforanyextensionTf0;1gofTs2,anywithjj62s2andanyn>s2,U6=Tn.ItisimmediatethatKTn>2p n. 3.6.2UpperComplexityBoundsInTheorem 3.6.1 weachievedalowerboundof)]TJ/F23 7.97 Tf 6.675 -4.977 Td[(7 6nfortheprex-freecomlexityofatreeTPofarandomclosedsetP.Itseemsplausiblethatwemightbeabletoachieveahigherboundof2n.Iftruethiswouldactuallyprovideandimmediatecharacterizationofrandomnessofclosedsetsintermsofprex-freecomplexityoftrees.Thatis,aclosedwouldberandomiKTn>2n)]TJ/F24 11.955 Tf 12.613 0 Td[(cforsomeconstantc.Toseethis,notethatfromxd2nwecancomputeTnuniformlysothatK3xd2n>KTn)]TJ/F24 11.955 Tf 12.307 0 Td[(bforsomeb.Howeverthefollowingtheoremprovidesanuppercomplexityboundlessthan2n,refutingsuchapossibility. Theorem3.6.4. ForanytreeTf0;1g,thereareconstantsk>0andcsuchthatKT`62`)]TJ/F15 11.955 Tf 11.955 0 Td[(2`)]TJ/F25 7.97 Tf 6.587 0 Td[(k+cforall`.Proof.Forthefulltreef0;1g,thisisclearsosupposethat=2Tforsome2f0;1gm.Thenforanylevel`>m,thereare2`)]TJ/F25 7.97 Tf 6.587 0 Td[(mpossiblenodesforTwhichextendandT`may 85

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beuniformlycomputedfromandfromthecharacteristicfunctionofT`restrictedtotheremainingsetofnodes.Thatis,xoflengthmanddeneaprex-freecomputerMasfollows.ThedomainofMisstringsoftheform0`1wherejj=2`)]TJ/F15 11.955 Tf 12.378 0 Td[(2`)]TJ/F25 7.97 Tf 6.586 0 Td[(m.MoutputsthestandardrepresentationofatreeT`suchthatnoextensionofisinT`andsuchthattellsuswhetherstringsnotextendingareinT`.ItisclearthatMisprex-freeandwehaveKMT`=`+1+2`)]TJ/F15 11.955 Tf 11.539 0 Td[(2`)]TJ/F25 7.97 Tf 6.587 0 Td[(m.ThusKT`6`+1+2`)]TJ/F15 11.955 Tf 11.539 0 Td[(2`)]TJ/F25 7.97 Tf 6.587 0 Td[(m+cforsomeconstantc.Now`+1<2`)]TJ/F25 7.97 Tf 6.586 0 Td[(m)]TJ/F23 7.97 Tf 6.586 0 Td[(1forsucientlylarge`andthusbyadjustingtheconstantc,wecanobtainc0sothatKT`62`)]TJ/F15 11.955 Tf 11.955 0 Td[(2`)]TJ/F25 7.97 Tf 6.586 0 Td[(m)]TJ/F23 7.97 Tf 6.586 0 Td[(1+c0: ThefollowingtheoremalsorefutesthepossibilitythatKT`>2`)]TJ/F25 7.97 Tf 6.587 0 Td[(cisacharac-terizationofrandomclosedsetsintermsofprex-freerandomness.Itshowsthatclosedsetswithsmallmeasure,suchasrandomclosedsetswhichhavemeasurezeroseeTheo-rem 3.5.1 ,aremorecompressible. Theorem3.6.5. If[T]<2)]TJ/F25 7.97 Tf 6.587 0 Td[(k,thenthereexistscsuchthat,forall`,KT`62`)]TJ/F25 7.97 Tf 6.587 0 Td[(k+1+c:Proof.Supposethat[T]<2)]TJ/F25 7.97 Tf 6.587 0 Td[(k.Thenforsomeleveln,Tnhas<2n)]TJ/F25 7.97 Tf 6.586 0 Td[(knodes1;:::;t.Nowforany`>n,T`canbecomputedfromthexedlist1;:::;tandthelistofnodesofT`takenfromtheatmost2`)]TJ/F25 7.97 Tf 6.586 0 Td[(kextensionsof1;:::;t.ItfollowsasintheproofofTheorem 3.6.4 abovethatforsomeconstantcandall`,KT`62`)]TJ/F25 7.97 Tf 6.587 0 Td[(k+`+1+c:Thusforlargeenoughsothat`+162`)]TJ/F25 7.97 Tf 6.587 0 Td[(k,wehaveKT`62`)]TJ/F25 7.97 Tf 6.587 0 Td[(k+1+c;asdesired. 86

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Weconjecturethataboundof)]TJ/F23 7.97 Tf 6.675 -4.977 Td[(4 3nwouldcharacterizerandomclosedsetsintermsofprefex-freecomplexity.Itwouldsuce,then,toshowthat)]TJ/F23 7.97 Tf 6.675 -4.976 Td[(4 3nisalowerboundandthatthisboundimpliesrandomness.Wealsostillseekupperboundsfor01orclosed02classestowardsestablishingprex-freecomplexitycharacterizationsoftheseclasses.Itseemsplausiblethat01classesaremorecompressiblei.e.necessarilyhavesmallerlowerboundsthanrandomclosedsetsandwewouldliketoexplorethisnotionfurther.3.7OtherNotionsofRandomnessforClosedSetsOthernotionsofrandomnessthatdependondierentprobabilitymeasures,ortheinclusionoftreeswithdeadendsintheencoding,mightalsobeconsidered.3.7.1RandomnesswithRegularProbabilityMeasuresForanyregularmeasure,wecandenethenotionofa-Martin-Loftestandtheresultingnotionofa-Martin-Lof-randomorjust-randomreal.Itiseasytoseethat-randomrealsexistforanyandhence-randomclosedsetsexist.Theresultsonghostcodesandjoinswillholdforanyregularmeasure.ThecorrespondingversionofLemma 3.3.8 willholdifisregularwithb0andb161 2.TheproofsofTheorem 3.4.14 andCorollary 3.4.15 ,thatnosubsetofameasure-zero01classisrandom,alsogothroughunderthisassumption.Someoftheresultsinthischaptermayalsobeobtainedforfwheref_i61 2fori=0;1.Forexample,withrespecttof,arandomclosedsetwillhavenoisolatedelementsanditwillalwayscontainarandomelement.Foranyregularmeasure,eithertheleftmostortherightmostpathwillbenonrandom,sinceeitherb0+b2>1 2orb1+b21 2.TheproofofTheorem 3.4.19 thateveryrandomclosedsethasmeasure0seemstorequire,forf-randomness,thatf_261 2forall.3.7.2RandomnesswiththeInclusionofTreeswithDeadsEndsReturningtothenotionofrandomnesswhichallowstreeswithdeadends,letb3nowbetheprobabilitythatagivennodehasnoextensionsandlettheprobabilityberegularasabove.Thenasimplerecursionshowstheprobabilitypofagivenclosedsetbeing 87

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emptysatisestheequationp=b3+b0+b1p+b2p2:Solvingforp,weobtainp)]TJ/F15 11.955 Tf 11.955 0 Td[(1b2p)]TJ/F24 11.955 Tf 11.955 0 Td[(b3=0:Thuseitherp=1orp=b3 b2.Itfollowsthatifb26b3,thenp=1,thatis,almosteveryclosedsetisempty.Supposenowthatb3
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ii Forn>2andanyQ1;:::;Qn2CTni=1Qi6Xf)]TJ/F15 11.955 Tf 9.298 0 Td[(1jIj+1T[i2IQi:;6=If1;2;:::;ngg:Thisisthealternatingofinniteorderproperty. iii IfQ=nQnandQn+1Qnforalln,thenTQ=limn!1TQn.Wewillassume,unlessotherwisespecied,thatTN=1foragivencapacityT. Denition3.8.2ComputableCapacities. AcapacityTiscomputableifitiscom-putableonthefamilyofclopensets.Itfollowsthatthecapacityofany01classisuppersemi-computable.Finally,thefollowingnotationaldenitionwillbeusedthroughout. Denition3.8.3TdQ,forQ2C. SupposeQ2C.DeneTdQ:=dVQ,whereVQisasub-basissetforthehit-or-misstopologyonCasgiveninsection 3.3.1 anddisagivenprobabilitymeasure.Thatis,TdQistheprobabilitythatarandomlychosenclosedsetmeetsQ.Wenowshow,inthefollowingtwotheorems,thatacomputablecapacityisalwaysobtainablefrom,oraconsequenceof,acomputableprobabiltymeasuredforsomed.Thefollowingtheorem,inparticular,iswell-known.Fordetailsoncapacityandrandomsetvariables,see[ 73 ]. Theorem3.8.4. IfdisacomputableprobabilitymeasureonC,thenTdisacom-putablecapacity.Proof.Thisiseasilyveried.CertainlyT;=0.Thealternatingpropertyfollowsbybasicprobability.Foriii,supposethatQ=nQnisadecreasingintersection.Thenbycompactness,QK6=;ifandonlyifQnK6=;foralln.Furthermore,VQn+1VQnforalln.ThusTQ=VQ=nVQn=limnVQn=limnTQn: 89

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ThecomputabilityofTiseasilyveried.Thatis,foranyclopensetI1[[Ikwhereeachi2f0;1gn,wecomputetheprobabilitydistributionforalltreesofheightnandaddtheprobabilitiesofthosetreeswhichcontainoneofthei. Thisresulthasaconverse,duetoChoquet.See[ 73 ]forthegeneralresult. Theorem3.8.5. IfTisacomputablecapacity,thenthereisacomputablemeasuredonthespaceofclosedsetssuchthatT=Td.Proof.GiventhevaluesTUforallclopensetsI1[[Ikwhereeachi2f0;1gn,thereisinfactauniqueprobabilitymeasuredontheseclopensetssuchthatT=Tdandthiscanbecomputedasfollows.SupposerstthatTIi=aifori<2andnotethateachai61anda0+a1>1bythealternatingproperty.IfT=Td,thenwemusthaved+d=a0andd+d=a1andalsod+d+d=1,sothatd=a0+a1)]TJ/F15 11.955 Tf 12.337 0 Td[(1,d=1)]TJ/F24 11.955 Tf 12.012 0 Td[(a1andd=1)]TJ/F24 11.955 Tf 12.011 0 Td[(a0.ThiswillimplythatT=Tdwhenjj=1.Nowsupposethatwehavedeneddandthatisthecodeforanitetreewithelements0;:::;n=andthusd_iisgivingtheprobabilitythatwillhaveoneorbothimmediatesuccessors.Weproceedasabove.LetTI_i=aiTIfori<2.Thenasaboved_2=da0+a1)]TJ/F15 11.955 Tf 11.955 0 Td[(1andd_i=d)]TJ/F24 11.955 Tf 11.955 0 Td[(aiforeachi. 3.8.2RegularMeasuresandCapacitiesofClosedSetsInthissectionallresultsarewithrespecttodxedastheuniformmeasurei.e.theregularmeasurewithb0=b1=b2=1 3;seeDenition 3.2.6 .Withthismeasure,wewillconsiderthecapacitiesofrandomclosedsetsandeectivelyclosedsets.WesaythatQ2Cisd-randomifxQisMartin-Lofrandomwithrespecttothemeasured. Theorem3.8.6. Fortheregularmeasuredwithbi=1 3,ifRisad-randomclosedset,thenTdR=0.Proof.Fixdasdescribedabovesothatd_i=d1 3andlet=d.Wewillcomputetheprobability,giventwoclosedsetsQandK,thatQKisnonempty.LetQn=[fI:2f0;1gn&QI6=;g 90

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andsimilarlyforKn.ThenQK6=;ifandonlyifQnKn6=;foralln.LetpnbetheprobabilitythatQnKn6=;fortwoarbitraryclosedsetsKandQ,relativetoourmeasure.Itisimmediatethatp1=7 9,sinceQ1K1=;onlywhenQ1=IiandK1=I)]TJ/F24 11.955 Tf 12.31 0 Td[(i.Nextwewilldeterminethequadraticfunctionfsuchthatpn+1=fpn.Thereare9possiblecasesforQ1andK1,whichbreakdowninto4distinctcases.CaseI:TherearetwochancesthatQ1K1=;.CaseII:TherearetwochancesthatQ1=K1=Ii,sothatQn+1Kn+16=;withprobabilitypn.CaseIII:TherearefourchanceswhereQ1=2NandK1=Iiorviceversa,sothatonceagainQn+1Kn+16=;withprobabilitypn.CaseIV:ThereisonechancethatQ1=K1=2N,inwhichcaseQn+1Kn+16=;withprobability1)]TJ/F15 11.955 Tf 12.118 0 Td[()]TJ/F24 11.955 Tf 12.118 0 Td[(pn2=2pn)]TJ/F24 11.955 Tf 12.118 0 Td[(p2n.ThisisbecauseQn+1Kn+1=;ifandonlyifbothQn+1IiKn+1=;forbothi=0andi=1.Addingthesecasestogether,weseethatpn+1=6 9pn+1 9pn)]TJ/F24 11.955 Tf 11.955 0 Td[(p2n=8 9pn)]TJ/F15 11.955 Tf 13.151 8.087 Td[(1 9p2n:Itfollowsthatthesequencehpnin2!iscomputableandwewillseethatthelimitiszero.Letfp=8 9p)]TJ/F23 7.97 Tf 13.605 4.707 Td[(1 9p2.Elementarycalculusshowsthatfhasxedpointsatp=)]TJ/F15 11.955 Tf 9.298 0 Td[(1andp=0andthatfor02)]TJ/F25 7.97 Tf 6.586 0 Td[(ng:LetCmbethenumberoftreesofheightmwithoutdeadends. 91

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ForeachQ2Am;nthereare2)]TJ/F25 7.97 Tf 6.586 0 Td[(nCmpossiblechoicesforKmsuchthatKmQm6=;andthusatleastDm;n=1 2)]TJ/F25 7.97 Tf 6.587 0 Td[(nAm;nC2mchoicesforK;Q2CCsuchthatKmQm6=;withQ2Am;nsinceeachpairmightbecountedtwice.Nowdeneacomputablesequencehmnin2!,sothatpmn<2)]TJ/F23 7.97 Tf 6.587 0 Td[(2n)]TJ/F23 7.97 Tf 6.587 0 Td[(1.Thenpmn>Dmn;n C2mn=2)]TJ/F23 7.97 Tf 6.586 0 Td[(n+1Amn;n:ItfollowsthatAmn;n62n+1pmn<2n+12)]TJ/F23 7.97 Tf 6.587 0 Td[(2n)]TJ/F23 7.97 Tf 6.586 0 Td[(1=2)]TJ/F25 7.97 Tf 6.587 0 Td[(n:LettingSn=[r>nAmr;ritfollowsthatSn62)]TJ/F25 7.97 Tf 6.587 0 Td[(naswell.NowletRbearandomclosedset.ThesequencehSnin2!isacomputablesequenceofc.e.opensetswithmeasure62)]TJ/F25 7.97 Tf 6.586 0 Td[(n,sothatthereissomensuchthatR=2Sn.Thusforr>n,fK:KmrRmr6=;g<2)]TJ/F25 7.97 Tf 6.587 0 Td[(randitfollowsthatfK:KR6=;g=limnfK:KmnRmn6=;g=0:ThusTdR=0,asdesired. Thisresultseemstodependonthemeasure.Fordierentregularmeasures,thecapacityofarandomclosedsetcanhavedierentvalues. Theorem3.8.7. Fortheregularmeasuredwithbi=1 3,thereisameasurezero01classQsuchthatTdQ>0:Proof.FirstletuscomputethecapacityofXn=fx:xn=0g.Forn=0,wehaveTdQ0=2 3.NowtheprobabilityTdXn+1thatanarbitraryclosedsetKmeetsXn+1maybecalculatedintwodistinctcases.LetKnbeasintheproofofTheorem 3.8.6 .CaseIIfK0=2N,thenTdXn+1=1)]TJ/F15 11.955 Tf 11.955 0 Td[()-222(TdXn2. 92

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CaseIIIfK0=Iiforsomei<2,thenTdXn+1=TdXn.ItfollowsthatTdXn+1=2 3TdXn+1 3TdXn)]TJ/F15 11.955 Tf 11.095 0 Td[(TdXn2=4 3TdXn)]TJ/F23 7.97 Tf 12.29 4.707 Td[(1 3TdXn2.Nowthefunctionfp=4 3p)]TJ/F23 7.97 Tf 13.812 4.707 Td[(1 3p2hasthepropertythatfp>pfor03 4=c0andforeachk,choosen=nk+1suchthatTdXn0;:::;nk;n>ck+1.Thiscanbedonesinceck+1limkck=1 2. Thisresultcaneasilybeextendedtoanyboundedmeasure. 93

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CHAPTER4RANDOMCONTINUOUSFUNCTIONSThefollowingchapterisjointworkwithGeorgeBarpalias,DouglasCenzer,JereyB.Remmel,andRebeccaWeberandwillappearintheArchiveforMathematicalLogicasanarticleentitledAlgorithmicRandomnessofContinuousFunctions[ 8 ].ApreliminaryversionofthisresearchwasoriginallypresentedattheThirdInternationalConferenceofComputabilityandComplexityinAnalysisinGainesville,Floridain2006byJ.B.Remmel.ThispreliminaryworkwaspublishedinthereferredconferenceproceedingsasRandomContinuousFunctionsP.Brodhead,D.Cenzer,andJ.B.RemmelinProceedingsofCCA2006D.Cenzer,R.Dillhage,T.GrubbandKlausWeihrauch,eds.,InformationBerichte,FernUniversitat06,pages76{89andinSpringerElectronicNotesinTheoreticalComputerScience,ElsevierScience16707[ 15 ].PortionsofthisworkwerealsopresentedbyP.BrodheadattheAMSFall2006EasternSectionalMeetingOctober2006,Storrs,CTandtheConferenceonLogic,Computability,andRandomnessJanuary2007,BuenosAires,Argentina.4.1OverviewInChapter 3 ,weconsideredanotionofrandomnessforclosedsets.Wedothesameforcontinuousfunctionshere.AnintroductiontorandomnessforrealsisprovidedinSection 3.2 .Thischapterisorganizedasfollows.InSection 4.2 ,weprovideadenitionofrandomnessforcontinuousfunctionsandshowthatitissound.InSection 4.3 ,weprovevariousresultsforimagesofrandomcontinuousfunctions{perfectness,non-injectivity,andinstancesofnon-surjectivity;wealsostudyimagesofcomputableelements.InSection 4.4 ,wetierandomclosedsetstorandomclosedfunctionsthroughimages:inverseimagesof0!arerandomclosedsets,butimages,ingeneral,arenot.Continuingonthethemeofinverseimagesof0!,inSection 4.5 weconsiderpseudo-distancefunctions.InSection 4.6 ,webrieyconsiderhowtheresultsofChapters 3 and 4 canberelativizedforn-randomness.Finally,inSection 4.7 ,wedescribesomedirectionsforfutureresearch. 94

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4.2DeniningRandomnessforContinuousFunctionsAfunctionF:2N!2Niscontinuousiithasaclosedgraph.Itseemsreasonable,then,todenecontinuousfunctionftoberandom,itheitsgraphGrF=fxy:y=Fxgisrandom.Howeverif[T]isthegraphofafunctionand2Thasevenlength,thenwemusthave_02Tand_12T.Thismeansthatthefamilyofclosedsetswhicharethegraphsoffunctionshasmeasure0inthespaceofclosedsetsandhencearandomclosedsetwillnotbethegraphofafunction.Weneed,therefore,adierentmethodtodenerandomnessforcontinuousfunctions.Wedothisbelow.4.2.1RepresentingFunctionsGivenacontinuousfunctionF:2N!2N,weareinterestedinrepresentingitinsuchawaysoastobeabletoconsideranotionofalgorithmicrandomness.WeshowbelowthatanysuchfunctionFmayberepresentedbyinnitelymanyrepresentingfunctionsoftheformf:f0;1g!f0;1;2g.Thiswillallowus,inthefollowingsection,tobeabletorepresentcontinuousfunctionaselementsof3!,so-calledrepresentingsequences,andtoconsiderasuchafunctionasrandomifitpossessesarandomrepresentingsequence.Informationoutput,thekeytorepresentation.ForanycontinuousfunctionFon2Nandany2f0;1g,thereisanaturalnumbernandbinarystringoflengthnsuchthatforallu2I,Fudn=.Inparticular,Fun=nforeverysuchu.Ingeneral,thelengthofmaybemuchlargerthann,sowemayhavetoextendbyseveralbitstogetuniformityofFudn+1withintheintervalaround'sextension.Representingfunctions.Takingtheaboveintoconsideration,wemayrecursivelyrepresentanycontinuousfunctionF:2N!2Nbysomefunctionf:f0;1g!f0;1;2gasfollows.SupposeFisgiven.Letf;=;.Forjj=m+1,havingdenedfdi=eiforalli6m,let=n1;:::;nkbetheresultofdeletingall2sfrome1;:::;em.Ifforallu2I,Fudk=_j,j2f0;1g,wemayletem+1=j.Ifnotwemusthaveem+1=2;evenifsoweallowem+1=2. 95

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Thecanonicalrepresention.Notice,fromtheabove,thatanycontinuousFhasinnitelymanyrepresentingfunctionsf:f0;1g!f0;1;2g.Therepresentationwhichusesasfew2saspossibleweshallcallthecanonicalrepresentation.4.2.2RepresentingSequencesWewanttocodetherepresentingfunctionasanelementof3Ntodiscussitsalgo-rithmicrandomness.Todoso,rstenumeratef0;1g=f;gas0;1;:::,orderedrstbylengthandthenlexicographically.Thus0=,1=,2=0,etc.Wedenerepresentingsequencesbelow. Denition4.2.1. i INF,Rem2LetINFequalthesetofy2f0;1;2gnsuchthatfn:yn6=2gisinniteand,fory2INF,letRem2ybetheresultofremovingfromxalloccurrencesof2. ii RepresentingfunctionsAfunctionf:f0;1g!f0;1;2grepresentsafunctionF:2N!2Nifforallx22N,thesequencey,denedbyyn=fxdnbelongstoINFandRem2y=Fx. iii RepresentingsequencesAsequencer2f0;1;2gNrepresentsthecontinuousfunctionFwrittenF=Frifthefunctionfr:f0;1g!f0;1;2g,denedbyfrn=rn,representsF. iv Labelled2!-treesGivenarepresentingsequencer,thefunctionfrgivesrisetoalabelled2!-tree.Weattach,orassociate,thevalueoffrwitheachnode. Example4.2.1The2!-treefortheIdentity,AGeometricIntrepretation. Theidentityfunctioncanberepresentedbyplacinganeonanynodewhichendsine.ThiscanalsobepicturedgeometricallyasrepresentingthegraphofFastheintersectionofadecreasingsequenceofclopensubsetsoftheunitsquare.Initiallythechoiceoffandfselectsfromthe4quadrants.Thatis,forexample,f==fimpliesthatthegraphofFisincludedinthebottomhalfofthesquareandf=;andf=impliesthatthegraphexcludesthelowerrighthandquadrant.SuccessivevaluesoffcontinuetorestrictthegraphofFinasimilarfashion. 96

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4.2.3ASoundDenitionInthissectionwedeneameasureonthespaceoffunctionsF:2N!2Nthatallowsustodenethenotionofrandomnessforfunctionson2N.Inshort,afunctionisrandomifitpossessesarandomrepresentingfunction,orequivalently,arandomrepresentingsequence.Wewillshowthatnocanonicalrepresentingfunctioncanberandom,sothatnecessarily,thedenitionofrandomnessforfunctionsisinthisexistentialformat.Itmaybe,however,thatnocontinuousfunctionhasarandomrepresentingfunction.Weshowthatthisisnotso.Infact,weshowthateveryrandomrepresentingfunctioniscontinuous.Clearlythen,randomcontinuousfunctionsexistand,infact,02randomcontinuousfunctionsexist.Thereforethedenitionissoundandthestructurebeginstomanifestitselfasrich.TheMeasureforRandomness.Themeasurewhichisusedtodenerandom-nessforcontinuousfunctionsistheLebesguemeasureonthespace3Nofrepresentingsequences.Thusforeachnewbitofinput,thereisequalprobability1 3thatfrgivesanewoutputof0forFr,givesanewoutputof1forFr,orgivesnonewoutputforFr.Thismeasurenowinducesameasure,say,onthespaceFofcontinuousfunctions. Denition4.2.2. AfunctionF:2N!2Nisrandomifthereisarepresentingsequencer23NforFthatisrandomwithrespecttothemeasure.Werstshowthateveryrandomrepresentingfunctioniscontinuous.Thefollowinglemmaisneeded. Lemma4.2.3. LetbeanitesetandletQNbea01classofmeasure0.ThennoelementofQisMartin-Lofrandom.Proof.Let=f0;1;2gwithoutlossofgenerality.LetQ=[T]whereTf0;1;2gisacomputabletreepossiblywithdeadends.Foreachn,letTn=Tf0;1;2gnandletQn=[fI:2Tng:Letgn=Qn=jTnj 3n.Thengnisacomputablesequenceandlimn!1gn=Q=0: 97

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ThisMartin-LoftestshowsthatQhasnorandomelements.AsobservedbySolovay,itissucienttohaveacomputablesequenceapproachingzeroratherthanthestrictertestwithasequenceofmeasuresgn62)]TJ/F25 7.97 Tf 6.586 0 Td[(n. Theorem4.2.4. i Thesetofrepresentingfunctionsfortotalfunctionshasmeasureone. ii Everyrandomfunctioniscontinuous.Proof.iLetr23Nandsupposethatfrdoesnotrepresentatotalfunction.Thenthereissomex22Nandsome2f0;1gsuchthatfrxdn=foralmostalln.Withoutlossofgeneralitywemayassumethat=;.LetAbethesetoffunctionsf:f0;1g!f0;1gsuchthatf=;forarbitrarilylongstringsandletp=A.Thencertainlyp65 9,sinceifrandrarebothinf0;1g,thenfr=2A.Consideringthe9casesfortheinitialchoicesoffandf,weseethatp=4 9p+1 9[1)]TJ/F15 11.955 Tf 11.955 0 Td[()]TJ/F24 11.955 Tf 11.956 0 Td[(p2];sothat1 9p2+1 3p=0,whichimpliesthatp=0.Thatis,thereare4casesinwhichjfij=1fori=0;1sothatimmediatelyf=2A,thereare4casesinwhichonlyoneoffi=;,inwhichcasetheremainingfunctiong,denedbyg=fi_mustbeinA,andthereisonecaseinwhichfi=;fori=0;1,inwhichcaseatleastoneoftheremainingfunctionsmustbeinA.Consequently,thesetofrepresentingfunctionsfortotalfunctionshasmeasureone.iiObservethatAisa01class,sincefr2Aifandonlyif8n92f0;1gnfr=;.ItfollowsfromLemma 4.2.3 thatnorepresentingfunctionon2forarandomfunctionon2NcanbeinA.Asallfunctionsrepresentingpartialfunctionson2NoccurinA,itfollowsthateveryrandomfunctionistotal.Sincethegraphofatotalfunctionisaclosedset,itfollowsthatrandomfunctionsarecontinuous. NowthesetofMartin-Lofrandomelementsoff0;1;2gNhasmeasureoneandthereexistsa02Martin-Lofreal.Hencewehavethefollowing. Theorem4.2.5. Thereexistsarandomcontinuousfunctionwhichis02computable. 98

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Wealsorstobservethatanycontinuousfunctionwillhavearepresentationwhichisnotrandom.Infact,thecanonicalrepresentationitselfcanneverberandom. Proposition4.2.6. ForanycontinuousfunctionF,thecanonicalrepresentationisnotrandom.Proof.Theideaisthatwheneverthecanonicalrepresentationlabelsanodewith2,thenthetwolabelsonthesuccessornodes_0and_1cannotbeboth0,orboth1.ThuswehavethefollowingMartin-Loftest.Assumebywayofcontradictionthatrisrandomandcanonical.LetSebethesetofr23Nsuchthatrhasatleasteoccurrencesof2andsuchthat,forthersteoccurrencesof2inr,thecorrespondingsuccessorvaluesarenotboth0orboth1.Sincerisrandom,itmusthaveinnitelymanyoccurrencesof2andsinceriscanonical,itmustbelongtoeverySe.ButeachSeisac.e.opensetandhasmeasure67 9e,sothatnorandomsequencecanbelongtoeverySe. Thetheorem,infact,demonstratestheneedfortheexistentialpartofdenitionofrandomfunctions.Inthefollowingsectionswewillobtainsomeadditionalpropertiesofrandomcontinuousfunctions.4.3RandomContinousFunctionsandImages4.3.1PerfectImages,ineveryinstanceInthissectionweshowthatallrandomcontinuousfunctionsalwayshaveperfectimages.Thisisaconsequenceofthefollowingtheorem. Theorem4.3.1. IfFisarandomcontinuousfunction,thentheimageF[2N]hasnoisolatedelements.Proof.LetfbetherandomrepresentingfunctionforFandletQ=F[2N].SupposebywayofcontradictionthatQcontainsanisolatedpathy.Thenthereissomenite@ysuchthatyistheuniqueelementofIQ.Fixsuchthatf=.Foreachn,letSnbethesetofallg2Fsuchthatforall1;22f0;1gn, 1. g_1iscompatiblewithg_2, 2. @g_1,and 3. @g_2 99

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Thenforanyeachm3 4.Proof.Theproofisbyinductiononjj.Withoutlossofgenerality,weassumethat=0n.Foreachn>0,letqnbetheprobabilitythatF[2N]meetsIn.LetfbetherepresentingfunctionforF.Forn=1,thereare9equallyprobablechoicesforthepairfandf,breakingdowninto4distinctcases.Case1.Iff==f,thenF[2N]doesnotmeetI.Thisoccursjustonce.Case2.Iff=0orf=0,thenF[2N]meetsI.Thisoccursin5ofthe9choices.Case3.Iffi=;andf)]TJ/F24 11.955 Tf 12.561 0 Td[(i=,thenF[2N]meetsIifandonlyifFi[2N]meetsI.Thisoccursin2ofthe9choices,withprobabilityq1.Case4.Iff=;=f,thenF[2N]meetsIifatleastoneofFi[2N]meetsI.Thisoccursin1ofthechoices,withprobability1)]TJ/F15 11.955 Tf 12.005 0 Td[()]TJ/F24 11.955 Tf 12.005 0 Td[(q12.Thatis,F[2N]failstomeetIifbothF[2N]andF[2N]failtomeetI. 100

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Puttingthesecasestogether,weseethatq1=5 9+2 9q1+1 9q1)]TJ/F24 11.955 Tf 11.955 0 Td[(q21;sothatq1satisesthequadraticequationx2+5x)]TJ/F15 11.955 Tf 11.956 0 Td[(5=0:Thusq1istheuniquesolutionin[0,1]ofthisequation,thatis,q1=p 45)]TJ/F15 11.955 Tf 11.955 0 Td[(5 2;whichisindeed>:75.Nowletqn=qandletqn+1=p.Onceagainweconsiderthe9initialchoices,nowbreakingdowninto6distinctcases.Case1.Iff==f,thenF[2N]doesnotmeetIn+1.Thisoccursjustonce.Case2.Iff=0=f,thenF[2N]meetsIn+1ifandonlyifatleastoneofFandFmeetsIn.Thisoccursjustonce,andwithprobability1)]TJ/F15 11.955 Tf 11.955 0 Td[()]TJ/F24 11.955 Tf 11.955 0 Td[(q2=2q)]TJ/F24 11.955 Tf 11.955 0 Td[(q2.Case3.Iffi=andf)]TJ/F24 11.955 Tf 12.023 0 Td[(i=,thenF[2N]meetsIn+1ifandonlyifFi[2N]meetsIn.Thisoccursin2ofthe9choices,withprobabilityq.Case4.Iffi=;andf)]TJ/F24 11.955 Tf 12.213 0 Td[(i=1,thenF[2N]meetsIn+1ifandonlyifFi[2N]meetsIn+1.Thisoccursin2ofthe9choices,withprobabilityp.Case5.Iff=;=f,thenF[2N]meetsIn+1ifatleastoneofFi[2N]meetsIn+1.Thisoccursjustonce,withprobability1)]TJ/F15 11.955 Tf 11.956 0 Td[()]TJ/F24 11.955 Tf 11.955 0 Td[(p2.Case6.Iffi=;andf)]TJ/F24 11.955 Tf 12.128 0 Td[(i=,thenF[2N]meetsIn+1ifatleastoneofthefollowingtwothingshappens.EitherFi[2N]meetsIn+1,orF)]TJ/F25 7.97 Tf 6.586 0 Td[(i[2N]meetsIn.Thisoccursin2ofthe9choices,withprobability1)]TJ/F15 11.955 Tf 11.955 0 Td[()]TJ/F24 11.955 Tf 11.955 0 Td[(p)]TJ/F24 11.955 Tf 11.955 0 Td[(q. 101

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Puttingthesecasestogether,weseethatp=2 3p)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 9p2)]TJ/F15 11.955 Tf 13.15 8.088 Td[(2 9pq+2 3q)]TJ/F15 11.955 Tf 13.15 8.088 Td[(1 9q2;sothatp=qn+1satisestheequationp2+3p+2pq)]TJ/F15 11.955 Tf 11.955 0 Td[(6q+q2=0:Wenotethatforp=q,thesolutionsarep=q=0andp=q=3 4.Thisexplainsthevalue3 4inthestatementoftheorem.Nowassumebyinductionthatq>3 4.Supposebywayofcontradictionthatp63 4.Itfollowsthat9 16+9 4+3 2q)]TJ/F15 11.955 Tf 11.955 0 Td[(6q+q2>0:Simplifying,thisimpliesthat16q2)]TJ/F15 11.955 Tf 12.138 0 Td[(72q+45>0.Butthisfactorsintoq)]TJ/F15 11.955 Tf 12.137 0 Td[(3q)]TJ/F15 11.955 Tf 12.138 0 Td[(15andisonly>0wheneitherq63 4orq>15 4.Sincethelatterisimpossible,weobtainthedesiredcontradictionthatq63 4. Theorem4.3.3. Norandomcontinuousfunctionisinjective.Proof.LetpbetheprobabilitythatanarbitarycontinuousfunctionFisinjective.ItfollowsfromTheorem 4.3.2 thatthereisa9 16chancethatFhasazeroinIandalsoinI,sothatp67 16.Nowingeneral,ifFisinjective,thenitmustbeinjectivewhenrestrictedtoIandwhenrestrictedtoI.Itfollowsthatp6p2,whichhappensonlyforp=0andp=1,giventhat06p61.Sincep67 16,itfollowsthatp=0,asdesired.ThiscanbereformulatedasaMartin-Loftestasfollows.FirstweobservethatFisinjectiveifandonlyif,theimagesofeachpairofdisjointintervalsIandIaredisjoint.LetD;=fF:F[I]F[I]=;:ThenD;isuniformlyc.e.sinceF[I]F[I]=;ifandonlyifthereexistsnsuchthatforallextensions0ofand0ofoflengthn,f0andf0areincompatible.NowletSm=fF:8;2f0;1gm6=!F2D;g: 102

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ItfollowsfromtheobservationabovethatFisinjectiveifandonlyifF2TmSm.TheargumentaboveshowsthatS167 16andthatSm+16Sm2andhenceSm67 16m:ItfollowsthatfSm:m2!gisaMartin-LoftestandthereforenorandomcontinuousfunctionmaybelongtoeverySmandhencenorandomcontinuousfunctioncanbeinjective. 4.3.3Non-surjectiveImages,ininstancesInthissectionweshowthatrandomcontinuousfunctionsarenotnecessarilyonto. Denition4.3.4F,therestrictionofFtoI. ForanyfunctionFon2Nandany2f0;1g,denetherestrictionFofFtoIbyFx=F_x:Clearlyanysuchrestrictionofarandomcontinuousfunctionwillberandom,butmorecanbesaid.RecallvanLambalgen'stheorem,Theorem 3.2.12 Proposition4.3.5. FisarandomcontinuousfunctionifandonlyifthefunctionsFandFarerelativelyrandom.Proof.LetrrepresentF.SupposerstthatFisrandom.Itfollows,asinCorol-lary 3.3.12 ,thatFFisrandomandhenceFandFarerelativelyrandombyvanLambalgen'stheorem.NextsupposethatFandFarerelativelyrandomandletrirepresentFifori=0;1.Letdbeanymartingale,whichwethinkofasbettingonr.Thenfori=0;1,wecandeneamartingalediwithoracler1)]TJ/F25 7.97 Tf 6.587 0 Td[(iasfollows.Wewillgivethedenitionford0andleaved1forthereader.Given=r0;:::;r0p+q)]TJ/F15 11.955 Tf 12.254 0 Td[(2where06q<2p,user1tocompute=r;:::;rp+1+q)]TJ/F15 11.955 Tf 12.133 0 Td[(2andthendeneditobetinthesameproportionasd.Thatis,di_j=di=d_j=dforj<3.ThusforanynodeontheleftsideofthelabelledtreeforF,d0ismakingthesamebetonthenextlabelthatdwouldhavemade,andsimilarlyford1andtherightside. 103

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SincetheFiarerelativelyrandomfori=0;1,itfollowsthatdidoesnotsucceedandhencethereexistupperboundsBiforfdiridngn2N.Butitfollowsfromtheabovedenitionsofdithatforanyp,drd2p+1)]TJ/F15 11.955 Tf 11.955 0 Td[(2=d0r0d2p)]TJ/F15 11.955 Tf 11.955 0 Td[(1d1r1d2p)]TJ/F15 11.955 Tf 11.955 0 Td[(1:Thisisbecausethemartingaledalternatesusingd0andd1andtheresultcanbeviewedineachalternationasmultiplyingthecapitalbysomefactor.Theningeneral,for0
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Dene^d=d,andfor2Band~thecorrespondingstringofA,^d_x=8><>:d~_x d~^dx2A^dx2B)]TJ/F24 11.955 Tf 11.955 0 Td[(AThefunction^disclearlyconstructive,sincedis.Toshow^disamartingale,considerthesumXx2Bd_x=Xx2Ad~_x d~^d+Xx2B)]TJ/F25 7.97 Tf 6.587 0 Td[(A^d=^dXx2Ad~_x d~+^djB)]TJ/F24 11.955 Tf 11.955 0 Td[(Aj=^d[jAj+jB)]TJ/F24 11.955 Tf 11.955 0 Td[(Aj]:Itremainstoshowthat^dsucceedsonX.However,thatisclear,asonbitswhichareinXbutnoteX,^dkeepsitscapitalconstant,andonbitsfromeX,itactsexactlyasdwould.ThereforesincedsucceedsoneX,^dsucceedsonXandXisnonrandom. Itiseasytoseethat,foranyrandomcontinuousfunctionFandanycomputablerealx,Fxisnotcomputable.Thisfollowsfromournextresult. Theorem4.3.8. IfFisarandomcontinuousfunction,then,foranycomputablerealx,Fxisnotcomputable.Proof.SupposethatFisrandomandletxandybecomputablereals.Foreachn,letSn=fG:Gxdnydng:ThenS0;S1;:::isaneectivesequenceofc.e.opensetsinF,andaneasyinductionshowsthatSn=2=3n.ThisisaMartin-LoftestanditfollowsthatF=2Snforsomen,sothatFx6=y. Wenowstrengthenthisresulttoshowthattheimageofacomputableelementisrandom. Theorem4.3.9. IfFisarandomcontinuousfunction,then,foranycomputablerealx,Fxisarandomreal. 105

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Proof.SupposethatFisrandomwithrepresentingfunctionfr,letxbeacomputablerealandlety=Fx.Denethecomputablefunctiongsothat,foreachn,gn=xdn:BytheVon-Mises{Church{WaldComputableSelectionTheorem,thesubsequencezn=rgnisrandominf0;1;2gN.Nowy=Fxmaybecomputedfromzbyremovingthe2's.ThusFxisrandombyProposition 4.3.7 WenotethatFouche[ 45 ]hasusedadierentapproachtorandomnessforcontinuousfunctionsconnectedwithBrownianmotion,rstpresentedbyAsarinandProkovskiy[ 5 ],andhasshownthat,underthisapproach,itisalsotruethatforanyrandomcontinuousfunctionF,Fxisnotcomputableforanycomputableinputx.ItfollowsthatarandomfunctionFcanneverbecomputablycontinuousandhencethegraphofFisnota01class.4.4RandomClosedSetsarisingfromrandomcontinuousfunctions4.4.1APositiveResult:InverseImagesof0!InthissectionweprovethatforanyrandomcontinuousfunctionF,thesetZF=fx:Fx=0gisarandomclosedset.ForanysubsetSofC,letZS=fF2F:ZF2Sg. Lemma4.4.1. ForanyopensetS,ZS6S.Proof.ItsucestoprovetheresultforintervalsS=I.WewillshowbyinductiononjjthatZI=1 4jj,whereasofcourseI=1 3jj.RecallfromCorollary 4.4.3 that02F[2N]withprobabilityexactly3 4.Forjj=1,therearetwodistinctcases.CaseISupposerstthat=i,wherei2f0;1g.ThenF2ZSifandonlyifFhasazeroinIiandhasnozeroinI)]TJ/F24 11.955 Tf 12.137 0 Td[(i.NowFhasazeroinIiiffi2f0;2gandiftherestrictedfunctionhasazero,whichgivesprobability2 33 4=1 2.ThusthecombinedprobabilitythatF2ZSis1 4.CaseIISupposenextthat=.ThenF2ZSifandonlyifFhaszeroesinbothIandI.ItfollowsfromtheargumentinCaseIthatZS=1 4. 106

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NoticethatZf;g=fF:Fhasnozeroesghaspositivemeasure1 4butf;g=0.Nowsupposejj=nandlet=_i;supposebyinductionthatZI6I.Interpretasthecodeforanitebinarytreeandlet2f0;1gbetheterminalnodeofthattreesuchthatiindicatesthebranchingof.Againtherearetwocases.CaseISupposerstthati2f0;1g.ThenF2ZIifandonlyifF2ZIandfurthermoreFhasazeroinI_iandhasnozeroinI_1)]TJ/F24 11.955 Tf 12.189 0 Td[(i.ItfollowsasabovethatZI=1 4ZI=1 4n+1.CaseIISupposenextthati=2.ThenF2ZIifandonlyifFhaszeroesinbothI_0andI_1.ItfollowsasabovethatZI=1 4ZI=1 4n+1.Anarbitraryopensetisadisjointunionofintervalsandthusthedesiredinequalitycanbeextendedtoopensets. Theorem4.4.2. ForanyrandomcontinuousfunctionG:2N!2N,thesetofzeroesofGiseitheremptyorisarandomclosedset.Proof.SupposethatGisarandomcontinuousfunctionwhichhasatleastonezero,andletS0;S1;:::beaMartin-LoftestinC.ThenthereisacomputablefunctionsuchthatSi=[nIi;n.WemayassumewithoutlossofgeneralitySi62)]TJ/F25 7.97 Tf 6.587 0 Td[(i)]TJ/F23 7.97 Tf 6.587 0 Td[(2andthateachSiisnotclopenandthat,foreachi,theintervalsIi;narepairwisedisjoint.WewilldeneaMartin-LoftestS00;S01;:::inthespaceFandusethefactthatGmustsatisfyfS0igi2!toshowthatZGsatisesfSigi2!.FixanintervalIinCandletC=ZI.ObservethatthereisaclopensetB2Nandacorrespondingniteset0;:::;k)]TJ/F23 7.97 Tf 6.587 0 Td[(1ofstringssuchthatB=[j
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iiFhasnozeroesoutsideofB.Let2N)]TJ/F24 11.955 Tf 12.06 0 Td[(B=[2AI.Bycompactness,FhasnozeroesoutsideofBifandonlyif9`82A80[j0j=`9mf0dm=1]:{1NotethatthemeasureofCmaybecomputeduniformlyfromgiventhecalculationfromCorollary 4.4.3 thatwheneverf2f0;2gjj,thentheprobabilitythatFhasazeroinIisexactly3 4.Foreach,wewilluniformlycomputeac.e.opensetSFsuchthatCBandsuchthatB62C.TherearetwostagesintheconstructionofB.StageI:LetUbethesetofcodes0forpartialfunctionsf0suchthat 4{1 holdswithf0inplaceoff,andsuchthatfurthermoreforeveryjand`suchthatf0isdenedonalllength-`extensionsofj,thereissuchawithf02f0;2g8.ItisclearthatforanyF2C,thereexists02UwithF2I0andhenceC[fI0:02Ug:Asusual,wemaythenuniformlycomputefromUasetU0suchthattheintervalsI0for02U0arepairwisedisjointinFand[fI0:02Ug=[fI0:02U0g:Foreach02U0,letQ0Ibethe01classinFconsistingofthoseextensionsof0whichactuallyhavezeroesineachIj.TheninfactwehaveC=[fQ0:02U0g:Asnotedabove,wecanactuallycomputethemeasureQ0uniformlyfrom0byexpressingQ0asaneectivedecreasingintersectionofclopensets.Thusforeach0,wecancomputeaclopensetB0suchthatQ0B0I0and 108

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B062Q0.LetB=[fB0:02U0g:ThenwehaveCBandB6C.Finally,foreachi,letS0i=[nB0i;n:ThenbyProposition 4.3.7 ,S0i62Si62)]TJ/F25 7.97 Tf 6.587 0 Td[(i)]TJ/F23 7.97 Tf 6.586 0 Td[(1andthereforethereexistssomeisuchthatG=2S0i,sinceFisrandom.ButthismeansthatZG=2SiandhenceZFmeetstheMartin-Loftest.ThusZFisrandom,asdesired. 4.4.2ANegativeResult:Images,ingeneralIngeneral,theimageofarandomcontinuousfunctionneednotbearandomclosedset.Toseethis,recallthestatementofTheorem 4.3.2 .Thatis,given2f0;1g,theprobabilitythattheimageofacontinuousfunctionFmeetsIisalways>3 4.Weobtainthefollowingcorollary. Corollary4.4.3. Foranyy22N, a fF:y2F[2N]g=3 4; b thereexistsarandomcontinuousfunctionFwithy2F[2N].Proof.aLetpbetheprobabilitythaty2F[2N].Itfollowsthatforeach2f0;1gn,theprobabilitythaty2F[I],giventhatfisconsistentwithy,alsoequalsp.ItfollowsfromtheproofofTheorem 4.3.2 thatp=3 4.bSincetherandomcontinuousfunctionshavemeasure1inCN,itfollowsthatsomerandomcontinuousfunctionhasyintheimage. Thisallowsustodemonstrateourresult. Theorem4.4.4. Theimageofarandomcontinuousfunctionneednotbearandomclosedset.Proof.ItwasshowninTheorem 3.4.12 thatarandomclosedsethasnocomputablemembers.LetFbearandomcontinuousfunctionwith0!intheimage,asgivenbyCorollary 4.4.3 .ThenF[2N]isnotarandomclosedset. 109

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4.5Pseudo-DistanceFunctionsInsection 4.4.1 weshowedthatifisarandomcontinousfunction,then)]TJ/F23 7.97 Tf 6.586 0 Td[(1!isarandomclosedset,ifitisnonempty.Thismotivatesthestudyofpseudo-distancefunctions. Denition4.5.1. :2N!2Nisapseudo-distancefunctionforQ2Nifiscontinuousand)]TJ/F23 7.97 Tf 6.586 0 Td[(1!=Q. Comment4.5.2Background. ThenamecomesfromamodicationofthedistancefunctiondistQ:2N![0;1]foraclosedsetQ.Forx22N,distQxisdenedtobeminfdx;y:y2Qgwheredisametricon2Ngivenbydx;y=8>><>>:0ifx=y;2)]TJ/F25 7.97 Tf 6.587 0 Td[(nifnistheleastsuchthatxn6=yn:Thismaybeviewedasacomputablemappingfrom2N2Ninto2Nbyrepresenting0as0!and2)]TJ/F25 7.97 Tf 6.587 0 Td[(nas0n10!.Fromthisviewpoint,wemayviewdistQsimilarly:distQx=8>><>>:0!ifx2Q;0n10!otherwise,wherenistheleastsuchthatxdn=2TQ:Everyclosedsethasacharacterizationintermsofpseudo-distancefunctions,asfollows. Theorem4.5.3. Q2Nisclosedithereispseudo-distancefunctionforQ.Proof.FirstsupposethatQisclosedandQ=[T].Deneamapb:2><>>:b_0if_i2T;b_1otherwise:Letting:2!!2!bedenedsothatXistheuniqueY2n[bXn],weobtaintherequiredpseudo-distancefunction. 110

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Nowsupposethat:2!!2!isapseudo-distancefunctionforQ.Thismeansthatthereissomeb:2
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4.7FutureWorkWeclosethischapternotingthatrandomBrownianmotionsasstudiedbyFouche[ 45 ]areaspecialcaseofrandomcontinuousfunctionsontherealline.Thisisanotherareaofinterestforfurtherresearch.Thatis,wewouldliketoextendthenotionofarandomcontinuousfunctiontofunctionsontherealunitinterval[0;1]andtherealline
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CHAPTER5CONTINUITYOFCAPPINGINCBTThefollowingchapterisjointworkwithAngshengLiandWeilinLiandwillappearintheAnnalsofPureandAppliedLogicasanarticleentitledContinuityofCappinginCbT[ 16 ].Forthisproject,P.BrodheadacknowledgessupportfromtheNationalScienceFoundationundergrantnumber0714151astheprincipalinvestigatortoconductthisjointworkinBeijingduringthesummerof2007aspartoftheEastAsiaandPacicSummerInstitutesEAPSI.ThisworkwaspresentedbyP.BrodheadattheFirstJointAMS-NZMSMeetingDecember2007,Wellington,NewZealand.5.1IntroductionGivensetsA;B!,wesaythatAisTuringreducibletoB,ifthereisanoracleTuringmachinesay,suchthatA=BdenotedbyA6TB.Furthermore,ifthebitsoforaclequeriesareboundedbyacomputablefunction,thenusingrecentnomenclaturefromSoare[ 88 ]wesaythatAisboundedTuringreducibletoB,writtenA6bTB.Theliteratureoftenreferstothisastheweaktruthtablereducibility,written6wtt.ATuringandaboundedTuringorbT,forshortdegreeistheequivalenceclassofasetundertheTuringreductionsandtheboundedTuringreductionsrespectively.Adegreeiscalledcomputablyenumerablec.e.,ifitcontainsac.e.set.LetCandCbTbethestructuresofthec.e.degreesundertheTuringreductionsandtheboundedTuringreductionsrespectively.Duringthepastdecades,thestudiesofthestructuresC;CbTfocusedonthatofthealgebraicproperties,leadingtomajorachievementssuchasthedecidabilityresultsofthe1-theoryofC,andthe2-theoryofCbTAmbos-Spies,P.Fejer,S.LemppandM.Lerman[ 3 ],andtheundecidabilityresultsofthe3-theoryofCLempp,Nies,andSlaman[ 63 ],andofthe4-theoryofCbTLemppandNies[ 62 ].Thisprogressbringsthedecidabilityproblemsofthe2-theoryofC,andthe3-theoryofCbTintosharperfocus,forwhichnewingredientsarewelcome. 113

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Intherecentyears,thestudyofthecomputablyenumerabledegreeshasfocusedonTuringdenabilityinthestructureC.Forinstance,Slamanaskedin1985ifthereareanyc.e.degreesthatareincompleteandnonzerowhicharedenableinthec.e.degreesC.ThisquestionofSlamanisstillopen.AnaturalapproachtothisproblemistondsomedenablesubstructuresofCthathavenontrivialminimal/maximaland/orleast/greatestmembers.Asaresult,topicssuchatthecontinuityofthec.e.degrees,startedbyLachlanin1967,haverenewedinterest.Inthischapter,wedemonstratethecontinuityofcappinginCbT.Thisrefutestheexistenceofamaximalnon-boundingdegree.Italsobringsthequestionofthe3-theoryofCbTintosharperfocus,asthestatementisoneof3-complexity.Tomotivatetheseideasfurther,webeginwithabriefhistoryofrelevantcontinuityresultsinSection 5.2 .Thismotivatesourmainresultandmethodofproof,describedinSection 5.2.2 .ThemainsubstanceoftheproofinvolvesdemonstratingthatTheorem 5.2.3 holds,thatlocalnoncappabilityholdsinCbT.Sections 5.3 { 5.6 aredevotedtoprovingthistheorem.5.2ContinuityResultsTheTuringdegreesformanupper-semilattice;thatis,eachpairofelementsa;bhasaleastupperboundorjoina_b.Agreatestlowerbounda^bmayormaynotexist.GivenaboundedTuringdegreea,wesaythataiscappableifabounded-Turingdegreeb6=0existssuchthata^b=0.Wesaythataiscuppableifthereisadegreeb6=00suchthata_b=00.ThestudyofcontinuitypropertiestheboundedTuringdegreesiswithrespecttomeetsandjoins,andrelatednotionssuchascappingandcupping.5.2.1ContinuityResultsinCIn1979,Lachlan[ 60 ]provedtheexistenceofanon-boundingc.e.degree{namely,anon-computablec.e.degreewithnominimalpairbelowit;Cooperdemonstratedin1974,thatnohighc.e.degreecanbenon-bounding[ 30 ].ReturningtotheSlamanquestion,Downey,Lempp,andShoredemonstratedin1993thatCooper'snon-boundingdegree 114

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couldbemadehigh2[ 41 ],leadingtothepossibilityofamaximalnon-boundingc.e.degree.Suchadegreecouldbeusedtoshowtheexistenceofadiscontinuity,whichcouldbeusedtoprovethedenibilityofac.e.singleton.Toseetheformer,notethatifb>0isnonbounding,thenforanyc.e.a>b,thereissomeminimalpairr^sbelowa.Theneitherb^r=0orb^s=0,butneithera^rnora^sequals0.However,Seetapunrefutedthispossibilityalbeitearlierin1991,demonstratingthenon-existenceofamaximalnon-boundingdegree[ 84 ].Welchprovedacomplimentaryresultin1981:thereisnomaximalboundingdegree,inthesensethatforalla6=00,thereisareb;csuchthatb^c=0andb;c66a[ 95 ].Continuingwithcappingresults,HarringtonandSoare[ 51 ]provedin1989,thenonexistenceofmaximalminimalpairs{thatis,foranynon-trivialminimalpaira;bofc.e.Turingdegreesa;b,thereexistsac.e.Turingdegreec>asuchthatc;bisstillaminimalpair.Seetapun[ 84 ]showedanevenstrongerresult,thecontinuityofcapping:foranyc.e.Turingdegreeb6=0;00,thereexistsac.e.Turingdegreea>bsuchthatforanyc.e.Turingdegreex,ifx6a,thena^x=0ifandonlyifb^x=0.Ambos-Spies,Lachlan,andSoare[ 4 ]provedthedualcaseoftheHarringtonandSoare'sresult:foranynon-trivialsplittingx;yof00,thereexistsac.e.degreea0,witha0beingtheuniquecomplementofa1intheinterval[0;a0_a1],suchthatifb
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anyxbsuchthatforanyc.e.bT-degreex,b^x=0$a^x=0.Forthis,itsucestoproveTheorem 5.2.3 below,ananalogoftheSeetapunlocalnoncappabilitytheoremforthec.e.Turingdegrees[ 84 ]. Denition5.2.2LocalNoncappability. Adegreeb6=0islocallynon-cappableifthereissomea>bsuchthatforallxbsuchthatifx6aisnoncomputable,thenx^b6=0.ProofofTheorem 5.2.1 .AssumingTheorem 5.2.3 ,wecanseeTheorem 5.2.1 .Givenb,letabethedegreeinTheorem 5.2.3 .Foraxedx2CbT,byTheorem 5.2.3 ,wecon-sideronlythecasewherex66a.Clearlyifx^a=0,thenx^b=0.Assumea^x6=0.Wecanchooseac.e.bT-degreeysuchthaty6=0andy6a;x.Therefore0bbethedegreeinTheorem 5.2.1 .WeclaimthattherearenobT-minimalpairsbelowa.Supposetothecontrarythatx;yisaminimalpairbelowa.Sincea^x=x6=0anda^y=y6=0,wecanchoosenonzerox1tobebelowbothxandb,andnonzeroy1belowbothyandb.Thenx1;y1isaminimalpairbelowb,acontradiction.Analternativeapproachistousethefactthatamaximalnon-boundingc.e.degreeisequivalenttoanon-boundingdegreewhichisnotlocallynon-cappable[ 49 84 ].Consequently,byTheorem 5.2.3 ,nomaximalnon-boundingdegreecanexist.Ourapproachtotheproofoftheorem 5.2.3 issimilartoSeetapun'sapproachforthec.e.Turingdegrees,butitisnon-obviousduetothecomputableboundsoforaclequery 116

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bitsinboththeconditionsandconclusionsofrequirements.Thatis,bT-reductionsarestrongerthanTuringreductions.SowhenwerequirethereductionsbeingbuilttobebT-reductions,wemustsatisfystrongerconditionsand,inthissense,theproblembecomeshardertosolve.Forexample,A.Li,W.Li,Y.Pan,andL.Tang[ 66 ]haveshownthatthesolutiontothemajorsub-degreeprobleminCbTi.e.thedualtothecontinuityproblemiscompletelydierentfromtheresultforCseeCooperandLi[ 31 ].TheyshowthatthestatementofthesolutioninCfailsbadlyinCbT:thereexistc.e.bT-degreesa;bsuchthat0a.Ourapproachmightnotbetheonlyone.KlausAmbos-Spiesprovedthatforanyc.e.set,itsTuringdegreeiscappableintheTuringdegreesiitsbT-degreeiscappableinthebT-degrees[ 2 ];wethankananonymousrefereeforpointingthisout.Therefore,anotherpossibleapproachmightbetoprove,ifpossible,thatforanytwoc.e.sets,theirTuringdegreesformaminimalpairintheTuringdegreesitheirbT-degreesformaminimalpairintheboundedTuringdegrees.Asconsequence,ourcontinuityresultwouldimmediatelyfollowfromSeetapun'scontinuityresult.Wecommentthatalthoughourcontinuityproofmightbenon-obvious,fromtheaboveperspective,oftentimesbT-degreescanbehandledmuchmoreeasilythanTuringdegrees[ 1 ].Ambos-Spiesprovidesvariousexamples[ 1 ].Forexample,densityofthec.e.bT-degreescanbeprovedbyaniteinjurypriorityargument,whereasthesameresultrequiresaninniteinjuryargumentforthec.e.Turingdegrees.TherestofthischapterisdevotedtoprovingTheorem 5.2.3 ,themainresult.Insection 5.3 ,weformulatetheconditionsofthetheorembyrequirements;insection 5.4 ,wearrangeallstrategiestosatisfytherequirementsonthenodesofatree,ormoreprecisely,theprioritytreeT.Insection 5.5 ,weusetheprioritytreetodescribeastage-by-stageconstructionoftheobjectsweneed.Finally,insection 5.6 weverifythattheconstructioninsection 5.5 satisesalloftherequirements,nishingtheproofofthetheorem.OurnotationandterminologyarestandardandgenerallyfollowSoare[ 86 ].Duringthecourseofaconstruction,notationssuchasA;areusedtodenotethecurrentapproximationstotheseobjects,andifwewanttospecifythevaluesimmediatelyat 117

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theendofstages,thenwedenotethembyAs,[s]etc.Forapartialcomputablep.c.,orforsimplicity,alsoaTuringfunctional,say,theusefunctionisdenotedbythecorrespondinglowercaseletter.Thevalueoftheusefunctionofaconvergingcomputationisthegreatestnumberwhichisactuallyusedinthecomputation.ForaTuringfunctional,ifacomputationisnotdened,thenwedeneitsusefunction=)]TJ/F15 11.955 Tf 9.298 0 Td[(1.Duringthecourseofaconstruction,wheneverwedeneaparameter,psay,asfresh,wemeanthatpisdenedtobetheleastnaturalnumberwhichisgreaterthananynumbermentionedsofar.Inparticular,ifpisdenedafreshatstages,thenp>s.5.3RequirementsandStrategiesInthissectionweprovidetherequirementsandstrategiesforprovingTheorem 5.2.3 .Werestateithereforconvenience.Theorem 5.2.3 LocalNoncappabilityinCbT.Foranyc.e.bT-degreeb,ifb6=0;00,thenthereisac.e.bT-degreea>bsuchthatifx6aisnoncomputable,thenx^b6=0.5.3.1TherequirementsGivenac.e.setB,wewillbuildac.e.setAtosatisfythefollowingrequirements: Pe: A6=eB_K6bTB Re: Xe=eA;B)167(!9c.e.De[De6bTXe;B&8iSe;i] Se;i: De6=i_Xe6T;_B6T;wheree;i2!,fe;e;Xe:e2!gisaneectiveenumerationofalltriples;;XofallboundedTuringbT,forshortreductions;,andofallc.e.setsX;fiji2!gisaneectiveenumerationofallpartialcomputationfunctions;andKisaxedcreativeset.Deforalle,arec.e.setsbuiltbyus.Leta;b;x;dbethebT-degreesofAB,B,X,D,respectively.BytheP-requirements,a>bunlessbwasalreadythedegreeof00,andbytheR-requirements,ifx6athereisadbelowbothxandbsuchthatd6=0unlesseitherx=0orb=0.Thereforetherequirementsaresucienttoprovethetheorem. 118

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Beforedescribingthestrategies,weintroducesomeconventionsoftheboundedTuringreductions.WewillassumethatforanygivenboundedTuringreductionor,theusefunctionsandwillbeincreasinginarguments.5.3.2AP-strategyAP-strategywilltrytosatisfyaP-requirement,Psaywedroptheindexinthefollowingdiscussion.Weuseanodeonatree,say,todenoteaP-strategy.ItaimstoensurethatifA=B,thenthereisaboundedTuringreductionsuchthatB=K.ThereforetheP-strategywilltrytobuildaboundedTuringreduction.willbebuiltbyan!-sequenceofcyclesk.EachcyclekofwillberesponsiblefordeningB;kasfollows:rstchoosesafreshwitnessakandwaitsforastage,vsay,atwhichwehaveB;ak#=0=Aak.Whenthisoccurs,wedeneB;ktobeKkwithusefunctionk=ak.Sinceispartialcomputable,soistheusefunctionof.WewillalwaysassumethatwheneverBchangesbelowthe-use,ksay,thecorrespondingcomputationB;kbecomesundenedsimultaneously.Supposethatatalaterstages>v,kisenumeratedintoK,andBhasnotchangedsinceB;kwaslastcreated,thenB;k6=Kk.Inthiscase,weenumerateakintoAsothataninequalityB;ak6=Aakiscreated.Thekeypointisthat,ifB;k6=Kkisapermanentinequality,soisB;ak=06=1=Aak.TheP-strategywillstartcycleskinincreasingorderofk.Cyclekactsonlyifthefollowingconditionsoccur: 1. Forallk0
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5.3.3AnR-strategyBeforedescribingtheR-strategy,weintroduceaconventionoftheboundedTur-ingreduction.Weassumethatforanyxandanys,ifxentersXatstages,thenA;B;x[s]#=1.GivenanR-requirement,Rsay,wedenethelengthfunctionofagreementasusual.Thatistosay:Atstages,thelengthfunctionofagreement`betweenA;BandXisdenedasthelargestxsuchthatA;BandXagreeonallvaluesy`[v].AtR-expansionarystages,anR-strategybuildsboundedTuringreductionsX;Bwithusefunctions,,respectivelysothatfortheleastundenedx<`:X;x#=Dxwithx=xandB;x#=Dxwithx=x{1whereistheuseofA;B,andDisac.e.set,whoseelementsareenumeratedintoitbylowerpriorityS-strategiesassociatedwithR,tosatisfytheR-requirement.SupposethatisanR-strategy.Asabove,theusefunctionsandoftheboundedTuringreductionsandbuiltbyaretheidentityfunctionandrespectively,sothatbothandareboundedTuringreductions.NoticethatA;BisaboundedTuringreduction,sothattheuseisapartialcomputablefunction.TosatisfyX=DandB=D,theR-strategywillimposethefollowingconstraintsonallS-strategieswiththesameglobalindexastheR-strategy:Foranys,andanyd,disallowedtobeenumeratedintoDatstagesonlyifbothX;dandB;dareundenedduringstages.WeassumethatfortheboundedTuringreductionsand,anycomputationwillautomaticallybecomeundened,whenevertheoraclechangesbelowthecorrespondinguse. 120

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Bythebuildingofand,andbytheconstraintsof,wehavethatifandarebuiltinnitelymanytimes,thenbothXandBaretotal,andbothequalD.HenceRissatised.ThereforethekeypointtowardsthesatisfactionofRisthatifthereareinnitelymanyR-expansionarystages,thenbothandarebuiltinnitelymanytimes.WethusdenethepossibleoutcomesoftheR-strategyby0
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setftobetotallyundenedthefproveswrong,soitiscancelled, droptheA-restraintbydeningrA=)]TJ/F15 11.955 Tf 9.299 0 Td[(1,and createalink;.[Noticethatatstagev,B;disundenedduetotheB-changeinthedomainoff.WeregardthisasaB-permissionfortheenumerationofdintoD.ThisB-permissionwillbekeptuntilthecurrentlink;iseithertravelledorcancelledsothat,ineithercase,thelinkisremoved.]Supposethatcreatesalink;atstagev.Thenthelink;willbetravelledatthenext-expansionarystages>v.Nowweconsidertwocases:Case1.ThereisanerrorbetweengandX.Inthiscase,thereisanx6dwhichhasenteredXsincestagev.ThereforeX;discurrentlyundened.TogetherwiththeconditionthatB;d",foundatthestagewecreatedthecurrentlink;,isqualiedtoenumeratedintoD.Sissatisedbyd=06=1=Dd.Case2.Otherwise,weknowthatgiscorrectduringthegap.ThereforewepreservegonitsdomainuntilopensanotherA-gap.Forthis,weimplement: foreveryy6d,iffyisundened,thendenefy=By,and denetheA-restraintrAoftobed.[NoticethatalthoughwehaveA-restraintatthisstage,XmaychangeduetoaB-changebelowd.ThedenitionoffatthisstageallowsustoimmediatelyopenanA-gaponcesuchaB-changeoccurs,inwhichcase,wedonotincreasethedomainofgbutresumewiththeoldcandidated.]ThereforetheS-strategyisagap/cogapstrategy.Itwillbuildpartialcomputablefunctionsfandgandwillproceedasfollows: 1. Deneapossiblecandidatecasfresh. 2. BuildingfWaitforastagevatwhich c#=0=Dc, X;c#=0=Dc,and 122

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B;c#=0=Dc:Thenforc=c, foreveryy6c,iffy",thendenefy=By, dened=c, setctobeundened,andgobacktostep1. 3. Creatingalink;Letsbethecurrentstage.SupposethatthereisabinthedomainoffthatentersBatstages.Noticethatthedomainoffispreciselyeverything6d=d,sothatB;dbecomesundenedatstages.Then: foreveryx6d=d,ifgxisundened,deneittobeXx, denetheA-restraintofbyrA=)]TJ/F15 11.955 Tf 9.299 0 Td[(1, setftobetotallyundened,and createalink;. 4. Travellingthelink;Wetravelthelink;atthenext-expansionarystaget>s.Therearetwocases:Case4a.SuccessfulclosureThereisanxsuchthatgx#=06=1=Xx.ThisxmustenterXsincethecurrentlink;wascreated.Then: enumeratedintoD,andstop.Case4bUnsuccessfulClosureOtherwise,then foreveryy6d,iffyisundened,thendenefy=By, deneanA-restraintofbyrA=d.ThePossibleOutcomesWeconsiderthefollowingcases.Case1.Case4aoccursatsomestaget.Inthiscase,limsd[s]#=d
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Noticethatgisneversettobetotallyundened,andthatforaxednumberd,disaxednumber,sothatBchangesbelowdonlynitelymanytimes,andsothatStep3occurswiththesamedonlynitelymanytimes.SinceCase4boccursinnitelymanytimes,wehavethatd[s]willbeunboundedoverthecourseoftheconstruction,andthatwheneverStep3occurs,webuildgontheinitialsegmentofthecurrentd.Thereforegisbuiltasacomputablefunction.Foranarbitrarilygivenx,weprovegx#=Xx.Letsbethestageatwhichgxiscreated.Supposethatsiareallstagess0>satwhichStep3ofoccurs,andthatforeachsi,ti2si;si+1isthestageatwhichthelink;createdatstagesiistravelledthroughCase4b.Bythechoiceofsi,s0=s.SinceCase4boccursatstaget0,andt0is-expansionary,wehavethat i gx=X[s0]xwillneverbevisitedatstages0. ii Foranys2[s0;t0],gx=X[s]x. iii gx=A;B;x[t0]=X[t0]x.BytheA-restraintrA[t0],andtheconventionof,wehavethatforanyt2[t0;s1, iv gx=A;B;x[t]=X[t]x.Supposebyinductionthatforn,wehavethat A Foranys2[sn;tn],gx=X[s]x. B gx=Xx[tn]=A;B;x[tn], C Foranyt2[tn;sn+1,gx=A;B;x[t]=X[t]x,and D gx=X[sn+1]x.ByC,Dforn,andbythechoiceoftn+1,Aholdsforn+1.ByAforn+1,andthechoiceoftn+1,wehaveBforn+1.ByBforn+1,bytheA-restraintatstagetn+1,andbytheconventionof,Choldsforn+1.Dforn+1followsfromCforn+1andtheassumptionthatisnotvisitedatstagesn+1.[Remark.WewillarrangetheconstructionsothatanA-gapcanbeopenedonlyatoddstages,andthatnoR-strategiescanbevisitedatthesestages.Thismeansthatno 124

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XforanyR-strategycanreceiveelementsatoddstages,sinceweassumethatXisenumeratedonlyatstagesatwhichisvisited.]Thereforeforanys>s0,eithergx=Xx[s]orgx=x[s].SinceA;B;xequalsXx,wehavethatgx=Xx.ThisisaglobalwinfortherequirementR,sothatwedon'tconsideranyotherS-requirementswiththesameglobalindexwiththerequirementR.Case3.Otherwise,andStep2occursinnitelymanytimes.Sincestep3occursonlynitelymanytimes,fissettobetotallyundenedonlynitelymanytimes.Letfbethenalversionoff.Bytheassumptionthatstep2occursinnitelymanytimes,fisbuiltasacomputablefunction.Bythechoiceoff,foranyx,oncefxiscreated,wehavethatfx=Bx.Biscomputable,contradictingtheassumptionofthetheorem.Soweassumethatthiscasewillneveroccur.Case4.Otherwise,thenbythestrategy,wehavethatlimsc[s]#=c
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Se;iwithi>e.LetP
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Proposition5.4.5. LetfbeaninnitepaththroughT.ThenforanyrequirementX,thereisanode0fsuchthateitherioriibelowholds, i Xissatisedatforanywith0f. ii Xisactiveatforanywith0f.Proof.LetfXi:i2!gbethepriorityrankingoftherequirementssothatXi
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Denition5.4.6. IfisanSe;i-strategyforsomee;i,thendenetoptobethelongestRe-strategysuchthat^h0i.5.5TheConstructionOurconstructionwillperformdierentactionsatevenandoddstages.Atevenstages,strategiesonthetreewillacttosatisfytherequirements.SupposethatBisenumeratedatoddstagesonly,andthatateveryoddstage,thereisexactlyoneelementthatentersB.GivenaB-permissionintheconstruction,wewanttoopenA-gapsforasmanyS-strategiesaspossible.ThisallowsustospecifyaP-strategysothatweenumerateitswitnessintoA.SowewillensurethatA-restraintsdropatoddstages,andalsothatAisonlyenumeratedatoddstages.Duringthecourseoftheconstruction,wemayinitializeanode,say,whichmeansthatalltheactionstakenbypreviously,arecancelled,orsettobetotallyundened.Precisely,ifanR-strategyisinitialized,thenbothandaresettobetotallyundened,Dissettobetheemptyset;,andalllinksassociatedwitharecancelled.IfanS-strategyisinitialized,thenbothgandfaresettobetotallyundened,parametersdandcarebothsettobeundened,andanylinkassociatedwithiscancelled.IfaP-strategyisinitialized,thenissettobetotallyundened,andallwitnessesofarecancelled.NoticethatanS-strategyopensitsA-gap,exactlyatstagesatwhichanerrorbetweenfandBoccurs,whichgivesaB-permissionforitscurrentcandidated.OurproblemistomakesurethatthereareinnitelymanystagesatwhichalltheS-strategiesonthetruepathorthecurrentapproximationofthetruepathdroptheirA-restraintssimultaneously.Givenanode,supposethat12nareallS-strategieswith^hgi.ToguaranteethatifisaP-strategy,thenthereareinnitelymanystagesatwhichforalli=1;2;;n,theA-restraintsrAiofidropto)]TJ/F15 11.955 Tf 9.299 0 Td[(1innitelyoften,proceedasfollows.Letfibefiforalli=1;2;;n.Wewillarrangethebuildingofvariousfisuchthat:foranys, 128

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1. foranyi,iffi[s]isempty,thenthecurrentA-restraintrAiis)]TJ/F15 11.955 Tf 9.299 0 Td[(1,and 2. foranyi
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letkbetheleastxsuchthatB;x#6=Kxandax62A, enumerateakintoA,and initializeallnodeswith>,andgotostages+1. 5. Otherwise,thengotostages+1.Stages=2n+2.Werstspecifytherootnodetobeeligibletoactatsubstaget=0.Ateachsubstaget,weallowthestrategywhichiseligibletoactatthissubstagetotakeaction,andtheneitherclosethecurrentstageorspecifyanewnodetobeeligibletoactatthenextsubstageofstages.Substaget.Letbethenodewhichiseligibletoactatsubstagetofstages.Ifhaslengths,theninitializeallnodes066,andclosethecurrentstage.Otherwise,therearethreecasescorrespondingtodierenttypesofstrategy.Case1.=isaP-strategy.Thenrunthefollowing:Program:willbuildaboundedTuringreduction,anddenewitnessesak.Forsimplicity,wedropthesubscriptioninthedescriptionoftheprogram. 1. Ifthereisannsuchthatanisdened,andlB;A6>an,thenlet^h1ibeeligibletoactnexti.e.atsubstaget+1ofstages. 2. Otherwise,letkbetheleastxsuchthatB;xisundened.Then: ifak#,thendeneB;k=Kkwithk=ak, otherwise,thendeneaktobefresh,and initializeallnodes>,andgotostages+1.Case2.=isanRe-strategyforsomee.RunthefollowingProgram: 1. Ifsisnot-expansionary,let^h1ibeeligibletoactnext. 2. Otherwise,andthereisalink;whichwascreatedandhasneverbeencancelledortravelled.Let0bethe<-leastsuch,andlet0beeligibletoactatthenextsubstage. 3. Otherwise,then, letxbetheleastysuchthateitherX;yorB;yisundened, 130

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ifX;x",deneX;x=Dxwithx=x, ifB;x",thendeneB;x[s]:=Dx[s]withusex:=x,whereistheuseofA;B,and let^h0ibeeligibletoactnext.Case3.=isanSe;i-strategyforsomee;i.Let=top.Weperformthefollowing,Program: 1. Ifhasalreadybeensatised,asdenedin2abelow,then^hdiiseligibletoact. 2. Travelalink;Ifalink;wascreatedandithasneverbeencancelledortravelledsinceitwascreated,thentravelthelink;bycases.Case2a.Successfulclosure9xgx#6=Xx.Noticethatgwascorrectatthestagewecreatedthecurrentlink;,sothiserrormustoccurduringtheA-gapoftheS-strategy.Then: enumerated,thelargestconrmedcandidateof,intoD, wesaythatissatisedatstages, initializeallnodes>,andgotostages+1.Case2b.UnsuccessfulclosureOtherwise.Then: deneu=maxfy:gy#g=d, setfd+1=Bd+1, setrA=u+1,and initializeallnodestotherightof^hgi,andgotostages+1.Ineithercase,thelink;isremoved.[Remark.Wehavethatifstep2ofprogramoccursatstages,thenisvisitedatstages.] 3. BuildingfIfc#=c,c#=Dc=0,X;c#,andB;c#then:Case3a.900^hgianddomf0[0;c].Then: initializeallnodes>^hwi,andgotostages+1.Case3b.Otherwise,then: 131

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foranyx6c,iffx",thendenefx#=Bx, setd=c;dissaidtobeconrmed, cancelc,sothatc",and let^hgibeeligibletoactnext. 4. Ifc",thendenecasfresh,initializenodes>^hwi,andgotostages+1. 5. Otherwise,let^hwibeeligibletoactatthenextsubstage.Thiscompletesthedescriptionoftheconstruction.5.6TheVericationInthissection,weverifythesatisfactionoftherequirements.Firstweinvestigatesomeglobalpropertiesthatholdattheendofanarbitrarystage.Thesepropertiesensurethattheconstructionisimplementedproperly. Proposition5.6.1. Letsbeastage. i Thereisatmostonelinkthatistravelledduringstages. ii Ifalink;istravelledatstages,thenbeforewetravelthelink,isvisitedandstep2ofprogramoccursatstages. iii Thereareno1;2;1;and2suchthat1212andbothlinks1;1and2;2existattheendofstages.Proof.Itiseasytoseethatbothiandiiholdbyobservingtheconstruction.Foriii,supposetothecontrarythatsistheleaststagesuchthatthereare1;2;1;and2with1212,andsuchthatbothlinks1;1and2;2existattheendofstages.Bytheminimalityofs,exactlyoneofthetwolinks1;1,and2;2iscreatedduringstages.Weconsidertwocases.Case1.Thelink1;1iscreatedatstages.Bytheconstruction,s=2n+1forsomen,andf1issettobetotallyundenedatstages.Weanalyzethelocationof2.If1^hwi2,thenbytheconstructionatstages,2isinitializedduringstages,sothelink2;2isremovedduringstages,contradictingthechoiceof2.If1^hgi2,thenbythedenitionoftheprioritytreeT,top26=2,sothatthereisnolink2;2whichcanbecreatedintheconstruction.If1^hdi2, 132

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thenthecurrentd2mustbedenedafter1createditsinequalityatargumentd1,afterwhichnolink1;1canbecreated,since1hassatiseditsrequirementthrough1d1=06=1=Dd1.Socase1doesnothappen.Case2.Thelink2;2iscreatedatstages.Lets11^hwiwereinitializedatstages1.Therefore2cannotbevisitedatanystage>s1unlessthelink1;1[s1]hasbeenremoved.Thiscontradictsthechoiceofs.iiiholds.ThePropositionfollows. Proposition5.6.2. i LetbeanS-strategy.Thenforanys;t,iffistotallyundenedatsubstagetofstages,thenrA=)]TJ/F15 11.955 Tf 9.298 0 Td[(1holdsattheendofsubstagetofstages. ii LetbeanS-strategy,andsbeastage.Lets)]TJ/F15 11.955 Tf 10.986 -4.338 Td[(bethegreateststaget
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Letb=domf[s],andlets0betherststageatwhichfwasdenedon[0;b].Byprogram,case3bofprogramoccurredatstages0.Bytheconstruction,therewasnolinkthatwastravelledbythesubstageatwhichwasvisitedduringstages0.Therefore,0wasvisitedatstages0,andcase3bofprogram0occurredatstages0.Bytheassumptionincase3bofprogram,wehavethatbothf0andfarenon-emptyattheendofstages0,anddomf0[s0]domf[s0].Bythechoiceofs,0hasnotbeeninitializedduringstages[s0;s],byiiiff0[s]isnotempty,thendomf0[s]domf0[s0],iiifollowsincase1.Case2.Case3bofprogramoccursatstages.Asthesameasthatincase1,byobservingtheconstruction,wehavethatforany,isvisitedatstages,sothat0isvisitedatstages,andfurthermore,case3bofprogram0occursatstages.Bytheassumptionofcase3bofprogram,thedomainoff0islargerthanthatoffattheendofstages.iiifollowsincase2.Thepropositionfollows. Denition5.6.3. SupposethatK66bTBandB66T;. i Letsbethelastnodewhichiseligibletoactatstages. ii DenethetruepathTP2[T]oftheconstructionbyTP=liminfss.Hereafterwheneverweconsideranodeonthetruepath,wewillusethenotation2TPratherthanTP. Proposition5.6.4. ExistenceofthetruepathSuppose2TP.Thenthereissomeasuchthat^haiisvisitedinnitelyoftenandinitializedonlynitelymanytimes.Hence^hai2TP.Proof.Weprovebyinductiononthelengthof.Supposebyinductionthatthepropositionholdsforall0and2TP.Lets0beminimalafterwhichwillneverbeinitialized.Bytheinductivehypothesis,willbevisitedinnitelyoften.Weprovethepropositionforbycases.Case1.=isaPe-strategyforsomee.Byprogram,therearetwosubcasestoconsider. 134

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Subcase1a.Thereareinnitelymany-expansionarystages.Bytheconstruction,step2ofprogramoccursinnitelymanytimes,sothatisbuiltinnitelymanytimes,andthatBisbuiltasatotalfunction.SinceK66bTB,thereissomemsuchthatB;m#6=Kmholdspermanently.Letnbetheleastsuchm,andletB;nbecreatedatstagev>s0.Bythechoiceofn,B;an[v]=0anditwillholdpermanently.Letu>vbethestageatwhichnentersK.Supposethat12lareallS-strategieswith^hgi.Bythechoiceofs0,fjwillneverbesettobetotallyundenedafterstages0byinitializationforanyj.Therefore,foreveryj2f1;2;;lg,fjissettobetotallyundenedafterstages0onlyifanerroroccursbetweenfjandB.Byinductivehypothesis,case3bofprogramjoccursinnitelymanytimes,sothatfjwillbebuiltinnitelymanytimesforallj2f1;2;;lg.Inparticular,flwillbebuiltinnitelymanytimes.BytheassumptionofB66T;,wecanchooseastages1>uatwhichthereisanumberbsuchthatflb#=06=1=Bboccurs.ByProposition 5.6.2 iii,foranyj2f1;2;;lg,iffjisnotemptyatthebeginningofstages1,thenthereisanerrorbetweenfjandBthatoccursexactlyatstages1.Wehavethatforeveryj2f1;2;;lg,iffj6=;atthebeginningofstages1,thenjopensitsA-gapduringstages1.ByProposition 5.6.2 i,foranyj2f1;2;;lg,iffjisemptyatthebeginningofstages1,rAj=)]TJ/F15 11.955 Tf 9.298 0 Td[(1holdsatboththebeginningandtheendofstages1.Byprogram,wehavethatrequiresattentionatstages1,andweletreceiveattentionbyenumeratingitswitnessanintoA.Bythechoiceofn,foranys>s1,wehavethatB;an=06=1=Aanholdsduringstages,contrarytotherebeinginnitelymanyS-expansionarystages.Thiscaseisimpossible.Subcase1b.Otherwise. 135

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Inthiscaseisbuiltonlynitelymanytimes.Lets1>s0beminimalafterwhichwillneverbebuilt.Bythechoiceofs1,^h1iwillneverbeinitializedafterstages1,andbyprogram,foranys>s1,ifisvisitedatstages,sois^h1i.Thepropositionfollowsincase1.Case2.=isanR-strategy.Observingprogram,weconsidertwosubcases.Subcase2a.Step3ofprogramoccursinnitelymanytimes.Then^h0i2TP.Bychoiceofs0,^h0iwillneverbeinitializedafterstages0.Furthermore^h0iisvisitedinnitelyoften.Therefore^h0i2TPandthepropositionholdsinthiscase.Subcase2b.Otherwise.SupposethatStep3ofprogramoccursatmostnitelymanytimessothat^h0iisvisitedatmostanitenumberoftimes.Wewillshowthat^h1i2TP.Toshowthat^h1iisinitializednitelyoften,rstnotethatbythechoiceofs0,onlynodes^h0icaninitialize^h1i.Lets1>s0beminimalafterwhichstep3ofprogramwillneveroccur.Thenforanys>s1,ifanode0^h0iisvisitedatstages,thenthereisanR-strategy0,andalink0;0whichistravelledatstages.Supposethat12n)]TJ/F23 7.97 Tf 6.587 0 Td[(1areallR-strategies0with0^h0i.Letn=.Weprovebyinductionthatforeachi6n,thereisastageviafterwhichtherewillbenolinksj;jthatcanbeeithercreatedortravelledforallj>iandforallj^h0i.Fori=n.Denebn=maxfnx[t]jt6s1;nx[t]#gForeverys>s1,dene 136

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pn[s]=maxfyjfy[s]#;top=ngBytheconstruction,wehavethatforeverys>s1, ifalinki;0istravelledforsomeis1,pn[s]6bn.Bytheconstructionatoddstages,alinkn;ncanbecreatedatastages>s1onlyifthereisanelementb6bnthatentersBatstages.Sincebnisaxednumber,Bchangesbelowbnonlynitelymanytimes.Thereforethereareonlynitelymanystagesatwhichwecreatelinksn;n.Sinceoncealinkistravelled,itisremovedimmediately,thereareonlynitelymanystagesatwhichalinkn;iseithercreatedortravelled.Letvn>s1beminimalafterwhichtherewillbenolinkn;nthatcanbeeithercreatedortravelled.Supposebyinductionthatvi+1isaminimalstageafterwhichtherewillbenolinkj;jwhichiseithercreatedortravelledforallj2fi+1;i+2;;ng,andallj^h0i.Denebi=maxfix[t]jix#;t6vi+1gForanys>vi+1,denepi[s]=maxfyjfy[s]#;^h0i;top=igBytheconstruction,itiseasytoseefromaninductiveargumentthatforalls>vi+1, ifalinkj;jistravelledforsomej
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ifalinki;iforsomei^h0iistravelledatstages,thenpi[s]6bi.Thisshowsthatforalls>vi+1,pi[s]6bi.Bytheconstruction,ifalinki;iforsomei^h0iiscreatedatastages>vi+1,thenthereisanumberb6biwhichentersBatstages.Sincebiisaxednumber,thecreationoflinksi;ifori^h0ioccursonlynitelymanytimes,sothatthereisastagevi>vi+1say,afterwhichtherewillbenolinkoftheformi;iforanyi^h0iwhichcanbeeithercreatedortravelled.Thereforethereisastagev1sayafterwhichnolinki;canbecreatedortravelledforanyi6nandany^h0i.Sothereareonlynitelymanystagesatwhichsomenode^h0iisvisited.Thus^h1iisinitializedonlynitelymanytimes.Bytheproofabove,thereareonlynitelymanystagesatwhicheitherStep2orStep3ofprogramoccurs.^h1iisvisitedatalmosteverystageatwhichisvisited.ThereforeinSubcase2b,wehavethat^h1iisinitializedonlynitelymanytimesandvisitedinnitelyoften.Case3.=isanS-strategy.Let=top.Subcase3a.alink;istravelledonceandsuccessfullyclosed.Then^hdiisvisitedinnitelyoftenandonlyinitializednitelymanytimes.So^hdi2TP.Subcase3b.;istravelledinnitelyoftenandunsuccessfullyclosed'sA-gap.AsinSubcase3a,^hgi2TP.Subcase3c.Otherwise.Bytheassumptionofthiscase,fisbuiltonlynitelymanytimes,sinceifthisisnottrue,thenfisbuiltasacomputablefunction,andf=B,contradictingthehypothesisB66T;.Byprogram,limsc[s]#=cs1,ifisvisitedatstages,sois^hwi. 138

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Therefore^hwiisinitializedonlynitelymanytimes,andvisitedinnitelyoften,^hwi2TP.ThepropositionfollowsinCase3. SincethetruepathexistsonlyifbothK66bTB,andB66T;holdasprovedinProposition 5.6.4 ,wealwaysassumethetwoconditionsfromnowon. Proposition5.6.5PossibleoutcomesalongTP. Given2TP: i If=isaPe-strategyforsomee,then^h1i2TPandA6=eB. ii If=isanR-strategy,then a if^h0i2TP,thenD=X=B; b if^h1i2TP,thenispartialorA;B6=X. iii If=isanS-strategy,thenfor=top,wehave: a if^hdi2TP,thenlimsd[s]#=ds0bethestageatwhichthecomputationB;nwascreated.NoticethateB;an[v]=0andBwillneverchangebeloweanafterstagev.Bytheproofincase1ofProposition 5.6.4 ,thereisastages1atwhichwecanenumerateanintoA.Bythechoiceofnands1,eB;an=06=1=Aanwillbepreservedforever.Acontradiction.ThereforewehavethatA6=eB.Forii.Let^h0i2TP. 139

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ByProposition 5.6.3 ,bothandarebuiltinnitelymanytimes.ByStep3ofprogram,itsucestoprovethatthefollowingconstraintsofaresatisedduringthecourseoftheconstruction:Foranyd,andanys>s0,ifdisenumeratedintoDatstages,thenbothX;dandB;dareundenedduringstages.ByProposition 5.6.1 i,thereisatmostonelinkwhichistravelledduringstages.LetdbeenumeratedintoDatstages.ThenthereisanS-strategysuchthattop=andcase2aofprogramoccursatstages,andd=d[s].ByProposition 5.6.1 ii,isvisitedatstages,andthereisalink;whichwascreatedatastages)]TJ/F15 11.955 Tf 7.085 -4.338 Td[(>s0s0afterwhichstep3ofprogramwillneveroccur.Howeverthereareinnitelymanystagesatwhichwetravelalink;forsome.Bythechoiceofs1,isaniteset,lets2bethestage>s1afterwhichBwillneverchangebelowmaxfx[s1]jB;x[s1]#g. 140

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BytheS-strategies,foranywithtop=,iffiscreatedafterstages2,thentherewillbenolink;whichcanbecreatedbyusingthedierencebetweenBandf.Thereforewecanchooseastages3>s2afterwhichthereisnolink;whichcanbecreatedforany.Bytheconstruction,oncewetravelalink;,itisremoved.Thereisastages4>s3afterwhichthereisnolinkfromtoanyS-strategywhichcanbeeithercreatedortravelled.Thiscontradictstheassumptionthatstep2ofprogramoccursinnitelymanytimes.WehavethatXe6=eA;B.Foriiia.If^hdi2TPthenthereissomestageswhereCase2aofprogramenumeratesd=limsd[s]intoD.Thenatallstagest>s,programenactsCase1.Furthermoresincedisonlydenedwhend=0;wehavethatd=06=1=Dd.Foriiib.Bythechoiceofs0,andbytheassumptionofthiscase,^hgiisneverinitializedafterstages0.Byprogram,case3bofprogramoccursinnitelymanytimes.Bythechoiceofs0,fbecomestotallyundenedatanystages>s0onlyifalink;iscreatedatstages,andthislinkwillcertainlybetravelledunsuccessfully,insteadofbeinginitialized.Foraxednumberd,disanednumber,sothatBchangesbelowdonlynitelymanytimes.Thereford[s]willbeunboundedintheconstruction.Bytheconstructionatoddstages,ifalink;iscreatedatstages,thend[s]isdened,andgisbuiltontheinitialsegmentd[s].Thereforgisbuiltasacomputablefunction.Noticethatgwillneverbesettotallyundenedafterstages0.Weprovethatforanyx,ifgxiscreatedatastagev>s0,thenforanys>v,gx=X[s]x.Givenanx,letv>s0bethestageatwhichgxiscreatedanddenedas0ifitis1,thengx=Xxtakesalreadythepermanentvalue.Supposethatv0vatwhichalink;iscreated,andlettibethestageatwhichthelink;[vi]istravelled.Thenv0=v. 141

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Bythechoiceoft0,thelink;[v0]isunsuccessfullytravelledatstaget0,thismeansthatforanyt2[v0;t0],gx=X[t]x.Sincet0is-expansionary,wehavegx=X[t0]x=A;B;x[t0].BytheA-restraintrA[t0],thedenitionoff[t0],andbytheconventionof,wehavethatforanys2[t0;v1,gx=A;B;x[s]=X[s]xispreserved.Supposebyinductivehypothesiswehave: 1. foranys2[vn;tn],gx=X[s]x. 2. gx=X[tn]x=A;B;x[tn]. 3. foranys2[tn;vn+1,gx=A;B;x[s]=X[s]x. 4. gy=X[vn+1].Sincethelink;[vn+1]isunsuccessfullytravelledatstagetn+1,holdsforn+1,andsincetn+1is-expansionary,2forn+1holds,andfurthermore,bytheA-restraintatstagetn+1,forn+1holds,holdssincewillneverbevisitedatoddstages,sotherearenoelementswhichenterXatoddstages.Thisprovesthatgx=Xx.Sincexisarbitrarilygiven,wehavethatforalmosteveryx,gx=Xx,Xiscomputable.iiibfollows.Foriiic.Bytheassumptioninthiscase,gisbuiltonlynitelymanytimes.Iffisbuiltinnitelymanytimes,thenthenalversionoff,denotedbyf,isbuiltasacomputablefunction,andf=B.Acontradiction.Thereforefisbuiltonlynitelymanytimes.Lets1>s0besuchthatfwillneverbebuiltatanystages>s1.Furthermore,wecanchooseastages2>s1afterwhichnoneofthesteps1,2,3,or4ofprogramwilloccur.Thereforelimsc[s]=cmustbechosenbeforestages2.Clearlyc62D.Sinceisvisitedinnitelymanytimes,theonlyreasonthatcase3bwillneveroccurafterstages2isthatc6=0.iiicfollows. Proposition5.6.6P-satisfactionProposition. Foreache,Peissatised. 142

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Proof.Givene,letbethePe-strategy2TP.ByProposition 5.6.5 i,necessarilyA6=eBsothatPeissatised.Thepropositionfollows. Proposition5.6.7. Foreache2!,ifXe=eA;BandXeandBarenotcomputable,thentheydonotformaminimalpair.Proof.ByProposition 5.4.5 ,letbethelongestRe-strategyonthetruepathTP.ByProposition 5.6.5 ii,^h0i2TPandD=X=B.ByProposition 5.6.5 iii,foranyS-strategy,iftop=and2TPtheneither^hwi2TPor^hdi2TP.Ineithercase6=DsothatXandBdonotformaminimalpair.Thepropositionfollows. 143

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PAGE 150

BIOGRAPHICALSKETCHPaulStevenBrodheadwasborninOakPark,ILin1980.HemovedtoRichlandCenter,WIin1989andcontinuedhisschoolingthereuntilhegraduatedfromRichlandCenterHighSchoolin1997.AsaChancellor'sscholar,PaulattendedtheUniversityofWisconsin-Madisonfrom1997until2000,whenheearnedaB.S.inmathematics.Duringthesummerof2000,PaulparticipatedintheNSF-fundedundergraduatemathematicsresearchexperienceSIMU,attheUniversityofPuertoRico-Humacao.In2003PaulstartedgraduateschoolattheUniversityofFloridaandearnedamaster'sdegreeinmathematicsin2005.Duringthesummerof2006,PaulwasagraduateassistantforanNSFSEAGEP-fundedundergraduatemathematicsresearchexperienceattheUniversityoftheVirginIslands.HewenttotheChineseAcademyofSciences,InstituteofSoftwareduringthesummerof2007asanNSFfellow,participatingintheEastAsiaandPacicSummerInstitutes.AsanNSFSEAGEPfellowduringthefallof2007,PaulwenttoVictoriaUniversityofWellingtonWellington,NewZealand.Asaninvitedvisitorandlecturer,PaulwenttotheUniversityofHawaiiatManoainthespringof2008.PaulearnedhisPh.D.inmathematicsfromtheUniversityofFloridain2008. 150


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