COMMON DERIVATIONS IN LOCALLY DETERMINED
NONMONOTONIC RULE SYSTEMS AND THEIR COMPLEXITY
By
AMY K. C. S. VANDERBILT
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
2000
I dedicate this dissertation to the memory of Sara Davidson Sartain, who put
her foot down. Most importantly, I dedicate it to Scott and Sabrina, for whom I
would accomplish anything and with whom I could accomplish anything.
ACKNOWLEDGMENTS
I would like to thank the University of Florida and the Department of Math
ematics for their encouragement and contributions to my education, experience, and
funding. Special and heartfelt thanks go to my committee, who from the first day
were supportive and helpful above and beyond the call of duty. Thanks to my parents,
Ronald and Linda Sartain, without whom I might never have taken my education this
far and without whom I might never have gone to conferences! An enormous thanks
to my advisor, Dr. Douglas A. Cenzer, who not only put up with me for these years,
but encouraged my tenacious approach and worked to help me achieve my goals, both
for this dissertation and for the years beyond. Lastly, the greatest thanks go to my
husband, Scott, for constant encouragement and support (and chocolate pie!), and
to my daughter, Sabrina, who (although not around for the entire undertaking) was
there with daily hugs and smiles and reminders of what life is really about.
TABLE OF CONTENTS
pUgQ
ACKNOWLEDGEMENTS ..................................................................... iii
A B STRA CT .................................. ......... ...... ............. ......... ............... vi
1 INTRODUCTION ............................................................................ 1
1.1 History ........................... ........................................................ 1
1.2 M otivation ..................................... ......................................... 3
1.3 Preliminary Theorems and Definitions: Nonmonotonic Rule
System s ............................................................................ 5
1.4 Common Derivations in Locally Determined Systems .......................... 16
2 THE FINITE CASE ......................................................................... 18
2.1 Finite Classical Nonmonotonic Rule Systems ..................................... 18
2.2 Finite Constructive Nonmonotonic Rule Systems ................................. 32
2.3 Characterizing the Set of Extensions ................................................ 36
3 THE INFINITE CASE ....................................................................... 38
3.1 Locally Finite Classical Nonmonotonic Rule Systems .......................... 38
3.2 Locally Finite Constructive Nonmonotonic Rule Systems ....................... 51
3.3 Characterizing The Set of Extensions ............................................... 59
4 LOCALLY DETERMINED NONMONOTONIC RULE SYSTEMS .............. 60
4.1 Locally Determined Nonmonotonic Rule Systems ................................ 60
4.2 Common Derivations in Locally Determined Nonmonotonic
R ule System s ..................................................................... 67
4.3 Characterizing the Set of Extensions ................................................ 69
5 COMPUTABILITY AND COMPLEXITY ISSUES.................................... 76
5.1 Finding Extensions with Low Complexity ......................................... 76
5.2 Computability and Complexity of Common Derivations ......................... 82
6 ALTERNATE FORMALISMS OF NONMONOTONIC LOGIC .............. 86
6.1 Default Logic: Preliminary Definitions and Theorems ........................ 86
6.2 Equivalence of Default Logic to Nonmonotonic Rule Systems ............. 90
6.3 Previous Results Through the Eyes of Default Logic ........................ 91
6.4 Logic Programming: Preliminary Definitions and Theorems ................ 114
6.5 Equivalence of Logic Programming to Nonmonotonic
R ule System s ................................................................... 120
6.6 Previous Results Through the Eyes of Logic Programming .................. 121
7 FUTURE DIRECTIONS ............................................................... 131
REFER EN CES .............................................................................. 133
BIOGRAPHICAL SKETCH .............................................................. 137
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
COMMON DERIVATIONS IN LOCALLY DETERMINED
NONMONOTONIC RULE SYSTEMS AND THEIR COMPLEXITY
By
Amy K. C. S. Vanderbilt
May 2000
Chairman: Douglas Cenzer
Major Department: Mathematics
Nonmonotonic Rule Systems are a general framework for commonsense rea
soning. U =< U, N > is a nonmonotonic rule system where the universe U is a
countable set, such as the set of sentences of some propositional logic; N is a set of
rules of inference under which a conclusion from U may be inferred from the pres
ence of premises and the absence of restraints. An extension of U is a set of beliefs
concurrent with the rules of inference. If q is a formula of the language, appearing
in every extension of < U, N >, then has a common derivation d1i, that gener
ates 0 in every extension. Further, every extension of < U, N > is an extension of
< U, N U {d10} >. These two sets of extensions may be equal, but this is not al
ways true. A constructive view enhances as well as simplifies the results. We explore
alternate forms of the common derivation d that ensure certain properties for the re
sulting nonmonotonic rule system < U, N U {d} > We then introduce the notion of
levels and locally determined systems. The previous questions are explored in terms
of these systems. Computability and complexity issues of the common derivations
are considered. Several conditions are detailed, based on the notion of levels, which
ensure that a nonmonotonic rule system will have a computable extension. These
results are refined to give conditions which ensure the existence of extensions of low
complexity (such as exponential time). If U is a recursive nonmonotonic rule system
with (effective) levels, there is a 1:1 degree preserving correspondence between the set
of extensions of U and the set of paths through a recursive finitely branching (highly
recursive) tree. The families of extensions of a nonmonotonic rule system with levels
are characterized. We find that for any closed (11H) family S of subsets of U having
levels, there exists a recursivee) nonmonotonic rule system U with recursivee) levels
such that S is the set of extensions of U.
CHAPTER 1
INTRODUCTION
In this chapter, we provide a motivation for the directions of the research as
well as necessary background definitions and preliminary theorems. Along with this,
we take a short exploration of the history of the subject and of the directions research
in the field is presently taking.
1.1 History
In mathematics, a conclusion drawn from a set of axioms can also be drawn
from any larger set of axioms containing the original set. The deduction remains
no matter how the axioms are increased. This is monotonic reasoning in that new
information does not affect the conclusions drawn from a set of premises. From the
time of Euclid, it has been the nature of this monotonic reasoning that makes math
ematical proofs permanent. Complete proofs of theorems are never later withdrawn
due to new information.
There is, however, another sort of reasoning that is not monotonic. In this
type of reasoning, we deduce a statement based on the absence of any evidence to
the contrary. Such a statement is therefore more of a belief than a truth, as we may
later come upon new information that contradicts the statement. Statistics is often
used as a tool for for deducing provisional beliefs, but not every problem comes with
its own set of distributions ready for use.
Such deductions may arise in situations where we are made to choose con
cretely between two actions in the absence of complete information. It may be that
we cannot wait for the complete information to show itself, or that we have no guar
antee that it will ever show itself. Thus beliefs are often accepted as truth based on
a lack of contradicting facts.
One of the standard illustrations of this commonsense reasoning is the "Tweety"
example. If we see only birds that can fly, then we deduce that birds can fly. If Tweety
is a bird, we conclude that Tweety can fly. If we later find that Tweety is a penguin,
then we are forced to retract our conclusion that Tweety can fly. We are back to
knowing only that Tweety is a bird, specifically a penguin.
Every vision of a nonmonotonic logic describing belief will be similar, depend
ing only on our definition of a lack of evidence against a conclusion. McCarthy [1980]
was one of several who initiated formalizations of nonmonotonic logic with his concept
of Circumscription. At the same time, Reiter [Rei80] created a formalization that
he termed Default Logic. Along with these are the Theory of Multiple Believers of
Hintikka (1962), the Truth Maintenance Systems of Doyle [Do79], the Autoepistemic
Logic of Moore (1985), the Theory of Individual and Common Knowledge and Belief
of Halpern and Moses (1984), and Logic Programming with Negation as Failure given
by Apt (1988). This is only a partial list. The first journeys into the nonmonotonic
aspects of logic can be traced to Reiter's Default Logic [Rei80]. This involved creating
a natural extension of classical logic that would easily handle negative information.
These nonmonotonic logics share many properties. Several translations be
tween them have been made by Konolige (1988), Gelfond and Przymusifiska (1986,
1989), Reinfrank and Dressler (1989), and Marek and Truszczyfiski (1989). It should
be noted that these translations are primarily for propositional logic.
Apt [A90], and Gelfond and Lifschitz [GL88] studied negation as failure in
logic programming. It has since been seen that each of these investigations were
in a common direction. Relationships were discovered by several, including Marek
and Trusczcyfiski (1989), who explored the precise nature of the connections between
Default Logic and Logic Programming.
Since then, many aspects of the computability and complexity of the exten
sions of these systems have been explored. Subsequently, algorithms for computing
the extensions of any one theory have been created. More importantly, a universal
system of nonmonotonic logic called Nonmonotonic Rule Systems was created by
Marek, Nerode, and Remmel [MNR90 and MNR92a]. This system has been shown
by Marek, Nerode, and Remmel and others to be equivalent to many other systems of
nonmonotonic logic including Reiter's Default Logic as well as Logic Programming,
and Modal Nonmonotonic Logic.
1.2 Motivation
We suppose that we have a sentence 4 appearing in some or all extensions
of a nonmonotonic rule system. We want to then construct a single rule that would
derive 0 in every extension of the system. The motivation behind this "common
derivation" is that it would tell us what is required, in a nutshell, to derive ( in
this system. Nonmonotonic Logic has a place in fields such as medicine and political
science where this kind of information would be useful.
In the world of medical diagnoses, we imagine that our system involves a
universe U of symptoms to be matched with a diagnosis. We would care to know
a concise list of what symptoms should be present and, just as importantly, what
symptoms should be absent in order to accurately diagnose a condition.
My ultimate goal is to manipulate Nonmonotonic Rule Systems to create a
mathematical system which would lend itself to applications in political science (and
possibly social science as a whole as well as advertising) and explore the subsequent
theory of that system.
First Order Logic is inherently monotonic. Human reasoning, however, in
volves not only the addition of beliefs, but the retraction of beliefs, based on new
information, i.e. it is nonmonotonic. For example, the study of Astronomy led to the
retraction of the belief of a geocentric universe. Thus, to formalize human reasoning,
a nonmonotonic logic is needed. I propose to consider Political Science as an area of
future application. This area is ripe for the use of Nonmonotonic Logic since Political
Science studies how groups of people react to a campaign and how they reason who
they will vote for. If human reasoning can be formalized, then the reasoning of the
electorate can be modeled.
Other types of mathematics used to model the electorate in the past include
statistics, game theory [Gol94] and measure theory [B90]. Some of these models
managed to get as accurate as 84 percent [Gol94], but had to be redone every time a
subset of the electorate changed its mind (which was often!). Secondly, the accuracy
of the model was only known after the election was already over. These two properties
made the models unreliable and expensive. The problem: these types of mathematics
are monotonic, so the conclusions they reach cannot be retracted upon receipt of new
information.
We imagine that we create one nonmonotonic rule system per category of
voters and we consider the extensions of each theory or system. We would like all
the extensions for a category to contain some conclusion such as, "I will vote for
candidate A." For this reason, we take great interest not only in the intersections of
extensions, but also in the process of deduction used to create those extensions. It is
for this reason that we explore the concept of a common derivation for a conclusion
that appears in some or all extensions. Having one rule that derives the conclusion
"I will vote for candidate A" in each extension allows us to see exactly what must be
concluded previously and what must not have been concluded in order to allow the
derivation to apply.
1.3 Preliminary Theorems and Definitions: Nonmonotonic Rule Systems
The following background is taken from Marek, Nerode, and Remmel's series
of papers [MNR90] and [MNR92a]. Here we introduce the notion of a nonmonotonic
formal system < U, N >.
Definition 1.3.1 A NONMONOTONIC RULE OF INFERENCE is a triple < P, R, >,
where P = {oi,...,Oan}, and R = {/3I,...,/1m} are finite lists of objects from U and
0 E U. Each such rule is written in the form r = 1,..., : /i,..., 3m/q0 Here
{a,..., an} are called the PREMISES of the rule r, {13,...,/3m} are called the RE
STRAINTS of the rule r, and cln(r) = c(r) = q is the CONCLUSION of r.
Either P or R or both may be empty. If R is empty, then the rule is monotonic.
In the case that both P and R are empty, we call the rule r an axiom. For a set A
of rules, let c(A) = {c(r)Ir E A}.
A NONMONOTONIC FORMAL SYSTEM is a pair < U, N >, where U is a
nonempty set and N is a set of nonmonotonic rules. Each monotonic formal system
can be identified with the nonmonotonic formal system in which every monotonic
rule is given an empty set of restraints. The standard example for U will be the
set Sent(V) of propositional sentences on a finite or countable set of propositional
variables (or atoms). Here we will frequently assume that the standard propositional
rules of deduction are implicitly included as monotonic rules in any nonmonotonic
rule system with universe U. In the constructive case, a proper subset of these rules
are assumed.
Now, if < U, N > is a nonmonotonic rule system, and S is a set of formulas
of the language, call a rule r E N GENERATING FOR THE CONTEXT S if the
premises of r are in 5, and no element of the restraints is in S. Let GD(N, S) be the
set of all rules in N that generate for the context S.
A subset S C U is called DEDUCTIVELY CLOSED if for every rule r e N,
we have that if all premises {oa,..., an} are in S and all restraints {/31,..., / m} are not
in S then the conclusion q belongs to S.
In nonmonotonic rule systems, deductively closed sets are not generally closed
under arbitrary intersections as in the monotone case. But deductively closed sets are
closed under intersections of descending chains. Since U is deductively closed, by the
KuratowskiZorn Lemma, any I C U is contained in at least one minimal deductively
closed set. The intersection of all the deductively closed sets containing I is called
the set of SECURED CONSEQUENCES of I. This set is also the intersection of all
minimal deductively closed sets containing I. Deductively closed sets are thought of
as representing possible "points of view." the intersection of all deductively closed
sets containing I represents the common information present in all such "points of
view" containing I.
Example 1.3.1 Let U = {a, b, c}.
(a) Consider U with N1 = {: /a; a : b/b}. There is only one minimal
deductively closed set S = {a, b}. Then S is the set of secured consequences of
.
(b) Consider U with N2 = {: /a; a: b/c; a: c/b}. Then there are two minimal
deductively closed sets, S1 = {a,b} and S2 = {a,c}. The singleton set {a} is the set
of secured consequences of < U, N2 >.
Part (b) of this example shows that the set of all secured consequences is not,
in general, deductively closed in the nonmonotone case. Note that if we implicitly
assume the rules of propositional logic we define Cn(S) to be the closure of the set
S under these implicit monotonic rules
Given a set S and an I C U, and SDERIVATION of 0 from I in the system
< U,N > is a finite sequence < 01,...,k > such that Ok = 0 and, for all i < k,
each Oi is in I, or is an axiom, or is the conclusion of a rule r E N such that all the
premises of r are included in {Oi,..., 1} and all restraints of r are in U S. An S
CONSEQUENCE of I is an element of U occurring in some Sdeduction from I. Let
Cs(I) be the set of all Sconsequences of I in < U, N >. I is a subset of Cs(I). Note
that S enters solely as a restraint on the use of the rules imposed by the restraints
in the rules. A single restraint in a rule in N may be in S and therefore prevent the
rule from ever being applied in an Sdeduction from I, even though all the premises
of that rule occur earlier in the deduction. Thus, S contributes no members directly
to Cs(I), although members of S may turn up in Cs(I) by an application of a rule
which happens to have its conclusion in S. For a fixed S, the operator Cs(*) is
monotonic. That is, if I C J, then Cs(I) C Cs(J). Also, Cs(Cs(I)) = Cs(I).
Generally, Cs(I) is not deductively closed in < U,N >. It is possible that all
the premises of a rule be in Cs(I), the restraints of that rule are outside of Cs(I),
but a restraint of that rule be in S, preventing the conclusion from being put into
Cs(I).
Example 1.3.2 Let U = {a,b,c},N = {: /a;a: b/c}, and S = {b}. Then, Cs(0) =
{a} is not deductively closed.
However, we do get the following result:
Theorem 1.3.2 (MNR90) If S C Cs(I) then Cs(I) is deductively closed.
We say that S C U is an EXTENSION of I if Cs(I) = S.
S is an extension of I if two things happen. First, every element of S is
deducible from I, that is, S C Cs(I) (this is the analogue of the adequacy property
in logical calculi). Second, the converse holds: all the Sconsequences of I belong to
S (this is the analogue of completeness).
8
In purely monotonic rule systems, extensions usually refer to larger theories
(under inclusion). This is definitely not the meaning of extension for nonmonotonic
rule systems. In fact, it will be seen that for any two extensions A and B of a
nonmonotonic rule system, it is never the case that A C B. An important family in
propositional logic is the set of complete consistent extensions, CCE(T), of a theory
T. Any such family can be obtained as the set of extensions of a nonmonotonic
rule system. We obtain this system by taking the monotonic rules as restraintfree
nonmonotonic rules and adding rules of the form: 0/0 and: 0q/q, for each sentence
(to secure completeness).
The notion of an extension is related to that of a minimal deductively closed
set.
Theorem 1.3.3 (MNR90) If S is an extension of I, then: (1) S is a minimal
deductively closed superset of I. (2) For every I' such that I C I' C S, Cs(I') = S.
Example 1.3.3 Let a, b, and c be atoms in the language L and let
U = {a,b,c}
and
N= {: {a}/b,: {c}/b,: {a}/c,: {c}/a}.
Then the nonmonotonic rule system < U,N > has two extensions:
S= {b,c}
and
S2 {b,a}.
Moreover, we find that
GD(N, S1) = {: {a}/b,: {a}/c}
and
GD(N, S2) = {: {c}/b,: {c}/a}.
Now, we may wellorder the rules of a nonmonotonic rule system < U, N > by
some wellordering <. We may then define AD< to be the set of all rules in N which
are applied when the wellordering is used to close 0 under the set of rules. This is
done in the following way: we define an ordinal 7.<. For every e < 77< we define a set
of rules AD, and a rule r,. If the sets AD,, e < a, have been defined but q,_ has not
been defined, then a rule r is applicable at stage a if the following two conditions
hold:
(a) c(U,
(b) c(U,
(1) If there is no applicable rule r E N \ UE<, AD,, then 7)< = a and AD =
UE
(2) Otherwise, define ra to be the <least applicable rule r e N \ U,< AD,
and set AD, = (U,<, AD,) U {r,}.
(3) Put AD = U,<, AD,.
Intuitively, we begin with the empty set of formulas and apply the rules of N
in their order. Each time we apply only the orderingleast rule which can be applied.
At some point (since the ordinal is welldefined [MT93a]) there will be no available
rules that may be applied. At this point, AD, is the set of all rules applied and 7<
is the number of steps needed to reach this stopping point.
Then, let T.< be c(AD.<). This is the theory GENERATED BY <. Then,
GD(N, T<) C AD< so that
c(GD(N,T_)) C T_.
Now, if is a wellordering of the rules and for every 3 in R(AD<), /3 c(AD<)
then T_< is an extension of < U, N >. That is, if
AD< = GD(N,T),
then
T = c(AD_)
is an extension of < U, N >. More precisely, if T< = c(GD(N, T<)), then T_< is an
extension of < U, N >. We now have that
(a) [MT93a] If S is an extension of < U, N > then there is some wellordering
of the rules in N such that
S =c(AD.) = c(GD(N, S)).
And,
(b) If S = T. = c(AD.<) = c(GD(N, S)) for some wellordering _, then S is
an extension of < U, N >.
Thus, S is an extension of < U, N > if and only if S = c(GD(N, S)).
It is important to note that a wellordering may not give rise to an extension.
The new rule r, may have a conclusion that contradicts the restraint of a previously
applied rule. Then, T. is not an extension. Consider the following example:
Example 1.3.4 Let < U, N > be the nonmonotonic rule system where U = {a, b}
and N = {: {a}/b; b: /a}. Then rule ri is applicable at stage one. At stage two, rule
r2 is applicable; however, the conclusion of r2 violates the restraint of rl. Thus, this
system has no extensions.
Thus, we must concede that if c(U,
rule r E UE
Let (< U, N >) be the set of all extensions of the nonmonotonic rule system
< U, N >. Call two nonmonotonic rule systems < U1, N1 > and < U2, N2 > EQUIV
ALENT, written < U1, N1 >=< /U2, N2 >, if they have exactly the same extensions,
i.e., if (< U/1, N1 >) = (< U2, N2 >) [MT93a].
Example 1.3.5 Let < U,N > be a nonmonotonic rule system where U = {a, b},
and N1 = {: /a; a : /b}. Then consider the nonmonotonic rule system < U, N2 >
where N2 = {: /b; b : /a}. These two theories are equivalent as they each have the
same single extension
S = {a,b}.
Theorem 1.3.4 (MT93) A nonmonotonic rule system < U,N > has an inconsis
tent extension if and only if Sent(L) is an extension and < U, N > has no other
extensions.
Theorem 1.3.5 (MNR90) The set of extensions of a nonmonotonic rule system
forms an antichain. That is, if S1, S2 are extensions of a nonmonotonic rule system
and S C S2, then Si = S2.
Though this is a widely known fact about all forms of nonmonotonic logic,
this is an important concept for the characterization of the family of extensions of a
nonmonotonic rule system The natural question here is under what conditions is an
antichain the set of extensions of a nonmonotonic rule system? Consider the following
example:
Example 1.3.6 Considering the monotonic theory T = Cn(a  b) and the set
V = {a, b} of atoms. The set of complete consistent extensions of this monotonic
theory can be seen as the set of extensions of the nonmonotonic rule system < V, N >
where N = {: ia/a;: a/,a;: b/b;: b/ib; a : /b}. This system has three extensions:
S1 = Cn({a,b}),
2= Cn( {ia, b}),
and
S3 = Cn({ia, b}).
It will not always be so easy to construct a nonmonotonic rule system whose
extensions are exactly what we want them to be. We address this in detail in
subsequent chapters.
Next we need to define the notions of recursive and highly recursive non
monotonic rule systems < U,N >. Without loss of generality, we may assume that
U C w and we will identify a rule r = a1,...,cn : /31,..., /am/ in N with its code
c(r) =< k,l,q > where Dk = {l,,...,an} and D, = {/3i,...,/m}. In this way, we can
think of N as a subset of w. We say that a nonmonotonic rule system < U, N > is
RECURSIVE if both U and N are recursive subsets of w. To define the notion of
a highly recursive nonmonotonic rule system < U, N >, we must first introduce the
concept of a PROOF SCHEME for 0 in < U, N >.
A proof scheme for 0 is a finite sequence p = < 0,ro, can(Go) >,..., <
OM, rm, can(Gm) > such that !m = 0 and
(1) If m = 0 then: (a) 00 is an axiom (that is, there exists a rule r E N, r =:
/o), ro = r, and Go = 0 or (b) is a conclusion of a rule r =: /31,..., A, r0 = r, and
Go =O{r..,A}.
(2) m > 0, << o,ro,can(Go) >,...,< m i, rm1, can(Gmi) > is a proof
scheme of length m and 0,m is a conclusion of r = o I,...,, : r/,..., 4/m where
io,...,i8
The formula 0fim is called the CONCLUSION of the proof scheme p and denoted
by cln(p), the set Gm is called the SUPPORT of p and is denoted by supp(p).
The idea behind this concept is this: any Sderivation, p, in the system U =<
U, N >, uses some negative information about S to ensure that the restraints of rules
that were used are outside of S. But this negative information is finite, that is, it
involves a finte subset of the complement of S, such that as long as G n S1 = 0, p
is an Sderivation as well. In the notion of proof scheme we capture this finitary
character of the Sderivation.
A proof scheme with the conclusion 0 may include a number of rules that are
irrelevant to the enterprise of deriving 0. There is a natural preordering on proof
schemes. Namely, we say that p < pi if every rule appearing in p also appears in pi.
The relation is not a partial ordering, and it is not a partial ordering if we restrict
ourselves to proof schemes with a fixed conclusion 0. Yet it is a wellfounded relation,
namely, for every proof scheme p there exists a MINIMAL PROOF SCHEME pi < p
[e.g. q is minimal if for every P2, if p2 < q, then q < P2.] moreover, we can, if desired,
require the conclusion of Pi to be the same as that of p.
We also set p = pi equivalent to p < p, A pi < p and see that = is an
equivalence relation and that its equivalence classes are finite.
We say that the nonmonotonic rule system < U, N > is LOCALLY FINITE if
for every 4 E U there are finitely many <minimal proof schemes with conclusion 4.
This concept is motivated by the fact that, for locally finite systems, for every 0 there
is a finite set, Dro, of derivations such that all the derivations of 0 are inessential
extensions of derivations in Dro. That is, if p is a derivation of 0, then there is a
derivation pi E Dr, such that pi < p. Finally, we say that the system < U, N > is
HIGHLY RECURSIVE if it is recursive, locally finite, and the map + can(Dro)
is partial recursive. That is, there exists an effective procedure which, given any
0 E U, produces a canonical index of the set of all <minimal proof schemes with
conclusion 0. We let (< U, N >) denote the set of extensions of the system.
For a nonmonotonic rule system < U, N >, define a rule r to be RESTRAINT
FREE if its set of restraints is empty. Define a sentence q E U to be TERMINALLY
RESTRAINTFREE if the last rule in each of its minimal proof schemes is restraint
free. Define a subset A of U to be terminally restraintfree if every sentence 0 e A is
terminally restraint free.
Definition 1.3.6 For a nonmonotonic rule system < U,N >, define a sentence
q e U to be PREMISEFREE if each of its minimal proof schemes has length one.
Define a subset A of U to be premisefree if every sentence 0 E A is premisefree.
Definition 1.3.7 For a nonmonotonic rule system < U, N >, define a rule r to
be NORMAL if the set of restraints consists only of the negation of the conclusion.
Define a sentence 0 E U to be TERMINALLY NORMAL if the last rule in each of
its minimal proof schemes is normal. Define a subset A of U to be terminally normal
if every sentence 0 E A is terminally normal.
To consider the relationship between sets of extensions and trees, we will need
the following background, as given by Cenzer and Remmel [CR99].
Let w = {0, 1, 2,...} denote the set of natural numbers and let <,>: w x
w + w be some fixed onetoone and onto recursive pairing function such that the
projection functions 7r1 and 7r2 defined by 7r(< x,y >) = x and 7r2(< x,y >) = y
are also recursive. We extend our pairing function to code ntuples for n > 2 by the
usual inductive definition, that is < Xi,...,Xn >=< X 1, < x2,...,Xn > for n z 2. We
let w<' denote the set of all finite sequences from w and 2
finite sequences of 0's and l's. Given a =< a,,..., an > and /3 =< 13i,... ,/3k > in
w
a =< a,,..., an > with its code c(a) =< n, < ai,..., an >> in w. We let 0 be the code
of the empty sequence 0. Thus, when we say a set S C w<' is recursive or recursively
enumerable, we mean the set {c(a)a E S} is recursive or recursively enumerable. A
TREE T is a nonempty subset of w<' such that T is closed under initial segments. A
function f : w + w is an infinite PATH through T if for all n, < f(0),..., f(n) >e T.
We let P(T) denote the set of all paths through T. A set A of functions is a 11% class
if there is a recursive predicate R such that A = {f E w' I(Vn), R(< f(0),..., f(n) >)}.
A 11i class is RECURSIVELY BOUNDED if there is a recursive function g: w + w
such that Vf E A, Vn, f(n) < g(n). A is a 11 class, if an only if A = P(T) for some
recursive tree T. We say that a tree T is HIGHLY RECURSIVE if T is a recursive,
finitely branching tree such that there is a recursive procedure which, given a =< a1,
..., an > in T produces a canonical index of the set of immediate sucessors of a in
T, that is, produces a canonical index of {f3 =< a,,..., an, k > I0 ET}. Here we say
the canonical index, can(X) of the finite set X = {xl <...< Xn,} C w is 2x1+...+2xn
and the canonical index of the empty set is 0. We let Dk denote the finite set whose
canonical index is k, that is can(Dk) = k. It is then the case that if A is a recursively
bounded 11 class, then A = P(T) for some highly recursive tree T. We note that
if T is a binary tree, then the set of paths through T is a collection of {0, 1} valued
functions and by identifying each function f E P(T) with the set Af = {xjf(x) = 1}
of which f is the characteristic function, we can think of the set of paths through T
as a 1101 class of sets.
We then have the following theorem:
Theorem 1.3.8 (MNR92a) Given a highly recursive nonmonotonic rule system
< U,N >, there is a highly recursive binary tree T such that there is an effective one
toone degreepreserving correspondence between the set of extensions of the system
and the set P(T) of infinite paths through T.
A bit more is required for the reverse implication, but it may be shown in the
following sense:
Theorem 1.3.9 (MNR92a) For any recursive binary tree, there is a highly re
cursive nonmonotonic rule system such that there is an effective onetoone degree
preserving correspondence between the set of extensions of the system and the set
P(T) of infinite paths through T.
The significance of these results is that we can apply recursive 11I classes to
obtain numerous corollaries [MNR92a, CR98].
Corollary 1.3.10 (CR99) Let S = (U, N) be a highly recursive nonmonotonic rule
system such that E(S) 0 0. Then
(i) If S has only finitely many extensions, then every extension E of S is recursive.
A new representation result is given in chapter four for locally determined
nonmonotonic rule systems.
1.4 Common Derivations and Locally Determined Systems
In the second chapter we consider the finite case in which a nonmonotonic rule
system has a finite language and/or finitely many extensions. Within this chapter
we consider various forms of the common derivation of a sentence appearing in all or
some of the extensions of a nonmonotonic rule system. Certain forms of the common
derivation are explored not only in the classical sense but also using a constructive
view.
The infinite case is explored in the third chapter in which we have an infinite
number of extensions and/or an infinite language. Within this chapter, as in the
second, we consider various forms of the common derivation for a sentence appearing
in all or some of the extensions of a nonmonotonic rule system. Each of these is
explored in the classical sense as well as with a constructive view. We consider the
problem of characterizing the set of extensions of a locally finite nonmonotonic rule
system.
Locally determined nonmonotonic rule systems are introduced in the fourth
chapter. When considering nonmonotonic rule systems with an infinite language, a
problem arises in being able to determine at what stage we are sure to either have
concluded q or (in the case of restraints) not have concluded 0 for some 0 E U.
To solve this problem, the notion of "levels" was devised. Intuitively, we require
that for each i (as we take U to be enumerated {I0o, 01,...}), there is a fixed n, such
that the inclusion or exclusion of Oi is determined by the nith stage. Hence the
term "locally determined" is used. This concept allows us to extend results from the
infinite case. The set of extensions of a locally determined nonmonotonic rule system
is characterized in general and in the effective setting.
In the fifth chapter, we consider the complexity of the sets of extensions and
explore the comparative complexity of the common derivations themselves. Within
these systems we explore both the finite and infinite settings, with the classical and
constructive views.
Alternate formalisms of nonmonotonic logic, specifically Default Logic and
Logic Programming, are discussed in the sixth chapter. We show the equivalence of
these formalisms to nonmonotonic rule systems and view some of the results of the
previous chapters in terms of these.
Lastly, in the seventh chapter we consider possible applications of the results
and future directions of the research.
CHAPTER 2
THE FINITE CASE
In this chapter we consider only those nonmonotonic rule systems which have
at least one but only finitely many extensions and/or a finite language. We consider
nonmonotonic rule systems in the classical view and also in the constructive view.
The underlying logic for a nonmonotonic rule system U =< U, N > can be any
form of logic desired. We will often take U to be the set of sentences of some propo
sitional language. The usual monotonic rules of propositional logic are implicitly
included in each nonmonotonic rule system unless otherwise stated. We will choose
only one representative for each sentence in some systematic way so that there are
not infinitely many representations for each sentence in U. For example, 'a V 'b is
identified with ,(a A b).
2.1 Finite Classical Nonmonotonic Rule Systems
We consider a single nonmonotonic rule system U that has a finite number
of extensions and/or a finite language. We intend to explore the following question:
If 0 is a formula of the language L, such that 0 E S for every extension S of U,
then what, if anything, can be said about the theory and/or the formula and its
derivation in each extension. We would like to find a single rule that generates 0 in
every extension. It would be most useful if the rule could tell us, through premises
and restraints, exactly what is required to derive q in the system. This rule need not
be in the set N of nonmonotonic rules, but it would be desirable to retain the same
set of extensions upon adding this rule into N.
We might first consider the simplest possible form of the common derivation,
namely taking q as an axiom : /. This will certainly generate 0 in every extension,
but we may gain new extensions when the rule is added to the set N of rules.
Example 2.1.7 Let U =< U, N > be the nonmonotonic rule system where U =
{a, b, c} and
N= {: {a, c}/b;: {b, c}/a; a V b: {a, b}/c}.
This system has two extensions S1 = {a}, and 52 = {b}. The sentence a V b appears
in each extension. The axiomatic common derivation for a V b will be : /a V b. Adding
this rule to the set N of rules will produce a third extension S3 = {a V b, c}.
This formulation does not accomplish the purpose of the common derivation
in that it tells us nothing about what premises and restraints are involved in deriving
0 in the various extensions of the original system. Thus, we consider other forms.
Definition 2.1.11 Let U =< U, N > be a finite nonmonotonic rule system. Then
U has finitely many extensions, S1, ..., Sm. Suppose that 0 E U appears in every 5,.
Define the supportbased common derivation d' for 4 in U by
d"=: 101A...A 3.0V S}/.
Example 2.1.8 Letting U be the nonmonotonic rule system where U = {a, b} and
N = {: 'b/b;: b/ib;: /a}
we will have two extensions, S1 = {a, b} and S2 = {a, 'b}. The intersection of these
is {a}. Thus there are eight sentences not in S and eight sentences not in S2. Hence
there are 64 sentences of the form 31 A /2, where /1i S and32 V S2. Some of
these will be trivially false, such as ,b A b. Some others will contain that which is
in neither extension such as a A 'a which will collapse to just a. The rest will
be restraints of d'. We see how the supportbased common derivation becomes very
large very quickly. This will be a disadvantage.
We nevertheless get the following result:
Theorem 2.1.12 Let U =< U,N > be a finite nonmonotonic rule system. Then U
has finitely many extensions. Suppose a sentence E U appears in every extension
of the system. Let dO be the common derivation of q as defined in Definition 2.1.11.
Then, d4' applies in every extension of the system U and
(U) =(< U,Nu{d} >).
Proof: Let U =< U, N > be a nonmonotonic rule system with finitely many
extensions S1, ..., Sm. Suppose a sentence 0 E U appears in every extension of
the system. Let d' be the common derivation of q as defined in Definition 2.1.11.
Consider an arbitrary extension Sj of U. Take any conjunction Oi3A...ALm such that
3i V Si. Then fj V Sj so that the conjunction 1A...ALm is not in Sj and thus none
of the restraints of d4' is in Sj. Thus d' applies in Sj and therefore it applies in every
extension of the system U. We now have that
C(U) C_ (< U, Nu{dO} >).
To show that we retain the same set of extensions, suppose that 5So is an extension
of < U, N U {dO} > that is not an extension of U. Then So is not S, for any
i E {1, 2,..., m}. Since So is not an extension of U the new rule must apply in So.
The set of extensions of U is noninclusive so that there exist 0i, ..., Om e So such
that Oi Si. Since we have implicitly included the rules of propositional logic, we
have that )1 A... A'm E 5So. By the definition of d', this conjunction is a restraint of
d'. Thus the rule dO does not apply in So contradicting that So is a new extension.
Thus,
(U) =E(< U, Ngu{d} >).
Similarly, we might construct a rule consisting only of premises and a conclu
sion. The advantage of this form of the common derivation is that it reveals, at least
in part, what is involved in deriving 0 in the extensions. The disadvantage is that
we must first know the extensions of the system before we can construct the rule.
Finding this rule takes considerably more time than would a rule which relied only
on the system U without having to find the extensions. Also, this rule would not
be possible in the infinite case. Thus we seek a form of the common derivation that
would be based on the rules of the system itself, not the extensions, and would also
be possible in the infinite case. We find the following form. For a sequence of sets
A1,.. Am of sets of formulas, let Ai Ai denote the set of conjunctions a, A ... A a,,
where each ai E Ai. Let AiAi denote the usual conjunction where the formulas of
A, are themselves conjuncted to form formulas. The notations Vi Ai and ViAi are
similarly defined.
Definition 2.1.13 Letting U be a nonmonotonic rule system with a finite language
L, we will have a finite number of extensions. List the extensions as {Si,..., Sm }
Let pi be the proof scheme deriving 1 in the extension Si and consider the last rule,
ri in each pi. Define the standard common derivation to be do be the nonmonotonic
rule
ViP(ri): A R(ri)/.
i
We will henceforth refer to the nonmonotonic rule system, < U, NU {Id1} >, as U+.
Theorem 2.1.14 Let U =< U, N > be a nonmonotonic rule system with a finite
language L. Then, for every sentence 0 E nl(U), there exists a standard common
derivation dp1 that generates 0 in every extension of U.
Proof: List the extensions of U as {SI,. . Sn}. Suppose that 4 is a sentence
of U that appears in every extension of the system. Since 0 appears in every extension
of the system, there must be a proof scheme pi for each Si deriving q in that extension.
List the proof schemes for 0 as {pi, ..., Pm}. Consider the last rule, ri = P(r,) : 01,
..., Q/ of each proof scheme pi. Let d1 be the nonmonotonic rule
VjP(r) :A R(ri)/O.
i
Let Si be any extension of U. We only have to show that d,1 applies in Si. Since Si
is an extension of U, we have that the last rule r, of the proof scheme pi applies in
Si. Thus, P(ri) E Si and 0ii', ..., ,i' V Si. Therefore,
P(ri) = VjP(r1) E Si.
For any 51A...A6m E Ai R(ri), 5i = f3 for some t, so 5i, Si and by the standard
propositional logic, d1 applies in Si and concludes c(do1) = 0.
The common derivation of any formula 0 in U may not be found in N as
illustrated by the nonmonotonic rule system of Example 1.3.3. Recall that
N = {: {a}/b,: {c}/b,: {a}/c,: {c}/a},
and that U has the two extensions S, = {b, c} and S2 = {b, a}. Also, as before,
GD(N, S1) = {: {a}/b,: {a}/c}
and
GD(N, S2) = {: {c}/b,: {c}/a}.
There is no rule of N that derives b in both extensions. However, we may consider
the common derivation of b in U which is
d'b=: {(a A c)}/b V b =: {(a A c)}/b.
Then, dlb will generate b in each of these extensions. To see this, note that dlb has
no premises, and a A c V S since in S1 we have that a S1 since : {a}/b applies in
51. In 52, c V 52 since : {a}/b applies.
Let U =< U, N > be an nonmonotonic rule system with a sentence 0 occurring
in every extension of the system. Let do' be the common derivation as defined in
Definition 2.1.13. We find that the sets of extensions of the original system U and
the set of extensions of the new system U+ will, under sufficient conditions, be the
same. However, U+ may have an extra extension, while still having as extensions all
the extensions of U.
Theorem 2.1.15 Let U =< U, N > be a nonmonotonic rule system with a finite
language L such that a sentence q E U appears in every extension of the system. Let
do1 be the common derivation of 0 as in Definition 2.1.13. Then,
E(U) C E(u+).
Proof: Let U =< U, N > be a nonmonotonic rule system with a finite lan
guage L and suppose that a sentence 0 E U appears in every extension of the system.
Let do' be the standard common derivation of 0 as in Definition 2.1.13. Let S be any
extension of U. Then, to show that S is an extension of U+, we need to show that
for any u E U, u E S if and only if there is a proof scheme p for u in the new system
such that all premises in p are in S and all restraints in p are outside of S. Suppose
first that u E S. Since S is an extension of the system U, there is a proof scheme po
for u such that all premises in P0 are in S and all restraints in P0 are outside of S.
It is clear that po is still a proof scheme for u in the new system U+. Next suppose
that p is an Sapplicable proof scheme in U+. It suffices to consider the last rule r of
the proof scheme p where the premises of r are in 5, the restraints of r are not in S,
and the conclusion of r is u. If r is the common derivation then u = q so the 0 E S
by the hypothesis. Thus, S is an extension of U+. Hence,
E(U) cE(< U, Nu{d4l} >).
We can generalize this result in the following way.
Theorem 2.1.16 Let U =< U, N > be a nonmonotonic rule system with a finite
language L such that a sentence 4 E U appears in every extension of the system. Let
r, be any nonmonotonic rule concluding ( such that r, applies in every extension of
U. Then,
E(U) C E(< U, Nu{ro} >).
Proof: The proof of this theorem mimics that of the previous theorem. The
key part of the proof showing that an extension S of U is closed under the expanded
system W+ follows as above since the conclusion 0 of the new rule ro is assumed to
be in every extension of U.
We see that the sets of extensions may or may not be equal in that the new
system U+ may or may not have one or more extra extensions. Recall Example 1.3.3.
Consider the nonmonotonic rule system U where U = {a, b, c}, N = {: a/b,:
c/b,: a/c,: c/a}, and U has the two extensions S, = {b, c} and S2 = {b, a}, and the
common derivation of b in U is d1b =: a A c/b. Adding this new rule to N, we see
that < U, N U {dbn} > has the same extensions Si and S2, since applying d'b does
not prevent the application of any rule in N or allow the application of any rules in
the system that were not previously applied.
To be precise, suppose there was a new extension S different from S, and
S2. Then, the new rule d'b must be generating for S. This implies that b E S and
(a A c) V S. It follows that either a V S or c V S. Thus, one of the original rules,
either : {a}/b or : {c}/b, will generate for S and either : {a}/c or {c}/a will also
generate for S. Then, either c E S or a E 5, but this means that either Si C S or
52 C S. By the noninclusive property, either S = Si or S = 52.
Example 2.1.9 Now consider the nonmonotonic rule system U where
U = {pi,p2,Ci,c2,a}
and
N = {Pi : /cl;p2 :/c2;: a,pl/p2;: a,p2/pl;:Pl,P2/Pl Vp2;Cl VC2 : C1,C2/a}.
This theory has two extensions,
S1 = {p2,c2},
and
S2 = {Pl,C1}.
The formula cl V c2 is in each extension and we find that
rl = P2 : /c2,
and
r2 = Pi : /Cl,
so that
dcIVc2 = Pl V P2 : /c1 V C2.
Then, S1 and S2 are both extensions of < U, N U {dlcc1v2} >, but this new nonmono
tonic rule system will have a third extension
S = {pL V P2, Cl V C2,a},
which is not an extension of the original system U.
In the previous example, S3 is the only new extension obtained by adding the
common derivation to the set of rules. To see this, let S be any new extension. The
common derivation must generate for 5, else S is an extension of the original system.
Thus, we have that p1 Vp2 E S so that the conclusion c1 Vc2 is in S as well. If p, C S,
then cl E S by the rule pi :/ci so that S$ C S, a contradiction to the noninclusive
property of extensions, since S is not an extension of the original system. Similarly,
if P2 E S, then c2 E S so that S2 C S, a contradiction. Thus, neither of pi or P2 is
in S. Also, if a is not in S, then the rule : {a,p2}/pl applies to conclude Pi in S.
Thus, a E S. From this we now have that S3 C S so that S3 = S since extensions
are noninclusive. Thus, there is only the one new extension.
Since the standard common derivation is based on the rules in the proof scheme
of q instead of the extensions of the system, it might be possible in an infinite system.
The good news is that the set of extensions of the new system contains the set of
extensions of the original system. The bad news is that these sets of extensions may
not be equal. We would prefer that they be equal.
We see that this form of the common derivation does not reach the goals that
we were after, although interesting. We strive further towards the goal of equal sets
of extensions in the next chapter.
If we have a formula appearing in some but not all the extensions of U, we
may in the same fashion as Definition 2.1.13 create a common derivation for 0 that
generates 0 in each extension of U in which it appears. However, we find that in this
case Theorem 2.1.15 becomes false, as the next example illustrates.
Example 2.1.10 Let U =< U,N > be the nonmonotonic rule system where
N = {: {ai, a2}/a3,: {a2, a3}/ai,: {ai, a3}/a2,
Pi {al}/cl,p2: {a2}/c2,a3: IP Vp2,a /p,1, 2: /p2}.
This theory has as three of its extensions,
S,= Cn({a3,pi Vp2}),
S2 = Cn({a2,P2,C2}),
and
S3 = Cn({ai,pi,cl}).
We see that the formula cl V c2 is in both S2 and S3, but is not in S1. We find the
standard common derivation for cl V c2 to be
dCivc2 = Pi V P2 {'(a1 V a2)}/Cl V c2.
Adding this new rule to the set of rules N, we find that both S2 and S3 are extensions
of the new nonmonotonic rule system < U, N U {dlcVC2} >' S1, however, is not an
extension of < U, N U {d1c Vc2} > since it it no longer closed under the set of rules.
It is important to note that S1 is a subset of {a3,pi Vp2, cl Vc2} which is an extension
o~f < U, N u flV'1 >
of.
Theorem 2.1.17 Let U =< U, N > be a nonmonotonic rule system with a finite
language L such that 0 appears in some but not all of the extensions and each gener
ating rule for each extension S is a generating rule for Su{I}. Consider d1, to be the
common derivation of 0 as defined in Definition 2.1.13. Then, for any extension S of
U, such that 4 E S there is an extension S+ of U+ such that S = S+. Furthermore,
any extension of U+ which is not an extension of U must contain 0.
Proof: Suppose that U is a nonmonotonic rule system with a finite language
L such that 0 appears in at least one, but not all of the extensions. Consider d1' to
be the standard common derivation of as defined in Theorem 2.1.13. If 0 appears
in only one of the extensions of U, then the common derivation d1' is found in N
so that < U, N >= /4+, and the theorem is trivially true. Thus, suppose that 0
appears in at least two extensions of the nonmonotonic rule system U, but does not
appear in every extension. Let Si be an extension of U. If Si is an extension of the
new system U+, we are done. Also, if 0 is in Si, then Si is clearly an extension of
< U, N U {d'0} > as in the proof of Theorem 2.1.15. Hence, consider the case in
which S is an extension of the new system U+, but is not an extension of the old
system U. If die does not generate for S, then
S = GD(N U {d1'}, S) = GD(N, S)
so that S is an extension of the system U, a contradiction. Thus, d1o generates for
Si so that
S= c(d'o) E c(GD(N U {d1}, S)) = S.
The added conditions of this theorem that require each generating rule for
each extension S to be a generating rule for S U {1} are necessary for the following
reason. If we consider the previous example, and add the rule a3 :/i(cl V c2) to the
set N of rules, S1 becomes
Si = {a3,p1 Vp2, C(1 Vc2)}.
Then, c1 V c2 cannot be added to S1.
Now we explore the question of what happens if we take the process further;
finding the intersection of the extensions, creating a common derivation, and then
adding that rule to the nonmonotonic rule system, over and over.
Suppose we let I, be the intersection of all the extensions of the original
nonmonotonic rule system U. Then, in Example 2.1.9
i = Cn({(p2 A c2)V (pi A cl)}).
We consider the new rule system
U/2 =< U, N2 >=< U, N U {dl1vcj 2}>
and the set 12 of all conclusions common to every extension of U2. We find that
12 = Cn({(pi V p2) A (ci V c2)}) C 1.
Moreover, we could choose a conclusion common to all the extensions of U42,
such as pi V P2. This has the common derivation
d2pivp =: {aVpi,a V p2, a,pi Vp2}/pi Vp2.
In any new extension S of
2
U3 =< U, N2 U {d P1V2I}>
that was not an extension of the theory U2, this common derivation must apply. Sim
ilarly to the previous argument, none of Pi, P2, C1 or c2 can be in S else we contradict
the noninclusive property of extensions, pi V p2 must be in S since the common
derivation applies. Thus, by the common derivation for c1 V c2, we conclude cl V c2 in
S and thus a is in S by the appropriate rule from N. This is a contradiction since the
common derivation requires that a not be in S. Thus, there are no new extensions,
that is, U2 has the same set of extensions as U43.
On a different note, we may force the intersections I, and 12 to be equal by
taking 4 to be AIi. This appears in every extension of U. Letting
N2 = N U {d1j},
we find that if S is any new extension, then 0 E S since the common derivation must
apply (else S is not new). We then get the following theorem.
Theorem 2.1.18 Let qS be the conjunction of the intersection I, of all extensions of
U. Then the intersection 12 of all extensions of U+ equals I,.
Proof: Clearly we have that 12 C I,. Now suppose that V I, and let S
be any extension of U+. If S is an extension of U then 0 E S since 4 IA. Thus,
suppose that S is an extension of U+ that is not an extension of the original system
U. This new extension must contain 0. Since is the conjunction of the intersection
of extensions I,, we have that 4 * 4. Thus the extension S must contain 4. Thus,
12 = Il.
We may consider any nonmonotonic rule system U, and the set
Ii =ne .
For every 0 in this set, we can find a common derivation d1, which generates 0 in
each extension S of U. Consider the nonmonotonic rule system U2 defined by
.
We may then repeat the procedure to find
/2 = n ),
and if this set is not empty, then we can find common derivations d2o for any 4 in
12 such that d2o generates 4 in every extension S of U2. Then, we may consider the
nonmonotonic rule system U3 defined by
.
We continue this construction by, assuming that Un is defined, let
I. = )
Then, letting dO be the common derivation for e In, in Un, we define Un+l to be
< U, N, U {an.1 E In} >.
This may be continued until Ik is empty for some k and/or Nk = N, for all 1 > k
for some k. This inductive definition begs the following questions. Is there a limit to
this process, i.e., is there always some k for which Ik is empty and/or Nk = NJ for
all 1 > k? Does it make a difference if we work in the case where N is finite versus
countably or uncountably infinite? These questions should be investigated.
Example 2.1.11 Consider the nonmonotonic rule system of Example 1.3.3. Using
this nonmonotonic rule system, we would find N2 to be
N U {db} = {: {a}/b,: {c}/b,: {a}/c,: {c}/a,: {(a A c)}/b}.
U2 has the same extensions as U so that 12 = I, = {b} and d2b = d1b. Then Nk = N2
for allk > 1.
Remark 2.1.19 By Theorem 2.1.15, we have that E(U) C (U2) C ... C e(Un) C
9(4+1) C ....
For a finite language, or just a finite I,, we must eventually have In+, = In
for some n. At this point, we find that we will have a common derivation in Nn for
each 0 E In, when we consider Nn to be defined using all of the common derivations
for conclusions in In,. Un will be said to HAVE common derivations.
Theorem 2.1.20 For any nonmonotonic rule system U with finite I, or with finite
N, there exists an n such that Un has common derivations, and Un is equivalent to
the nonmonotonic rule system Un+k for any k.
For an alternate approach, we might consider the nonmonotonic rule system
U and the set I, as before. However, we may choose one element q6 from I, and define
< U, N2 > to be the nonmonotonic rule system
U+
as seen for Example 6.3.26. We may then consider the set of all formulas which
appear in every extension of < U, N2 >, choose some V) among those, and define
to be
.
We may continue in this way with the same result as in Theorem 2.1.20.
2.2 Finite Constructive Nonmonotonic Rule Systems
Again, consider a single nonmonotonic rule system U with a finite set of ex
tensions and/or a finite language. In this section, we consider the same question
about nonmonotonic rule systems as in the previous section, but use a constructive
approach. That is, we consider that pVq is not derived unless one ofp or q is derived.
Note that we will no longer implicitly assume the propositional axiom p V 'p. Had we
chosen to keep this axiom, while considering a constructive view, all the extensions of
every system would be complete, and consequently uninteresting. We will, however,
keep the propositional rules for conjunctions as we did in the classical case.
Again, we consider 0, a formula of the language L such that E fl(U).
Then 0 E S for every extension S of U. Since each extension S is of the form
c(GD(N, 5)), there must be, as before, a proof scheme pi deriving 0 for each extension
Si. The advantage of the constructive approach is that it allows us to view each set
of restraints as a single formula instead of as a set of formulas. Then, instead of
asking if each element of the set of restraints is not in a context, we would ask if the
restraint itself is not in the context. As it stands, the last rule ri in each proof scheme
pi has as its restraints the set {fil, ...,0k} for some O3j formulas of the language. We
may instead let R(ri) = ,iV ... V3ik. Then R(ri) is in the extension Si if and only
if ij3 E Si for some j, a contradiction since ri generates for Si. We find that several
of the theorems of the last section still hold.
Theorem 2.2.21 For any formula 0 of the finite language L such that 0 Ef 8(/U),
0 has a common derivation d1o which generates 0 in each extension S of U where U
is a constructive nonmonotonic rule system.
Proof : Suppose U is a constructive nonmonotonic rule system with a finite
language L. Suppose further that is a sentence of U that appears in every extension
of the system. List the extensions of U as Si,..., Smn. Then for each extension Si of
U there is a proof scheme pi deriving 0 in Si. Consider the last rule, r, = P(ri) : 01,
..., /3i// of each proof scheme pi. Let do1 be the nonmonotonic rule
ViP(ri}) : AR(r)/0.
i
Let Si be any extension of U. We only have left to show that do applies in Si. Since
Si is an extension of U, we have that there is some proof scheme Pi that derives in
Si. Since this proof scheme derives in Si, each rule in the proof scheme applies in
Si so that ri applies in Si. Thus, P(ri) E Si and ji4, ..., ti V Si. Thus,
P(ri) = ViP(ri) E S,
and any 61A...A6m e Ai R(ri) will not be in Si. Thus, d1 applies in Si and concludes
c(d41) = 0. Thus, for any formula 0 of the language such that E A (u), 0 has a
common derivation d1X which generates 0 in each extension S of U. Note that dIO
may be in N, but then U+ will have the same set of extensions as U.
Example 2.2.12 Consider the nonmonotonic rule system of Example 1.3.3. Under
the approach of this section, we let
N = {: a/b,: c/b,: a/c,: c/a}.
Then the nonmonotonic rule system U has the two extensions
S, = {b,c}
and
S2 = {b,a}.
Also,
GD(N, SI) = {: a/b,: a/c}
and
GD(N, S2) = {: c/b,: c/a}.
Just as before, there is no rule of N that derives b in both extensions. However, we
may consider the common derivation of b in U by the methods of this section, which
is
d'b =: (a A c)/b.
Then, d1b will generate b in each of these extensions. Compare this version of the
common derivation of b in U to that of the previous section where we had
d'b =: {(a A c)}/b.
We see that the two methods produce very similar results, but that the common deriva
tion produced by the methods of this section is computationally simpler.
By the same argument as in Theorem 2.1.15, we still have that
(U) C_ (< U, NU{d } >)
for any e ME(U). Equivalently, E(U) C E(< U,NU{dlo} >), for any fml(U).
Also as before, the converse of this theorem is false, as illustrated in example 2.1.9.
We must then ask if there might be a better formulation of the common
derivation that would ensure that same set of extensions when added to the system.
There is, but it is not without strings attached.
Theorem 2.2.22 For any formula q of a finite language such that E Fl 8(U),
define the strong common derivation d*, to be
VYP(ri) : UR(ri)/O.
i
Then, d*0 E GD(NU {d*},ne(u)).
Proof : Let 4 be a formula of the language such that 0 E fN (U). Consider
d*0 as defined above. Clearly d*, E N U {d*0}. Now, for every extension Si in E(U)
we have that some proof scheme p, generates 0 in Si so that P(ri) E Si. Thus,
P(d*o) = ViP(ri) E nA(U). Lastly, we have that R(d*o) = Ui R(r) C fl(U) if
and only if R(ri) C fnE (U) for all i. However, this is if and only if R(ri) C Si for
some extension Si where pi generates 0 for Si. Then, R(ri) E Si, a contradiction. So
we have that R(d*) is not a subset of nfg(U). Thus d*0 generates 0 in mf(U).
Note that although this new rule generates for the intersection of all the ex
tensions, it may not generate for each particular extension.
Example 2.2.13 Consider the nonmonotonic rule system U where
N= {: {a}/b,: {b}/a}.
This rule theory has two extensions,
and
S2 = {a}.
We have that the formula a V b is in both of the extensions and we find that
d*aVb =: {a, b}/a V b.
By the theorem, d*avb generates for the intersection of the two extensions. However,
it does not generate for either of the extensions Si or S2 since b E S1 and a E S2,
respectively.
We may again inductively define the nonmonotonic rule system Un as we did in
the last section. Investigations into the questions raised by this definition are further
warrented in this setting as the common derivations are computationally simpler and
the constructive view may make the difference in what the answer to those questions
will be.
Example 2.2.14 Consider the nonmonotonic rule system of Example 1.3.3. Using
this rule system, and the methods of this section, we would find N2 to be
N U {d\b} = {: a/b,: c/b,: a/c,: c/a,: (a A c)/b}.
Again, U2 has the same extensions as U so that 12 = 1, = {b} and d2b = d'b. Then
Nk = N2 for all k > 1.
2.3 Characterizing the Set of Extensions
Now, since the intersections of families of extensions are of particular interest,
we might ask which families of extensions have intersections which yield a specific
theory. Knowing this, given a theory which we desire to see as the intersection of
extensions we may create the nonmonotonic rule system whose extensions intersect to
yield that theory. Note that we will not implicitly assume any rules of propositional
logic for the case of this construction.
Theorem 2.3.23 Given any finite family of sets
F = {T,,T2,...,T,},
in the language L there is a nonmonotonic rule system U whose set of extensions
(U) is exactly F.
Proof: Let
F={T,, T2,.. ., T,},
be any finite family of sets. We create the system U in the following way. Let U be
the set of atoms of the language L. Let
N ={: {:jTjlj 5 i}/ A Till < i < n}.
Then, U will have F as its set of extensions.
For example, for the family {{a, b, c}, {a, c, d}, {a, b, d}} we would choose the
nonmonotonic rule system < U, N > where
N = {: {aAbAc, aAcAd}/aAbAd;: {aAbAc, aAbAd}/aAcAd;: {aAbAd, aAcAd}/aAbAc.
This system will have exactly the three extensions: {a, b, c}, {a, c, d}, and {a, b, d}.
So we see that any given theory may be derived as the intersection of a set of
extensions by determining what family of sets intersects to yield the desired theory,
and then constructing the appropriate nonmonotonic rule system whose extensions
are that family.
The constructive case will be different only in that the number of possible
extensions that are acceptable is greatly reduced. This is explored in detail in terms
of Default Logic in Chapter 6.
CHAPTER 3
THE INFINITE CASE
In this chapter we consider nonmonotonic rule systems with infinitely many
extensions and an infinite language. This necessitates the concept of a locally finite
system as seen in the introduction. We first consider nonmonotonic rule systems in
the classical view and then in the constructive view. Lastly, we consider the task of
characterizing the set of extensions of a locally finite nonmonotonic rule system.
3.1 Locally Finite Classical Nonmonotonic Rule Systems
Considering locally finite nonmonotonic rule systems, we can have results for
the infinite case similar to those of the finite case. We should note, however, that the
supportbased common derivation dO as shown in the finite case is no longer possible
in the infinite case since it would result in an infinite conjunction in the restraints of
the rule. Still applicable, however, is the standard common derivation d1.
Theorem 3.1.24 Let U =< U, N > be a locally finite nonmonotonic rule system.
Then, for every sentence q e n c ((U), there exists a standard common derivation d61
that generates q in every extension of U.
Proof: Suppose U is a locally finite nonmonotonic rule system. Suppose
further that is a sentence of U that appears in every extension of the system. Since
the system is locally finite, we have that there are only finitely many minimal proof
schemes for . List the proof schemes for ( as Dr, = {pl, ..., pm}. Consider the
last rule, ri = P(ri) : 01i, ..., I3ti'/ of each proof scheme Pi. Now, since 0 appears
in every extension of the system, we have that for each extension S, there is a proof
scheme that derives 0 in S, call it Ps. Since each rule of the proof scheme applies in
S, we know that there must be a minimal proof scheme pi, for some i E {1, ..., m},
contained in Ps that derives 0 in S. Let do1 be the nonmonotonic rule
VjP(ri) :AR(ri)/O.
i
Let S be any extension of U. We only have left to show that do1 applies in S. Since
S is an extension of U, we have that there is some minimal proof scheme pi E Dr
that derives 0 in S. Since this proof scheme derives 0 in S, each rule in the proof
scheme applies in S so that ri applies in S. Thus, P(ri) e S and /?i, ..., Otii V S.
Thus,
P(ri) = ViP(ri) E S
and any 61A...A&m E Ai R(ri) will not be in S. Thus, d1 applies in S and concludes
c(d') = .
We must ask several questions concerning how this new rule effects the system
U. First, if the system U is locally finite, then what happens when we add in the
new rule to consider the system U+. Is it still locally finite? In terms of the sets of
extensions, will we find that
C(U) = e(U+)?
We find that the expanded system may no longer be locally finite, as seen in
the following example.
Example 3.1.15 Let U =< U,N > be the nonmonotonic rule system where
U = {ai, a2, b, d, e} U {cili E w}
and
N = {ai : /b; a2 : /b;: a,, d/a2;: a2, d/ai;: a,, a2/a1 V a2; a, V a2, b: a,, a2/d}
U{a, V a2, b: a,, a2/Cii E w}
U{c : ai,, a2/eli E }.
This system has two extensions, S1 = {al, b}, and S2 = {a2, b}. The atom b appears
in each and has the standard common derivation d1b = a, V a2 : /b. Upon adding this
rule to the set N, we find that the new system provides an infinite number of minimal
proof schemes for e. Thus the new system is not locally finite.
The problem arises that there may be infinitely many distinct rules concluding
a sentence V) E U that are not applied in any extensions and therefore do not enter
into the set of minimal proof schemes for V). However, the addition of the common
derivation to the set of nonmonotonic rules may allow these rules to be applicable
and thereby the set of minimal proof schemes to be infinite. Then the new system
is not locally finite. To mend this problem, we must rid the system of rules that are
not applied in any extension prior to the addition of the common derivation.
Definition 3.1.25 Call a nonmnotonic rule r e N ACTIVE if it applies in some
extension of the rule system U; call the rule DORMANT, if it does not apply in any
extension of the system. Call a nonmonotonic rule system U DORMANT if every
rule in N is dormant. Call the system PARTIALLY DORMANT if N has at least
one dormant rule and at least one active rule. Finally, call the system ACTIVE if
every rule in N is active.
[Note that if a system is dormant, then it has no extensions.]
Theorem 3.1.26 Let U =< U, N > be a locally finite active nonomonotonic rule
system such that q appears in every extension of the system. If d,1 is the standard
common derivation for q in U, then U+ is locally finite and active.
Proof: Suppose U is a locally finite active nonomonotonic rule system such
that 0 appears in every extension of the system. Let do1 be the standard common
derivation for 0 in U. Consider the nonmonotonic rule system, U+ created by adding
the common derivation to the set of nonmonotonic rules:
U+.
Given that the original system is active, we have that every rule in N applies in some
extension of U. Since the system < U, N U {dol} > will have the same extensions
as the original system, and any extra extensions obtained are such that the common
derivation applies, we have that the new system is active. Further, U is locally finite
so that the set of minimal proof schemes for any sentence V) e U is finite. These sets
of minimal proof schemes will remain finite unless applying the common derivation
allows the application of an infinite list of rules that conclude some 4 E U. Suppose
this is true. This infinite list of rules must be from N, as the common derivation is the
only rule added to obtain the new system. Thus, these rules are active so that each
rule is applied in some extension of U. Thus, each rule provides a proof scheme for
V) in U. Since each rule is distinct and concludes 0, we have that none of these proof
schemes is contained in any other so that there are infinitely many minimal proof
schemes for 4 and U is not locally finite, a contradiction. Thus, the new system, U+
is locally finite.
Let U =< U, N > be a nonmonotonic rule system with a sentence 0 occurring
in every extension of the system. Let d1' be the standard common derivation as
defined in Theorem 3.1.24. We find that the sets of extensions of the original system
U and the set of extensions of the new system U+ will, under sufficient conditions,
be the same. However, U+ may have an extra extension, while still retaining all the
extensions of U.
Theorem 3.1.27 Let U =< U, N > be a locally finite nonmonotonic rule system
such that a sentence q E U appears in every extension of the system. Let do' be the
standard common derivation of q as defined in Theorem 3.1.24. Then,
O(U) C OU+}).
Proof: The proof for this theorem is similar to that of Theorem 2.1.15.
We can generalize this result in the following way, as in the finite chapter:
Theorem 3.1.28 Let U =< U, N > be a locally finite nonmonotonic rule system
such that a sentence q E U appears in every extension of the system. Let ro be any
nonmonotonic rule concluding such that ro applies in every extension of U. Then,
(U) C E().
Proof: The proof of this theorem mimics that of the previous theorem.
We see that the sets of extensions may or may not be equal in that the new
system U+ may have one or more extra extensions. Consider the following example:
Example 3.1.16 Consider the nonmonotonic rule system U from Example 3.1.15.
This system has the two extensions S1 = {al, b}, and S2 = {a2, b}. The atom b
appears in each and has terminal rules
r, = a : /b,
and
r2 = a2 :/b,
so that the standard common derivation is
d1b = a, V a2 : /b.
Upon adding this rule to the set N, we find that the new system aquires and extra
extension. S and S2 are both extensions of < U, N U {d1b} >, but we will have a
third extension
S3 = {ai V a2, b, d, e} U {cji E wl},
which is not an extension of the original system U.
In the previous example, 53 is the only new extension obtained by adding the
common derivation to the set of rules. To see this, let S be any new extension. The
common derivation must generate for S, else S is an extension of the original system.
Thus, we have that a, V a2 E S so that the conclusion b is in S as well. If a, E S,
then b E S by the rule a, : /b so that S1 C S, a contradiction to the noninclusive
property of extensions, since S is not an extension of the original system. Similarly,
if a2 S, then b E S so that S2 C S, a contradiction. Thus, neither of a, or a2 is in
S. Then the rules : a,, a2/ai V a2 and a, V a2 : a,, a2/ci for each i apply so that cq E S
for each i. From this we now have that S3 C S so that S3 = S since extensions are
noninclusive. Thus, there is only the one new extension.
Further, we might try considering active nonmonotonic rule systems, in hopes
that this would cure the new system of its extra extensions. This does not help, as
seen in the next example.
Example 3.1.17 Consider the nonmonotonic rule system U where
U = {p1i,p2,j1,j2,c}
and
N = {pi : jl/c;p2 : j2/c; : pl/p2; : p2/pI; c: 'jl/j2; c: j2/jl }.
This system can be taken to be active and has two extensions,
S1 = {Pl,C,j2},
and
S2 = {P2,c,jl }.
c appears in both extensions and we see that
dc = PI Vp2 : ' ji A'j2/c.
Adding this rule to the set N, we find that the new system, < U, N U {d'e} >, has
both S1 and S2 as extensions but also has two other extensions, S3 and S4 where
S3 = {p,l c},
and
S4 = {p2,*j2,c}.
At this point, all possible versions of the common derivation that carry over
from the finite classical case have been exhausted. Yet we crave a form of the common
derivation that may have better luck retaining the same set of extensions when added
to the original set N of rules. We turn then to the following more complex rules.
These are constructed with more intricate restraints that are meant to rule out the
causes of extra extensions in U+.
Definition 3.1.29 We define two new versions of the common derivation, d, (0) and
d2(0) similarly to the definition of do in that we consider the last rule in each of the
minimal proof schemes of :
Suppose U is a locally finite nonmonotonic rule system. Suppose further that
0 is a sentence of U that appears in every extension of the system. Since the system
is locally finite, we have that there are only finitely many minimal proof schemes
for 0. List the proof schemes for as Dro = {pi, ..., Pmo}. Consider the last rule,
ri = P(ri) : /31i, ..., 1/4/1 of each proof scheme pi. Now, since appears in every
extension of the system, we have that for each extension S, there is a proof scheme
that derives 0 in S, call it Ps. Since each rule of the proof scheme applies in S,
we know that there must be a minimal proof scheme pi, for some i e {1, ..., m},
contained in Ps, i.e. p, < Ps, that derives 0 in S.
Define the refined common derivation dI (O) to be the nonmonotonic rule
VjP(r) : A R(ri) U V(P(ri) A [V R(ri)])/O.
i i
Further, define the strong refined common derivation d2(0) to be the nonmonotonic
rule
VjP(r2) : A R(r,) U A(P(rt) A [V R(r,)])/O.
i i
Taking each rule in its turn, we find different properties. Since q belongs
to each extension of the system, it follows as usual that < U,N U {d1 (0)} > and
< U, N U {d2(0)} > each retain the extensions of < U, N >. Considering the rule
d, () we obtain the following results:
Theorem 3.1.30 Let U =< U, N > be a locally finite active nonomonotonic rule
system such that 4 appears in every extension of the system. If d1 (0) is the refined
common derivation for q in U, then < U, N U {d (0)} > is locally finite and active.
Proof: Suppose U is a locally finite active nonomonotonic rule system such
that appears in every extension of the system. Let dl () be the refined common
derivation for in U. Consider the nonmonotonic rule system created by adding the
refined common derivation to the set of nonmonotonic rules:
< U, Nu {dj()} >.
Given that the original system is active, we have that every rule in N applies in some
extension of U. Since the system < U, N U {d1 (0)} > will retain the extensions of
the original system, and any extra extensions obtained are such that the common
derivation applies, we have that the new system is active. Further, the system U is
locally finite so that the set of minimal proof schemes for any sentence 0 E U is finite.
These sets of minimal proof schemes will remain finite unless applying the common
derivation allows the application of an infinite list of rules that conclude some 7 E U.
Suppose this is true. This infinite list of rules must be from N, as the common
derivation is the only rule added to obtain the new system. Thus, these rules are
active so that each rule is applied in some extension of U. Thus, each rule provides
a proof scheme for V in U. Since each rule is distinct and concludes '0, we have that
none of these proof schemes is contained in any other so that there are infinitely
many minimal proof schemes for V and U is not locally finite, a contradiction. Thus,
the new system, < U, N U {d1(0)} > is locally finite.
However, using this version of the common derivation, we may not have that
d1 () applies in every extension of the system. The reason for this is that the dis
junction in the restraints,
V(P(ri) A [V R(ri)]),
i
cannot be found in any extension of the system. For this to be true, none of the
(P(ri) A [V R(ri)])
may appear in any extension of the system. This is too much to ask, as we can only
be certain that for each extension there is at least one rule r, that applies. There
may be one or more rules that do not apply; i.e. there may be a rule rj that does
not apply such that P(r,) is in the extension, but some restraint of rj is also in the
extension. In that case, P(r,) A [V R(ri)]) is in the extension so that the disjunction
V(P(r,) A [V R(r)])
is in the extension and the common derivation d1i() does not apply.
Considering the strong refined common derivation, d2(0), we get just the
opposite problem. We find that this version of the common derivation applies in
every extension of the system, but may create new extensions when added to the set
of nonmonotonic rules.
Theorem 3.1.31 Let U =< U, N > be a locally finite nonmonotonic rule system
such that a sentence 0 E U appears in every extension of the system. Let d2(0) be the
strong refined common derivation of as defined in Definition 3.1.29. Then, d2(0)
applies in every extension of the system U and
(U) C (< U, N U {d2()} >).
Proof: Let U =< U, N > be a locally finite nonmonotonic rule system such
that a sentence 4 E U appears in every extension of the system. Let d2(0) be the
strong refined common derivation of 0 as defined in Definition 3.1.29. Recall that
d2(0) = ViP(r,) : AR(ri) U A(P(r,) A [V R(r1)])/O.
i i
We need to show that the premise of the rule is in every extension of the system U
and that no restraint is in any extension of the system. Let S be any extension of U.
First, consider the premise, ViP(ri). Since S is an extension and thus q is in S, we
have that there is a minimal proof scheme, pi E Dro that applies in S and concludes
q. r, is the last rule of this proof scheme and thus it must apply in S so that its
premise, P(ri) is in S. Thus, ViP(ri) is in S. Now we consider the set of restraints,
A R(r)U A(P(r,) A [V R(r,)]).
i i
Since r, applies in S we have that no restraint in the set R(ri) is in S. Any a E
Ai R(ri) has the form /31J A ... A /3" where / i E R(ri) for each i. Since O3'i V S, it
follows that a V S. Thus, no restraint in the set A, R(ri) is in S and P(r1) A [V R(ri)]
is not in S. Thus, Ai(P(ri) A [V R(ri)]) is not in S so that no restraints of d2(0) are
in S. Thus, d2(0) applies in every extension of the system U and therefore
&(U) C6(< U,N u{d2(q5)}>)
by Theorem 3.1.28.
Due to some of the same problems as before, we may still gain extra extensions
when the common derivation is added to the set N of rules. We now must ask if it
is possible to make sure that the sets of extensions will be the same, and if so, what
restrictions will be needed. We find that we can ensure that the sets will be equal,
and there are several options to do so.
Theorem 3.1.32 Let U =< U, N > be a locally finite premisefree nonmonotonic
rule system such that a sentence 0 E U appears in every extension of the system. Let
d2(0) be the strong refined common derivation of 0 as defined in Definition 3.1.29.
Then,
(U) = C(< U, N U {d2()} >).
Proof: Suppose that U be a locally finite premisefree nonmonotonic rule
system and let E U be a sentence that appears in every extension of the sys
tem. Let d2(0) be the strong refined common derivation of 0 as defined in Theorem
3.1.31. Then d2() =: A(V R(ri))/O. By Theorem 3.1.31 we know that the common
derivation applies in every extension of U. Hence, by Theorem 3.1.31, we have that
(U) C e(< U,NU{d2(0)} >).
We have only to show the reverse inclusion. For any consequence '0 in some extension
S in the new system, there must be an Sapplicable proof scheme for 0. If the last
rule in the proof scheme is anything other than d2(0), then it is a rule of the original
system and therefore the proof scheme is valid in the original system. Otherwise,
d2(0) is the terminal rule of the proof scheme for 0 and hence 0 = and this rule
applies in S so that Ai(VR(ri)) is not in S. That is, for some i, (VR(ri)) is not in
S so that the premisefree rule r, applies in S. Hence, the proof scheme pi applies in
S. In either case, we see that any proof scheme in the new system is a proof scheme
in the original system. Therefore S is an extension of U, and
(U) = (< U,Nu{d}2()} >).
Note that in a premisefree system, all minimal proof schemes have just one
line. However, the requirements on the rule system need not be this strong, to ensure
that the sets of extensions will be the same. There are other options.
We then gain the following less restrictive results:
Theorem 3.1.33 Let U =< U, N > be a locally finite nonmonotonic rule system
such that any sentence in the intersection of all the extensions of the system is
premisefree. Suppose that a sentence 0 E U appears in every extension of the system.
Let do be the standard common derivation of 0 as defined in Theorem 3.1.24. Then,
e(U) = E(u+).
Proof: Let U =< U, N > be a locally finite nonmonotonic rule system such
that any sentence in the intersection of all the extensions of the system is premise
free. Suppose that a sentence q E U appears in every extension of the system. Let
do' be the standard common derivation of 0 as defined in Theorem 3.1.24. Then,
since the last rule in each proof scheme for q are the only rules considered in the
construction of the common derivation, we may follow the proof of the case of the
system U itself being locally finite and premisefree.
We may instead place the requirements on the sentence in question, i.e. on
the sentence 0 which appears in every extension of the system.
Theorem 3.1.34 Let U =< U, N > be a locally finite nonmonotonic rule system
such that a sentence 0 E U appears in every extension of the system and is premise
free. Let do be the standard common derivation of q as defined in Theorem 3.1.24.
Then, do' is premisefree and
(U) = 8(U+}).
Proof: Suppose that U =< U, N > is a locally finite nonmonotonic rule
system such that a sentence 4 E U appears in every extension of the system and is
premisefree. Let do' be the standard common derivation of 0 as defined in Theorem
3.1.24. For any element a of the restraints of do1, a has the form 3'i A ... A jjm
where flj, E R(ri) for each i, and no O'j1 A ... A f0m% is in S, for any extension S
in which the rule applies. We claim that for some i, none of the flj, is in S so that
the rule r, applies. Supposing not, we may choose for each I some 61 = f1l in S.
However, we then have that delta1 A ... A 6k S, a contradiction. Thus, any proof
scheme in the new system may be replaced with a proof scheme in the original system
and hence
e(U) = E(U+).
Next, we consider iterating the common derivation as in Chapter 2.
Considering the language to be infinite, and taking the common derivation
to be either the standard, refined or strong refined common derivation, we still have
that I, 12 ; 13 2D.... We may define I* to be the intersection of all the I,, define
N* to be the union of all the Nn, and define * to be the union of all the sets of
extensions of the rule systems < U, Nn >. We then find the following:
Theorem 3.1.35 I* = e*
Proof: Suppose that E I*. Then, 0 E In, for all n so that
E fn ()
for all n. Letting S be any extension in E* we have that S is an extension of < U, N" >
for some n so that E S and thus 4 E N *. For the reverse containment, suppose
that we have 4) E fE*. Then, E S for any extension S E * so that E S for any
extension S of any < U, Nn >. Hence, E In,,, for any n so that 0 E n In = I*. Thus
the equality holds.
Theorem 3.1.36 8* C (< U, N* >).
Proof: Let S E F* so that S is an extension of some < U, Nn > and an
extension of any < U, Nm > for m > n. We want to show that S is an extension of
< U, N* >. To do this, suppose that a rule r E N* generates for S. r E Nm for some
m, and hence for some m > n. Since S is an extension of < U, Nm > we have that
the consequence of r is in S so that S is an extension of < U, N* >.
One might ask at this point if we can show that N* has common derivations
or if it does not. This is a point to be investigated.
3.2 Locally Finite Constructive Nonmonotonic Rule Systems
In this section, we consider the same question about nonmonotonic rule sys
tems as in the previous section, but use a constructive approach. That is, we consider
that p V q is not derived unless one of p or q is derived, and extensions are not closed
under the propositional rule p V ip, as in the finite constructive case. Say that an
extension S is constructive if p V q E S implies that p E S or q E S. Note that if a
set S conatins p V ip and is constructive, then S is complete.
Again, consider 0, a formula of the language L such that 46 E6 fg(U). Then
E S for every extension S of U. As before, there is a finite list of minimal proof
schemes Dro = {pl,... ,pm} that derive q in the extensions of U. For each i let r, be
the last (a.k.a. terminal) rule in pi. The advantage of the constructive approach is
the same as in the finite constructive case. It allows us to view each set of restraints
as a single formula instead of as a set of formulas. Then, instead of asking if each
element of the set of restraints is not in a context, we would ask if the restraint itself
is not in the context. As it stands, each rule ri has as its restraints the set {fil,
...,0it} for some ij sentences of the language. We may instead let R(ri) = lV ...
V/3't. Then R(ri) is in an extension S if and only if i'j e S for some j. We find that
several of the theorems of the last section still hold.
Theorem 3.2.37 For any formula ( of the language L such that E nfC(U), q has
a common derivation d1, which generates 0 in each extension S of U where U is a
locally finite constructive nonmonotonic rule system.
Proof : Suppose U is a locally finite constructive nonmonotonic rule system.
Suppose further that q is a sentence of U that appears in every extension of the
system. Since the system is locally finite, we have that there are only finitely many
minimal proof schemes for 0. List the proof schemes for 0 as Dro = {Pl, ..., pm }.
Consider the last rule, r, = P(ri) : /1i', ..., )ti1/ of each proof scheme pi. Now, since
4 appears in every extension of the system, we have that for each extension 5, there
is a proof scheme that derives 0 in S, call it Ps. Since each rule of the proof scheme
applies in S, we know that there must be a minimal proof scheme pi, for some i E {1,
..., m}, contained in Ps that derives 0 in S. Let do be the nonmonotonic rule
ViP(ri) : AiR(ri)/O.
Let S be any extension of U. We only have left to show that d4' applies in S. Since
S is an extension of U, we have that there is some minimal proof scheme p, e Dr,
that derives in S. Since this proof scheme derives 0 in S, each rule in the proof
scheme applies in S so that ri applies in S. Thus, P(ri) E S and/3i', ..., 3t2 S.
Thus,
P(ri) = ViP(ri) E S
and any 6iA...AMm E A^R(r1) will not be in S. Thus, do' applies in S and concludes
c(do1) = 0. Thus, for any formula of the language such that E (E(U), has a
common derivation d1, which generates in each extension S of U. Note that dI
may be in N, but then U+ will have the same set of extensions as U.
As in the finite case, the classical and constructive locally finite versions of the
standard common derivation d1' produce very similar rules. The difference is that
the constructive version is computationally simpler. We may also simplify the strong
common derivation.
Theorem 3.2.38 For any formula q of the language such that E 6 (U), define
d*0 to be VjP(r1) : ViR(ri)/O. Then,
d* E GD(N U {d}, (U)).
Proof : Let 0 be a formula of the language such that 0 E f (mu). Consider
d*0 as defined above. Clearly d*, E N U {d*0}. Now, for every extension S in (U)
we have that some minimal proof scheme Pi generates 0 in S so that P(r,) S.
Thus, P(d*o) = VjP(r1) e nf.(U). Lastly, we have that R(d*,) = ViR(ri) E fl(u)
if and only if R(ri) e f (u) for all i. However, this is if and only if R(ri) is in every
extension. Now let 5o be an extension where the rule ri generates. Then R(ri) E So,
a contradiction. So we have that R(d*o) is not in fE(U). Thus d*0 generates 0 in
ne(u).
By the same argument as in Theorem 3.1.28, we still have that E(U) C (S(+)
for any ( E NE((U). Also as before, we may gain extra extensions, i.e., the contain
ment may be strict.
Further, we might try considering active nonmonotonic rule systems, in hopes
that this would cure the new system of its extra extensions. This does not help, just
at it did not help in the finite case, and for the same reasons.
However, all is not lost. The mission to attain equal sets of extensions for U
and U+ is salvaged by returning to the refined common derivation d1(0). The only
required restrictions are that the system be locally finite and constructive.
Theorem 3.2.39 Let U =< U, N > be a locally finite constructive nonmonotonic
rule system such that a sentence 0 E U appears in every extension of the system. Let
di () be the refined common derivation of as defined in Definition 3.1.29. Then,
E(U) =(< U,NU{d1(0)} >).
Proof: Let U =< U,N > be a locally finite constructive nonmonotonic rule
system such that a sentence 0 E U appears in every extension of the system. Let
di(k) be the common derivation of q as defined in Definition 3.1.29. that S E {(U).
Then S is the set of Sconsequences in U of a subset I0 of U. To show that S E
E(< U,N U {d1(0)} >) we need to show that S is the set of Sconsequences in
< U, N U {d 1(q)} > of some subset I of U. We let I = I0. This is true as previously
since 0 is in S and the only new rule concludes . Thus, we have only to show the
reverse inclusion. For any consequence 0 in some extension S in the new system,
there must be an Sapplicable proof scheme for V). If the last rule in the proof scheme
is anything other than di(o), then it is a rule of the original system and therefore
the proof scheme is valid in the original system. Otherwise, di(o) is the terminal
rule of the proof scheme for 0i and hence 0 = 0 and this rule applies in S so that
Ai(V R(ri)) is not in S. That is, for some i, (V R(ri)) is not in S. Thus, no restraint
of di(O) is in S. Considering the premise of the rule, ViP(R,), and knowing that the
premise is in S since this rule applies, we have that one of the P(Ri) is in S. This
is due to the system being constructive. Thus, the rule ri applies in S. Hence, the
proof scheme pi applies in S. In either case, we see that any proof scheme in the new
system is replaceable by a proof scheme in the original system. Therefore S is an
extension of U, and
E(U) = (< U,Nu{d2(0)} >).
Now we begin to question the usefulness of the standard common derivation
d1. This version still has its place. Under appropriate restrictions the less com
plex (as compared to d1(q)) standard common derivation will yield the equal set of
extensions sought.
Theorem 3.2.40 Let U =< U, N > be a locally finite restraintfree constructive
nonmonotonic rule system such that a sentence q E U appears in every extension of
the system. Let do' be the standard common derivation of 0 as defined in Theorem
3.1.24. Then,
&(U) = {u+).
Proof: Suppose that U is a locally finite restraintfree constructive nonmono
tonic rule system such that a sentence 0 E U appears in every extension of the
system. Let do1 be the common derivation of 0 as defined in Theorem 3.1.24. In this
case, d1 = Vi P(ri) : /. By Theorem 3.1.24 we know that the common derivation
applies in every extension of U. Hence, by Theorem 3.1.28, we have that
,(u) C (U+).
We have only to show the reverse inclusion. For any consequence 4 in some extension
S in the new system, there must be an Sapplicable proof scheme for 40. If the last
rule in the proof scheme is anything other than d', then it is a rule of the original
system and therefore the proof scheme is valid in the original system. Otherwise, de1
is the terminal rule of the proof scheme for 0 and hence 0 = 0 and this rule applies
in S. Considering the premise of the rule, ViP(R1), and knowing that the premise is
in S since this rule applies, we have that one of the P(RP) is in S. This is due to the
system being constructive. There are no restraints to worry about. Thus, the rule
r, applies in S. Hence, the proof scheme pi applies in S. In either case, we see that
any proof scheme in the new system is replaceable by a proof scheme in the original
system. Therefore S is an extension of U, and
(U) =(/(+).
It is important to note here that being restraintfree automatically makes the
rules monotonic, but that this case is still interesting in the context of what we are
trying to accomplish.
Theorem 3.2.41 Let U =< U, N > be a locally finite constructive normal non
monotonic rule system such that a sentence E U appears in every extension of
the system. Let dI be the standard common derivation of q as defined in Theorem
3.1.24. Then,
E(U) = E(U+).
Proof: Suppose that U is a locally finite constructive normal nonmonotonic
rule system such that the sentence 0 E U appears in every extension of the system.
Let do1 be the common derivation of 0 as defined in Theorem 3.1.24. Then, d01 =
ViP(ri) : 0'/. By Theorem 3.1.24 we know that the common derivation applies in
every extension of U. Hence, by Theorem 3.1.28, we have that
E(u) c E(u+).
We have only to show the reverse inclusion. For any consequence V in some extension
S in the new system, there must be an Sapplicable proof scheme for iV. If the last
rule in the proof scheme is anything other than d41, then it is a rule of the original
system and therefore the proof scheme is valid in the original system. Otherwise, d1
is the terminal rule of the proof scheme for o and hence ip = 0 and this rule applies in
S so that ,0 is not in S. Considering the premise of the rule, VjP(Ri), and knowing
that the premise is in S since this rule applies, we have that one of the P(Ri) is in S.
This is due to the system being constructive. Thus, the rule ri applies in S. Hence,
the proof scheme pi applies in S. In either case, we see that any proof scheme in the
new system is replaceable by a proof scheme in the original system. Therefore S is
an extension of U. Thus,
S(U) = E (U).
However, the requirements on the rule system need not be this strong, to
ensure that the sets of extensions will be the same. As in the classical case, there are
other options.
Theorem 3.2.42 Let U < U,N > be a locally finite constructive nonmonotonic
rule system such that the intersection of all the extensions of the system is terminally
restraintfree. Suppose that a sentence q E U appears in every extension of the
system. Let do' be the standard common derivation of qj as defined in Theorem
3.1.24. Then,
E(U) = E(u).
Proof: Suppose that U is a locally finite constructive nonmonotonic rule
system such that the intersection of all the extensions of the system is terminally
restraintfree. Suppose further that a sentence 0 e U appears in every extension of
the system. Let do be the common derivation of 0 as defined in Theorem 3.1.24.
Then, since the last rule of each proof scheme for q are the only rules considered in
the construction of the common derivation, we may follow the proof of the case of
the system U itself being locally finite, constructive and restraintfree.
Theorem 3.2.43 Let U =< U, N > be a locally finite constructive nonmonotonic
rule system such that the intersection of all the extensions of the system is terminally
normal. Suppose that a sentence q E U appears in every extension of the system. Let
do' be the standard common derivation of q as defined in Theorem 3.1.24. Then,
e(U) = E(u+).
Proof: Let U =< U,N > be a locally finite constructive nonmonotonic rule
system such that the intersection of all the extensions of the system is terminally
normal. Suppose that a sentence 0 e U appears in every extension of the system.
Let do1 be the standard common derivation of 0 as defined in Theorem 3.1.24. Then,
since the last rule in each proof scheme for 0 are the only rules considered in the
construction of the common derivation, we may follow the proof of the case of the
system U itself being locally finite, constructive, and normal.
Also as in the classical case, we may instead place the requirements on the
sentence in question, i.e., on the sentence 0 which appears in every extension of the
system.
Theorem 3.2.44 Let U =< U, N > be a locally finite constructive nonmonotonic
rule system such that a sentence ( E U appears in every extension of the system
and is terminally restraintfree. Let do1 be the standard common derivation of 0 as
defined in Theorem 3.1.24. Then, do1 is restraintfree and
e(u) = E(u+).
3.3 Characterizing The Set of Extensions
In the infinite case, we must again ask if, given a family of sets, we can con
struct a nonmonotonic rule system whose set of extensions is exactly that family. The
unfortunate answer to this is no. Although the set of extensions of any nonmonotonic
rule system is noninclusive (i.e., forms and antichain), not every noninclusive family
of sets is the set of extensions for a nonmonotonic rule system. Consider the following
example.
Example 3.3.18 Let = {{lu} : i = 0,1,...}, that is, the family of all singleton
sets. This is clearly noninclusive. Now suppose that E were the set of extensions of
some rule system U. For the extension {uo}, there must be a proof scheme with
finite support S and conclusion Uo. Now just choose some Uk V S. Then 0 is also
applicable in {uk}, which means that {uk} is not deductively closed and is not an
extension.
The remedy for this is explored in the next chapter, where we are able to
successfully characterize the set of extensions of a locally determined nonmonotonic
rule system.
CHAPTER 4
LOCALLY DETERMINED NONMONOTONIC RULE SYSTEMS
In this chapter we consider a special nonmonotonic subcase of the infinte case.
We let U be a locally determined nonmonotonic rule system. This has a large impact
on the results.
4.1 Locally Determined Nonmonotonic Rule Systems
We may obtain a similar result in the case of U having infinitely many exten
sions, if we make some adjustments. To do this, we will first need some preliminaries.
Let U =< U, N > be a countable nonmonotonic rule system. For simplicity,
we will assume that U is a subset of the natural numbers w = {0,1, 2,...}. Let
U = {uo < ul <... }.
The informal notion of a locally determined rule system U is one in which the
existence of a proof scheme for a sentence u, (or the lack of existence thereof) can be
determined by examining only rules or proof schemes involving some initial segment
of U.
Given a proof scheme or a rule p, we write max(p) for the maximum of the
set {ilui occurs in 0}.
We shall write Nn for the set of all rules r E N such that max(r) < n and we
shall write Un = {uo,..., un} and refer to the nonmonotonic rule system made up of
these two sets as < Un, Nn >.
Definition 4.1.45 Let U =< U, N > be a computable nonmonotonic rule system.
Say that n is a LEVEL of U if for every subset S C {u0, ...,un} and all i < n + 1, if
there exists a proof scheme p such that the conclusion c(p) = ui, and R(p) n S = 0,
then there exists a proof scheme q such that c(q) = ui, R(q) n S = 0, and max(q) <
n+ 1.
Theorem 4.1.46 Suppose that n is a level of U and suppose that E is an extension
of U. Then, En = E n {u0,..., un} is an extension of < Un, Nn >.
Proof: Suppose that n is a level of U and suppose that E is an extension of
U. Let En = En {uo,..., un}. Since E is an extension of U, then for any ui E Sn, there
is a proof scheme p such that the conclusion of p is ui and the support of p has empty
intersection with E. Thus, in particular, the support ofp has empty intersection with
4n so that since n is a level, there exists a proof scheme po such that max(po) < n,
P0 has conclusion ui and the support of Po has empty intersection with En. Thus, po
is a proof scheme of < Un, Nn >. In the other direction, if i < n and ui V 4n, then
there can be no proof scheme po of < Un, Nn > such that po has conclusion u, and
the support of po has empty intersection with En since this would violate the fact
that E is an extension of U. Thus, n is an extension of < Un, Nn >.
Corollary 4.1.47 Suppose that n is a level of U and suppose that S C {uo,...,un}
is not an extension of < Un, Nn >. Then, there is no extension of E in U such that
En {uo,...,Un}=S.
Further, say that U is LOCALLY DETERMINED or "has levels" if there
are infinitely many n such that n is a level of U. For a system U that is locally
determined, we let lev(U) be the set of all n such that n is a level of U and we write
lev(U) = {lo < 11 <... }.
Suppose that U is a recursive nonmonotonic rule system. Then we say that
the system HAS EFFECTIVE LEVELS if there is a recursive function f such that
for all i, f(i) > i and f(i) is a level of U.
Recall that a nonmonotonic rule system is said to be LOCALLY FINITE if for
each q E U, there are only finitely many <minimal proof schemes with conclusion 0.
Recall also that U is HIGHLY RECURSIVE if U is recursive, locally finite, and there
is an effective procedure which, when applied to any 0 E U, produces a canonical
index of the set of all codes of <minimal proof schemes with conclusion 0.
Now, a nonmonotonic rule system U that is locally determined is not neces
sarily local finite, but the set of extensions of U will nevertheless correspond to the
set of infinite paths through a finitely branching tree and similarly the extensions of
a nonmonotonic rule system with effective levels will correspond to the set of infinite
paths through a highly recursive tree.
Theorem 4.1.48 Let U =< U,N > be a recursive nonmonotonic rule system.
1. If U has levels, then there is a recursive finitely branching tree T and a
onetoone degree preserving correspondence between the set of extensions E(U) of U
and the set [T] of infinite paths through T. Also,
2. If U has effective levels, then there is a highly recursive finitely branching
tree T and a onetoone degree preserving correspondence between the set of extensions
6(U) of U and the set [T] of infinite paths through T.
Proof: There is no loss of generality in assuming that U = w so that u0 =
0, ul = 1, u2 = 2,.... Next, observe that for each n, < Un, Nn > has only finitely
many minimal proof schemes so that we can effectively list all of the minimal proof
schemes po < pi <..., so that
1. If max(pk) = i and max(pl) = j and i < j, then k < 1. (This says that if
i < j the the proof schemes whose maximum is i come before those proof schemes
whose maximum is j.)
2. If max(pk) = max(pt) = i, k < I if and only if code(pk) < code(p1) where
code(p) denotes the index assigned to a proof scheme p under some effective Godel
numbering of the proof schemes.
We shall encode an extension S of the system U by a path 7rs = (7ro, 71,...)
through the complete wbranching tree ww as follows.
first, for all i > 0, Tr2i =s (i). That is, at the stage 2i we encode the information
if i belons to the extension S. Next, if 7r2i = 0 then 7r2i+1 = 0. But if 7r2i = 1 that
is, if i E S, then we put 7r2i+1 equal to that qs(i) such that Pqs(i) is the first minimal
proof scheme in our effective list of minimal proof schemes such that the conclusion
of Pqs(i) is i and the support of Pqs(i) has empty intersection with S.
Clearly S is turning reducible to 7rs. For it is enough to look at the values of
7rs at even places to read off S. Now, given an Soracle, it should be clear that for
each i e S, we can use an Soracle to find qs(i) effectively. This means that 7rs is
turning reducible to S. Thus, the correspondence between S and 7rs is an effective
degreepreserving correspondence. It is trivially onetoone.
The construction of a recursive tree T C wW such that the set [T] of infinite
paths through T equals the set {7TslS is an extension of < U,N >} is given in
[MNR92b]. The key fact needed to establish the branching properties of the tree T
is that for any sequence a E T and any i, either a(2i) = a(2i + 1) = 0 or a(2i) = 1
and a(2i + 1) codes a minimal proof scheme for i. We just note that when a proof
scheme p = u(2i + 1) does not correspond to a path 7rs, then there will be some k
such that a has no extension in T of length k. This will happen once we either find
a smaller code for a proof scheme or we find some u > i in the support of the proof
scheme p such that all possible extensions T of a have r(2u) = 1.
Let Lk = max({ilmax(pi) < k}). It is easy to see that since the system U is
a recursive nonmonotonic rule system, we can effectively calculate Lk from k.
We claim that the tree T is always finitely branching and that if the system
U has effective levels, then the tree T is highly recursive.
Clearly the only case of interest is when 2i + 1 < k and a(2i) = 1. In this
case we will let a(2i + 1) = c where the conclusion of the proof scheme Pc is i and
the support of Pc has empty intersection with I, and there is no a < c such that the
conclusion of the proof scheme Pa is i and the support of pa has empty intersection
with I,. Now suppose that q is a level and that i < q. Then, by definition, there
must be a minimal proof scheme p such that max(p) < q, the conclusion of p is i and
the support of p has empty intersection with I,. Thus, p = Pt for some I < Lq. It
follows that c < Lq where q is the least level greater than or equal to i. Thus, the
tree T is always finitely branching. Now, if the system U has effective levels which is
witnessed by the recursive function f, then it will always be the case that c < Lf(i)
so that the tree T will be highly recursive.
Corollary 4.1.49 Suppose that the system U is a nonmonotonic rule system with
levels such that there are infinitely many n such that < Un, Nn > has an extension
Sn. Then U has an extension.
Proof: Now consider the tree T constructed for the system U as in Theorem
4.1.48. Here we can again construct our sequence of minimal proof schemes po, pi, ...
recursive in U just as we did in Theorem 4.1.48. We can only conclude that the tree
T is recursive in the system U but in any case, T will be a finitely branching tree.
Now fix some level n and consider some m > n such that < Urn, Nm >
has an extension Sn. Then by the exact same argument as in Theorem 4.1.46,
Sn = Sm n {0,..., n} will be an extension of < Un, Nn >. Now consider the node
a = (a(0),... cr(2n + 1)) such that
1. a(2i) = 0 if i V Sn,
2. a(2i) = 1 if i E Sn,
3. a(2i + 1) = 0 if a(2i) = 0,
4. a(2i + 1) = c where c is the least number such that max(pc) < n, the
conclusion of Pc is i and the support of Pc has empty intersection with S,.
It is easy to see from our construction of the tree T that a E T. It follows that
T is infinite and hence T has infinite path 7r by Konig's Lemma. But the proof of
Theorem 4.1.48 shows that S, is an extension of the system U.
The next question is how to ensure that we can get recursive extensions. In
particular, suppose that the set of extensions of U is a decidable [z class, that is, the
set of infinite paths through a computable tree T with no dead ends. Then every node
of the tree has an infinite recursive extension. If we only assume that the set of dead
ends of T are computble, the any node with an infinite extension has a computable
infinite extension.
Definition 4.1.50 1. Let U =< U, N > be a nonmnotonic rule system with levels.
Suppose that {nln is a level of < U,N >} = {lo < 11 <...}. Then we say that
U HAS THE LEVEL EXTENSION PROPERTY if for all k, if Sk is an extension
of < Ui,,Nik >, then there is an extension of Sk+l of < UI+1,Nlk+l > such that
Sk+l n {uo,...,UI} = Sk.
2. A level n of U is a STRONG LEVEL of U if for any level m < n of U and
any extension Sm of < Urn, Nm >, if there is an extension Sn of < Un, Nn > with
Sn n {Uo,...,Um} = Sm, then there is an extension S of U with S f{uo,...,um} = Sm.
3. U HAS STRONG LEVELS if there is a computable function f such that,
for each i, i < f(i) and f(i) is a strong level.
The level extension property provides a way to construct an extension of the
system U by extending from level to level. The following result is immediate:
Theorem 4.1.51 If E is a decidable Il"i class of subsets of U with levels, then there
exists a nonmonotonic rule system U with the level extension property such that E is
the set of extensions of U.
Definition 4.1.52 We say that a recursive nonmonotonic rule system U with levels
has IMMEDIATE WITNESSES if for any levels m < n of U, whenever there is a
set V C {uo,...,Um} such that there is an extension S ofU with Sn {uo,..., um} = V
but there is no extension So of U such that So n {uo,..., Um+i} = V, then either
(i) there is a proof scheme p with max(p) < m such that the support of p is a
subset of {uo,..., un} V and p concludes Un+l or
(ii) there is a proof scheme p with max(p) < n + 1 such that the support of p
is a subset of {uo,...,un,un+i}V and the conclusion of p is in {uo,...,un,Un+I }V
(note: it is easy to see that it must be the case that un+1 is in the support of p.)
One can extend this concept as follows:
We say that a recursive nonmonotonic rule system U HAS WITNESSES OF
DELAY k if for all n whenever there is a set V C {uo,..., un} such that there is an
extension S of U with S n {uo,..., un} = V but there is no extension So of U such
that So n {uo,..., un+i} = V, then either
(i) there is a proof scheme p with max(p) < n + 1 such that the support of p
is a subset of {uo,..., un} V and p concludes Un+i or
(ii) for all sets T C {un+2,..., Un+k}, there is a proof scheme PT with max(prT) <
n + k such that the support of pT is a subset of {uo,..., Un+k} (T U V) and the
conclusion of p is in {uo,..., un+k} (T U V).
Theorem 4.1.53 Suppose that U is a recursive nonmonotonic rule system which has
witnesses of delay k for some k > 1 and which has at least one extension. Then the
lexicographically least extension of the system is recursive.
Proof: We can construct the lexicographically least extension S of the system
U by induction as follows.
Suppose that for any given n we have constructed Sn = Sn {uo,..., un}. Then
Sn+, = Sn unless either
(i) there is a proof scheme p of level n such that the support of p is a subset
of {uo,..., un} Sn, and p concludes un+1 or
(ii) for all sets T C {un+2,..., Un+k}, there is a proof scheme PT of level n + k
such that the support ofpT is a subset of {uo,..., Un+k} (T U Sn) and the conclusion
of p is in {uo,...,Un+k} (TU Sn).
in which case Sn+, = Sn U {un+}. Note that since there are only finitely
many minimal proof schemes of any given level, we can check conditions (i) and (ii)
effectively. Since there is an extension, it is easy to see that our definitions insure
that Sn is always contained in the lexicographically least extension of the system U.
Thus, S = UJn Sn is recursive.
By putting suitable effective bounds on the effective levels and/or the effective
witnesses, one can readily come up with conditions that force U to have exponential
time, NP, or Ptime extensions. This is a topic of current research.
4.2 Common Derivations in Locally Determined Nonmonotonic Rule Systems
Let U =< U, N > be a nonmonotonic rule system with infinitely many ex
tensions that is locally finite, locally noninclusive and locally determined (or "has
levels"). Suppose 0 E U appears in every extension of U. Since the system is com
putable, we have that 0 = uj for some j. Since the system is locally determined,
there are infinitely many n such that n is a level of U so that we may choose n(o) to
be the least level of U greater than or equal to j. When we restrict each extension
of the system to the set {u0, ...,Un(0)} we then have only finitely many extensions,
call them Si, ..., Sm. Then we may define a common derivation d, for 0 in U to be
d, =: {JlAA...AA,3,lVi,i {uo, ...,U(O)},i A Si}/.
We then get a result similar to the finite case.
Theorem 4.2.54 Let U =< U, N > be a locally finite, active nonmonotonic rule
system that is locally noninclusive and locally determined. Suppose a sentence 0 U
appears in every extension of the system. Let d, be the common derivation of 0 as
defined above. Then, d! applies in every extension of the system U and
(U) =(< U,Nu{dt,} >).
Proof: Let U =< U, N > be an active nonmonotonic rule system with in
finitely many extensions. Suppose a sentence 0 E U appears in every extension of
the system. Let d, be the common derivation of 0. Consider an arbitrary restricted
extension Sj of U. Take any conjunction /31A ...A/3m such that hi 0 Si. Then /3j 5,j
so that the conjunction /3iA...A/3m in not in Sj and thus none of the restraints of do is
in Sj. Thus d' applies in Sj and therefore it applies in every extension of the system
U. By Theorem 3.1.28 we now have that
E(U) C_(< U, Nu{d,} >).
To show that we retain the same set of extensions, we need only to show the reverse
inclusion. Suppose that So is a restricted extension of < U, N U {d.} > that does not
extend to an extension of U. Then, since U is locally noninclusive, there exist 1, ...,
V)m E So such that V. 5' S. Then, 01iA...Am is a restraint of d,. Thus, ViA...A'bm
is not in So since d, applies in So, a contradiction. Thus,
,(U) =(< U,Nu{d,} >).
The other notions of the common derivation, the standard common derivation,
the refined common derivation and the strong refined common derivation developed
in chapter three still apply in locally determined nonmonotonic rule systems by the
same theorems.
4.3 Characterizing the Set of Extensions
In this section, we shall provide a characterization of the set of extensions of a
recursive locally determined nonmonotonic rule system U = (U,M). Let (U) be the
set of all extensions of U. Our first observation is that C(U) is closed in the natural
topology on subsets of U.
Proposition 4.3.55 If U is locally determined, then (U) is a closed set.
Proof: Let El, E2,... be a sequence of extensions with limit E in the usual
product topology on sets. Suppose that ui E E. Then there must be some K such
that ui E Ek for all k > K. Thus for each k > K, there is proof scheme Ok such that
Eknsupp(Ok) = 0. Now let I be the least level > i. Then since U is locally determined,
it follows that for each k > K, there is a proof O5k of Ut such that Ek n supp(k) = 0.
But there are only finitely many possible support sets for such proof schemes in U1
so that infinitely many of the k have the same support S. However since S n Ek =0
for infinitely many k, it must be the case S n E = 0 and hence u, E E. Vice versa,
suppose that ui V E. Thus there must be some K such that for all k > K, u, V Ek.
Suppose by way of contradiction that there is proof scheme with cln(4) = ui and
supp(4) = S with S n E = 0. Since S is finite, there must be some M such that
S n Em = 0 for all m > M. But this would mean that 0 would witness that ui Em
for all m > M, contradicting our previous assumption. (This direction applies even
if U is not locally determined.) We should note however that, in general, S(U) is not
a closed set.
Example 4.3.19 Let U = {uo, u, . .} and let N consist of the following set of rules:
:u2k(2+l) : Vn, k}. This means that for any extension E and any k, Uk E E if and
I uk
only if at least one of the set {u2k(2n+l) : n = 0,1,...} is not in E. It is not hard to see
that for any k, there will be an extension E of U which contains all of {uo, U1l,..., Uk}.
Thus if (U) were closed, then U itself would be an extension, which is clearly false,
since none of the rules are Uapplicable. Hence E(U) is not a closed set.
We say that a family of sets is noninclusive if for any two sets A, B E ,
neither A C B nor B C A. A second key property of the set of extensions of a
nonmonotonic rule system is that it must be noninclusive. That is, the following
holds.
Lemma 4.3.56 For any nonmonotonic rule system U, the set (U) is noninclusive.
However, not every noninclusive family of sets can be the set of extensions of
a nonmonotonic rule system. Recall the example given previously.
Example 4.3.20 Let E = {{ui} : i = 0,1,...}, that is, the family of all singleton
sets. This is clearly noninclusive. Now suppose that C were the set of extensions
of some rule system U. For the extension {uo}, there must be a proof scheme 0
with finite support S and conclusion uo. Now just choose some uk S. Then is
also applicable in {uk}, which means that {uk} is not deductively closed and is not
an extension. Thus = {{ui} : i = 0,1,...} cannot be the set of extensions of a
nonmonotonic rule system.
However, by combining the two ideas of closure and noninclusivity, we can
define a condition which guarantees that a family of sets is the set of extensions of a
nonmonotonic rule system U = (U, N) with strong levels. Given a family S of subsets
of U, let Sn, = {E n {uo,..., u,,} : E E S}.
Definition 4.3.57 Let S be a family of subsets of U.
1. We say that n is a level of S if Sn is mutually noninclusive.
2. We say that S has levels if there are infinitely many n such that n is a level of
S.
3. We say that S has effective levels if there is a recursive function f such that,
for all i, f(i) > i and f(i) is a level of S.
There are many examples of families of set of U = {uo, Ul,...} with effective
levels. For example, consider the family S of all sets S such that for all n, IS n
{uo,... ,u2n}I = n. It is easy to see that for all n, 2" is a level of S. For a more
general example, let U be the set of all finite truth table functions on a countably
infinite set {ao,a1,...} of propositional variables. That is, for each sentence V of
propositional calculus, U contains exactly one sentence logically equivalent to V.
These are listed in order of the maximum variable ak on which the sentence depends.
Thus, Uo and ul are the (constant) True and False sentences; u2 and u3 are the
sentences a0 and a0, u4,..., u15 list the sentences depending on a0 and a,, and so
on. Now let F be any consistent set of sentences and let S(F) be the set of complete
consistent extensions of F. The levels ofS = {UnS : S : S(F)} are just the numbers
22 1. This is because if two sets in S disagree on the first 22k 1 sentences, then
there must be some i with i < k such that they disagree on ai, which means that
one of the sets contains a, but not 'ai whereas the other set contains ai but not ai.
Thus the two sets are mutually noninclusive.
Theorem 4.3.58 If S is a closed family of subsets of U with levels, then there exists
a nonmonotonic rule system U with strong levels such that S is the set of extensions
of U. Furthermore, if S is a decidable HI? class and has effective levels, then U may
be taken to have effectively strong levels.
Proof: Given the family S, we shall directly construct a nonmonotonic rule
system U = (U, N) such that .((U) = S. First, if S is empty, we let N consist of the
single rule `0. It is easy to see in this case that U has no extensions.
u0
Thus we assume that S 0 0 and hence that each Sn is nonempty as well. We
then create a set of rules for every level n of S. For each level n, let E ,..., En be the
list of all sets of the form En {uo,..., Un} for E E S. Then for each such En and each
c E E,, N will contain a rule ri,c = 13i,.Am where {,..., I= {uo,..., Un} E.
C
It is then easy to see that each E' is an extension of Un and that n is a level Un.
Moreover it easily follows that the set of extensions of U is exactly S.
For the second part of the theorem, we use the same nonmonotonic rule sys
tem. We note that decidable nIclass of sets has the property that there is a highly
recursive tree T contained in {0, 1}* such that set of infinite paths through T corre
spond to the characteristic functions of elements of S and T has no dead ends, i.e.
every node r7 E T can be extended to an infinite path through T. Thus for any level
n of S, the sets E',..., En described above will just correspond the set of nodes of
length n in the tree T. Because each of the nodes of length n can be extended to
infinite path through T, it follows that each extension E of Un can be extended to
an extension E of U such that E n {uo,..., Un} = E It then easily follows that n
is a strong level of U. Thus U will have effectively strong levels in this case.
One problem for nonmonotonic rule systems is to determine which sets of
extensions can possibly result from some recursivee) nonmonotonic rule system. It is
well known that any set of extensions must be mutually noninclusive, that is, if S
and So are two different extensions of the system U, then we can never have S C So
or So C S. We will sometimes say that the set of extensions has the noninclusive
property. On the other hand, for infinite languages, not every mutually noninclusive
set of extensions can be realized.
Nonmonotonic rule systems with levels may be used to produce a large family
of possible sets of extensions. Given a family F of subsets of U, let Fn = {S n
{uo,..,un}S E f}.
Definition 4.3.59 Let F be a family of subsets of U.
1. We say that n is a level of F is Fn is mutually noninclusive.
2. We say that F has levels is there ar infinitely many n such that n is a level
ofF.
3. We say that F has effective levels is there is a recursive function f such
that for all i,i < f(i) and f(i) is a level of F.
Theorem 4.3.60 1. If F is a closed family of subsets of U with levels, then there
exists a nonmonotonic rule system U with levels such that F is the set of extensions
of U.
2. If F is a decidable [1 family of subsets of U with effective levels, then
there exists a nonmonotonic rule system U with effective levels such that F is the set
of extensions of U.
Proof: Observe that ifF is empty, then there is a single rule: {uo}/Uo with no
extensions. Thus, we may assume that F is nonempty and thus each Fn is nonempty
as well.
For each level n, we have a set of rules as follows. For each S1, ..., Sk list the
family of intersections S n {uo,... Un} for S F. For each partial extension Si and
each c e Si, we will have a rule ric with conclusion c and with restraints b for each
b V Si. We know that the rule ri,c must exist since Sj is not a subset of Si by the
noninclusive property.
We claim that each level n of the family F is also a level of the proof system
U. To see this, let T be a subset of {uo,..., un}, let j < n and let p be a proof scheme
with conclusion uj = cn such that p has empty intersection with T. Since the rules
of U have no premises, we may assume that p consists of a single rule r = ri,c. Thus,
there exists S E F and a level m of F such that Si = S n {uo,...,Um}. If m < n,
then max(p) < m < n as desired, on the other hand, if m > n, then consider the
restricted partial extension Sk = S nl {uo,..., un} and the corresponding rule rk,c as
level n. It is clear that Sk C Si and that Sk n {Uo,..., un} = i5, n {uo,...u}. Then the
support of the rule rk,c has empty intersection with S since max(rk,c) < n and the
support of rk,c is equal to {uo,.., un} Sk = {uo,.., un} Si which is a subset of the
support of ri,c.
We need to show that the set of extensions of U is exaclty F. Let us first show
that any element S of F is an extension of the system U. Given c E S, the rule ri,c
has no premises, has restraints all not in S, and has conclusion c. Thus, c has a one
line proof scheme. Next suppose that S admits some rule r = rk,d. If k = i, then of
course d E S, so that S is closed under r. If k 5 i, then by the noninclusive property,
r has a restraint b E 5i Sk so that S does not admit r.
Next we show that U has no other extensions. Let S*o be any extension of U
and suppose that it differs from each S E F. Since F is closed, there must be a level
n such that S* = S*0 o l{Uo,...,un} differs from each S n {uo,...,Un}. Otherwise, there
would exist, for each level n, some Sn such that S* n {Uo,...,Un} = E nn {uo,..., un.
But then S* = limnEn would be in F, since F is assumed to be closed.
Now there is at least one S E Fn and S*O* : S by the choice ofn. Furthermore,
S* n {uo,..., un} is an extension of < Un, Nn >, by Theorem 4.1.48, since < Un, Nn >
has levels. It follows that S* n {uo,..., Un} is not a subset of S, by the noninclusive
property, so that S* n {uo,..., u,} is nonempty. Thus there is some c eS* n{uo,..., un}
and therefore S*o* admits some rule ri,c where Si is an extension of Fn. But then it
follows that S* n {uo,..., un,} C S, again violating the noninclusive property.
Suppose now that F is a decidable 101 class and has effective levels. Then
for any n, we can compute the set of intersections S n Un for S E F, which will just
be the set of paths of length n in the tree T. Now we can effectively compute an
increasing sequence of levels and use the set of intersections S n Un to compute the
rules at level n for the desired rule system.
It follows from the proof of Theorem 4.1.46 that if U has effective levels and has
the level extension property, then the family of extensions of U may be represented
as a decidable 111 class. Thus, we have the following results:
Theorem 4.3.61 1. Suppose that U is a recursive nonmonotonic rule system with
effective levels and has the level extension property. Then for every level n and
extension Sn of < Un, Nn >, there is a recursive extension of S of U such that
S n f{uo,...,Un} = Sn.
2. Suppose that U is a recursive nonmonotonic rule system with strong levels.
Then for every level n and extension Sn, of < Un, Nn >, if there is an extension S of
U with S n {uo,..., un} = Sn, then there is a recursive extension of S of U such that
S n {uo,...,Un} = S.
CHAPTER 5
COMPUTABILITY AND COMPLEXITY ISSUES
In this chapter we show a few results concerning the computability and com
plexity of the common derivations and their associated nonmonotonic rule systems.
5.1 Finding Extensions with Low Complexity
Throughout this section, we shall assume that U = (U, N) is a recursive
nonmonotonic rule system such that U = w = {0, 1, 2, *}. Moreover if U is locally
determined, we let {lo < 11 < ...} denote the set of levels of U and if U has strong
levels, then we let {So < Sl < ... } denote the set of strong levels of U.
In this section, we shall show how we can use the notions of levels and strong
levels to provide conditions which will ensure that U has an extension which has
relatively low complexity even when U is infinite. We shall distinguish two represen
tations of U, namely the tally representation of U, Tal(U), and the binary represen
tation of U, Bin(U). In the tally representation of U, we shall identify each n E U,
with its tally representation, tal(n), and in the binary representation of U, we shall
identify each natural number n with its binary representation, bin(n). Given a rule
r = i,,ani...,rm, we let the tally and binary representations of r be given by
tal(r) = 2tal(al)2.. .2ta1(an)3tal(l3)2.. .2tal(3m)3tal(p) (5.1)
bin(r) = 2bin(a)2... 2bin(an)3bin(3i)2 .. 2bifn(0m)3bin(p). (5.2)
We then let Tal(N) = {tal(r) : r E N} and Bin(N) = {bin(r) : r E N}. Similarly
given a proof scheme = (51,..., qk), we let the tally and binary representations of
V be given by
tal(V)) = 4tal(0j)4.4tal(On)4 (5.3)
bin(O) = 4bin(0i)4.. .4bin(n)4 (5.4)
Finally given a finite set of proof schemes proof F = {1,..., s}, we let the tally
and binary representations of F be given by
tal(r) = 5tal(4i)5... 5tal(O,)5 (5.5)
bin(r) = 5bin(Vi)5.. .5bin(V,)5 (5.6)
Definition 5.1.62 We say that the nonmonotonic rule system U is polynomial
time locally determined in tally, if the nonmonotonic rule system Tal(U) =
(Tal(w), Tal(N)) has the following properties.
1. There is a polynomial time function g such that for any i, g(tal(i)) = tal(lk,)
where ki is the least number k such that lk > i.
2. There is a polynomial time function h such that for any i, h(tal(i)) = tal(Fi)
where Fr is the set of all proof schemes Ulk whose conclusion is i where tal(lk) =
g(tal(i))
Similarly we say that U is polynomial time locally determined in binary, if definition
(5.1.62) holds where we uniformly replace all tally representations by binary repre
sentations. We can also define the notions of U being linear time, exponential time,
and polynomial space in tally or binary in a similar manner.
This given, we then have the following:
Theorem 5.1.63 1. Suppose that U is polynomial time locally determined non
monotonic rule system in tally which has the level extension property. If I is a
level of U and El is extension of U1 such that there is a unique extension E of
U with E n {0,..., 1} = El, then E e NP.
2. Suppose that U is polynomial space locally determined nonmonotonic rule sys
tem in tally which has the level extension property. If I is a level of U and El is
extension of UHi such that there is a unique extension E of U with El{0,..., l} =
El, then E E PSPACE
3. Suppose that U is polynomial time locally determined nonmonotonic rule system
in binary which has the level extension property. If I is a level of U and El is
extension of U1 such that there is a unique extension E of U with En{0,..., l} =
El, then E E NEXPTIME.
4. Suppose that U is polynomial space locally determined nonmonotonic rule sys
tem in binary which has the level extension property. If I is a level of U
and El is extension of U1 such that there is a unique extension E of U with
E n {0,..., l} = El, then E E UC>o(DSPACE(2nc).
Proof: For (1), suppose that I = It where recall that the set of levels of U is
{lo < 11 < ...}. Then for any i > It, consider the level lk, where g(tal(i)) = tal(lk,).
By the level extension property, it follows that there is an extension of U4 Elk such
that Elki n {0,..., 1} = El. Moreover, it must be the case that En {0,... lk} = Elk
since otherwise we could use the level extension property to show that there is a
sequence of extensions {E, : j > ki} such that for each j > ki, El, is an extension of
U1, where E, nf{0,...,ljj} = El_,. One can then easily prove that E' = UjkEI is
an extension of U such that E' n {0,..., l} = El contradicting our assumption that
E is the unique extension of U such that E n {0,..., l} = El.
It follows that to decide if i E E, we need only guess Elk,, verify that it
is an extension of Ulk, and check whether i E Elk,. We claim that this is an NP
process. That is, we first guess the sequence XEIk (0) ... XEik (lk) where XEk is
the characteristic function of El,. Note that our conditions ensure that there is
some polynomial p such that lk, < p(Ital(i)I). It follows that we can compute lko =
g(tal(O)),lk1 = g(tal(1)),...,lk,k = g(tal(lk,)) in polynomial time in Ital(i)I. Since
for each j, lk is the least level greater than or equal to j, it follows that lko < 1k, <
K.. <: lk, = ki* Thus if ki = s, we can find 10 < 11 < ... < 1I is polynomial
time Ital(i)I. Note by assumption, t < s. Now consider Er = Elk, n {0,... ,r} for
r = t,t + 1,.. s. By our definition of levels, it must be the case that each Er is
an extension of Ul,. That is, if x V Er, there can be no proof scheme 0 of U1, such
that cln(Vb) = x and supp(o) n Er = 0 since otherwise V) would witness that x E E,.
Vice versa, if x E Er, then x E E, and hence there is a proof scheme 0x of 1, such
that cln(O.) = x and supp(Ox) n E, = 0. But since I is level, there must be a proof
scheme V such that max(Vx) < lr, cln(V)) = x, and supp(x)n((En{O,..., lr}) = 0.
Thus 0., is a proof scheme which witnesses that x e Er. Note that if it is the case
that Ir1 < x < lr, then Ox in Fx. Thus since we can also compute h(tal(O)) =
tal(ro),h(tal(1)) = tal(rF),...,h(tal(lk,)) = tal(r,,) in polynomial time in Ital(i),
it follows that to check that Elk is an extension, we need only verify that that for each
X > It, XE, (x) = 1 iff there is a proof scheme / E FLx such that supp(VP,) n E k = 0.
It is easy to see that for each such x our codings of proof schemes and rules is such
that we can decode tal(rx) and check if there is such a ?Px is polynomial time in 1k.
Thus we can verify that Ek is an extension of Ul, in polynomial time in Ital(i)l.
Hence it follows that E E NP.
The proof of part (2) is similar. However since in this case, the length of the
sequence XE1 (0) 'XE, (1k,) is bounded p(\tal(i) I) for some polynomial p, we do not
have to guess it. That is, in p(i) space, we check all strings of {0, 1 }'k, to see if they are
the characteristic function of an extension E* of Uk, such that E* n {0,... 1} = El.
Since there is only one such extension with this property, we can search until we find
it. Thus our computations will require only polynomial space.
The proof of parts (3) and (4) uses the same algorithms as in parts (1) and
(2). However in this case the string XE,,i (0) XE, \ i (1k,) may be also long a 2P' Ibil1
for some polynomial P. Thus the algorithm could take on the order of 2Ibln(i)lc steps
in case (3) and require 2lIl(i)1c space in case (4).
We should note that if we replace the hypothesis of polynomial time and
polynomial space by linear time and linear space in parts (3) and (4) of Theorem
(5.1.63) respectively, then we get the following.
Theorem 5.1.64 1. Suppose that U is linear time locally determined nonmono
tonic rule system in binary which has the level extension property. If I is a level
of U and El is extension of U1 such that there is a unique extension E of U with
Enl {0,... ,1} = Ei, then E E NEXT.
2. Suppose that U is linear space locally determined nonmonotonic rule system
in binary which has the level extension property. If I is a level of U and E1 is
extension of4 U such that there is a unique extension E of U with En {0,... .,l} =
El, then E E EXPSPACE.
It is easy to show that we can weaken the hypothesis in Theorems (5.1.63) and
(5.1.64) that there is a unique extension E of U extending El to the assumption that
there are only finitely many extensions of U extending El and obtain the conclusion
that all of the extensions of El are in the same corresponding complexity classes.
However if we do not make any assumption about the number of extension of U which
extend El, then the only thing we can do is try to construct the lexicographically
least extension of El. One can see that in cases (2) and (4) there would be no change
in the conclusion. However in case (1), the computations could take 2n'C steps and in
case (3), the computations could require 22n steps.
Finally we note that similar results can be proven using strong levels instead of
the level extension property. We state the appropriate definitions and results without
proof. Recall that if U has strong levels, then we let {so < s1 < ... } denote the set
of all strong levels of U.
Definition 5.1.65 We say that the nonmonotonic rule system U has polynomial
time strong levels in tally, if the nonmonotonic rule system
Tal(U) = (Tal(w), Tal(N)) has strong levels and the following properties.
1. There is a polynomial time function g such that for any i, g(tal(i)) = tal(sk,)
where ki is the least number k such that Sk > i.
2. There is a polynomial time function h such that for any i, h(tal(i)) = tal(r)
where Fi is the set of all proof schemes U, k whose conclusion is i and tal(lk,) =
g(tal(i))
This given, we then have the following.
Theorem 5.1.66 1. Suppose that U is a nonmonotonic rule system which has
polynomial time strong levels in tally. If I is a level of U and El is extension of
U1 such that there is a unique extension E of Ut with E fl {0,..., l} = El, then
EENP.
2. Suppose that U is nonmonotonic rule system which has polynomial space strong
levels in tally. If I is a level of U and El is extension of U, such that there is a
unique extension E of U with E n {0,..., l} = EL, then E E PSPACE
3. Suppose that U is nonmonotonic rule system which has polynomial time strong
levels in binary. If I is a level of U and El is extension of U1 such that there is
a unique extension E of U with E n {0,..., l} = El, then E E NEXPTIME.
4. Suppose that U is a nonmonotonic rule system which has polynomial space
strong levels in binary. If I is a level of U and E1 is extension of U1 such
that there is a unique extension E of U with E n {0,..., l} = El, then E E
U,>o(DSPACE(2nc)).
5.2 Computability and Complexity of Common Derivations
Corollary 5.2.67 Let be a highly computable nonmonotonic rule system.
Then, for every sentence 4 E N E, there exists a computable standard common
derivation do1 that generates 0 in every extension of < U,N >.
Proof: Using the standard common derivation defined as in Theorem 3.1.24,
the definition of the rule yields an algorithm for constructing it. Since the system
< U, N > is computable we have that each of U and N is computable so that the
set of minimal proof schemes, Dr, for any sentence V e U is computable. Hence the
common derivation is computable.
Theorem 5.2.68 Let < U,N > be a highly exponential time (polynomial time, etc.)
nonmonotonic rule system. Then, for every sentence 4 E fN E(< U, N >), there
exists an exponential time standard common derivation de' that generates 4 in every
extension of < U,N >.
Proof: We use the same definition for the standard common derivation as in
Theorem 3.1.24. Since the rule system is highly exponential time, we have that the
set of minimal proof schemes, Dr, may be computed in exponential time. Once this
is done, we need only to list the last rule in each proof scheme and construct the
common derivation. Since there are only finitely many proof schemes, the common
derivation may be computed in exponential time, i.e., in the time required to compute
the set of minimal proof schemes.
Theorem 5.2.69 Let < U, N > be a highly computable active nonomonotonic rule
system such that 4 appears in every extension of the system. If do1 is the standard
common derivation for ) in < U, N >, then < U, N U {d,1} > is highly computable
and active.
Proof: Suppose < U, N > is a highly computable active nonomonotonic rule
system and 0 appears in every extension of the system. Let do1 be the standard
common derivation for in < U, N > as previously defined, and consider the system
< U,N U {d41} >. This system is active and locally finite by Theorem 3.1.26. To
show that it is highly computable we have left to show that it is computable, and that
there exists an effective procedure which given any sentence V e U produces the set
Dr of minimal proof schemes for V). The set U has not changed since we have added
no new sentences to the system by the addition of the common derivation. Thus,
U is still computable. Since the set N is computable, and the common derivation
is computable, we have that the set N U {de1} is computable. Thus, the system
< U, N U {do1} > is computable. Since the original system < U, N > is highly
computable we have that there exists an effective procedure which given any sentence
,0 e U produces the set Dr, of minimal proof schemes for V). This procedure is still
effective upon the addition of the common derivation to the set of nonmonotonic
rules of the system. Thus the new system < U, N U {do1} > is highly computable
and active.
Lastly, we find the following results as to the computability of the common
derivations dl (O) and d2(0).
Theorem 5.2.70 Let < U,N > be a highly computable nonmonotonic rule system.
Then, for every sentence E flE(< U, N >), the common derivations d1(0) and
d2(0) for 0 are computable.
Proof: The definition of each rule yields an algorithm for constructing it.
Since the system < U, N > is computable we have that each of U and N is computable
so that the set of minimal proof schemes, Drp, which is finite for any sentence V) E U,
is computable. Since these rules are constructed from the last rule in each proof
scheme, we have that they are computable.
Theorem 5.2.71 Let < U, N > be a highly exponential time nonmonotonic rule
system. Then, for every sentence 0 e f E(< U, N >), the common derivations di ()
and d2(0) for 0 are exponential time.
Proof: Since the rule system is highly exponential time, we have that the set
of minimal proof schemes, Dr, may be computed in exponential time. Once this is
done, we need only to list the last rule in each proof scheme and construct the rules
d1(O) and d2(0). Since there are only finitely many proof schemes, the rules may
be computed in exponential time, i.e., in the time required to compute the set of
minimal proof schemes.
Theorem 5.2.72 Let < U, N > be a highly computable active nonomonotonic rule
system such that 0 appears in every extension of the system. If dl (O) is the common
derivation for 0 in < U, N >, then < U, N U {d1()} > is highly computable and
active.
Proof: Suppose < U, N > is a highly computable active nonomonotonic rule
system and 0 appears in every extension of the system. Let d1() be the common
derivation for 0 in < U, N > as previously defined, and consider the system <
U, N U {d1(0)} >. This system is active and locally finite by Theorem 3.1.26. To
show that it is highly computable we have left to show that it is computable, and that
there exists an effective procedure which given any sentence V E U produces the set
Dr of minimal proof schemes for V. The set U has not changed since we have added
no new sentences to the system by the addition of the common derivation. Thus,
U is still computable. Since the set N is computable, and the common derivation
is computable, we have that the set N U {di (q)} is computable. Thus, the system
< U, N U {d1 (0)} > is computable. Since the original system < U, N > is highly
computable we have that there exists an effective procedure which given any sentence
85
SE U produces the set Drp of minimal proof schemes for V. This procedure is still
effective upon the addition of the common derivation to the set of nonmonotonic
rules of the system. Thus the new system < U, N U {d1 (0)} > is highly computable
and active.
CHAPTER 6
ALTERNATE FORMALISMS OF NONMONOTONIC LOGIC
In this chapter we will discuss how the results of the previous chapters trans
late into two particular formalisms of nonmonotonic reasoning, specifically Default
Logic and Logic Programming. We will treat each separately, beginning with the nec
essary definitions and preliminary theorems. Continuing, we show how each theory is
equivalent to nonmonotonic rule systems. Finally we will see how some of the results
of the previous chapters look through the eyes of these formalisms.
6.1 Default Logic: Preliminary Definitions and Theorems
Definition 6.1.73 A DEFAULT THEORY is a pair (D, W), where W is a set of
formulas of the language L, and D is a set of DEFAULTS of the form d = A :
B1,...Bn/C where A, B1,...,Bn, and C are all formulas of the language L and the
default is interpreted as saying, "If A is true and B1, ..., Bn are not contradicted,
then conclude C." Call A the PREREQUISITE of d, and write p(d) = A. Call
B1,...,Bn the JUSTIFICATIONS of d, and write J(d) = {B1,...Bn}. Lastly, call C
the CONSEQUENT of d, and write c(d) = C. Note that it is possible for a default
to have no prerequisite and/or no justifications.
If E is a set of formulas of the language L, let Cn(E) denote the closure of E
under semantical consequence.
Now, if (D, W) is a default theory, and S is a set of formulas of the language,
call a default d GENERATING FOR THE CONTEXT S if p(d) is in S, and for all
3 E J(d), "0 V S. Let GD(D, S) be the set of all defaults in D that generate for the
context S.
Most simply, an EXTENSION, S, of a default theory (D, W) is a set of for
mulas of the language L that is the smallest set T (by inclusion) such that T is
theoretically closed, i.e., T = Cn(T), T contains W, and for all defaults d E D such
that p(d) E T and V/3 E J(d),,03 S, c(d) E T [BDK97]. This last property is best
described as being "closed under the defaults of D."
Example 6.1.21 Let a, b, and c be atoms in the language L and let
D = {: {a}/b,: {c}/b,: {a}/'c,: {c}/'a}.
Then the default theory (D, 0) has two extensions:
Si = Cn({b, c})
and
S2 = Cn({b, a}).
Moreover, we find that
GD(D, SI) = {: {a}/b,: {a}/,c}
and
GD(D, S2) = {: {c}/b,: {c}/,a}.
Definition 6.1.74 Now, we may wellorder the defaults of a default theory (D, W)
by some well ordering <. We may then define AD.< to be the set of all defaults in D
which are applied when the wellordering is used to close W under the set of defaults.
This is done in the following way: we define an ordinal 7.<. For every e < 7< we
define a set of defaults ADE and a default d,. If the sets ADe, e < a, have been
defined but _< has not been defined, then
(1) If there is no default d E D \ Ue<, ADe such that:
(a) W U c(U,<, ADe) p(d)
(b) W U c(U,<,0 AD,) V/ 3, for every f3 E J(d)
then r7< = a.
(2) Otherwise, define d0 to be the <least such default d E D \ U,
Then set AD, = (U,<, AD,) U {d,}.
(3) Put AD< = U<, AD,.
Intuitively, we begin with the set W of formulas, and apply the defaults of D in
their order. Each time we apply only the orderingleast default which can be applied.
At some point (since the ordinal is welldefined [MT93]) there will be no available
dafaults that may be applied. At this point, AD< is the set of all defaults applied and
77< is the number of steps needed to reach this stopping point.
Then, let T_ be Cn(W U c(AD._)). This is the theory GENERATED BY <
Then, GD(D, T<) C AD< [MT93] so that
Cn(W U c(GD(D, T_))) C T<.
Now, if < is a wellordering of the defaults and for every 3 in J(AD.<), 03 Cn(WU
c(AD <)) then T_ is an extension of (D, W) [MT93]. That is, if
AD.< = GD(D,T<),
then
T_ = Cn(W U c(AD.<))
is an extension of (D, W). More precisely, if T. = Cn(W U c(GD(D, T))), then T_
is an extension of (D, W). We now have that
(a) If S is an extension of (D, W) then there is some well ordering _ of the
defaults in D such that S = Cn(W U c(AD<)) = Cn(W U c(GD(D,S))) [MT93].
And,
(b) If S = T = Cn(W U c(AD.<)) = Cn(W U c(GD(D, 5))) for some well
ordering <, then S is an extension of (D, W).
Thus, S is an extension of (D, W) if and only if S = Cn(W U c(GD(D, S))).
Let E(D,W) be the set of all extensions of the default theory (D, W). Call two
default theories (D1, W1) and (D2, W2) EQUIVALENT, written (DI1, Wi) = (D2, W2),
if they have exactly the same extensions, i.e., if E(D1,W1) = E(Dt,W,)
Example 6.1.22 Let (D, W) be a default theory where D = {: /a * b} and W =
{a}. Then consider the default theory (D',0) where D' = {: /a + b,: /a}. These two
theories are equivalent as they each have the same single extension
S = Cn({a + b,a}).
Theorem 6.1.75 (MT93) Let S C L. Then, S is an extension of (D, W) if and
only if S is an extension of (D U D0o, 0), where Do = {: /201 E W}.
Proof : Suppose S is an extension of (D, W). Then,
S = Cn(W U c(GD(D, 5)))
which is clearly equal to Cn(0 U c(GD(D U Do, 5))) since each new default in Do
will generate for any context S. Thus, each extension of (D, W) is an extension of
(D U Do, 0). For the same reasons, the converse also holds. Since they have exactly
the same extensions, the theories are equivalent.
Theorem 6.1.76 (MT93) Let S and S' be be two extensions for the default theory
(D, W). Then, if S C S', then S = S'.
Theorem 6.1.77 (MT93) A default theory (D, W) has an inconsistent extension
if and only if Sent(L) is an extension and (D, W) has no other extensions.
6.2 Equivalence of Default Logic to Nonmonotonic Rule Systems
The equivalence of Default Logic to Nonmonotonic Rule Systems has been
shown by Cenzer, Marek, Nerode, Remmel [1999] and others. Here, we give the
equivalence as shown by Cenzer and Remmel. Let be the propositional language
underlying the given default logic. With L fixed, all our nonmonotonic rule systems
will have the same universe, namely the set of all wellformed formulas of .. We now
show how to interpret a given default theory as a nonmonotonic rule system.
Let (D, W) be a default theory. For every default rule r,
a : 01...,/3k
r =
7
construct the following nonmonotonic rule dr
a: ll, .,ik
dr =   
Next, for every formula O E L, define the rule
do=
and for all pairs of formulas X1, X2 define
Xi, Xi D X2:
mPx ,x =  ^
X2
Now define the set of rules ND,W as follows:
ND,W = {dr : r E D} U {dp : V E W or V is a tautology} U {mpx,,x2 X1, X2 E }.
We have the following result:
Theorem 6.2.78 [MNR90] Let (D, W) be a default theory. Then a set of formulas
S is a default extension of (D, W) if and only if S is an extension of nonmonotonic
rule system (U, ND,w).
Theorem 6.2.78 says that at a cost of a simple syntactic transformation and additional
encoding of logic as (monotonic) rules, we can faithfully represent default logics by
means of nonmonotonic rule systems.
6.3 Previous Results Through the Eyes of Default Logic
We will consider only those deafult theories which have at least one, but only
finitely many extensions. We will consider both the classical case and the constructive
case.
We consider a single default theory (D, W) that has a finite number of ex
tensions and where D and W are at most countable sets. We intend to explore the
following question: If is a formula of the language L, such that E S for every
extension S of (D, W), then what, if anything, can be said about the theory and/or
the formula and its derivation in each extension.
Let Do be the set {: /1p E W}. Let D1 = D U D0. We consider now the
default theory (D1,0). This theory is equivalent to the original theory (D, W) since
they have the same set of extensions.
Theorem 6.3.79 For any formula of the language L such that q E fN E(D,w),
has a common derivation d1', which generates 0 in each extension S of (D, W).
Proof: We prove the theorem using the equivalent default theory (D1,0).
Then, since (D, W) and (D1,0) have the same extensions, the theorem holds for
(D, W). Suppose that 0 is a formula of the language L such that ( E f (D1 ,0).
Then 0 E S for every extension S of (D1,0). Since each extension S is of the
form Cn(c(GD(D1, S))), there must be, for each S, a finite set of defaults {dsi, ...,
don} C D1 such that {c(d81), ..., c(d'n)) I q. Define d(o,,) by
Ap(dat) : U J(di,)/Ac(ds:).
t i i
Where we simply omit from the prerequisite conjunction those i for which dsi has no
prerequisite.
Claim 1: For any extension S of (D1, 0), d(o,8) E GD(D1 U {d(O,,)}, S)
Proof of claim 1: Let S be an extension of (D1, 0). Consider d(o,s) as defined
above. Clearly d(o,s) E DlU{d(s,,)}. Also, p(d(o,,)) = Aip(di) E S since each p(d',) is
in S since each dAi must be applied to deduce 4 and therefore each ds, is in GD(D1, S).
Now, for every f3 in J(d(s,8)) = U1 J(d'1),  S since no (3 in any of the J(d'i) is
in S. Thus, d(4,,) E GD(D1 U {d(o,s)}, S). Moreover, c(d(o,,)) = Aic(di) deduces 0
since {c(dai), ..., c(d'n)} I 0. Thus, d(o,s) deduces 0 in S.
Now we have for each extension S of (D1, 0) a default d(o,,) which deduces 0
in S. For any sets of formulas J1, ...,Jm define JiV...VJm to be the set of disjunctions
{jiV...Vjm :ji E Ji,i = 1,...,m}. We then define d'l to be
k k k
V i=l( =))" lJ(d(@, ))/V i=ic(d(0,.)),
where S1,...,Sk are all the extensions of the default theory (D, W). Call d1o the
COMMON DERIVATION of in (D1, 0).
Claim 2: d1, E GD(D1 U {dl1}, S) for every extension S of (D1,0).
Proof of claim 2: Suppose that 0 is a formula of the language such that 0
nA (Dl,0). Consider d1l as defined above. Clearly d1' E D1 U {d'0}. Also, for any
extension i5, of (D1, 0), for all / in J(d(o,8,)), V Si so that for every /3 E J(d1l),
0 = jlV...Vjk where j, E J(d(o,j)) for each i = 1,...,k. Then, for any extension Si
of (D1,0), ,/3 e Si if and only if 'jlA...A'jk E Si, which implies that 'ji e 5,i, a
contradiction. Thus, ,3 V S for any extension S of (D1,0). Lastly, p(d1o) e S for
any extension S of (D1,0) since for any 5, p(d(o,s)) E S and p(dl) = Vp(d(,s)).
Thus, d1o E GD(D1 U {d'1}, S) for every extension S of (D1, 0). Furthermore,
c(d10) = V, c(d(o,s)) deduces 0 in every extension S since each c(d(s,,)) deduces 0 in
S. Thus, for any formula q of the language such that 4 En N (Dl,0), 0 has a common
derivation d1p which generates 0 in each extension S of (D1, 0). Note that d1i may
be in D1, but then (Di U {d1l}, 0) will have the same set of extensions as (D1, 0), i.e.
(D1 U {d'o}, 0) will be equivalent to (D1,0). Further note that since d'i generates
for each extension S of (D1,0), we have that it generates for the intersection of all
the extensions, that is, it generates for N E(D,W). This completes the proof of Claim
2 and thus of the theorem.
Theorem 6.3.80 For any formula 0 of the language such that En f E(D,W), define
d*% to be
Vp(d(,) UJV d
Then, d*, E GD(D1 U {d*,},NE(D,w)).
Proof: Again, we prove the theorem using the equivalent default theory
(D1, 0). Since (D, W) and (D1,0) have the same extensions, the theorem holds for
(D, W). Let q be a formula of the language such that 0 E fN E(D,,0). Consider d*%
as defined above. Clearly d*0 E D1 U {d*O}. Now, for every extension S in Z(Di,0) we
have that d(o,5) generates for S so that p(d(O,s)) E S. Thus, p(d*o) = VSp(d(d,s))
A (E(D1,0). Lastly, for every extension S in E(Di,0) we have that VO E J(d(o,,)), 13 V S.
Now, V/3 E J(d*o) = U, J(d(s,8)), 0 E J(d(o,s)) for some extension S so that /3 V S.
Hence, o/3 N E(D1,0). Thus d*0 generates for f E(D,,0).
Note that although this new default generates for the intersection of all the
extensions, it may not generate for each particular extension.
Example 6.3.23 Consider the default theory (D, 0) where
D= {: {ia}/b,: {ib}/a}.
This default theory has two extensions,
Si = Cn({b}),
and
S2 = Cn({a}).
