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## Material Information

Title:
Theorems for finite automata
Creator:
Wright, Reverdy Edmond, 1933-
Publication Date:
1971
Language:
English
Physical Description:
vii, 74 leaves. : ; 28 cm.

## Subjects

Subjects / Keywords:
Automata ( jstor )
Equivalence relation ( jstor )
Homomorphisms ( jstor )
Ions ( jstor )
Isomorphism ( jstor )
Mathematical congruence ( jstor )
Matrices ( jstor )
Monoids ( jstor )
Semigroups ( jstor )
Submonoids ( jstor )
Dissertations, Academic -- Mathematics -- UF
Finite automata, Theorems for ( lcsh )
Machine theory ( lcsh )
Mathematics thesis Ph. D
Sequential machine theory ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

## Notes

Thesis:
Thesis -- University of Florida.
Bibliography:
Bibliography: leaves 73-74.
Also available on World Wide Web
General Note:
Manuscript copy.
General Note:
Vita.

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Full Text

i" ::: f!i 1111
'ii:~illlii

To L~ydia

The author thanks the mncrbers of his supervisory

committee, Dr. R.C efigD. A. R. ednarck,

Dr. r!. E. Thomas, andi Dr. K. Il. Sipgmon, for their

assistance and cncouragcrent. Par t icular thanks are due

Professor Selfridgye, the connrittee chairmanr~r, and Professor

De dns erk., the c h a irnn of nc7t h em!at ic s, for their many

This pape~r w:as ty'ped byI the ATS/;GO. This research

wars suPported in part bya C~raduat Fel~l 1owsh ip from

The~ University' of Florida andl .i art by a Trainrteship

from~ the Inational1 Aeron~autics and! SpaIce AdnJrinlS';tjr)I.Lin

Page

introduction 1

Chapter Section

1. Prelim~inaries 4

2. Semimachines and Semiautomata '11

Section 1. General Discussion 11

Section 2. Semimachines 16

Section 3. Sem i automk~t a 22

3. 1-'achines and Automa~ta 33

Section 1. General Discussion 33

Section 2. Auitomata 37

Section 3.. F!, chi:nes 39)

Section O. Semimachiines and Semiautomata 42

II. rl1ul1t ip ro gr.a mm ing E;I!I

Section 1.. General Discussion qrl

Section 2, rMachines with Fre-e Semi groups 62

Bibliography 73

10DL 111 llotation

f: A--B

AMB

(S,X,., Y, *)L'T,:',.,Z1, *)

K(j,Rr)

K(j)

e ~x

f is a mapping from
a set A into a set B

h:S--Tf, j:X--W

A is isomorphic to E

(S,X,.) is semi automaton
isomorphic to (T,Pi,.)

(SX,.p is semima~ch irne
isomorphic to (T,\!,.,o)

(S,Y,.,Y,*) is automaton
Isom~orphic to (T,WI,.7,*)

inHput right congr!:n nce
relative to a subset T
of a state space

quasikernel of a mapoing
j and a relation R

kernel of a mapoine ]

the equivalence class
containing the identity
of a semigroup

semii ringq on the power
set of a semi~erono X
and el1emen t\li se
multiplication

semi ring of matrices over
H1{X) with analoanus
matrix operations

the subscriPt of 1 in b
where b is an elementary
basis element of a vector
space of n-tuples

Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the R~equirem~ents For the Degrce of
Doctor of Philosophy
at the University of Florida

THEOREIS FOR
FINITE AUTIOHATAI

B3y

Roverdy E~dmond \'right

June 1971l

Cha i rman: Ralph C. Selfri dge
Iajor- Departmnent: rrathemastics
Minor rDepartment: Industrial and Sylstems Engineering

The automata of this paper are finite in that they

have a finite state space and finitoly generated

semigrouips. Definitions of these systcns and! useful

r.1ap i ngs be t e en them are developed and used to examine

their properties.

Au~tomata without output are shown to be isomorphic

to som~e ?!ith state spcces that are right congruences whecn

a start state is designated or mappings froml~ subsets of the

inteGcrs into the integers for state spaces otherw~ise.

SEm I r ing of matr i ces ovra seml r ing isused in the

development of the~ latter.

Automata with output are shown to be homnomorphic to

some wI~histh-tat spaces that are sets of functions from the

i npuit s em ig rlu p to mappi~ngs from input to output. These

homomo rph~i;sms i nvolve i den t ity miapp ings on the input and

output and are shown to indicate "black box" equivanence

between automata.

The idea of' mul1t i p rog ramm ing is def inedi wit hou t

recourse to products of automata. Certain products are

shown to be multi programming automata. In some cases, the

produce t is the mi n imal1 automaton which can accomplish a

Thle ability of a given automaton to act as a

multi programming au~tomaton is investigated. The existence

of input elements satisfying the condition that, except for

the identity, no power of one is a powel~r of the other is

shown to be necessary but not sufficient. A further

condition is shown to be sufficient.

In the cast" of free semligTroups i t is s how1n that

the generators of the input: semigroups need not bie the samie

in the simulated automaton as in the multiprogramming

Sautomaton.

vil

INTRODUCTION

This dissertation treats a special class of

automata, finite sequential automata with finitely

generated semigrouos of input and output. Because the

terms used in the Theory of Sequ~ential Automata are defined

differently by various authors, they must be defined at the

outset of this paper. Although all semi groups are assumed

here inI to have i dent ity elements, this fact will be

emphasized by referring to them as monoids from time to

time. No topology need be assumed For any semigroup or

state space.

The topic is developed by starting w~ith- a class

(semimachines) of sequential automata with designated start

states and no output. The development continues with an

investigation of the class (somiautomata) of sequential

automata with neither start state nor output. In each

case, homomorphisms are investigated and canonical forms

developed.

A semi ring of matrices w~ith subsets of sem;igr oups

for- entries is generated during the investigation of

semiautomata. Each semiautomaton is shown to have a finite

subset of these matrices corresponding to it. Any w

sem i autopa ta with the same input semitroupJ are ;somorphic

if and only if they correspond to the same set of these

matrices.

In Chapter 3, the classes (automata and machines)

of sequential automata with output, first without and then

wri th des ignated start states are i nve s tigatred .

Homomorphism is shown to be a sufficient condition for

"black-box" equivalence of sequential automata. Canonical

forms are developed for minimal sequenti al aut oma ta, those

writh the smallest number of states which have given input

to output properties. Finally, with the help of the

canon ical forms for semi mach lines and sen iau toma ta,

canonical forms are developed for those automata and

machines which are not necessarily minimal.

In Chapter 4, the idea of multiproeramming for

sequential automata is explored; first for automata and

machines as developed in the previous chapters and then for

the special case where the input semigfroulp is a finitely

generated free monoid. The notion of product is not used

in the defi n it ion, but product automata are shown to be

examples of multiprogramming automata. The principal

result might loosely be stated, "The most efficient way to

build a machine to do the job of two machines is to put

these two machines into a single box unless there is a more

efficient way to build at least one of the two machines."

Doth necessary conditions and sufficient conditions for a

machine to be a multipror~ramning machine are investigated.

An example s howi ng: that, even in the case of free

sem i;'rPou psilll' of i ripu t and output, the generators of

semigroups for simulating and simulated machines needl not

be the same is also important.

CHAPTER 1: PRELIMINARIES

Def i nit ion 1. A SemIa u toma ton is a triple

(S,X,.) consisting of a non-empty finite~ set S, a finitely

generated semigroup X, and a function fromn SxX into S

with the property that s.(xy)=(s~x).y for all s in S and

all x and y in X. Th~e set S Is -icalled the state space.

The semigroup X is called the input semigroup. Frequently

a finite set of generdsors .bf X will be f81ected and called

the input alphabet. In particular the generators of a free

monoid are thus designated. When an input alphabet is

designated, each generator other than the identity is said

to have length 1, the identity is said to have length 0,

and length is defined for all other elements of X by the

shortest product of generators. The function is called the

trans it ion funct ion and its property s(xy)=(sx)y, the

sequential property. The image under the transition

function will usually be denoted by juxtaposition.

De~f in i tion 1. (1. A Semimach ine is a quadruple

(S,,.r)where (S,X,.) is a semiautomaton and r is an

ingress, a state such that, for all s in S, there exists an

x In X such that: ser.x. There may be other ingresses bu t

only the one designated is the start state.

Definition LO],. An Automaton is a quinituple

(S, X, ,Y,*) whe re ( S, X,. ) is a sir m~i I utoma ton, Y is a

semigroup and the output function is a function from SxX

into Y with the property that, for all s in S and all x and

z in X, s*(xz)=(s*x)(sxxz) and Y is the monoid generated by

the range of the output function.

DeFi n it ion 1_ S}. A M~ach ine is a sextuple

(S,,.,,Y,) here (S, X,., Y, *) is an au toma ton and

(SX.,r)Is a semimachine.

The definitions of the systems are generalizations

of those used by S. Ginsburg (Gl) and A. Ginzburg (G2 and

03). Ginzburg's definition of semiautomaton is restricted

to finitely generated free monoids. Ginsburg's complete

sequential machine and Ginzburg's Me!caly machine or Mea ly

automaton are essentially this paper's automaton restricted

to finitely generated free semigroups of input and either

free monoids or right zero semigroups of output.

Ginsburg's quasimachine is still more general in that its

state space may be infinite. His abstract machine is less

general in that its output semigroup must be left-

cancellative. (G2 and G3) (Gl) (G2 and G3) (Gl)

Ginzburg's cyclic semiautomaton and its generator

are restrictions of this paper's semimachine and its start

state. An extension analogous to that from semiautomaton

to automaton leads to the definition of machine.

Definition 1.05. A Semiautomaton Homomorphism is

a pair of mappings (h,j) such that h:S--T, j:X--Z, j is a

semi group homomorphism, and (sx)h=(sh)(x)). 'f both h and

j are one-to-one and onto, t~he n the pa ir (h, j) is a

semiautomaton isomorphism.

Definition 1.06. A Semimachine Homomorphism is a

semiautomaton homomorphism(h,j) such that the image of the

start state of the first machine is that of the second. If

(h,j) is a semimachine homomorphism and a semiautomaton

isomorphism, then it is a semimachine isomorphism.

Defi nation 1.07. An Automaton Homomorphism from

(S,X,.,Y,*) to (T,Z,.,W~,*) is a triple of mapping (h, j,k)

such that (h,J) is a semi automaton homomorphism and such

that k:Y--l is a semi group homomorph ism and

(s +x)k-( h) *( x)) for alli s inl S and all x~ in X coice

thiat neither the sequential property on the output nor that

k is a semigroup homomorphism implies the other. If k is

a semigrouip isomor-phism onto W and (h,j) is a semiautomaton

isomorphism, then (h,j,k) is an auitomaton isomorphis~m.

ef i n It ion 1.08. A Fra c h i ne Homomo rp h ism is an

aut oma ton homnomorph ism (h, j, k) such that (h, j) is a

semimachine homnomorphism. If (h,j) is a semimachine

isomorphism and (h,j,k) is an automaton isomorphism, then

(h~~k)is a machine isomorphism. W!hen the meaning is

clear from the context, I frequently be used to denote

the identity mapping on one of the state spaces or

semligroups. For example, (,,) (S,XY,.,Y, *)--(S,Z,.,Y, *)

woul d mean a triple consisting Iof the identity mapping cn

S, a mapping from X Into 2, and thle identity mapping on Y.

!Jomomro r phi smr i s de f inedr for sie~m i automa ta by

Ginzburp. (G;3) in an analogous fashion. It has also been

defined by otes (e.IIIIIIIIIIIL g. Foarris (N1)) for other types of

automata w~ith only inpu!ts and states. Isomorphism, but not

homomorphism, is defined for automata with output by

Hartmanis and Stearns (HS). Ginsburg= (Gl) likewise defines

only the former. Arbib (KFA) defines homomorphism slightly

differently but makes little use of his definition.

Definition 1.09. A Subsemiautomaton of a

semi a ut oma ton (SX) sem ima c h ine (SX) aut ona to n

(S,,.Y,),or machine (X.,Y,)is a semiaut-omaton

(S',X',.) such that S' is a subset: of S, Xi is a finitely

generated subsemigfroup of X, and the transition functions

agree on the elements of S'xX' and the restricted range is

within S'

Definition 1.10. A Subsemimachine of one of these

same systems is a semimachine (Sl,X;',.,q) whifere q is some

element of S (not necessarily r), X' is a finitely

generated subsemigroup of X, and the transition functions

agree on S'xX'.

DeFi n it ion 1.11i. A Subautomaton of an automaton

( S, X ,.,IY ) or machine (,.rY,)is an au toma ton

(S'X',,Y'*)such that (S',X',.) is a subsemiautomaton,

the output Functions agree on S'xx', and the range of the

restricted output function generates Y'.

Def ini t ion 1.12. A Submrachine of an automaton or

mac zh i ne is a mach ine the a ut ona t on of lh ic h i s a

s~ub a utoma ton ;arn i the s em ima ch~in e of wh ic h is

subsemimachine.

Djef~i n it ion 1 .13 The Internal Prorduct

Semin !u t oma ton of the f ini te set of semciautoma~ta (Si,X:i ,. i)

whecre !=,.n and the Xj a re s ub5s cm i rou ps or a

semigroup X such~ that, for any 1, the mapp~ing; i which maps

each element of Xi onto itself but mosps all other elements

of the union of the XYi onto thec identity can be ex:tendied to

a semi Irou p hc~omomorph ism, is a semisutomaton (TA,.) where

T Is the Cartesia~n pr-oduct of the Si, W' is the subse-migroup2

of X generated by the union of the Xi, an-d the transition

function is defined For each element x of the union of thel

Xi by coaord~inatewrise tran~sition onr all coordin~ates whi~~~er x

is in the respctive Xi and by1 the identity tralnSitionl on

2ll otle r coor~d inates andl th~e tr ans i ti on functi on i s

defined for all other' elements by the sequecntial property.

The i nt'e nal pr~oduc t semi automa toni of a: set of

seli rl)c7h i nes, au tomata, or' m~achines is decfinedi as that: of

the i : scm i aut~omaJta.

if ea~ch input mnonilid Xi is d'isjoint: 'r om eachl of

the o the rs, this product is esse~ntially rGinibuir.E's direct

p~roduc t ().IF thenre is a pair of inrput monld wi th

ele-mentIs other than t~he identity in their inerein th r

eFffct of an! Inout Fr-o' this intersection is seenr in :raore

than one of: the coo~rdina~tes. Of p~alrticurlar interest is the

case whenlr :cX1=...=X~n whenc; eachn input'i may7 affect ev er y

coordC1ir.a t?. An exam~ple of; this is seen inl thle

represenc;tativi e semiaach~ine defined~ beclow~.

Defniton1_1. he Internal Product S rm imach in e

of the fini te set of semlmachines (Si,Xi,. i,ai ), where

i=1,,...,n and the Xi have the same property as in the

p rev i ous def in it ion, i s the semimachine (T,lU,.,q) wher~e

(T,,.)is the appropriate subau toma ton of the i nte rnal

product au tomlaton of the semimnachines and the start state

Definition 1.15. The Intcrrnal Product Automaton of

the f in i te set of au toma ta (SiiY~w) here

i=1,...,n, the Xi have the same property as in the previous

definition, and the Yi are isomorphic to subsemigroups of

a semigroup Y, is the automaton (T,1N,.,7,*) w,here(T,)

is the internal product semi automaton of the semiautomataa

(Si,Xi,.i) and Z is the subsemigroup of Y generated by the

imageof TxX: under the mapping defined by extending the

*i such that (...,si,...)*x corresponds to (si)(*i)x For

iall x in Xi, provided the output sequential property holds.

The internal product automaton of a set of machines is that

of their automnata.

Definto .6 The Internal Product IMachine of

the f ini te set of mach lines (Si.,iY~i,where

l=1..,nand the Xi and Yi have the same propr~t?!ies as in

the previous definition, is the machine (T,H!,,,q,Z, ) where

(TW.,)is the internal product sem~i imac h in e and

(T,,,,,* is a sub7u tomariton of the internal product

au tomnaton.

Definition 1.17. The Representative Semimachine of

a semiautomaton is the internal product semimachine of the

subsemimachines of the semi automaton starting with each of

its states in turn.

Def ini tion 1 11. The Representative Machine of: an

automaton is the internal product machine of the

s ubmac h lines of the au toma ton starting wi~ith each of its

states in turn where the output semigroup is the Cartesian

product of the output semigroups of the respective

submachines.

cHAPrTER 2?: SEMINACHINES AND SEr~__AUTOMATA:~!

Section _1: Genelral Discucs i on

The presentation in this chapter doe s not f oll1ow

the development usually made for semiautomata with finitely

generated free semigroups of input but does touch on

familiar results. The usual treatment of homomorphisms is

either to restrict the semi group homomorphism on the input

to a semigroup isomorphism or to the homomorphism on free

semigroups induced by a mapping of generators into

generators.

Th~e relations on the input semigroup~ and

equivalence classes are frequently mentioned in connection

writh the class of regular subsets of a s em i g rou~p. The

classic works for regular subsets of finitely generated

free semigroups are those of Kleene (K) and of Rabin andr

Scott (RS). M~crnight (Mc) has generalized Kleene's

results. A very lucid 'treatment of regular sets is made by

Cinzburg (G3).

Def in iti on 2. 01 The Input Right Congruence

relative to a subset T of the state space S of a

semimachine or semiauto~maton is an equiivalence relation

E(T) on this input semigroupp such that (x,z) is in E(T) when
11

tx=tz for all t in T. Clearly this is a right congruence

relation.

Proposition 2.0~2. I;'f T is a subset of T', then

E(T') is contained in E(T).

Proof: Follows immediately from the definition. The

converse does not necessarily hold.

Definition 2.03. The input Congruence of a system

with state :pace S is E(S).

Defi n it ion 2.a4. The Input Right Con,7ruence of a

semimachine with start state r is E(r), the input right

-congruence with respect to the singleton set of th~e start

state r. Notice that any subset T of the state space S

which contains an element s has the property that F(T)

contains E(S) anid is contained in E(s). E(S) is clearly

the intersection over all s ini S of the E(s).

Definition 2.0)5. For any mapping j from a set X to

a set Y and any relation R on Y, the set of all pairs

(X,x') in X with the property that ()xj is in R is

cal led Kt,) the ~u a s i k e rn l of j and R, or KUj), the

Kernel of j if R is the i dent i ty relation. FMorr is (N)

defines the kernel in this way. The definition of

quasikernel is an obvious Extension.

Proposition 2.06. For any semimachine (S,X:,.,r) or

semi autom~aton (S, X,. ) the pai r of mnappings ( i,j wh ere

is the identity mapping on S, j is the natural mapping from

the elements of X to the elements of X/E(S), and the

transition function is the naturally induced s(x))*sx, is

a semimachine or semiautomaton homomorphism.

SxX S

Sx(X/E(S)) ~-- S

Proof: If (y,y') is in E(S), then for all s in S sx Is

also in S and (sx)y=(sx)y'. Hence E(S) is a congruence

rela t ion, X/E(S) is a semigroup, and j is a semierouap

homomorph!Iismi.

This result is a special case of a result of

Bednarck and yJallace (8W2). The semi group of

transformations induced on S by X is isomorphic to X/F(S)

and is called by Ginzburrg (G2) the Semigroup of the

semimachine or semiautomaton.

Prop~os it ion 2. 07 if {,)and (h,' are

semimachine or semiautomaton homomorphisms front (S,X,.,r)

to (S', X',., r') and thence to (S",X",.,r") respectively or

from (SX) to(S,'. and thence to (S",X",.), then

(hh',jj') is a semimachine or semiautomaton homorp.hism from

(S,X,,.,r) to (S",X",.,r) or from (S,X!,.) to (S",X",.).

SxX -----~ S

(h", J")S' xX '~eS' h

SIIXX' J, '

Proof: (sx)(hh')=((sx)h)F'=C((sh)(xj)))h'=((s)hlh((xj))]'

S( s( hh ')) ix(j j ') and in the case of the s emimrac h ini

homomlor ph ism r(hh')=( rh)h'=r'h '= r". Norris (N) p'rovesd this

for acts as a corollary to a lemma.

Proposition 2.n8. If (h,j):(S,X,.,4)--( T,Y,., r) is

a semimachine homomorphism, then (h, i) is a semimachine

homomlorph ism from (S,X,,,q) to (TX,.,r) if the transition

function on the latter is de ined by (sh)x=(sx)h and

fu~rt he r (i :TX.r)-TY.r is a s em imach ine

h omomo rph ism. L ikew is e, th ere is a d ecom po s it ion of a

semiautomaton homomorphism.

SxX S

(b, j ) TxX ---T h

TxY *T

Proof: ((sh)x)i=(sh~x=(sx)h=(sh)(x))=((sh)i)(x)) and in

the case of the semimachines qh=r and ri-r.

Proposition 2.n9. I ij:SW.-(Y. is a

semiautomaton homomorphism and ] maps W onto X and I is the

i den t ity map ,i ng on S, then W/ E(S)M X/E(S) and

(s,w/E(S),.)u(S,X/E(S),.).

Sx Sxx-;- S

(iJ")' 1\

Sx( ~/E:(s)) s

Sx(X/E(S)) .--- S

Proof: Let j"' be the naturall' mapping from W to WI/E(S) and

j' the natural mapping from X to X/E(S). Further let w and

w' be in W~. If ('' is in E(S) then

s(w~(jj '))=s((w))]')==s( w)j= sw=s(w)")Il=s(w' j")=,..=s(w' (jj'))

for all s in S. I(ww)Is not: in F(S), then for some s

in S s(wl(jj '))=s(~, w)" p( lll)-s(w'J")s(w'jj'))

Since j, j', and j" are all onto there is exactly one

element of X/E(S) corresponding to any v in NI/E(S) and that

is wU)(j') where w~ is any element of: W such that w)"=Lv. r]ow

s(w(jj '))=s(w)")ll=sv.

Sct ion i: SFemimachi nes

A semimachine has the simplest structure of all of

the systems here described. It is a basic part of all

systems since for each state a of any system there is a

corresponding subsemimachine (S',X,.,g).

Definition 2.10. The Right Congruence Semimachine

is defined For any right congruence relation with a

fini te set of equivalence classes on a semrigroup .XF to be ai

semimachine (T, Y,.,e) such~ that: The states are equ lvalience

classes of E, the start state e is the equivalence class

containing the identity element a of X!, and the trans it ion

function takes (g,x) to the equivalence class containing x

and (s,x) to the equivalence class containing zrx where z is

any element of s.

Proposition 2.11. For any em Ima ch in e (,,,

the pair of mappings (h,1), where 1 is the identity on X

and h maps each s in S to the set of all x such that rx-s,

is a s emiima ch ine isomoriphism from (S,X,.,r) to the right

congruence semimachine of E(r).

SxX .--- S

(h,) h

(X/E(r))xX' -T--/E~r)

17

Proof: For all x in X, x is an element of (rx)h. If x is

an el 1empn t of both sh and th, then s-rx=t. x and y are

elements of sh if and only if rx-s=ry. Since re=r, e is an

element of rh. if z is an element of sh, then since z is

an element of sh zx is an element of (sh)x and since

sx=(rz)x=r(zx) zx is an element of (sx)h.

Propos it ion 2.12. Let E a~nd E' be right congruence

relations with finite numbers of equivalence classes ove~r

a finitely generated semigroup X. L~et i be the identity

mapping on Xr. If E IS a subset of Fthen there is a

semimachine homomorphism (h,i) from the right congruence

semimachine of El to that of E. If E is not a subset of

E', then no pair of mappings (5,i) is a semimachine

homomorphism from the right congruence semimachine of E to

that of E'.

Proof: Assume (x,y) is in E but not in E'. The qualities

(rx)h=(rh~x and (ry)h=(rh)y must hold and rh must be the

start state of its semimachine for (h,i) to be a

semimachine homomorphism, but (rh)x/{rh~y while rx-ry.

Assume E. is a subset of E'. Since X/E refines X/E', h can

be defined such that s is a subset of sh for any s in X/E.

Clearly e is an element of both e and clh. Since the

transition function of each machine is defined by right

mul1t ipli cat ion, sx is a subset of (sh)x. Since sx is a

subset of (sx)h and X/E' is a partition, (sh)x=(sx)h.

FIPropos it ion ] 1. For any s em ima ch i n e(SX.

there exit 6,(Ll pd;pping b from X/E{S) to S such that thel

18

pa ir (h, i) is a semimachin~e homomor ph ism f rom the r i ght

congruence semi mach ine of the input congruence where i is

the identity on X. Further, for any x in an equivalence

class t, rx=th.

(X/E(S))xX -- X/E(S)

(h,) h2

SxX *S

Proof: Since E(S) is contained in E(r), by proposition

2.12 there is a semimachine homomorphism (h,)from

(X/E(S),X,.,e) to(/Er,.e. From proposition 2.11

there is an i somorphi sm (h,)from X/r),.e)to

(S,,., ). By propos it ion 2.0!7 ( i)( h" i) is the

required semimachine homomorphism. Since any t in X/E(S)

is a subset of th' in X/E(r), any x in t is also in th'

wJhich is the set of all z such that rz=th'h". Hence rx=th.

The representative machine of (S,Xv,.,r) is clearly

i somo rph ic to the ia c hin e (/S,,.) The

representative machine is maximal in the sense that any

machine having the same semi group of transformations is one

of its homomorphic images. Although there is at least one

machinef with as few~ states as any other w,ith the sarme

s em i erou p of t r an s forma t ion s, there may be none for which

some semimlachine homomorphism can be found from any other

w~ith the sameI semi group.

Pro~ogiton 210. Let (S,X,.,a) and (T,X,.,r) be

two semnimachines with the same inpuit semirroioo. The inpuit

19

right congruence of (S,X,.,a) refines that of (T,X,.,r) if

and only if there exists a semimac=hine homomorphism (h,i)

from (S,X,.,q) to (T,X,.,r) with i the identity on X.

Proof: By proposition 2.11I, each machine is isomorpMiC to

the right congruence semimachine of its input right

congruence. Since semimachine homomorphisms compose, there

is a semimachine homomorphism from one semimachine to the

other if and only if there is a corresponding semimachine

homomorphism between right congruence semimachines of their

input r igh t congruences. Th i s in turn, is equivalent to

the refinement of E(r) by E(q).

Proposition 2 11. Let E(q), E(r), and E'(r) be the

respective input right Congruences of the machines

(SX.,q, (TY,.r),and (T,X,.,r) of proposition 2.08.

Proof: For any x and z in X,

rx=(qh)x=(qx)h=(ah)(x))=r(x)). and

rz=(qh)z=(qzh-~)=(q)ZJ)=r~zj). Hence rx=rz if and only if

Pr opos i tion 2. 16 Let (S,X, .,q) and (T,Y, ., r) be

semimachines. If there is a semimachine homomorphism (h,j)

from the former onto the latter, then the input right

congrucene of the former refines the quasikernel of j and

the input right congruence of the latter. If there is a

semigroup homomorphism from X onto Y such that the input

right congruence of the former refines the quasikernel of

j and the input right congruence of the latter, then there

is a mapping h f rom S on to T such that (,)is a

semlmnachine homomorphism.

Proof: Assume that there is a semigroup homomorphism )

such that E(q) refines the quasikernel of j and E(r). The

mapping b defined by (rxlh=0(xj) for each rx in S is the

mapping such that (h,j) is a semimachine homomorphism. The

remainder of the proof follows directly from propositions

2.08, 2.14, and 2.15.

Proposition 2.1.7. I (i ):S ,,r S, .r)is

a semimachine homomorphism, then H!/E(S)UX/E(S) and

(W/E~r)x(W/E(S))-------W/E r

Proof: Since any state in S can be expressed as rw or rx

j is necessari ly onto. The rest of the proof follows

directly from~ 2.09 and 2.11.

Right congruence semimachines can be considered

canonical forms for semimachines. Each semimachine is

isomorph~ic to the right cong:ruence semimachine of its input

right congiruence. All of the properties of a semimachine

qua semimachine are those which are preserved by the

isomorphism from a s em ima ch ine to i ts =c;ano i c al fo rm.

21

Propos it ion 2.16 gives a criterion for the existence of

homomorphismns in terms of: right congruences.

22

Section 3: Semiautomata

A semiautomaton with at least one ingress can be

investigated by treating it as a machine with one of its

ingresses as start state. Decomposing a semiautomaton into

semimachines does not preserve all of the characteristics

of a semiautomaton unless correspondences are defined

between the states of each semimachine and the set of

semimachines.

Cons ide r the foll1owri ng ill 1ust r at ion of the

d iff icul1ty: Let X be the commutative semigroup w~ith three

elements such that ee=e, ea-ab-a, and eb=aa=bb=b. Let the

t ransi t ion func t ion for the machines ({1,2, 3,ri ),X,.) and

((5,6, 7,8e 1, X,.) be given by the Cayley tables:

eab eab
111 2 1 515 6 5
212 1 2 616 5 6
313 1 2 717 6 5
414 1 2 818 5 6

{{c, b },{(a }=X/(1E (1 ) =X/iE ( 2) = X/F_( 5)i= X/ ( 6) ,

{{ ),{a)(J~ ,{(b}})=XC/ E( 3) =X / E([4 ) =X/F:( 7)=X / E~( 8>.

But no isomorphism exists since 2a=3a=0af1a while

6a=8af~a Sa.

In pursuit of a canonical form similar to the right

congruence s~emim;;achine it will be convenient to examine the

foll 1ow ing system expressed in terms of finite dimensional

vector spaces:

Let F; be a basis of a finite dimensional vector space Vr and

A be a semigroup of linear transformations of V with the

pr-operty that, for all (b, a) in RxA, be is in B. Clearly

(B,A,.) Is a semiautomaton.

Proposition 2.18. For any semiautomatoni (S,X,.)

with n states, any basis R of an n-dimensional vector space

V', and any one-to-one mapipiling h fr-om S to B, the mapping j

fromn X into the set of linear transformations on V defi ned

by (sh)(xj))(sx)h is a semigroup homomorphism. A fortiort:

(h,j) is a semiautomatoin homnomorphism and the semiautomaton

homomorphism (hi:SX.-(,,)is a semiautomaton

isomorphisml.

Proof: For each x in Xthere is a unique linear

tr a nsf o rma t ion c a -rry i n each sh of I8 to (sx;)h. Let j be

the mapping which carries each x in Xto this

transformation. That is to say: (sx)h={sh)(x)). For any

x and z in X, (x)j)(zj)={xz)j since for each s in S

(sh)(x)){zj)={(sx)h){Cz))={(sx)z)h=(s~ x)h=s)(x))

The existence of the semiautomaton homomorphism (h,i)

-follows from proposition 2.08. Since b is one-to-one andi

onto, {hI)s a semriautomaton isonlorphism.

ain the vectcor space is the set of n-tuples over

a field and the basis is the elementary basis, then the

matrices of the linear transformations give a peculiarly

graphic picture of the semiautomaton. This representation

is well known in some circles. (A) In this case,

multiplication of an n-tuple by a matrix amounts to

selecting a row of the matrix since each n-tuple consists

of a single 1 and several O's. Hence, each row of every

matrix must be a member of the elementary basis. Since the

only elements of the field that are used in this

construction are the zero and the one, the set of n-tuples

over any semi ring with zero and one which cori-esponds to the

elementary basis of a vector space and these same linear

transformations are isomorphic to this construction.

In this case the mapping j is a representation of

X by a Rees matrix semi group over a group w~ith zero since

the set with elements 0 and 1 under multiplication is such

and only one element of each row is nonzero. The

representation is faithful when X"X/E(S).

The following semi ring of matrices over a semi ring

of subsets of a semigroup provides a useful tool for the

next step in the development of a canonical semiautomaton.

Consider the power set of a semiffroup X under the

operations of union and setwise multiplication H1(X). Roth

operations are associative. Union is commutative.

Mlul t ipli cat ion d is tr ibutes ove r union. The empty set is

the zero and the singleton set of the identity of X is the

identity of H1(X).

it igh 3.2,3]

For Iny monold X, H1(X) is thus~l

de f ined.'

Since the distributivity of the us ual mat r ix

product over addition depends only on the above properties

of a semiring, the n by n matrices Uin(X) ove r HI(X) wi th

union defined coordinatewise and multiplication defined by

AB(i;k)= UA(i;j)8(j; k) is a semiring.

Defini t~ion.2 2.29. Hn(X) is thus defined.

Kameda (Ka) has independently developed several

matrices' over sets which relate regular express ions of

input to regular events over the output alphabet. One of

these matrices resembles matrices of Hn(X).

Investigation of all of the properties of Hn(X) is

beyond the scope of this paper. Some properties which are

not necessary for proofs will be mentioned to provide a

mo re graph ic picture of the development of a canonical

form. Definitions may be considerably more general than

absolutely necessary.

Definition 2.,21. MiulItiplication of an n-tuple s

over H1(X) and a matrix z in Hn(X) is defined such that

sz(j ) is the union of thle s(i)z(idj) where i =1,...,n. Thi s

is analogous to multiplication of a vector by a matrix when

both are over a field.

Prqposition Zall. A matrix P is a right unit if

and only if every entry except one in each row and in each

column is the empty set and every nonempty entry is a set

of right units which have a common right inverse.

26' "1r"";

Proof: Let Q be a right inverse of P. Each diagonal entry

of PQ is the singleton set of the identity of X. Hence for

each i there must be a i such that P~;)~~)is this

singleton set;hence neither P(i;j) nor Q(j;i) can be empty,

but rather all of the elements of P(i;j) must be left

Inverses of all the elements in Q(j;i). Since for all kfi

P~k;)Q(~i)must be empty and n(j;i) Is not empty P(k;j)

must be empty. Since the number of rows is the same finite

number as the number of columns, only one j can correspond

to a single i in the previous statements.

In H1l(X) there will be one-sided units if X has

one-sided units. In this case the definition of similarity

of matrices usually formulated does not necessarily give an

equivalence relation. For example, if X is the bicyclic

semigroup with ab=etba and y is said to be similar to x if

there are elements p and q such that pg-e and xs=oy. Since

p must be some power of a and q must be the same power of

b, the set of elements of X similar to any given x will

include x, axb, aaxbb, etc. For any x in X, x is simi lar

to bxa but bxa is not similar to x. To avoid this

difficulty the definition of similarity will be restricted

to twPo-sided units.

Definition 2.23. The matrices A and P in H~n(X) are

Similar if there are matrices P and 0 in Hn(X) such that

P=QrP=I where Iis the identity of Un(X) and AQ=QB.

It should be noted that Hn(X) has no one-sided

units if X has none. This is because

Pci;j)Q(J;i)=Q(J~i)P~i;J) Is either empty or the singleton

set of the identity of X.

Definition 2.24. The matrices A and n in Hn(X) are

Strictly Similar if there are matrices P and 0 in Un(X)

such that each nonempty entry of P is the singleton set of

the identity of X and POl=i and A=PBQ). clearly B=QAP and nl

is the transpose of P. Strictly similar implies similar.

If, as in the case of free semigroups, there are no units

ot her than the identity, then strict similarity and

similarity are equivalent.

Definrition 2.25. In Hln(X), the partial order I is

defined by AFB when A~UB=B.

Proposition 2.26. If A
Proof: (AC)U(BC)=(AUB)~eC= and (CA)U(CB)=C(AUB)=CB.

Propos i tion 2.27. Let (S,X,.) be a sem~i aut omaton.

Let Bbe the elementary basis of n-tuples over a field.

Let j be a semigroup homomorphismn as defined in proposition

2.18. Define m':Y.--Hn(X() such that every entry of xm' is

an empty set or a singleton set of x as thle corresponding

entry of x) is a zero or one respectively. m' is a

semigroup homomorphism from X into the multiplicative

sem~i roup of Pn(X).

Proof: (xy)m' has singleton sets of xy and empty sets in

exactly those places that (xy)j has ones and zeros

respect ive ly. The product of xm' ; and ym'' has s ingleton

sets of xy and empty sets in exactly those places that the?

product of x) and yj has ones and zeros re seciivel1y.

Since {xy)j=(x))(yj)) it must follow that

(xy)m'=(xm')(ym').

Proposition 2.28. L~et (S,X,.) be a s aliau toma ton

Let m' be the semigroup homomorphism defined in proposition

2.27. Let m:H1(X)--Hn(X) be defined so that for any non-

empty subset a of X am=U~xm'IXE a~and m maps the empty set

to the n-by-n matrix of empty sets. m is a semirinr?

homomorphism.

Proof: ( aUb )m = U!Ixm' jxe( aUb )} = ( UIxm' xrea})U)L(UE xm ixcb})

=(am)U(bm). Since (xy)m'*(xm')(ym') by proposition 2.27,

(ab)m= {xm'ixcab} = ((xy)m'[xca,yeb} = {{xm')(ym')jxca~yeb}

={xm'lxca}{ym'|yeb}={a)(m)(b)

Proposi t~ion 2.29. Let (S,X,.) be a s em iau toma ton .

Let G be a finite set of generators for X. Let m and m' be

defined as in proposition 2.28. Define M(0) to be the

image under m of the identity element of X. Define M(n) to

be the image under m of the set of all words in X of length

less than or equal to n with respect to the input alphabet

G. The n-th power of M(1) is M~n).

Proof: The case when n=13 is trivial. Assume th~e case! for

n-1 and calculate the product M~(n-1)M(1). If x in X is of

length n or less, then there is a y of length n-1 or less

and a z of length I or less such that x=yz. xm'AM(n) only

if ym'iM(n-1) and zm'AM(1).

Conversely, if ymr'ltdtn;-1) and zl'(AM(1), then (yzjll~meF' -n).

Proposition 2.30. In the notation of propositions

2.28 and 2.29, Xm is the limit as n increases without bound

of M~(n).

Proof: Every x in X 1s a product of a finite sequence of

generators. Therefore XmAUM~(n) S since each M(n) is

clearly contained in Xm, Xm=UM(n).

Proposition 2.31. Continuing the notation of

propositions 2.27 through 2.30 ndadding the notation:

I?=Xm and b is the integer such that b(h)=1, Ml~sh;ft) is the

set of all x in X( such that sx=t.

Proof: IMultiplication by sh is equivalent to selecting the

sh- th row of a matrix. The sh-th row of x) will be th if

and only if sxet. In other words the th-th entry of the

gh-th row of x) is 1 and the same? entry of the same rowr xm'

is th~e singleton set of x if and only if sx~t. M~(sh;th) is

by definition the union of the corresponding entries for

all x in X but only those x for which sx=t have a non-emipty

entry in this position.

PLQposition 2.32. Let k' now be defined to be any

one-to-one mapping of the state space of (S,X',.) onto the

integers 1 through n wJhere n=ISI. Let k now be defined so

that it is that mapping from S into the set of functions

from X to the integers 1 through n such that the image of

x under the mapping sk (notation: x:(sk)) is (sxik'. The

pair of: mappings (kc,i) where i is the identity mapping on

X is a semi automation i somorphi sm from (S, v,. ) to (Sk,X,.)

writh thle naturally induced transition function (sk)x=(sx)k.

30

Proof: if sft, then (se)k'=sk' ftkC'=(te)k' and hence

e:(sk)fe:(tk). The remaining properties of a semiautomaton

isomorphism follow directly from the hypotheses.

Proposition 2.33. Let the notation be as in

proposition 2.32. For all x and y in X

(xy):(sk)=y:((sx)k).

Proof: (xy):(sk)=(s(xy))k'=((sx)y)k'=y:1(sx)k).

Propo~s ition 2.311. If k and k' are d~efined as in

proposition 2.32 and sk'=sh in the notation of Proposition

2.31, then x is an element of Il(sh; x:(sk) )

Proof: 9y proposition 2.32, x:(sk)=(sx)k'=(sx)h. "y

proposition 2.31, x is an element of M?(EE (x;)

Prop~os it i on 2.35. Let the mappinffs h and m and the

matrix I: be those defined in propositions 2.27, 2.28, and

2.29 relative to (,.)and h", mi", and r" be those

relative to (T,X,.). Let n=|Slin"=1TI. The existence of

an n" by n matrix P over H1(X) and] its transpose n such

that every column of P has n"-1 empty sets and one

singleton set of the identity e of Y. for entries and PM0.=M"

is a necessary and sufficient condition for the existence

of a ma7ppings g from S onto T such that (e,i) is a

semi au toma ton hlomomnorph isml.

Proof: Assume that (_s,i) is a semimachine homonorphism

from (S,Y.,.) onto (T,X,.). Since g is onto and h and h"

are both onto and one-to-one, there is a mapniing p from the

first n positive integers to the first n" such that

shp=gsph" for all s inl S. Let P be defined so thatPj,"

__

is the singleton set of a "CIIfI~l~ j" ~llj and is the empty set

ot he rw:ise. Let x bk ;any e''lleen t ofllli M"I( th";vb") There

exists an element s of S such that sA=t. Since g is a

s em imach i ne homomo rph ism, ( sx )X=v, The mapping p laSS

defined so that _shp=th" and sxhp=vb". x is an element of

Mr(sh: ; xh)= P(th'' h)MA(Sh; SxhT)Q(sxh;_ght) A PMQl(th"l; v") .

H-ence I"(PM40.

Assume nowr that x is not an element of n!(th"';vb"). if

sg=t, then (sx)gfy and sxhrp vb". For all s such that spw.=t,

x is only in r:(sh;sxh) and P(th";sh)M(sh; sxh)0(sxh; vb") is

empty.

Hence POM.It has nowr been shown that M!"=~PEH if: (g,i)

is a semimachine homomorphism.

-!ext assume the existence of the matrix P with the stated

properties. Let p be defined such that. P(jp;j) is a

singleton set for each jUn. Let g; be defined such that

sho=sgh" for all s in S.

Since P(.sbp;sh)M~[sh; sxh )n,(sxh~;sxhp

M~"(5ph!;gh) S since x Is in Fl"(th";.yh") only if tx=v,

(ss)x=(sx)g and hence (g,1) is a semi automaton

homomorphism.

Propos i tion 2.36. Let the n ot at i on be as i n

proposition 2.35. M? is strictly similar to M~" if and

only if there 15 a mapping g frorm S to T such that (s,i) is

a semimachine isomoirphism.

52-"

This is the special case of proposition 2.35 where

The mappingq p is a per~mutation. The matrix P is

of Hln(X).

Proof:

n=n".

a unit

Any semiautomaton which is the image under the

semiautomaton isomorphism (k,i) of proposition 2.32 can be

considered a canonical form. The canonical form is unique

up to permutations of 1 through n. The properties of a

sem iau tom t on qua s em iau toma ton are preserved b y the

isomorphism from a semiautomaton to its canonical form.

E~ven as the transition function of a canonical formi of a

sem imach ine could be d e3te rm in ed f rom the sta~tes, the

transition function of a canonical form of a semi automaton

can be determined from its states.

Proposition 2.34 ties each canonical form to a

matrix over H1(X). Proposition 2.35 then pr~ovides a

characterization of homomorphism in terms of these

na t r i c s. For s em i a utora t a w~i t h different i npu t

semigr~oups, there must be a semi groupl homomiorphismn j such

that each equivalence class of the quasikernel of J and of

the matrix of the image automaton is the corresponding

entry of the matrix PM~Q of the preimage automaton.

CHAPTER 3: MblACHINf~ES AlND.=AUlJC~ilTMAT

SP~cti,ion I: Gene ral ni~s~cussiono

The first diff~erncce between the result ts of this

chapter and analogous results is that the definitions are

more general. Wilaen thl t~aset of f;i n itely ifgeneratedl free

semigiroups of input and output is treated, the egdulk
automaton is isoorhi to a minimal automaton and also has

in its definition all information about the transition and

output functions. Ginsburg (Gl) and Booth (B) make clear

presentations of~ the mlore usual treatment: of mi nimizati on.

Two states s and t are equivalent if there are map~pings c

and c' with the property of c in definition 3.06 from their

respective state spaces into the state space of some

automaton such that sc-sc'.

Promosition 3.01. if (bi) is a machine

homomorphism from (S,X,.,a,Y,*) onto (T,X,.,r,Y,*) or an

automaton homomorphism from (S,X,.,Y, *) onto (T,X,.,Y, *),

then for each state s in S and each x In X, sex=(sh)*x.

Proof: (sh)*x=(sh)*(xi)=(s*x)i-s~x.

Proposition 3.02. I h3k and (h,'k)are

machine or automaton homomorphisms from (S,X,.,r,Y,*A) to

(S'X',,rY'*)and thence to (S", X",., r", Y", *)

rcs pe c t iviely or f'rom SX,, )to (Sf' ,X ',., Y ', *) and

33

thence to (S",Xr",.,Y",*), then ( hh ', i ',kk 1) is a macoh Ine

or automaton homomorphism from (S,X,.,r,Y,*) to

(S",*X"I,.,r",~Y",*) or from (S,X,.,Y,*) to (S",X",.,Y",*).

S xX *l Y

h" S' .---- S'xX' a Y' k"

Proof: By proposition 2.07, (sx)(hh')=(s(hh'))(x(jj')) and

in the case of the machine homomorphism r(hh')=r".

By the definitions, (s~x){kk')=((s*x)k)kkr=(sh)*(x)))k'

={(sh)h')*((x))j')=(s(hh'))*(x(jJ')).

Proposition 3.03. If (h j,k):(S / .q Y *)--

(TW.,r,,*)is a machine homomorphism which is onto, then

(!,~k)is a machine homomorphism from (S,X,.,q, Y, *) onto

(S,X,.,s,Z,x) if the output function on the latter is

defined by s~x=(sex)k and further(hJ):S,,qZ)-

(TW.,r,~w)is a machine homomorphism. There is a like

decomposition of an automaton homomorphism.

S .SxX. -- Y

h S .SxX Z k

1 (h, j)
T TxU! Z

Proof : Csix)i=s#*x (s*x)k= (sh)*;(x)) and in the case of

machines gi-(l and g~h=r.

Proposition 3.04. If (hj):SXq2)-

(TW.,r,,*)is a machine homomorphism which is onto, then

(h,i,i) is a machine homomorphism from (S,X,.,a,7,I) onto

(T,X,.,r,Z,a) if the transition and output functions on the

latter are defined by (sh)x=(sx)h and (sh)*x=s~x and

further (ji) is a machine homomorphism from

(T,X,., r, Z,*) onto (T,W~, ,,r, Z,*).

S SxX TY

S SxX Z

b T .TxX Z

1,j)i

T TxW Z

Proof: Proposition 2.08 provides the proof for the

transition functions and, in the case of the machines, the

start states. For the output functions

(ssx)i=srx={sh)*x=(sh)*(xi) and

((sh)*x~i=(s~x)i={sh)*(x))=((sh)i)*(x)).

Definition 3.05. A machine (S,X,.,q,Y,*) is

M.i n imal1 if there is no mach Ine (TX.rY* such that

ITI<|SI and (ax)*y={rx)*y for all x and y in X.. A minimal

machine is as small as any machine which, having read in r,

maps y to (qx)*y.

Definition 3.06. An automaton(SX.Y) is

Mini mal i f there i s no automation (T,',.,', *) \i th mappi ng

36i

c from S into T such that ITI(ISI and (sx)*y=((sc)x)*y for

all s. in S and all x and y in X.

Section 2: Automata

Let T be a finite set of mappings from a finitely

generated semigroup X onto a finite set of functions from

X; to a finitely generated semigroup Y such that T has the

followingc properties: Y Is the semi group generated by the

union of the ranges of the functions in the range of the

mappings in T. For each t in T and each x in X there is a

t' in T such that, for all y ini X, yt'=(xy)t. Thle image

(yz):xt of yz in X under the mapping Xt is equal to the

product in Y of y:xt and z:(xy)t.

Definition 3.0l7. An Egdulk Automaton is an

automaton (T,X,.,Y,*) where T i s a set of mappi ngs as

described above and the transition function is defined

such that, for all y in X, y(t.x)=(x~y)t and the ou~tpuLt

function is defined by t*xzx:et.

The sequential property is demonstrated by: For all z in X:,

z~t.(xy))=((xy)z)t={x(yz))t=(yz)(t.x)=z((tx.)

The output sequential property is demonstrated by: Since

(ex~t = (xe)t = e(tx),

t*(xz)=(xz):et=(x:et)(z:(ex)t)=(txx)(2:e tx)tx(xz.

~raopos it ion 3.08. Let (S,X,.,Y,a) be an automaton.

Let b be a mapping from S Into the set of mappings from X

into the siet of functions from X lato Y with the property

that anry x le~, X is. mapped by sh to the function which maps

38

any y in X to y:x(sh)=(sx)*y. (ii)is an automaton

homomorph i sm f rom (S, X,.,Y,*) onto the egdulk automaton

(Sh,X,.,Y,*).

Proof: Each i is a semieroup homomorphism. For each s in

S and x in X., (sxx)i=(se)*x=x:e(sh)=(sh)*x and for each y

anid z in X z:y((sx)h) = ((sx)y)*z = (s(xy))*Z = z:(xy)(sh1)

= z:y((sh)x) = z:y((sh)(xi)).

Proposition 3.09. Let (S,X,.,Y,t) and (S',X.,.,Y,*)

be aultomata and h and h' be the respective mappings from

them to egdulk automata. Let h" be a mapping from S to S'.

If (h",i,i) is an aut~omatoni h~omomorphisml, thecn h=h"h'.

Proof: For all s in S and all x and y in X, y:x(s(h"h'))

=y:x((sh")h') = ((sh")x)*y = ((sx)b")*y = ((sxlh")*(yi)

((sx)*y)i = (sx.)*y = y:x((sh),

Proposition 3.10. An automaton is minimal if and

only if the homomorphism (h,i,i) of proposition 3.?8 is an

automaton isomorphism.

Proof: If (h,i,i) is not an isomorphism, then IShl~lSI.

Ey proposition 3.01, (S,X,.,Y, +) is not minim~al.

Assume that (S, X,.,Y,*) is not minimal and let c be the

mapping of definition 3.06. Let s and s' be distinct

elements of S such that sc=s'c. Then h is not one-to-one

since y:x(sh)=(sx)*y=((sc)x)*y=((s'c)x)*y=(s'x)*~~~')

Sectin f: Ma~chines

De~fini tion 3.11. An Egdulk State is a mappi ng r

from a finitely generated semigroup X. onto a finite set of

functions from X to a finitely generated semigroup Y with

the follow ing p roper t ies: The Unbo~Fn of: the ranges of the

functions in the range of r is a set of generators of Y.

For all1 x, y, and z in X, (yz):xr is the product i'n Y of

y:xr and z:(xy)r.

Propos it ion 3. 12. Every state t of an egdulk

automaton (T,X,.,Y,*) is an egdulk( state which maps X into

a set of functions fromt X Into a subsemigroup Y' of Y.

Proof: Let: t be any state in T. For all x, y, and z in X,

(yz):xt is the product of y:xt and zr:(xy)t from definition

3.07. Let A be a finite set of generators of X;. Since

there are a finite number of distinct functions xt wher~e x

is in X and each z:xt can be written as a finite product of

a:xt, b:{xa)t, c:(xab)t,..., and g:(xab...f)t where

z=abc...fg and a, b, c,.,f, and g are elements of A, the

set of all elements a:xt such that a is in A and x ini X is

a f ini te set of generators of: Y'. H~ence Y' is finitely

generated.

Proositon 313.If r is an ensdulk state and T is

the st;t of all functions t such that: there exists an x in

X for which yt=(xyir for all y in X, then (T,X,.,Y,*) is an

egdulk automaton when y(r.x)=(xy)r and rx*y=y:xr for all x

and y in X.

Proof: For each t in T there is an x in X such that for

each y in X vt=(xy)r. The set of ranges of the functions

in the ranges of the mappings in T is eaual to the set of

ranges of the functions in the range of r. The function t'

with the property that: zt'=((xy)z)r for all z inX is an

element of T. For all z in X, zt'={{xy)z>r=(x(yz))r=(yz)t.

For all z and w in X., (zw):yt=(zwl):(xy)r which is the

product in Y of z:(xy)r and :(xlzr and further

z :(x.Yy ) r= z:y wI hi le w :((xi Y ) z) r'\w: i(x yz) ) r=wI:(Iz jt .

Therefore T is a set of mappings for which the transition

and output functions of definition 3.07 can be defined.

Definition 3.14. An Egdulk M~achine is a machine

(TX,.,rlY,*) where r and (T,.,.,,Y,*) are as in proposition

Proposi t ion 3.15. Let (S, X,., q,Y, n) be a machine.

Let r be the mapping from X: to the set of functions From X

to Y defined by y:xr=(ax)*y for all x and y in X. Let h be

defined such that plh-r and y((ex)hi)=(xy)r for all x. and y

in Thle mapping r is an egdulk state and (h,i,i) is a

machine homomorphism.

Proof: For any state s in S there is az2 in X. such that

s=qz. y:x(sh) = y:x((sz)h) = y:(2x)r = (qzx)*y (sx)*y

for all xv and y in 7. ( Sh,XY.,., Y, *) is an au1t oma t on and

(h~~i)is an automaton homomor~phism by proposition 3,n8,

6y proposition 3.12, r is an enduik state. By proposition

3.13 and def in itioan 31,(Sh, X,., r,Y, *) is an egdulk

machine. Since ahir, (h,I,1) is a machine homomorphism.

Propopiti~on T4.26. Let (S,X,.,q,Y,*) and

(S',X,.,q',Y,*) be machines and h and h' be the respective

mappings from them to egdulk machines. Let h" be a mapping

from S to S'. I hii is a machine homomor ph ism, then

h=h"h'.

Proof: Proposition 3.0J9.
Proostin ,.,.. A machine is minimal if and only

if the' homomorphism (i,)of proposition 3.15 is a

machine isomorphism.

Proof: If(~~)is not a machine isomorphism, then

IShl~lS|. By proposition 3.01, (S,X,.,q,Y,*) is not

minimal.

Assume that ( S, X,., q,Y, *) is not mi n imal1. Let

(T,,.rY*) be a machine such that ITJIlSI and

(qx)*y=(rx)*y for all x and y in X. Let x and z be

elements of X such that qx and oz are distinct while rx-rz.

y:x(qh) t (qx)*y t (rx)*y = (rz)*y = (qZ)*y P y:z(Qh) for

all y in X. The mapping h is not one-to-one.

li~i~llllllllllllllllllllllllllF''

Section 4: Semimachi~nes and Semiaultomata

The definitions provide that (S,X,.) is the

semlautomaton, (S,X,.,a) is the semimachine, and

(S,,.,,*)is the automaton of the ma ch ine(,X.,Y)

wyh ile ( S,. ,.) i s the s em i au tona t on of the a urt oma ton

(S,X,.,Y,*). The propositions of Chapter 2 can be applied

to the machines and automata of Chapter 3 w~ith the help of

proposition 3.18.

Proposition _3.18. If (h,i) is a semimachine or

sem iau tomnaton i somorph ism and (sh)*y-ssy, then (h,i i) is

a machine or automaton isomorphism.

PrCoof : (s5 x.) 1= s ~=(s h ) *x-( s h) i( x i) The rest follows f'rom

the definitions.

Each machine defines an input right congruence E(r)

and an input congruence E(S). A machine is isomorphic to

a machine with a right cong~ruence semimachine for its

semimachine by proposition 2.11 and is a homomorphic image

of a machine which has as its semimachine the right

congruence semimachine of the input congruence of the

former machine by proposition 2.13.

Proposition 3.19. The input right congruence of

any machine refines that of its egdulk machine.

Proof : (ii)of proposition ; .1 is a mach ine

homromor ph ism. ( h, i) is a semi mach i n homomorph ism. Sy

proposition 2.14b, the former input right congruence refines

the latter.

The semiautomaton of each automaton is isomorphic to a

semiautomaton of the type described in proposition 2.32.

Using proposition 3.18, a canon ical form with funct ions

into the integers for states can be defined.

CHAPTB@.A : MULtTIPROGRAF!AMING

section 1: Gene~ral Di~scussion

Definition rr.01. A Multiproarammine. ~ Fachinee is a

machine (S' rX ',.,n, Y ',*) wi th submach ine (S,X~,., p,Y, *) such

that there exist machines (V,A,.,q,9,*) and (W',C,,,r,0,*)

where A and C are contained in X and 8 and D in Y and

s emhi grou p homomorph isms Ja1X--A2, k:-8 ':X--C, and k':Y

-[) such that aj =a and aj'=o for all a i n A, bk=b and bk'=e

for all~ b in B, c)=e and c)'=c for all c in C, dk=o and

dk'=dr for all d in D, and such that (alxj))*y)*f(y))={{o)*y

and (r(x)'))k(yj')=(( px)*y~k' for all x and y in X\. T'he

first machine is said to simulate the last twPo.

Definition U.02. A Multiprotramming Automaton is

an au tom~a ton (S',Xi,.,Y',*) wi th subautomaton (S, X, .,Y,*X)

such that there exist automata (V, n,.,tB, ) and(W,,D)

where A and C are contained in X and 8 and D in Y and

semigroup homomorphisms j:X--A, kY-,Jl:X--C, and k':Y-

-D and relations G and H in SxV and SxW such that aj=a and

aj'=0 for all a in A, bk=b and bk'=o for all b in B, c) =e

and c' for all c in C, dk-r and dkl=d for all d in D,

there exists an s in S for each (v,w) in VxWr such that

(s,v) is in C, and (s,w) is in P', ((sx)ty)k=v(tlx)))*(y)1

for all x and yin X and All (s,v) in C, and

((sx)*y)k'=(w(x)'))*(y)') for all x and y in )( and all

(S,w) In H!. The first automaton is said to simulate the

last tw~o.

in less precise language, a multi programming

automaton is one which can s imulia te two other au tomato a

operating independently, starting at any given state and

each receiving its inputs while disregarding: inputs to the

other. O~ne or both of these ma y i n turn be a

multi programming automaton so that one automaton may be

capable of simulating any Finite number of automata.

There may be states in S which are not related by

C or H- to any states of V or I. There may be inputs in X

or outputs in Y' that are not in the semi groups generated

by AUC and BUD. In this way, a multiprogramming automaton

may be able to do more than simulate two automata operating

independently. In particular, there may be control inputs

which may interact with the elements of A and C.

This section treats the case where S=S', X=X', and

Y=Y'. The addition of extra states does not change the

properties of the other states. Adding elements to the

input semigroup may affect several of the properties of the

system. Additional output elements may be included wirth

added input elements. Tehypotheses w~ill include any

needed preferences to these semi groups.

Propos it ion 4 .03 L E t (,.,,*)ard

(H,!C,.,ir,0,*) be machines. Let X be the free product

monoid of A and C, that is, every elovent of X can he

written uniquely as a product of elements of AUC no two

adjacent terms in the product being both from A or both

from C where the identity element of X is in both A and C.

Let Y be the free product monold of R and D. The machine

(VxJ, X,., (ollr),Y, *), the internal product mach ine of the

first two, is a multi programming machine which simulates

them.

Proof: Let j and j' be the mappings from X into Aand C

defined such that ajra, cJe (xc)j=(cx))'=x),

(ax)j=(aj)(x)), (xa)j=(x))(aj), aj're, c~'=C,

(cx) '=( j') x)' (x )J' (x) )(c '),and

(xa)j'=(ax)j'=x)', for all a in A, c in C, and x in X.

Both j and j' are clearly semiglroup homomorphisms. .S since

(v,w)ax=(va,w~x=(v(aj),w)x and (v,w)cx=(v,wc)x=(v,w(c)'))x

for all (v,w) in VxW, a in A, c in C, and x in X,

(q, r)x=(4(x)), r(x)')) and (v,wcr)x=(v(x)),w(x) ')) for all x

in X and (v,w) In VxW.

Let k and k' be the mappings from Y into R and D defined

such that bklb, dkle, (yd)k=(dy)k=yk, (by)k=(bk)(yk),

(yb)k=(yk)(bk), bk'=e, dk'=d, (dy)k'=(dk')(yk'),

(yd)k'=(yk')(dk'), and (by)k'={yb)k'=yk', for all b in n,

d in D, and x in X. Both k and k' are clearly semigroup

homomorphisms. Since (v,w)*(acx)

((v,w)*a)(((v,w)a)*c)(((v,w)ac)*x)

( (v, w ) *a)( (vya ,w) *c)>( (va ,wc ) *x) = ( v *a ) (wIc ) ((va, wc ) x ) and

similarly (v,w)*(cax) = (wr*c){v*a)((va,wc)*x) while

(vea)k=ven, (vea)k:' e=(w.*c)k~, and (w~+c)k'=wr~c for all (v,w)

In VxWl, a in A, c in C, and x in X, it followrs that

( ( (qlr)x)*y)k = ( (o(x) r(x ) )* y)k = (a(x )))*(y ) wlh ile

(((ql,r)x)*y~k' = (r~x)'))*(yjr),

Proposition 4.04. Let (V,A,.,B,*) and(,,.,)

be automata. Let X be the free product monoid of A and C

and Y the free product monoid of P and n. The automaton

(Vx:, X,.,Y, *), the internal Product automaton of the first

twio, is a multi programming automaton which simulates them.

Proof: Let G; be the set of all ((v,w),v) and Y be the set

of all ((v,w.),w) with (v,w~) in VxWl. Def ine the semi group

homomor~i .Ph i sms j, j ', k, and k'as i n the proof of

proposition 4.03. If (s,v) is in r, then there is a wr in

W such that s=(v,w~). P'ence, by proposition 4.033, ((sx)*y)ke

= ((v~~x)y~k= (~x))*(])in the subm~nachine with start:

state s. Similarly, there is a v in V! for each (s,w) in I'

such that s=(v,w) and ((sx)*y)k' = (((v,w)*y)(:'=

(w~)')*(y').The output function on a submachine is the

restriction of the output function of the automaton to the

states in the submachinc.

Pro pos it ion Il. 5. Lt V,,aP*) and

(WC.,r,,*) e machines suIch that A and r ar~e disjoint

subsemigroups of a semi group X and 8 and are disjoint

subsemigroups of a semi group Y. If there is a machine

(S',X,.,p,Y'r,*) that has Ffewer states than Vx: and canl

simulate the first two machines, then at least one of the

first two is not minimal.

118

Proof: Let j, 3 l, k, and k' be as in definition 4.01.

h be the mapping from V/ to endulk states and h' be

mapping from Wto epndulk states such that (h,i,i)

(h',i,i) are machine homomorphisms to epgdulk machines

the type in pr oposi ti on 3.15. Since S' has fewer st

Let

the

and

of

ates

than Vxr, there is at least one pair

a'c' in X. such that pac=pa'c' in S' wl

in Vxl. If qaifqa', then b is not (

any x and y in X, (y)):(x))((aa)h) =

=(q((ac)j)(x)))*(y)) = (o((acx)])

((pa'c'x)*y)k = (q~((a'c'x)])j)*(y]) =

=(y)):(x))((o((a'c')j)h) = (yj):(x)

thecn h' is not one-to-one, since it

similar manner that (rc)h'=(rc')h',

homomorphisms is not an isomorphism

3.16, one of the twro machines is not r

Props it ion 4. 06. Let (V,AE,

of products

ac and

while (ca,rc)/(aa',rc')

one-to-one, since, for

)*(yj) = ((oacx)*y)k =

(al Ca'c.: )j(x))) *(yj)i

)((qa )h), If PCfrC',

can be shown in a

He~nce, one of the two

and, by proposition

minimal.

.,n,*) and (W~,C,.,D,*)

be automata such that A and r are disjoint subsemirroups of

a semigroup X and a and D are disjoint subsemigroups of a

semigroup Y. If there is an automaton (S',X,.,Y,*) that

has fewer states than VrxW and can simulate the first two

au toma t then at least onp of the f i rst two! is not

minimal.

Proof: Le~t ), j',kk' C, and i? be as in definition

4.02. Let h be the mapping from V to egdulk states and h'

be the map~ping from 0 to egpdulk states such that (h, 1, i)

and (h,~)are automaton homemorphisms of the type ;in

propos~i ti on 3.08. Since S has fewecr states than VxW~, there

is at: least one distinct pai r of elements of vx!, (v,w~) and

(v',w'), such that there is one element s in S for which

both ( sIv) and (s,v') are in I; and both (s,w) and (s,wl')

are in Hl. If viv', then b is not one-to-one, since, for

any x and y in X, I y)):(x))(vb)=(hx)*y) = ((sx)*y)l:

= (v (x) )*()) (y) :(x )(v h). If w~fw', then h' is not

one-to-one, since, for any x and y in X, ())())w'

= (wJ(x) '))*(y)') = ((sx)*y)k' = (w{x')*y)) Hnce,

one of the two homomorphisms is not an isomorphism and, by

proposition 3.10, onie of th~e PhJo automata is not minimaol.

Propositions 4.05 and 4.06 say that no

m~ultiprogramming system is more efficient than the product

of efficient systems. While propositions 4.?3 and rr.04

showed the existence of a multi programming system

simulating any two given systems as an internal product

w,hen its semi,7roups are free products of those of the

simulated systems. the following propositions examine some

of the properties of systems with other semigroups.

Proposition 4.'17. If the automata (S,X,.,Y,*),

(V,A,.,C, *), and (WI,C,.,'l,*) as defined in definition 4.n2

are egdulk automata hiavinR the property that s~a is in S

and s~c is in D for each s in S, a in A., and c in r where

X is Fgrenerated by AUC and P(v) and PC\w) are, respectively,

the sets of all s such that (s,v) is in G; and (s,u) is in

1 for somei u in Wi~ and such that (s,w) is in H! and (s,u') i

in G for some u' iA V,, it e6 for e,~ch wr~ in Wi (P(wr),A,.,n,*)

is i somorph ic to (V, A,.,RP, *) wh ile for each v in

(P(v),C,.,D, *) is isomorphic to (W,C,.,D,*).

Proof: If s is in the intersection of P(v) and P~v') where

v and v' are in V, then vv'V since for all a and a' in A

a'Y'ay = va~a' = (v(aj))*a'j) = (sasa~')k = (v'(aj))*(asj

v'a*a' = a':av'. If s is in the intersection of P(w;?) and

P(w'~) where w~ and w are in W, then w=w'. There is at

least one s in the intersection of each P(v) and each P(w).

Assume that there is a (v,w) in VxW such that s and t are

in the i nte rs ec tion of P(v) and P (w,). If s and t are

distinct, then there !< some x; and some y in X such that

y:xsfy:xt, and since y is in the semigroup generated by

AUC, there are y' and y" in X. and z in AUC such that

Y=y''zy" and z:(xy')siz:(xy't If z is n ,then z:(x~y')s

= {z:(xy')s~bk = (zj):((xy')j)v = (2:(xy')t)': = z:(xy')t.

Similarly, if z is in C, then z:(xy')s = z:(xy')t. Hence,

s and t cannot be distinct. For any wr in 0~, define the

mapping h fromi P(!) to V by restricting: r toPwxV For

any v in V, define the mapping h' by restricting H to

P(v!~xU. The triples of mappings (h,j,k) and (h!',j',k') are

automaton homomorphisms. The triples of mappings

(h, j A, k l) and (h',j'iC,k('(D) are automaton homomorphisms

which, since they are one-to-one, are isomorphisms.

.P.C.Qggsi~tion 4. 08. If the mnac hi n es ( S,XY,.,p, Y, j

(VA.a"*,and (H, C,., r,D, *} as delcf ined i n def in it ion

4.01 ar-e egdulk: machines having th~e property that: s*a is in

B and sac is in D ror each s in S, ;a in A\, and c in C where

51

X. is generated by AUC, then for each c in C the submachine

(T,A~,., pc, P, *) of (S, X,., p, Y, ) is is omoroh ic to

(V,A~,., q, P, ) wh ile for each a i n A the submnch i ne

(TC.paD,)of (S, X,., p, Y,*) is is omor ph ic to

Proof: The relations 0 and H1 of def ini t ion 4.02~ and

subsets P(v) and P(wr) of proposition 4.0l7 can be defined.

For each wr in WI (P(\,),A,.,B,*) is auertomaton i somor ph ic to

(V,A,.,B, k) and for each v in V. (P~v),C,.,D,*) is autom~aton

isomnorphic to (W,C,.,D,*) by proposition 4.07. Let c be

(rc(x)'))*(y)')

some element

of C.

Since

(rc ')x ')* y) ) = ( ( x) )) (j ) = ((pcx)*y)k',

(DC,rC) is in r and pc is in P(rc). Since(rx))*y)

=- (r cI(j ) x ) ( j ) = ( ( c x J ) y '

((pax*y~',pea is in P' rc) for all ain A. Since

(P(rc),A,.,P,*) is automaton isomorphic to(V,.,)

which is the automaton of a machine, any element of P(rc)

can be expressed as pcJ. in particular, since (a(x)))*(yj)

=(q(cj)(x)))*(yj) = (rl((cx)j))*(yj) (nex*y)k:, (Dc)h=a

and each~ automaton isomorphism is a mlachi ne i somorphi sm.

SimInila rly, each au toma ton i somo rph ism ( h' j' IC, k D) a s

defined using proposition 4.07 is a machine isomorphism

with (paJ)h'=r.

Although the previ ous t w~o propositions may seem

obvious and the conditions on the output function

unnecessary, there! are egedulk automata which have their

state spaces contained in the first projections of both of

52

the relate ons G; and H and yet have more Jltates than VxWI.

For example, for some s and t in S and a in A, the ou t pti

mapping might be defined so that s*C=d'bd and tea-b where

b is in 3 and d' and d are mutually inverse in D.

Proposition 4.09. Let the automata, mappings, and

relations be as defined in definition 4.02. If S is in the

first projections of both and H-, then there exist

au toma ta ( V' ,A, ., B, a), (iC.D, and(Vx',,,)

and semi group homomorph isms f":Y--Z, f:Z.--B, and f':Z--D

and m~appings h:V--V', h:W-, h":S--V'xW!', 8Vx'-

and e':V'xW'-- W' such that yf"'F-yk and yf"f'=yk' for all y

in Y, (b~d)f"'=(d~b)F' f~or all b in S and dl in 0

(v',wJ')g=v'and (v', w')gr. r' for all vw)in Yx'

sh"'=(vb,wrh' for all (s,v) in G and (s,w) in i', and the

triples of mappings (~,)(,,,,)-V,,,,)

(h", i, f"):(S,X~,.,v, *)--(V' xJ' ,X,., Z, *),

(h'I~i:(WC,.D,*--(',C.,D*) are all1 au tomaj"ton

homomorphisms.

(S,X,.,Y,a)

(V,,.,,*)(h", i~f") (WjC,.,n, *)
(h i~ ), V' ,X Z, *) (h' i )

(V',,.,Bw) (~j~f (g'j',f ) (f', C,.,D, *)

Proof: i.et E be the intersection of K(k) and UK('), L,=X/E,

and f" be the natural homromorphism from Y to Z. for all

in R and d in D, s in ce (b~d 3k={ (b kf( k)3= be =;ill (k ) (bhk)= -( Fh

and (bd)k'=ed=de3=(db)k', it fotllows t-h~at (bd,db~) is in E,

(bd)f"=(db)f", and E=K~f:"). Since K(f") is a subset of

K(k) there is a unique homomorphism f such that ykr=yf"f for

all y in Y (CP). There is likewise a unique homomorphism

f' such that yk'=vf"f' for all y in Y.

Let (V,,,,)and (IJ',C,.,D,*) be egdulk. automata and

(h,i,i) and (h',i,i) the respective automaton homomorphisms

from (V,A,.,Y, *) and (W, C,.,D,*) as; def ined in propos it ion

3.08. Lect g be the projection from V'x'' to V' and a' the

projection to 0!'. If (s,v) and (s,v') are both in r, then

vb=v'h because for all a and a' in A

a':a(vb)=va~a'=((sa)*a')E=v'a*a'=a':aIv'h) Similarly

wh'=w'h' if (s,w) and (s,w') are both in H!. Thne mapping h"

is defined by sh"=(vh,wh') wrher-e (s,v) is in G and (s,w) in

For the transition and output functions of (V'xgy ,,.7 4

define (vb, h ')x=((vb)(x)), (wh')(x) ')) and

(vh,,h ')*rx=((v7)*(xj)))((w.h')*;(x)')). Thlat (v h,whl')*(xy) =

((vb *((x )])) (wh' *((x )]')

((vb)*(x)))((vh)(x) )*(yj))((wh')*(x ) '))((wh')(x)' )*(y)'))

=((vb)*k(xj)))((wr )*(x)'))((vb)(xj))( yj))((wh')(x)')*(v] '))

=((vh, wh ')(x)((vh,weh ')x ay) for all x and y in 7,

(vh,w~h')*a = ((vh)r(aj))((wh~' )(aj')) = ((vb)*a)T! = (vb)*a

for all a in A, and (vh,wh')*c = (wh')*c for all c in C

wrhenever (vh,wh') is i 'W shows that (V'xH'~,X,.,Z,*) is

the i nte rnal1 p rodu ct au toma ton of (V,,,,)and

(WrC,,D,).Sinice j and g arle semigroup homiomornhisms

54

and ((vh,wh ')*x)f = (((vh)*(xj)))((Lh '*(x)')))f

(((vb)*(xj)))f)((( wh')*(x)'))f) = ((vh)*(x)))e=

((vh,wh')g)*(x)), (e,J,f) is an au!tomaton homomorohism.

The triple(.'3,) is also an automaton homomornhism.

Since (sh")*x = (vh,wh')*x = ((vb)*(x)))((wh') *(x)'))=

(s*x)f" for all1 (s,v) in C,(s,w) in H1, and x in X,

(h", i,f") is an automaton homomorphism.

Proposition 11.10. Lst the machines and mappings be

def ined as in def ini tion 1. 01. There exist machi nes

(V', ,., h,8 *),(W', ,., bC *),and

(V'xW',XI,.,(ph, rhj,2, *) and semniEroup homom~orphisms f"', f,

and f', and mappings h, h', h", g, and e' as in proposition

4.09 such that yf"f-yk and yf"f'=yk' for all y in u,

(bd~f"=(db)F" for all b in R and d in 0, (v',w)p:=v' and

(v'w'a'w' for all vw) in V' xWl'

(px>]h"=((n(x)))h,(r(xj)'))) for all x in Xw, and the triples

of ma ppi ngs(h i ):V ,.q ,* -V ,,. h, )

(h", i, f"):(S, X,., p,Y, *)--(S,X,.,(ah, rh'), 7,*),

(h'i~i :(W C,,r,,*) -(W ,P .,r ',D*) are all machine

homomorphisms.

(S,X,.,p,Y,*)

(V,A,.,ai,*,)("if) WC.rP*
(h~i~i) (Vx X .,(rh, ch'), D,*)~(' (h' i,i

(V', ,., hC *) R~j f) ( ',3 ,f' fW ,C..,q ', r, *)

P roof:~""i; Le t G be t he se-t: of all (px,q(x))) and H- theI selt ofllll

all (px,r(x)')) such that x is in X. S is the first

p roj ec ti on of both G and H and (S, X, .,Y, *) is aiii;

mult ip r o gr amm i ng a ut oma to n s'imulia t ina ( V, A,., ,* ) a~nd~i

(w,C,.,D,*). The existence of the automata and automaton

homomorphisms follows from proposition 4.0l9. Since (o,4)

is in G; and (p, r) is in 8~, ph"={vYh,wh'). By propos it ion

3.15, (ii)and (h,~)are machine homomorphisms.

Si nce ( ph"'')g=-(c3h,r h ') g=a h and ( ph ")y' = ( ah, r h ') R'=r h ',

(h,i,i) and (h',i,1) are machine homomorp~hism-s.

Propofs i"t ion .1 if (S,'.Y,) is a

mnult i p rog~r amm i n au toma ton s imula t i n ( V, A, ,,*) and

(W,,.,~w)and is a subautomaton of (S",X",.,Y",I), then

(S", X"',., Y", *) simulates (V, A,.,R, *) and (H~, C,.,D, *).

Proof: The automaton (S,X,.,Y,*) of the definition is also

a subatomaton of (S",X",.,Y",*). The mappinas ferom S, X,

and Y and the relations G and H serve to show that any

automaton with this subautomaton simulates (V,A,.,C,*) and

(W,C,.,D,*).

Proposition 4.12. if (S',X',.,p,Y',*) is a

m~ult ip ro r amm ing machine s imu a t in q ( V, n,., q, *) and

(WI,C,.,r,D,r) and is a submachine of (S",X",.,r>,Y",*), then

(S",X",.,p,Y",*) simulates (V,A\,.,q,R,-k) and (W,C, .,r,D,*).

Proof: The machine (S,X,.,p,Y,*) of the definition is also

a submach~ine of (S",X",.,p,Y",*) w~ith the same start state.

The may' En~gs from S, X, and Y serve to show? that any

machine with the same start state which has this machine? as

a submachine simulates (V,A\,.,a,R~,*) and (W,C,.,r,C,*).

Propos it ion 4. 13. I f (V,'.48,and

(W'C'.,rD'*) are res pecti vely submach lines of

(V, A, .,q, B, x) and (W, C,., r,Dn.*) and the latter pa ir of

machines is simulated by (S',X',.,p,Y',*), then the former

pair of machines is also simulated by the same

multiprogramming machine.

Proof: Let X," be the subsemicroup of X' generated by the

union of: A' and C'. Lect (S",X",.,p,Y",*) be a submachine

of (,X,,Y'). Since X" is a subsemigroup of v,,

(S",",.pY,*)is a s ubma ch in e of the(SX.,Y)

def ined i n def ini tion 4.01. The restrictions of j and j'

to X" and of k and k' to Y" are the mappings wh ich show

that (S F,X' I,., pY ', *) si mul1ates (V,'.qR,)and

Pro posi t ion 4 .14. If (V,',R,) and

(W',',.,',*)are respectively subautomata of' (V,A,.,R,k)

and (C,,w)and the latter Pa ir of au toma ta is

s imula tcd by S'X.Y,) then the fo rme r pai r of

automata is also simulated by the same multi programming

automaton.

Proof: Let X" be the subsemi~rrou of 7.' Renerated by the

unlon of A' and C'. Since X" is a subsemicroup of X(, there

is a subsemigroup Y" of Y such that (S,X",.,Y",*) is a

subautomaton of (S,X,.,Y,*) as defined in definition 4.012.

The relations G and 0' and the restrictions of j and j' to

X"! and of k and k' to Y" show that (S',X',.,Y',*) simulates

Proposition b.15. Any machine is trivially a

multi programming machine.

Proof: Let (S,X,.,P,Y,*)=(V,A,.,qP,,*) and let (Id,C, .,r,*)

be a single state machine writh input and output semigfroups

having only single elements. j and k are identity mnappingqs

on X and Yr while j' and k' map all elements of each to its

identity element.

Proposition 4.16. Any automaton is trivially a

mul1t i p ro r anmmi ng a utoma ton.

Proof : 1.et (S, X,., Y, *)=(V', A,.,S, *) and l et (~, C,.., ,=) be

a single state machine with input and output semig~roups

having only single elements. j and k are identity mappingFs

on X. and Y while jr and k' map all elements of eadch to its

identity element. G is the identity relation on S while HI

is the set of all (s,wr) such that: s is in S and w is the

element of EI.

Definition Ir.17. AI Trivial tMonold is a monoid w~ith

a single element.

Def in it ion I4. 18 A Trivial Aa c h in e, Au t ola t on,

Semimachine, or Semiautomaton is one with a trivial input

semipgroup.

As a result of proposition-4.13, the investigation

of the exIstence of a pai r of nontrivial machines which a

given machine (S',X',.,p,Y',*) can stimulate can be limited

to machines with cyclic subsemigroups of X'for input.

Thi s i nvest iga ti on can be further limited to nontrivial

cycl1ic monolds wi thout proper mi nimal nontri via\$IIIIIIII~lno smood.

Propos it ion 4 .19. Eve ry -n~ont ri v ial monold

contains a submonold where the identity of the sutdmibnold is

that of the monold of one of the following types: a 2-

element semi lattice, a group of prime order, or an infinite

order cyclic monoid. Each of these is nontrivial but

contains no nontrivial proper submonolds.

Proof: Any element other than the identity of a monold

generates a cyclic submonoid. A cyclic monoid contains an

identity element and all powers of some element b. If the

monold is infinite, any submonold is either the trivial one

or a monoid generated by some power of b. Neither type is

both minimal and nontrivial. If the monold is finite, some

power of b is an idempotent. If this idempotent is not the

i dent ity, the monold consisting of the idempotent powerr of

b and the identity is a -2-element Semilattice every

submonoid of which is trivial. if this idempotent is the

identity, let the n-th power of b be the identity. If n is

prime, there is only one submonold and it is trivial. If

n is not p r ime and n/m i s p r ime, the m-th power of b

generates a nontrivial submonoid of order n/m.

Proposition 4.20J. If (S,'.Y,) is a

multi programming automaton simulating two nontrivial

automata or if (S',X,',.,p,Y') is a multiproprammlng; machine

simulating two nontrivial machines, then there are in X'

two elements a and c which generate monoids of prime or

infinilte order such t~hiat~ the only powerF? of a whi ch is a

power of c is the identity.

Proof: If (S,'.Y,)simulates nontrivial automata

(V,An,.,P, *) and (II,C,.,DI, I), A and C are nontrivial and

there are, by proposition 41.19, an a in A and a c in C

which generate submonoids of prime or infinite order. if

x is a power of' both a and c, then x)=x because it is a

power of a while x=x)=e because it is a power of: c.

Propoos it ion 4.27 I f (S',X',.,Y',*)i is an

au toma ton such that: there? are elements a and c in X' such

that the monoid gener-at~ed by each~ is either infinite or a

group of prime order or a' semi lattice of' order tw~o and the

natural homomorphism from the monoid X generated by a and

c Lo X/E whrtere E is the congruence on X: generated by

(ac, ca) is one-to-one w~hen restrict-ed to the se tw!ise

product of the cyclic monoid generated by a and that

gene ra ted by c, t hen there are nontrivial auIt omat a

(V,,.,,*) and (WrC,., D, +) such that ('X,,'*

simulates them.

Proof: Let (S,X,.,Y,*) be any subautomaton such that X( is

generated by a and c. Let A be the submonoid generated by

a and C that generated by c. Letf be the natural

homomorphism from X to X/f. For any x. in X there is a

unique a' in A and a unique c' in C such that xf=(a'c')f.

Define 3 and j' surch that X~j=a' and x)'=c' for each x in X.

These meanpi ngs are..endon~or.,phisms since for any x and y in7

X (xf)(y f) ;= (()' ) )) ( y'(y )

((x))(x)')(y))(y)'))f = (x)y)x))v') =

((xy)j(xy)j')f (xy)f. Let k and k' be the constant

mappings from Y to the identity of Y. Let the machines

(S, A,., B, ) and (S, C,.,D, b) be defined with transition

fulnctionss agreeing with(SXY) and all1 ou tput the

identity. Let C and H! be the identity relation on S. For

all x and y in X and all (s,s) in G; ((sx.)*y)k=(s(xj))) (y)) .

For all x and y in X and all (s,s) in H

((sx)*y)k'=(s(x)'))e(y)').

Proposition 4.22. if (S',X',.,p,Y',*) is a machine

such that there are elements a and c in X' such that the

monoid generated by each is either infinite or a group~ of

prime order or a semi lattice of order two and the natural

homomorpzhism from the monoid X' generated by a and c to X/E

where E is the congruence on X generated by (ac,ca) is one-

to-one wlhen restricted to the setwJise product of the cyclic

ro~onold generated by a and that generated by c, theni there

are nontriv/ial machines VA.,,,)and(WCrP)

such that (S',7.',.,p,Y',*) simulates them.

Proof: Cy proposition 4.21, the automa~ton of: this machine

simulates two nontrivial machines. Fy proposition 11.14, it

simulates the automaton of any submachinle of them. Any

submachine of a nontrivial1 automaton which has the same

input monoid is likew~ise nontrivia~l. In particular, the

automaton of any submachine starting with a state a such

that (p,co) is in C; or state r such that (n,r) is in is

s imnula Te d as a u tma ton by the automaton (S',X', .,Y',* .

le n ~cl ((ox)1*y ik = ( q x )))*( y j) a nd ((oxpr) *y )k '= (r [x )' ) ) *( v )
and they are simulated as machines.

Section 2.: Free Monoids of Inou~t and rDutou~t

Definition 4.23. The relation C(P.) is defined for

any relation P on the generators of a finitely generated

free monoid X to be the congruence relation rcenerated by

the set of all pairs (@h,ba) such that a and b are

generators of .y but neither (a,b) nor (b,a) is in n.

Proposition Ir.24.A necessary and sufficient

condition for C(0) to be a subset of C(R) is that P be a

subset of thre union of n, the inverse relation of 0,and

the identity relation.

Proof: Assume that for every (a,b) in 9 either a-b, (a,b)

is in 0, or (bla) is i Let x and y be tw!o element' of

the finitely generated free monoid v. such that (x,y) is in

C(Q). Either there is a finite sequence of words

x',x",...,y",y' su:ch that x differs from x', y' differs

from y, and each wrord in the srcquence differs from the next

only in the transposition of two distinct generators ic

are not related by 0 or the inverse of n or x differs from

y only in the same manner or x=y. in the first case each

of the pai rs(xx) (x',x"), ..., (y",y'), and (y',y) i s

in the congruence relation \$(R) since the transposed pair

of rener;;tors w~as not in either or its inverse and hence

not in R. In the second case (x.,y) is similarly in c(R.).

In the last case (x,y) is in C(R) because it is an

equivalence relation.

Assume now that there is a pai r (a,b) ofe di sti nct

generators which is in Rl but in neithLe~'r 0 nor its inverse.

The pair of words (ab,ba) is in C(Q). Since there is no

word w~hich differs from ab only in the transoositioln of two

generators not related by D, (ab,ab) is the only pair in

C(R) w~ith ab for an entry. Since abtfba, (a~, ba) is not in

C(R).

Proposition b.25. If and are equiivalence

relation=, then a necessary and sufficient condition for

C(Q) to be a sulbset of Cl(R) is that R be a sublset of P.

Proof: This is a corollary of proposition 4.25r since 0] is

its own invecrse and contains the identity relation.

Prooos it ion i. 36. if .X is the f r e monnid

generated by the finite set X', then for any R in X'xX'

C(X'xV')
and 0 the empty set.

Proof: Since 1UF=tll<0sUIL<'xu', this is another corollary

of proposition 4.2ri.

PLED~os it ion r4.?7. if P. is an Pauivalence relation

on the genrac~tors of a finitely generated free monoid X and

P(1)0(2)...,~m)are the equiivalence classes orf P, then

for each x in Yl there are elnments a,b,...,d respectively

of the free m~onoids generated by P(l),r(2),...,Plr?) such

that (x,ab...d) is in C(R).

64

Proof: The proof is trivial for words of length 0 or 1

since e=ee...e and for each generator z z=c...eze...e.

Assume for any m greater than 1 that the proposition is

true for all -words of length less than m. Let x be any

word of: length m. Since x: is the product of two, words, z'

of length m-1 and z of length 1, (z',ab...d) and

(x,ah...dz) are in C(R). If z is in the monold generated

by P(m), so is dz. Othe-rwise if ab...d=y'y where 2 i in

P(m') y' is a product of elements from D(1) through P(n'-1)

and y a product of elements From P(m'+1) through P(m)

(y?,zy) is in C(R) because y=n or the sequence of words

w~ith z transposed with each of the generators of y has each

01ement C!P) related to yz. F finally (x, ab...cz...d) where

oz is in the monald generated by P(n').

Proposition .. If the semiqroups 4 and C of

definition 4.01 or 4.02 are finitely generated free monalds

and X: Is their free product and P is the eauivalence

relation on the generators of Y with the Penerators of A in

one class and those of: C in the other, C()is the

intersection of the kernels v(j) and K(j'). If the outpuit

semigrouips E, D, and Y are similarly related and n is the

analogous relation on the generators of Y, r(0) is the

intersection of the kernels 1(4) and K(k~').

Roof: Each kernel of a homomorphism is a congrunnce

relation. The intersection of congruence relations is

congruence relalion. r each a in Aand c in C

(ac)j=(aj)(cj)=3o=Ea=(c))(aj)=(cs).1 and

(ac)j'=ec=ce=(ca)j'. Since each pair (ac,ca) and (ca,ac)

whe re a i s in A and c in C is in the i nte rsec t ion of the

kernels K(3) and K(j') and C(R) is the conpvruence Reneratedl

by such pairs, C(R) is a subset of the intersection of

kernels. For each x in X there are a in A and c in C such

that a=aj=x) and c=c)'=x)j'. If (x,y) is in the

intersection of the kernels, a=y) and c=yji. From

proposition 4.14, there are a' in A and c' in C such that

(a'c',x) is in C(R). Since C(R) is in the intersection of

kernels, a'=(a'c')j=x)=a and c'=(a'cl)j'=x)'=c and (x,ac)

is in C(R). Similarly (y,ac) is in C(R). Hence the

intersection of K(j) and K(j') is in C(R). The

intersection of K(k) and K:(k') is shown to be ealal to C(0.)

by a simillar argumecnt.

Propos it ion 4.29. If the monoids A, R, C, Y,

and Y and the mappings j, j', k, and k' are defined as in

proposition 4.28, then For all x In ? and y in Y x)'=0 only

if x)=x, x)=e only if x)'=x, yl:'=e only if yk=y, ndyk=0

only if ykc'=y.

Proof: Py proposition 4.27, for each x in there are a in

A and c in C. such that (x,ac) is in C(R) where R is defined

as in proposition 4.14. Since C(R) is in Mij'), if x)'=e,

then e=(ac)j'=C~~(cj')(cj')=(c)'=c Since ac=ae is C(R)

related only to~itself, x-ae=a and x)=aj-x. The proof of

each of the other parts of this proposition is similar.

66

Proposition 4.30. If (S,X(,.,Y,*) is an automaton

w~ith Y a finitely generated free monald, then s*P=e for all

s in S.

Proof: For any s in S, s*P=s*(se)=(s~e)(sexe)=(s*e)(s*e).

Since the only idemnpotent in a free monold is the identity,

s*e=e.

Proposition 4r.31. If the machines (S,X,.,p,Y,k),

(V,A,.,qr, ,*), and (,.rD,)as defined in definition

4.01 are egdulk machines and A, C, 0, and Y are free

monolds where the set of Penerators of X is the union of

the sets of generators of A and C and the set of generators5

of Y is the union of the sets of Fenerators of and D,

then for each c in C the submachine (T,A,.,pc,s,*) of

(S,v,.,p,Y,*) is isomorphic to (V,A,.,cl,?,*) while for each

a in A thie submachine (T',C,.,pa,0,*) of (S, u,., rv,Y,) is

isomor~lphic to (W',C,.,r,D,*r).

Proof: Since (se*a!k'=(s(n '))*(aj')=se*9=0 by proposition

0.30, ve~a=(sewa)k-se~a by proposition 4.?6 for all a in A.

Similarly we*C=se*Cc for all c in C. Proposition 4.n8

completes the proof.

Proposition 4.T2. I f the aut o7a ta (SF, 7,.,Y, j),

(V,,.B,),and (W,C,.,n,ic) as defined in definition rr.n2

are eedulk automata and A, a,, C, D], y, and Y are free

monoids where the set of generators of X is the union of

the sets of generators of A and C and the set of generators

of Y/ is the union of the sets of generators of B and D and

P(v) and Pt!,) are defined as in proposition 4.t17, then for

each w in Wf (P~w),A,.,R,*) Is isomorphic to (V,A,.,R,*)

while for each v in V (P(v),C,.,D,*) is isomornhic to

(W,C,.,D,*).

Proof: Since (se*a)k'=zse*Fle by propos it ion 4.30 and

ve*a=(se*a)k=se*a by proposition 4.29, the result is

immediate from proposition 4.07.

Proposition 4.33. Any machine (ey e.

with X' a noncyclic free monold is, nontrivially, a

multiprogramming machine.

Proof: Let a and c be any two distinct generators of X'.

The submonoid X generated by a and c is the free product

monoid of the cyclic submonoids generated by a and by c.

Let E be C(I). C(I) is the congruence relation generated

by (ac,ca). If a' and a" are powers of a and c' and c" are

powders of c and (a'c',a"c") is in rt I), then a'=a" and

c'=c". Hence, the natural homomorphism onto .Y/E is one-to-

one from the se twJi se product of the cycl1i c monoids

generated by a and by c in that order. Sy proposition

4.22, there a re non-trivi al mach lines (V, A, ,aR, *) and

(W, C,., r,D, *) such that (S',X',.,p,Y',*) simulates them.

Proposition 4.34. Any automaton ('X,,'*

Ii t h X' a noncycl ic f ree mono id is, nontrivial ly, a

multiprogramming automaton.

Proof: Once more let a and c be distinct generators of X

and E be C(1) where Iis the identity relation on the set

with elements a and b. Sy proposition 4.21 there are

nont ri v Ial au toma ta (V, A,., B, *) and (W, C,., n, ) such that

(S',X',.,Y',*) simulates them.

ProPosi t ion 4.35 Let (S,'.p,'* e a

machines where X' and Y' are both noncyclic free monoids and

each pxex' such that x is an element of X' and x' is a

generator of X' is one of the generators of Y'. The

existence of generators a and c of 5' such that px*alpx'*c

for all x and x' in the submonold rene~rated by a and c is

a necessary and sufficient condition for the existence of

machines (V,A,.,q,8,*) and (,.r,,) where

(S'X'.,Y,* si mul1atres them and! each Rener'a to r of

either A or C is a generator of Y' and! for each Renerator

a of Aand c of Cl and each element x of the free product

m~onoid of A and C: q(xj))a and r(x)')*c: are ,cenertorsT of
to I

P~r oo f: Lot (S',X:',.,p,Y',*) be a machi ne which simulates

machines (V,A,.,q, B,*) and (r,C,.,r, P, ) w~i th the stated

properties. For any generators a of A and c of C and

element x of the free product monoid of A and C

q(x))*a=a(xj(j))*(pj)=(x+) is a generator of Y'. Since

(px~l=(x'c)kx(xj)*(c)=(x'j)*e=e for any x' of the free

product: mono id of A and C, px'Jec can not be equal to Dx~a

for any x' in the froo product monoid.

Conversely, let a and c be PRenerators of X' such that

px~alpx'*c for all x and x' in the submonoid u Renerated by

a and c. Each pxxa anid pxec are generators of Y'. Sy

propose it5ion rl. 33, there are nont ri v ial mac i nes wh ich

( S', Y',., p, Y ') s imul 1a te s as a mrul1t i ro~rammini p mach ine.

Let (V,A,.,a) (nd (WiC,.,r) be the semi mach lines of these

machines and define new output functions on them by

v*a=b=p*a for any v in V and w~c~d=o*c for any w in W~. if

a' is a power of a and c' Is a power of c, then

q*(a'a)=(s*a')s +)(ze*al)={*a')(a and

r*(c'c)=(r*c')(rC'*c)={"C)r*c')(c) Hence, if a' is the m-

th power of a and c' is the n-th power of c, then o*a' Is

the m-th power of b and rec' Is the n-th power of d. Since

the submonold generated by all of the elements of Y' of the

form px*a and pxec is a free monoid on the subset of the

generators of Y' and the px*a are distinct from the pxec

and b and d gene rate free submonoids, there a re

homomorphisms k and k' such that (px*a)k=b,

(pxec)k-e=(pxtc)k', and (px*c)k'=d. For any x and x' in Y,

(n(x'J))*(xj))=o*(x)=(p*(x ))k=~(~ )k=(p~ )()) and

(r(x'j'))(x '-*(x)')=r(x))(*(xj)))k'=pp'j)*x)))' The

machine (S,'.pY,) simulates the machines

(V,A,.,q, B,*) and (W,C,.,r,*) where R is generated by b and

D is generated by d.

There are machines which have px*a=px'*c for some

generators a and c and x and x' in the submonoid they

generate but can simulate machines with free monoids for

both input and output where all outputs from generators of

the input monald are generators of the output monoid. The

following example of a machine with twlo generators for each

of the input and output rmonolds will serve to illustrate

this point.

Let the transition and output functions be as follows:

.t t' t" u) U' u1"
Of t' t t u' u u
1-1 t"' u t U" t u"

t t' t" U u' u"
01 a 0 1 0 1 1
11 1 1 01 0 01

The machine (S,'.tY,)has 1=t*0=tl1:l=t"-1, but the

su bmach ine (S,X,.,t, Y,*) wlhereX is Renerated by 00l, n1,

and 10) has the following transition and output functions:

.t n
001 t u
01 uI t
101 t u

101 11 11.

S in~ce3 01':t *r01t*10 =u+10 r=11j-u;* 01 = 00, (SS.tY* simu.l at:Fs

machines of the desired tyPe.

In particular:

O V
n01 ql v
rAll v q

301 r

Q V1
001 00 01
011 01 00

r
101 11

The homomnorphism~s are defined such that (00)j=r)0, (01 ) i=n 1,

(10)j'=10,

(n0)j'=(01)j'=(ln)j=e,

(?0)k=n0,

(rl1)k=01,

(11)E'=11, and (10)E'=(01)k'=(11)E=0.

Propnos i tion r. 36. L-et (S, ,Y,) be an

au toma ton where Y' and Y' are both noncyclic free monoid~s

and each six such that s is in S and x is one of the

generators of X' is one of the generators of Y'. The

existence of a state p in S' and of generators a and c of

X' such that px*alpx'*c for all x and x' in the submonoid

generated by a and c is a necessary and sufficient

condition for the existence of automata (V,4,.,4,*) and

(W,,.,,* whre S'X',,Y'*)simulates them and each

generator of either A or C is a generator of Y' and for

each v' ini V and w in I via and wle are pgenerator-s of Y'.

Proof: Let (S',X',.,Y',*) be an automaton which simulates

au toma ta (A,,,)and (:, C,., D, i) Ji t h the r stated

properties. Let q and r he elements of V and

respectively. There is an element p of S' such thlat (o,q)

is in r; and (p,r) is in H. The submachine wlith! start state

p simulates the submach~ines with start states q and r. By

propos it ion 11.35, there exist generators a and c suich that

px*a/px'*c for all x and x' in the submronoid generated by

a and c.

Conversely, let p, a, and c be such that px*afnx'*Ce for all

x and x' in the submonoid generated by a and c. By

proposition 4.35, there are machines(VA.,,) and

(HC.,r,,*)such that the submachine of (S',7',,.,Y',*)

with start state p simulates them. The! relation n can be

defined as the set of all (nx,qx) and the relation II as the

72

set of all (px, rx) where x is in the free product of A and

C.

The automata of the machines in the previous

example, with (t,9) and (t,v) in C and (t,r) and (u,r) in

H and the same homomorphisms, illustrates that an automaton

might simulate tw~o automata with free semi groups of input

and output where each output from any generator of the

input is a generator of the output even though it can not

s imul1a te any with these same p rope rt ies Jhe reth

generators of the monolds of: the s imu 1a ted automate are

generators of the monoids of the simulating automaton.

11111111111111111111111111111111111

RBSIBOGRAPHY

(A) D. N~. Arden. Delayed Log~IC and Fi~nite S~tate
Machines, Ouarterly Proeress Report, R.L.E.,
M.IT.62 (1961) 1319

(B) T. L. Booth. Sequential rMachines and Automata
Theory, W'iley (1967).

(BW1)> A. R. Pednarek and A. D. W~allace. Enu ivya lances
gon Maclhie Sta~te Spaces, Mathematick)
Basopis 17 (1967) 3-9.

(BW12) A. R. Bednarek and A. D. Wlallace. Finite
Approximants of Compact, Totally Di~sconneczted
Irachines, M~ath. Systems Theory 1 (1967)
209-216.

(CP) '-A. H. C i fford and G. R. Preston. The Alpebraic
Theory of SemiRroups, Amner. M4ath. Soc.
Surveys 7 (1961 and 1967).

(01) S. Ginsburg. An Introduction t12 Miathematical

(G2) A. Ginzburg. Six Lectures on Alpebraic Theory of
Automata, Carnegie Institute of Technology
(1966).

(G3) A. GinzburR. Alprhraic Theory of Automata,

(G4) Y. Give'on. On Some Propertiees of the Free rMonoids
with Application to Automata Theory, Jour.
of Computer and System Sciences 1 (1967)
137-154.

(05) V. M. Glushk~ov. Introduction in Cybernetics,

(HS) J. Hartmanis and R. E. Stearns. Aleebraic
Structure Thqory of Seauential Machines,
Prentice-Hall (1966).

(KFA) R. E. Kalman, P. L. Falb, and Ml. A. Arbib. Topics
in Mathematical System Theory, M~cGraw~-lll
(1968).

(Ka) T. Kameda. Cosneralized Transition M~atrix sf a
Sequential Machine and Its A\nalications,
Information and Control 12 (1968P) 259-275.

(K) S. C. Kleene. Representation of Fvents In r! rve
Engs and Finite Automata, Auitomata Studies,
Ann. of IPath Studies, '34 (1956) 3-41.

( L) YE. S. i Lya pi n. Se~mirrou~ns, Translations of FMath~.
Monographs 3 (1363).

(M~c) J. D. rMcl~n i ht, J!r Kleene nuotie~nt Theorems,
Pacific Jour. of M~ath. 14 (196rr) 13113-1352.

(M~i) M1. :insky. C~omoutation: Finite and InfinitP
Ma~c h ine s rentice- ~all ( ~196 7) .

(M~o) E. F. rdoore (Ed.). Sec~uential Mtachines - Selected
Papers, Add i son -'~s le y ( 1 9 6r) .

(ii)E. r. orris. Some Stru~cture Theorns for
Tooolovical tPachines, Dissertation, Univ. of
Florida, (1960).

(RS) M". n. Plabin and D. Scott. Finite Automa~ta and
their Decision Problems, IBM Jolur. of Res.
and Dev. 3 (1959) 1114-125.

BIOGRAPHICAL SKETCH

Reverdy Edmond Wirirght was born 5 AIugrust 1933 in

Sa rasota, Flor ida. He was graduated from Sarasota High

School in June 1951. From lSeptember 1951 until January

1956, he attended the Massachusetts institute of

1958, he served

Technology.

From June 1956 through !ay

w,ith the U. S. Ar my. After attendinga the U~n

Florida from September 1958 through January

received w,ith honors the degree of: Rachelor of

mathematics. From JanuarJy 1959 through June

woredas a prgrmmr or th're Un ive rs ity

Statistical Laboratory. From July 1960 to Janua

served with the U. S. Army. From !a n ua ry 1961 t;

1963, he was a programmer For the Statistical

the Un1i vers ity of Flori da's AP:ri cul tu ral

Stationi. From rMay 1963 through Aufiust 1955,

systems surperv~isor for the U~nlversity aF Florida

diversity of

1960, he

Science in

19601, he

of Florida

ry 1961, h

to Ap~ril1

Selction of

rxpe-rime~nt

he was the

Computinff

Center. F-om September 1965 through August 1000, he was a

pgraduate students ;n the department of m~athemratics. Drn

the 19 89-701 and 1.970-71 academic years, he has been an

assistant professor of conouter science~ at the Virginia

Polytechinic Institute.

Revcrdyl Edmolndl IPr i ht: is mar~pr i edC to the forr!.1er'

Lydiz RoTake. Th~ey have three children, Tana ra, 9 r`anwen,

andl En id. H~e is a miember of the A~ssociation for rComputinp:

Macineythe rrne r Ic an As scia t ion of Uiest

Professors, and the Amonti~ican; As s o i a tion slor the

Certify that I have read this study and that inl
r.y opinion it conform~s to acceptable standards of' scholarly
p re sentatS:i on andt is fu'lly adequate, in scop~e andr q~ua i ty,
as a diissertation for the degree of Dloctor of Ph~ilosophy.

Rl C. .elfld, Chasi rman
Pro eso of Pthematics

Icertrfy that I have read this stud~y and that in

presentation andl is fully adeqiuate, in scope cinti uliy
as a r.issertation for the degree of Doctor of Philosophy.

Professor of ath~enattics

s cer~tIfy that have rcadl this studyl a~nd: that i
my;~ opi;nicai ; conforn~s to accejptab~le standardcS of~ sch~olaril
as ;; dissortati;onl fr~ the- degre-c of Dotr "Ciospy

M~ich el j ho!u
P'rofes~sor- ofIdsr!11:
Sy;stenrs Eng ineie r ing

Scriythat: I have readc this stul~dy ;Indc tha1t Iin
r.:y o~pinion it conformns to accep!Lable sta~ndarrds ofi s!rlo3rly
presen;i-tation anrd Is fuLly adeq~ua~te, nsoe ndqli,
aS a issortation fo~r' t~h degree of D~ctorl or i:hilosoo:h:.

Assistant P'rrof;essor of ;'c-athemaitic

This dissertation Was subm'itted to the Dean of the Collee
of Arts and Sciences and to the Graduata Council, and w;jas
accepted as partial fulfillment of the requirements for the
degree of Coctor of Philosophy.

Dean, College of : ~rts and Sciences

June, 1971

Full Text

PAGE 1

THEOREiMS FOR FINITE AUTOMATA by Rcvercly r;drnond ''/ri.^ht A Di !:,?,erl-.at ion Presented to ther,r.?dt.;ate T'curicn r.f The University of Florida in Partial Ful f i 1 la\-^nt of the Pequ i rp.rrf^nts for ti-.n Pegroe of Doctor of Philosophy UMIVERSITY OF FLORIDA 19 71

PAGE 3

To Lydia

PAGE 4

The author thanks the menbers of his supervisory committeey Dr. R. C. Sel fridge. Dr. A. R. Bednarek, Dr. M. E. Thomas, and Dr. K. M. Simmon, for their assistance and encouragenen t . Particular thanks sro due ProfosGor r.elfridf^e, the connittee chairman, and Professor Bednarek, the chairman of mathematics, for their many helpful suggest ions . This paper \;as typed by the ATS/3G0. This research v/as supported in part by a Graduate Fellowship from The University of Florida and in part by a Trainoeshlp from the flational Aeronautics and Space Administration. * f * I i t

PAGE 5

TABLE OF CONTENTS Page Introduction 1 Chapter Section 1. Pre! imi naries Â•Â» 2. Semimachines and Semlautomata 11 Section 1. General Discussion 11 Section 2. Semimachines 16 Section 3. Semiautomata 22 3. Machines and Automata 33 Section 1. General Discussion 33 Section 2. Automata 37 Section 3. f'achines 39 Section U. Semimachines and Semi automata '2 U. Multiprogramming ^k Section 1. General Discussion kh Sf.'Ction 2: Machines v/ith Free Semigroups 62 Bib! i v'j-;^raphy 7 3 IV

PAGE 6

Key to ^'otation f: A--B f is a mapplnf^ from a set A into a set B (h, j):(S,X, .)--(T,V;, .) h:S--T, j:X--V/ (h,j,k) :(S,X, .,Y,*)--(T,V/, .,7,*) h:S--T, j:X--'./, k:Y--Z A^B A is isomorphic to E (S,X, .)Â«(!, W, .) (S,X,.) is semiautomaton isomorphic to (T,V;, .) (S, X, . , p)y (T, V,', . , q) (S,X,.,p) is senimachine isomorphic to (T,'.', .,q) (S,X, .,Y,*)y(T/>7, .,Z,*) (S,y,.,Y,*) is automaton isomorphic to (T,^i,.Z/*) E(T) input ripht congruence relative to a subset T of a state space K(j,R) quasikernei of a mapoinp; j and a relation R K(j) kernel of a mappinK j e. the equivalc^nce class containinp the identity of a semigroup inCX) * senirin.r: on the pov/er set of a semi.p:roup X with union for addition and elementv'ise mul t i pi i cation Hn(X) semirin?< of matrices over Hl(X) v/ith analogous matrix operations b. the subscript of 1 in b v/herc b is an elementary basis element of a vector space of n-tuples

PAGE 7

Abstract of Tissprtation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy at the University of Florida THEORF.nS FOR FINITE AUTOi'lATA By Rcvordy Ednond V'right June IS '/I Chairman: Ralph G. Selfridge Major Department: fat hematics Minor Department: Industrial and Systems Engineering The automata of this paper are finite in that they have a finite state space and finitely generated semigroups. Definitions of these systems and useful mappings between them are developed and used to examine thai r properties. Automata without output are shown to be isomorphic to some with state spaces tfiat are right congruences when a start state is designated or mappings from subsets of the integers into the integers for state spaces otherwise. A semiring of matrices ovor a semiring is used in the development of the latter. VI

PAGE 8

Automata with output are shov;n to be homonorphic to some v/i th state spaces that are sets of functions from the input semigroup to mappings from input to output. These homomorph i sms involve identity mappings on the input ano output and are shown to indicate "black box" equlvanence between automata. The idea of multiprogramming is defined without recourse to products of automata. Certain products are shown to be multiprogramming automata. In some cases, the product is the minimal automaton v/hich can accomplish a given multiprogramming task. The ability of a given automaton to act as a mul 1 1 program.ml ng automaton is Investigated. The existence of input elements satisfying the condition that, except for the identity, no power of one is a pov/er of the other is shown to be necessary but not sufficient. A further condition is shov/n to be sufficient. in the case of free semigroups, it is shov/n that the generators of the input semigroups need not be the same in the simulated automaton as in the multiprogramming automaton. VI 1

PAGE 9

INTRODUCTION This dissertation treats a special class of automata, finite sequential automata v^ith finitely generated semigroups of input and output. Because the terms used in the Theory of Sequential Automata are defined differently by various authors, they must be defined at the outset of this paper. Although all semigroups are assumed herein to have identity elements, this fact v;ill be emphasized by referring to them as monoids from time to time. No topology need be assumed for any semigroup or state space. The topic is developed by starting with a class (semimachi nes) of sequential automata with designated start states and no output. The development continues with an investigation of the class (semi automata) of sequential automata with neither start state nor output. in each case, fiomomorph i sms are investigated and canonical forms developed. A semiring of matrices v/ith subsets of semigroups for entries Is generated during the investigation of semiautomata. Each semi automaton is shov^/n to have a finite subset cf these matrices corresponding to it. Any tv;o semiautomata vnth the same input semigroup are Isomorphic 1

PAGE 10

2 If and only if they correspond to the same set of these matrices. In Chapter 3, the classes (automata and machines) of sequential automata with output, first without and then with designated start states are investigated. Homomorphism is shov^rn to be a sufficient condition for "black-box" equivalence of sequential automata. Canonical forms are developed for minimal sequential automata, those with the smallest number of states v/hich have given input to output properties. Finally, v;ith the help of the canonical forms for semimachines and semi automata, canonical forms are developed for those automata and machines which are not necessarily minimal. In Chapter h, the idea of multiprogramming for sequential automata is explored; first for automata and machines as developed in the previous chapters and then for the special case v^here the Input semigroup Is a finitely generated free monoid. The notion of product is not used In the definition, but product automata are shov/n to be examples of multiprogramming automata. The principal result might loosely be stated, "The most efficient way to build a machine to do the job of two machines is to put these tv/o machines into a single box unless there is a more efficient way to build at least one of the two machines." Both necessary conditions and sufficient conditions for a machine to be a multiprogramming machine are investigated. An example shov/ing that, even in the case of free

PAGE 11

semigroups of Input and output, the generators of semigroups for simulating and simulated m.achines need not be the same is also important.

PAGE 12

CHAPTER 1: PRELIMINARIES Def i nl tion 1.01 . A Semlautomaton is a triple (S/X/,) consisting of a non-empty finite set S, a finitely generated semigroup X, and a function . from SxX into S with the property that s. (xy)=(s.x) . y for all s in S and all X and y in X. The set S is called the state space. The semigroup X is called the input semigroup. Frequently a finite set of generators of X will be selected and called the input alphabet. In particular the generators of a free monoid are thus designated. V/hen an input alphabet is designated, each generator other than the Identity is said to have length 1, the identity is said to have length 0, and length is defined for all other elements of X by the shortest product of generators. The function Is called the transition function and its property, s(xy)=(sx)y, the sequential property. The image under the transition function will usually be denoted by juxtaposition. Def ini ti on 1. 02 . A Semimachine is a quadruple (S,X, .,r) where (S/X,.) is a semlautomaton and r is an ingresS/ a state such that, for all s in S, there exists an X in X such that s=r.x. There may be other ingresses but only the one designated is the start state. Def ini t i on 1. 03 . An Automaton is a quintuple (S,X, .,Y,*) where (S,X,.) is a semlautomaton, Y is a k

PAGE 13

5 semigroup and the output function * Is a function from SxX Into Y with the property that, for all s in S and all x and 2 in X, s*(xz)=(s*x) (sx*z) and Y Is the monoid generated by the range of the output function. Def Inl t ion 1. OU . A Machine Is a sextuple (S,X, ., r, Y,*) v/here (S/X/.,Y/*) is an automaton and (S,X, .,r) Is a semimachlne. The definitions of the systems are generalizations of those used by S. Glnsburg (Gl) and A. Ginzburg (G2 and G3). Ginzburg's definition of semi automaton is restricted to finitely generated free monoids. GInsburg's complete sequential machine and Ginzburg's Mealy machine or Mealy automaton are essentially this paper's automaton restricted to finitely generated free semigroups of input and either free monoids or right zero semigroups of output. GInsburg's quaslmachine Is still more general in that its state space may be infinite. His abstract machine is less general In that its output semigroup must be leftcancellative. (G2 and G3) (Gl) (G2 and G3) (Gl) Ginzburg's cyclic semi automaton and its generator are restrictions of this paper's semimachine and its start state. An extension analogous to that from semi automaton to automaton leads to the definition of machine. Def I ni ti on 1. 05 . A Semi automaton Homomorphism is a pair of mappings (h,j) such that h:S--T, j:X Â— 1, j Is a semigroup homomorphism, and (sx)h = (sh) (xj ) . f both h and J are one-to-one and onto, then the pair (h,j) is a

PAGE 14

6 semi automaton isomorphism. Def i ni ti on 1.06. A Semimachine Homonorphism is a semi automaton homomorphi sm (h, j ) such that the image of the start state of the first machine is that of the second. If (h,j) is a semimachine homomorphism and a semi automaton isomorphism, then it is a semimachine isomorphism. Def i ni ti on l_.j07.. An Automaton Homomorphism from (S,X,.,Y,*> to (T,Z,.,W,*) is a triple of mappings (h,j,l<) such that (h,j) is a semi automaton homomorphism and such that k:Y-"V.' is a semigroup homomorphism and (s*x)k-(sh)*(xj ) for all s in S and dll x in X. r'cjLice tiiat neither the sequential property on the output nor that k is a semigroup homomorphism implies the other. If k is a semigroup isomorphism onto V.' and (h,j) is a semi automaton Isomorphism, then (h,j/k) is an automaton isomorphism. Dof ini ti o n 1. 08 . A f''achine Homomorphism is an automaton homomorphism (h,j,k) such that (h,j) is a semimachine homomorphism. If (h,j) is a semimachine isomorphism and (h,j,k) is an automaton isomorphism, then (h,j,k) is a machine isomorphism. V/hen the meaning is clear from the context, i will frequently be used to denote the identity mapping on one of the state spaces or semigroups. For example, (i,j,i): (S,X, .,Y,*) Â— (S,Z,.,Y,*) would mean a triple consisting of the identity mapping en S, a mapping from X into Z, and the identity mapping on Y. Homomorphism is defined for semi automata by Ginzburg (G3) in an analogous fas!. ion. It has also been

PAGE 15

7 defined by others (e. g. f'orris (M)) for other types of automata v/lth only inputs and states. Isomorphism, but not homomorphi sm, is defined for automata v/i th output by Hartmanis and Stearns (HS). Ginsburp; (Gl) likewise defines only the former. Arbib (KFA) defines homomorphism slightly differently but makes little use of his definition. Def i ni ti on . 1. 09 . A Subsemi automaton of a semiautomaton {S,y.,.), semimachine (S,X/.,r), automaton (S,X,.,Y,*), or machine (S, X, . , r, Y, * ) is a semiautomaton (S',X',.) such that S' is a subset of S, X' is a finitely generated subsemigroup of X, and the transition functions agree on the elements of S'xX' and the restricted range is v/ithin S'. Def i ni t i on 1.10 . A Subsemi mach i ne of one of these same systems is a semimachine (S',X',.,q) v/Iiere q is some element of S (not necessarily r), X' is a finitely generated subsemigroup of X, and the transition functions agree on S'xX ' . Def i ni t i on 1.11 . A Subautomaton of an automaton (S,X,.,Y,*) or machine (S, X, . , r, Y, *) is an automaton (S' , X ' , . , Y' , *) such that (S',X',.) is a subsemi automaton, the output functions agree on S'xX*, and the range of the restricted output function generates Y'. C ef i ni t i on 1. 12 . A Submachine o^ an automaton or machine is a machine the automaton of which is a subautomaton and the semimachine of which is a subsemlmachlne.

PAGE 16

8 .Df}f i ni t. i o n 1.13. The Internal Product Semi automaton of the finite set of somi automata (Sl^Xi^cI) where 1=1,... ,n, and the Xi are subscnl groups of a semigroup X such that, for any I, the mapping j which naps each element of Xi onto itself but maps all other elements of the union of the XI onto thie identity can bo extorided to a semigroup homomiorphl sm, is a semi automaton (T,V.', .) where T is the Cartesian product of the Si, V/ is the subsenii group cf X generated by the union of the Xi, and the transition function is defined for each elemerit x of the union of the Xi by coord i natewi se transilion on all coordiriatos where x is in the respective Xi and by the identity transition on all other coordinates and tlie transitiori function is defined for all other elements by the sequential property. The internal product soni automaton of a set of semimacbi ncs, automata, or machines is defined as that of their semi automata. if eacli input monoid X' is disjoint ftoni each of the ctiiers, this product is essentially Cinzburg's direct product (G3). If tiiero is a pair of input monoids with elements other than the identity in their intersection, the effcci of an Input from this intersection is seen in moi'e than one cf the coordinates. Of particular interest is the case v.hen X"Xl"...-Xn 'when each input may affect evary coordinate. An example of this is seen in the reprcser; tat I ve sern i. na citi ne defined beiov/. r:

PAGE 17

9 Def i ni t i on 1. lU . The Internal Product Semimachine of the finite set of semimachi nes (SI ,X I , . i , qj ), v/here 1=1,... ,n and the XI have the same property as In the previous definition/ is the semimachine (T/V/^./q) where (T/V/, .) is the appropriate subautomaton of the Internal product automaton of the semimachi nes and the start state q Is (nl/ . . . , qn) . Def i ni 1 1 on 1.15. The internal Product Automaton of the finite set of automata (S I ,X i , . I , Y i , * I ) , where 1=1/... ,n, the XI have the same property as in the previous definition/ and the YI are isomorphic to subsemi groups of a semigroup Y/ is the automaton (T/W/./Z/*) where (T/VJ/.) is the internal product semi automaton of the semi automata (Sl/Xi/.l) and Z is the subsemi g;roup of Y generated by the Image of TxX under the mapping * defined by extending the *i such that ( . . . /S i / . . . )*x corresponds to (si)(*i)x for Â•-all X in Xi/ provided the output sequential property holds. The internal product automaton of a set of machines is that of thei r automata. D ef ini tl on 1.16 . The Internal Product Machine of the finite set of machines (S i /X 1 / . I / qi /YI , *!)/ where I=l/.../n and the XI and Yi have the same properties as in the previous definition/ is the machine (T,V.'/ ./ q, 7/ *) where (T/'7/./q) is the Internal product semimachine and (T/ v.'/ , / 7, *) is a subautomaton of the internal product automaton .

PAGE 18

10 Def i n i ti on 1 . 17 . The Representative Semirriachi ne of a semi automaton is the internal product semimachine of the subsemimach ines of the semi automaton starting with each of its states in turn. Def i ni ti o n 1.18 . The Representative Machine o^ an automaton is the internal product machine of the submachines of the automaton starting with each of its states in tL:rn where the output semigroup is the Cartesian product of the output semigroups of the respective submachi nes.

PAGE 19

C.MAPThR 2: SEMI MACH I f'FS AMD S FM i AUTOMATA Sectio n 1.: General Pi scuss i on The presentation in this chapter does not follov/ the develor>"ient usually made for semi automata v/ith finitely generated free semigroups of input but does touch on familiar results. The usual treatment of homomorph i sms is either to restrict the semigroup homomorphism on the input to a semigroup isomorphism or to the homomorphism on free semigroups induced by a mapping of generators into generators. The relations on the input semigroup and equivalence classes are frequently mentioned in connection with the class of regul ar subsets of a semigroup. The classic v;orks for regular subsets of finitely generated free semigroups are those of Kleene (K) and of Rabin and Scott (RS). McKnight (Mc) has generalized Kleene's results. A very lucid treatment of regular sets is made by Ginzburg (G5). Def in? ti on 2. 01. The Input Right Congruence relative to a subset T of the state space S of a semimachlne or semi automaton is an equivalence relation [:(T) on ti^e input semigroup such ',:hat (x,z) Is in E(T) when 11

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12 tx = tz for all t In T. Clearly this is a right conp:ruence relati on. Proposi tion 2. 02 . If T is a subset of T', then E(T') is contained in E(T) . Proof: Follows immediately from the definition. The converse does not necessarily hold. Def ini ti on .2 . 05 . The Input Congruence of a system with state cpace S Is E(S). Def ini tion 2. Qtt . The input Right Congruence of a semimachlne with start state r is F(r), the input right congruence with respect to the singleton set of the start state r. Notice that any subset T of the state space S which contains an element s has the property that F(T) contains E(S) and is contained in F(s). E(S) is clearly the intersection over all s in S of the E(s). Def ini t i on 2. 05 . For any mapping j from a set X to a set Y and any relation R on Y, the set of all pairs (x,x') in X with the property that (xj,x'j) is in R is called K(j,R), the Quaslkernel of j and R, or K(j), the Kernel of j if R is the identity relation. Morris (N) defines the kernel In this way. The definition of quaslkernel is an obvious extension. Propositi on 2.06 . For any semimachlne (S,X,.,r) or semiautomaton (S^X,.) the pair of mappings (i,j)/ v/here i Is the Identity mapping on S, j Is the natural mapping from the elements of X to the elements of ''/E(S), and the transition function is the naturally induced s(xj)=sx, is

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13 a semlmachine or semi automaton homomorphi sm. SxX ; Â»S (UJ) I Sx(X/E(S)) Â— ^S Proof: If (Y/Y') is in E(S), then for all s in S sx is also in S and (sx)y= (sx)y ' . Hence E(S) is a congruence relation, X/E(S) is a semigroup, and j is a semigroup homomorph i :>iTi. This result is a special case of a result of Bednarek and '-'a 11 ace (BV/2). The semigroup of transformations induced on S by X is isomorphic to X/F(S) and is called by Ginzburg (G?) the Semigroup of the semimachine or semi automaton. Proposition 2.07 . If (h^j) and (h',j') a\e semlmachine or semi automaton homomorphi sms from CS,X,.,r) to (S',X',.,r') and thence to (S",X",.,r") respectively or from (S,X,,) to CS',X',.) and thence to (S",X", .), then (hh'^jj') is a semimachine or semi automaton homorphism from (S,X,.,r) to (S",X",.,r) or from (S,X,.) to (S",X",.). SxX ; -. S .(h,j) S'xX' "(h'.j') h S"xX" Proof: (sx)(hh')-((sx)h)h' = ((sh)(xj))h'-((sh)h'((xj)J '^ = (Â£(hh' )) '.'xC j j ' ) ) and in the case of the semimachine (h",j")

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Itl homomorphism r (hh ' ) = ( rh)h ' =r ' h ' = r". T'orris (N) proved this for acts as a corollary to a lemma. Proposition 2.08 . If (h, j ) : (S, X, . , q) -Â•(T, Y, . , r ) is a semimachine homomorphism/ then (h^i) is a semimachine homomorphism from (S,X, .,q) to (T,X,.,r) if the transition function on the latter is defined by (sh)x=(sx)b and further ( i , j ) : (T, X, ., r )--(T, Y, . , r ) is a semimachine homomorphism. Likewise, there is a decomposition of a semi automaton homomorphism. (h,j) TxY Proof: {(sh)x)i=(sh)x=(sx)h-(sh)(xj)-((sh)i)(xj) and in the case of the semimachines qh=r and ri=r. Propositio n 2.^9 . If ( i , j ) : (S, W, . )-( S, X, . ) is a sem.i automaton homomorphism and j maps W onto X and i is the identity mapping on S, (S/W/E(S),.)y(S,X/E(S),.). then W/i:(S)yx/E(S) and Sxi; -* S -Â— s Sx(X/E(S))-^ Â— Â— S

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15 Proof: Let j" be the natural mapping from V/ to W/H(S) and j' the natural mapping from X to X/E(S). Further let w and w' be in W. If (v-/,w') is in E(S) then ' UC III Â»l Â• II \ vv ^ vÂ» / 1 J III I\ w s (w( j j Â• ) ) -s ( (wj ) j ' ) = s (v/j ) =sw = s (v.j ") = s (w ' j ") = ... =s (w '( J j ') ) for all s in S. If (v/,v/') is not in F(S), then for some s In S s(v'(jj'))=s(v/j")^s(w'j")-s(w'(jj')). Since j, j', and j" are all onto there is exactly one element of X/E(S) corresponding to any v in V//E(S) and that is v/(jj') \vhere vi is any element of Vf such that v;j" = v. Mow s(w(j j ' ) )=s(wj")=sv.

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16 Section 2: Somlmach! nes A semlmachine has the simplest structure of all of the systems here described. It is a basic part of all systems since for each state q of any system there is a corresponding subsemimachi ne (S',X,.,q). Def ini ti on ? . 10 . The Right Congruence Semimachine is defined for any right congruence relation F with a finite set of equivalence classes on a semigroup X to be a semimachine (T,X,.,e.) such that the states are equivalence classes of E, the start state e. is the equivalence class containing the identity element e of X/ and the transition function tal
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17 Proof: For all x In X, x Is an element of (rx)h. If x is an element of both sh and th, then s=rx=t. x and y are elements of sh If and only If rx-s=ry. Since re=r^ e Is an element of rh. If z Is an element of sh, then since z Is an element of sh zx Is an element of (sh)x and since sx=( rz)x=r (zx) zx Is an element of (sx)h. Prppos i 1 1 on 2.12 . Let E and E' be rl^ht congruence relations with finite numbers of equivalence classes over a finitely generated semigroup X. Let I be the Identity mapping on X. If E is a subset of F', then there is a semlmachine homomorphism (h, i) from the right congruence semlmachine of E' to that of E. if E Is not a subset of E'/ then no pair of mappings (h/l) is a semlmachine homomorphism from the right congruence semlmachine of E to that of E'. Proof: Assume (x,y) is in E but not in E'. The equalities (rx)h=(rh)x and (ry)h=(rh)y must hold and rh must be the start state of its semlmachine ^or (h,i) to be a semlmachine homomorphism, but (rh)x^(rh)y while rx=ry. Assume E Is a subset of E'. Since X/E refines X/F', h can be defined such that s Is a subset of sh for any s In X/E. Clearly e is an element of both e and e.h. Since the transition function of each machine Is defined by right multiplication, sx is a subset of (sh)x. Since sx Is a subset of (5x)h and X/F' is a partition, (sh)x=(sx)h. Proposition 2. l.'?. For any semlmachine (S,X, .,r) there exists a mapping h frorr X/E(S) to S such that the

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18 pair (h, i ) is a semimachine homomorphi sm from the right congruence semimachine of the input congruence v/here i is the identity on X. Further, for any x in an equivalence class t, rx=th. (X/E(S))xX -^Â— X/E(S) (h, 1) / hi SxX Â• -. ^ S Proof: Since n(S) is contained in F(r), by proposition 2.12 there is a semimachine hononorphism (h',1) from (X/E(S) , X, . ,e) to (X/E ( r)/ X, ., e) . From proposition 2.11 there is an isomorphism (h",i) from (X/E( r) , X, . ,e) to (S,X,.,r). By proposition 2.07 (h, i )-(h ' h", i i ) is the required semimachine homomorphi sm. Since any t in X/E(S) is a subset of th' in X/E(r)/ any x in t is also in th' which is the sot of al 1 z such that r2=th'h". Hence rx=th. The representative machine of (S,X,.,r) is clearly isomorphic to the machine (X/E(S) , X, . .. e) . The representative machine is maximal in the sense that any machine having the same semigroup of transformations is one of Its homomorphic images. Although there is at least one machine v,'i th as fev/ states as any other v;ith the same semigroup of transformations, there may be none for v;hich some semipiachine homomorphism can be found from any other with the same semigroup. Proposition 2.1U . Let (S,X,.,q) and (T,X,.,r) be tv/o semimachines with the same input semigroup. The input

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19 right congruence of (S,X,.,q) refines that of (T,X/.,r) If and only If there exists a semlmachlne homomorph I sm (h,I) from (S,X,.,q) to (T,X,.,r) with I the identity on X. ProofBy proposition 2.11/ each machine is isomorphic to the right congruence semlmachlne of its Input right congruence. Since semlmachlne homomorphi sms compose, there Is a semlmachlne homomorphism from one semimachine to the other if and only if there is a corresponding semimachine homomorphism between right congruence semimachines of their input right congruences. This, In turn. Is equivalent to the refinement of E(r) by E(q). Proposition 2.15 . Let F(q)/ ECr), and E'(r) be the respective Input right congruences of the machines (S/X/.^q), (T/Y,.,r)/ and (T,X,./r) of proposition 2.08. E'(r) is K(j,E(r)). Proof: For any x and z in X, rx=(qh)x=(qx)h=(oh) (xj )=r(xj ) . and rz=(qh)z=(qz)h=(qh) (z j )=r(zj ) . Hence rx=rz if and only if r(xj)=r(zj). Proposi tion 2. 16 . Let (S^X^.^q) and (T/Y/.^r) be semimachines. If there is a semimachine homomorphism (h/j) from the former onto the latter, then the input right congruence of the former refines the quasikernel of j and the input right congruence of the latter. If there is a semigroup homomorphism from X onto Y such that the Input right congruence of the former refines the quasikernel of j and the input right congruence of the latter, then there

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20 Is a mapping h from S onto T such that (h,j) is a semlmachlne homomorphl sm. Proof: Assume that there is a semigroup homomorphism j such that E(a) refines the quasikernel of j and E(r). The mapping h defined by (rx)h=n(xj) for each rx in S is the mapping such that (h,j) is a semimachine homomorphism. The remainder of the proof follows directly from propositions 2.08, 2.114, and 2.15. Proposition 2.17. If ( i , j ) : (S//, . , r ) -(S, X, . , r ) is a semimachine homomorphi sm, then V.'/E(S) ^X/ECS) and vv/ r. <. r ;, i'.7tn.S) , .,e; = k a/E t.r;, X/F(b;/ ./ e) . Sxh' ; S ( i , j ) SxX (V//E(r))x(W/E(S)) u (X/E(r))x(X/E(S)) . * X/E(r) Proof: Since any state in S can be expressed as rw or rx j is necessarily onto. The rest of the proof follows directly from 2.09 and 2.11. Right congruence semimachines can be considered canonical forms for semimachines. Each semimachine is isomorphic to the right congruence semimachine of its input right congruence. All of the prooertles of a semimachine qua semimachine are those which are preserved by the Isomorphism from a semimachine to its canoiilcal form.

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21 Proposition 2.16 gives a criterion for the existence of homomorph i sins in terms of right congruences.

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22 Spcti on 3.: Semi automata A semiautomaton with at least one ingress can be Investigated by treating it as a machine with one of its ingresses as start state. Decomposing a semiautomaton into semimachines does not preserve all of the characteristics of a semiautomaton unless correspondences are defined between the states of each semimachine and the set of semimachi nes. Consider the following illustration of the difficulty: Let X be the commutative semigroup with three elements such that ee=e, ea=ab=a, and eb=aa=bb=b. Let the transition function for the machines ({ 1, 2, 3/ H }, X, . ) and ({ 5, 6, 7, 8 }, X, . ) be given by the Cayley tables: e a b e a b 111 2 1 515 6 5 212 12 61656 313 12 717 6 5 li|fil2 8185 6 {{e,b},{a}}=X/F(l)=X/E(2)=X/E(5)=X/E(6), {{e }, {a} , {b}} =X/E(3)=X/E(it)=X/F:(7)=X/F(8). But no isomorphism exists since 2a = 3a = '4a?*la while 6a = 8ai*7a-5a .

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?3 In pursuit of a canonical form similar to the rij^ht congruence semimach i ne it v/i 1 1 be convenient to examine the following system expressed in terms of finite dimensional vector spaces: Let B be a basis of a finite dimensional vector space V and A be a semigroup of linear transformations of V with the property that, for all (b,a) in ^xA, ba is in B. Clearly (B,A, .) is a semi automaton. ProDos i t i on 2.18 . For any semi automaton (S,X,.) v;Ith n stateS/ any basis P of an n-dimensional vector space V, and any one-to-one mapping h from S to B, the mapping j from X into the set of linear transformations on V defined by (sh) (xj )=(sx)h is a semigroup homomorphi sm. A fortiori: (h^j) is a semi automaton homomorphism and the sen! automaton homomorphism (h, i ) : (S, X, . )--( B, X, . ) is a ser^i automaton isomorphi sm. Proof: For each x in X there is a unique linear transformation carrying each sh of B to (sx)H. Let j be the mapping which carries each x I n X to this transformation. That Is to say: (sx)h=( sh) (xj ) . For any x and z In X, (xj ) (zj )=(xz) j since for each s in S ( sh ) (xj ) (z j ) = ( ( sx ) h ) (z j ) = ( ( sx)z ) h = ( s ( xz ) ) f( sh) ( ( xz ) j ) . The existence of the semi automaton homomorphism (h/i) -follows from proposition 2.08. Since h is one-to-one anci ontO/ (h, i ) is a semi automaton isomorphism. V.'hen the vector space is the set of n-tuples over a field and the basis is the elementary 'o3siS/ then the

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2k matrices of the linear transformations give a peculiarly graphic picture of the semi automaton. This representation is well known In some circles. (A) In this case, multiplication of an n-tuple by a matrix amounts to selecting a rov; of the matrix since each n-tuple consists of a single 1 and several O's. Hence, each rov/ of every matrix must be a member of the elementary basis. Since the only elements of the field that are used In this construction are the zero and the one, the set of n-tuples over any semiring with zero and one which coi .esponcfe to the elementary basis of a vector space and these same linear transformations are Isomorphic to this construction. In this case the mapping j is a representation of X by a Rees matrix semigroup over a group v/ith zero since the set with elements and 1 under multiplication is such and only one element of each row is nonzero. The representation is faithful when X^y^/FAS) . The following semiring of matrices over a semiring of subsets of a semigroup provides a useful tool for the next step In the development of a canonical semi automaton. Consider the power set of a semigroup X under the operations of union and setwise multiplication Hl(X). Both operations are associative. Union Is commutative. Multiplication distributes over union. The empty set is the zero and the singleton set of the Identity of X is the identity of HKX).

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25 Definition 2.19 . For any monoid X, Hl(X) is thus def i ned . Since the distr ibut i v i ty of the usual matrix product over addition depends only on the above properties of a semiring/ the n by n matrices Hn(X) over Hl(X) with union defined coord inatewise and multiplication defined by AB(i;l<)= U A(i;j)B(j;I<) is a semiring. Definition 2.20 . Hn(X) is thus defined. Kameda (Ka) has independently developed several matrices over sets which relate regular expressions of input to regular events over the output alphabet. One of these matrices resembles matrices of l-ln(X). Investigation of all of the properties of Hn(X) is beyond the scope of this paper. Some properties which are not necessary for proofs will be mentioned to provide a more graphic picture of the development of a canonical form. Definitions may be considerably more general than absolutely necessary. Def in i t ion 2.21 . Multiplication of an n-tuple s over HKX) and a matrix z in Hn(X) is defined such that sz(j) is the union of tlie s(i)z(i;j) where i=l,...,n. This is analogous to multiplication of a vector by a matrix when both are over a field. Proposition 2.22 . A matrix P is a right unit if and only if every entry except one in each row and in each column is the empty set and every nonempty entry is a set of right units which have a common right inverse.

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26 Proof: Let Q be a right inverse of P. Each diagonal entry of PQ Is the singleton set of the identity of X. Hence for each i there must be a j such that P(i;J)Q(j;i) is this singleton set; hence neither P(i;j) nor 0(j;i) can be empty, but rather all of the elements of P(i;j) must be left inverses of all the elements in Q(j;i). Since for all kj^I P(k; j )Q(J; i ) must be empty and 0(j;i) is not empty P(k;j) must be empty. Since the number of rows is the same finite number as the number of columns/ only one j can correspond to a single i in the previous statements. In HKX) there will be one-sided units if X has one-sided units. In this case the definition of similarity of matrices usually formulated does not necessarily give an equivalence relation. For example, if X is the bicyclic semigroup with ab=eji*ba and y Is said to be similar to x if there are elements p and q such that pq=e and xq=ay. Since p must be some power of a and q must be the same power of b, the set of elements of X similar to any given x will Include x^ axb, aaxbb, etc. For any x in X, x is similar to bxa but bxa is not similar to x. To avoid this difficulty the definition of similarity will be restricted to two-sided units. Def ini tion 2.23 . The matrices A and P in Hn(X) are Similar if there are matrices P and in Hn(X) such that PQ=QP=I where I is the identity of Hn(X) and AO=QB. It should be noted that Hn(X) has no one-sided units if X has none. This is because

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27 P(i ; j )Q( j ; i )=Q( J; i )P( i ; J ) Is either empty or the singleton set of the identity of X. Def ini tion 2 . 2 'i The matrices A and B in Hn(X) are Strictly Similar if there are matrices P and in '-"nCX) such that each nonempty entry of P is the singleton set of the identity of X and PC! and A=PBO. Clearly B=QAP and Is the transpose of P. Strictly similar implies similar. If, as in the case of free semigroups, there are no units other than the identity, then strict similarity and similarity are equivalent. Def Ini tion 2 . 25 . In Hn(X), the partial order <, is defined by A^B when AUB=B. Proposi tion 2.26 . if A<.B, then AC<.BC and CA<.CB. Proof: (AC)U(yC)=(AU!OC=BC and (CA)U(CB)=C(AUB)=CB. Proposi ti on 2.27 . Let (S,)', .) be a semi automaton. Let B be the elementary basis of n-tuples over a field. Let j ba a semigroup homomorphism as defined in proposition 2.18. Define m':X--Hn(X) such that every entry of xm' is an empty set or a singleton set of x as the corresponding entry of xj is a zero or one respectively. m' Is a semigroup homomorphism from X into the multiplicative semigroup of Hn(X). Proof: (xy)m' has singleton sets of xy and empty sets in exactly those places that (xy)J has ones and zeros respectively. The product of xm' and ym' has singleton sets of xy and empty sets in exactly those places that the

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28 product of xj and yj has ones and zeros respectively. Since (xy) j =(xj ) (yj ), it must follow that (xy)n-!' = (xm' ) (ym' ) . Proposition 2. 28. Let (S,X,.) be a semi automaton. Let m' be the semigroup homomorphism defined in proposition 2.27. Let m:Hl(X)--Hn(X) be defined so that for any nonempty subset a of X am=U{xm'|xe a}and m maps the empty set to the n-by-n matrix of empty sets. m is a semirin,!^ homomorphi sm. Proof: (3Ub)m = l){ xm' I xe ( aUb)} = (U{ xm' | xca} )U(U{ xm' | xeb} ) = (am)!J(bm). Since (xy)rn' = (xm' ) (ym' ) by proposition 2.27/ (ab)m= {xm'Ixeab} = { (xy)m' | xea, ycb} = { (xm' ) (ym' ) | xea, yeb} ={xm' I xea} { ym' | yeb} =(em) (bm) . Proposition 2.29 . Let (S/X,.) be a semi automaton. Let G be a finite set of generators for X. Let m and m' be defined as in proposition 2.28. Define M(0) to be the Image under m of the identity element of X. Define f'Un) to be the Image under m of the set of all words i n X of length less than or equal to n with respect to the input alphabet G. The n~th power of M(l) is M(n). Proof: The case when n=l is trivial. Assume the case for n-1 and calculate the product f.(n-3)M(l). If x in X is of length n or less, then there is a y of length n-1 or less and a z of length 1 or less such that x=yz. xm'<.M(n) only If ym'a.Un-i) and zm'<.M(l). Conversely, if ym'
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29 Proposition 2.30 . In the notation of propositions 2.28 and 2.29, Xm is the limit as n increases v/ithout bound of M{n). Proof: Every x in X is a product of a finite sequence of generators. Therefore Xn^UM(n). Since each M(n) is clearly contained in Xm, Xm=UM(n). Proposi t i on 2.31 . Continuing the notation of propositions 2.27 through 2.30 and adding the notation: M=Xm and b is the integer such that b(h.) = l, M(sjh;th.) is the set of all X in X such that sx=t. Proof: Multiplication by sh is equivalent to selecting the sji-th rov/ of a matrix. The sh.-th row of xj will be th If and only if sx = t. in other v/ords the ^-th entry of the ^-th row of xj is 1 and the same entry of the same row xm' is the singleton set of x if and only if sx = t. M(sJi;tJ].) is by definition the union of the corresponding entries for all X in X but only those x for which sx=t have a non-empty entry in this position. Proposi ti on 2.32. Let k' nov/ be defined to be any one-to-one mapping of the state space of (S,X,.) onto the integers 1 through n where n=iS|. Let k now be defined so that it is that mapping from S into the set of functions from X to the integers 1 through n such that the image of X under the mapping sk (notation: x:(sk)) is (sx)k'. The pair of mappings (k,i) v/here i is the identity mapping on X is a semi automaton isomorphism from (S,X,.) to (Sk,X,.) v.'ith the naturally induced transition function (sk)x = (sx)k.

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30 Proof: If s?^t, then (se)k ' =sk Vtk' = ( te)k ' and hence e: (sk)j*e : ( tk) . The remaining properties of a semi automaton isomorphism follow directly from the hypotheses. Proposi tion 2.33 . Let the notation be as in proposition 2.32. For ail x and y in X, (xy) : (sk)=y: ((sx)k) . Proof: (xy):(5k)=(s(xy))k'=((sx)y)k'=y:((sx)k). Proposi ti on 2. 3't . If k and k' are defined as in proposition 2.32 and s k ' = s h in the notation of proposition 2.31/ then x is an element of M(^h.; x:(sk) ). Proof: By proposition 2.32, x; (sk) =(sx)k '= (sx)h . Py proposition 2.31, x is an element of M(sJi; (sx)h ) . Proposi ti o n .? ..35. Let the mappings h and m and the matrix f' be those defined in propositions 2.27, 2.28, and 2.29 relative to (S,X,.) and h", m", and M" be those relative to {T,X,.). Let n= | S I >.n"= |T| . The existence of an n" by n matrix P over Hl(X) and its transpose such that every column of P has n"-l empty sets and one singleton set of the identity e of X for entries and PMQ=M" is a necessary and sufficient condition for the existence of a mapping g from S onto T such that (g,i) is a semi automaton homomorphi sm. , Proof: Assume that (g/i) is a semimachine homomorphism from (S,X,.) onto (T,X,.). Since g is onto and h and h" are both onto and one-to-one, there is a mapping p from the first n positive integers to the first n" such that shp= sgh" for all s in S. Let P be defined so that P(.),j")

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31 is the singleton set of e if j"p=j and is the empty set otherwise. Let x be any element of M"( th" ; vh" )^ There exists an element s of S such that sp: = t. Since g is a semimachine homomorphi sm, (sx)g=v. The mapping p v;as defined so that shp= th" and sxh p= vh" . x is an element of M(sh;sxh)= P( th"; sh)f/(sh; sxh)0(sxh; vh")< PMQ( th" ; vh") . Hence f'"<.PMO. Assume now that x is not an element of r'( th"; vh") . If sg = t, then (sx)g/*v and sxh PT^ vh" . For all s such that sg = t, x is only in rUsh;sxh) and PC^h^; sh)M(^; SiLh)0( sxh; vji!!) is empty. Hence PMO^M". It has now been shown that ^'" = PNiO if (g, i) is a semimachine homomorph i sm. Mext assume the existence of the matrix P with the stated properties. Let p be defined such thatP(Jp;j) is a singleton set for each j<.n. Lot g be defined such that ,sho = sgh" for all s in S. Since P(shp;sh)M(sh;sxh )Q(,sxh; sxhp)
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32 Proof: This is the special case of prooosition 2.35 where n=n". The mapping p is a permutation. The matrix P is a unit of Mn(X). Any semi automaton which is the ima;
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CHAPTER 3: MACHIMES AND AUTOMATA Spct i on 1: General Dl scuss ion The first difference between the results of this chapter and analogous results is that the definitions are more general. V'hen the case of finitely generated free semigroups of input and output Is treated, the egdulk automaton Is Isomorphic to a milnimal automaton and also has in its definition all information about the transition and output functions. Ginsburg (Gl) and Booth (B) make clear presentations of the more usual treatment of minimization. Two states s and t are equivalent If there are mappings c and c' with the property of c In definition 3.05 from their respective state spaces Into the state space of some automaton such that sc=sc'. ProDos I ti on 3. 01 . If (h,i,i) is a machine homomorphism from (S,X, . ,q, Y, * ) onto (T, X, . , r, Y, * ) or an automaton homomorphism from (S,X,.,Y,*) onto (T,X,.,Y,*), then for each state s In S and each x In X, s*x=(sh)*x. Proof: (sh)*x=(sh)*(xi)=(s*x)I=s*x. Propositio n 3.02 . If (h,j,k) and (h',j',k') arc machine or automaton homomorph I sms from (S, X, . , r, Y, * ) to (S',X', .,r',Y',*) and thence to ( S", X", . , r", Y", * ) respectively or from (S,X,.,Y,*) to CT ' .. X ' , . , Y ', *) and 33

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3U thence to (S", X", ., Y", *), then (hh ', j j M
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35 Proposition 3.0'> . If (h, j , i ) : (S, X, . , q, 7, # )-(T^ V/, . , r, Z, *) is a machine homomorphism which is onto, then (h,i,i) is a machine homomorphism from (S, X, . , q, Z, # ) onto (T, X, ., r, Z, *) if the transition and output functions on the latter are defined by (sh)x=(sx)h and (sh)*x=s#x and further (i/j/i) is a m.achine homomorphism from (T,X, .,r,Z,*) onto (T,W, . , r, 7, * ) . Proof: Proposition 2.08 provides the proof for the transition functions and, in the case of the machines, the start states. For the output functions (s#x)i=s'^x=(sh)*x=(sh)*(xi ) and ((sh)*x)i=(s#x)i=(sh)*{xj)=((sh)i)*(xj). Def ini ti on 3. 05 . A machine (S, X, . , q, Y, *) is Minimal if therv-^ is no machine (T,X, . , r, Y, *) such that |T!<|S| and (ex) *y= ( rx) *y for all x and y in X. A minimal machine is as small as any machine v/hich, having read in x, maps y to (ax)*y. Def ini ti on 3. 06 . An automaton (S,X,.,Y,*) is Minimal if there is no automaton (T,X,.,', *) vi th mapping

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36 c from S Into T such that |T|<|S| and (sx)*y=( (sc)x)*y for all s. in S and all x and y in X.

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37 Sect! on 7.' Automata Let T be a finite set of mappings from a finitely generated semigroup X onto a finite set of functions from X to a finitely f!;enerated semigroup Y such that T has the following properties: Y Is the semigroup generated by the union of the ranges of the functions in the range of the mappings in T. For each t in T and each x in X there is a t' in T such that, for all y in X, yl'=(xy)t. The image (y2):xt of yz in X under the mapping xt is equal to the product in Y of y:xt and 2:(xy)t. Def i ni ti on 3.07 . An Egdulk Automaton is an automaton (T,X,.,Y,*) v/here T is a set of mappings as described above and the transition function . is defined such that, for all y in X y(t.x)=(xy)t and the output function is defined by t*x=x:et. The sequential property is demonstrated by: For all z in X, z(t. (xy))=((xy)z)t=(x(yz))t=(y2)(t.x)=z((t.x).y). The output sequential property is demonstrated by: Since (ex)t = (xe)t = e(tx), t*(xz)=(xz):et=(x:et)(z:(ox)t)=(t*x)(z:e(tx))=(t*x)(tx*z). Proposi ti on 3.08. Let (S,X,.,Y,*) be an automaton. Let h be a mapping from S into the set of mappings from X Into the set of functions from X into Y with the property that any x in X is napped by sh to the function which maps

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38 any y in X to y :x(sh) =(sx)*y. (h,l,l) is an automaton homomorph I sm from (S,X/.,Y,*) onto the egdulk automaton (Sh,X, .,Y,*). Proof: Each i is a semigroup homomorphi sm. For each s in S and X in Y., (s*x) i = (se)*x = x :e(sh) = (sh) *x and for each y and z in X 2:y((sx)b) = ((sx)y)*7: = (s(xy))*2 = z:(xy)(sh) =2:y((sh)x)=z:y((sh)(xi)). Pr oposition 3.09 . Let (S,X,.,Y,*) and (S',X,.,Y,*) be automata and h and h' be the respective mappings from them to egdulk automata. Let h" be a mapping from S to S'. If (h",i,i) is an automaton homomorphi sm, then h=h"h'. Proof: For all s In S and all x and y in X, y:x(s(h"h')) = y:x((sh")h') = ((sh")x)*y = ((sx)h")*y = ( (sx)h")* (y i ) = ((sx)*y)i = (sx)*y = y:x(sh). ProDosi ti on 3.10 . An automaton is minimal if and only if the homomorphism (h,i,i) of proposition 3.08 is an automaton isomorphism. Proof: If (h,i,i) is not an isomorphism, then |ShI<|SI. By proposition 3.01, (S,X,.,Y,*) is not minimal. Assume that (S,X,.,Y,*) is not minimal and let c be the mapping of definition 3.06. Let s and s' be distinct elements of S such that sc=s'c. Then h is not one-to-one since y:x(sh)=(sx)*y=((sc)x)*y=((s'c)x)*y=(s'x)*y=y:x(s'h).

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39 S ect? on 3.: Mach I nes Def i ni ti on 3.11 . An Egdulk State is a mapping r from a finitely generated semigroup X onto a finite set of functions from X to a finitely generated semigroup Y with the following properties: The union of the ranges of the functions in the range of r is a set of generators of Y. For all X, y, and z in X, (yz):xr is the product in Y of y:xr and z:(xy)r. Proposi ti on 3.12 . Fvery state t of an egdulk automaton (T,X,.,Y,*) is an egdulk state which maps X into a set of functions from X into a subsemigroup Y' of Y. Proof: Let t be any state in T. For all y, y, and z in X^ (yz):xt is the product of y:xt and z:(xy)t from definition 3.07. Let A be a finite set of generators of X. Since there are a finite number of distinct functions xt where x is in X and each z:xt can be written as a finite product of a:xt, b:(xa)t, c : (xab) t, . . . , and g:(xab...f)t where 2=abc...fg and a, b, c, . . . ^ f, and g are elements of A, the set of all elements a:xt such that a Is in A and x in X is a finite set of generators of Y'. Hence Y' is finitely generated. ProDosi ti on 3. 13 . If r is an egdulk state and T is the set of all functions t such that there exists an x In X for which yt = (xy)r for all y In X, then ''T,X,.,Y/*) is an

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egdulk automaton when y(r.x)=(xy)r and rx*y=y:xr for all x and y in X . Proof: For each t in T there Is an x in X such that for each y in X yt=(xy)r. The set of ranges of the functions In the ranges of the mappings in T Is equal to the set of ranges of the functions in the range of r. The function t' with the property that 2t'=((xy)2)r for all z In X Is an element of T. For all z in X, zt ' = ( (xy)z ) r = (x(yz) ) r = (yz ) t. For all z and w in X, (zv/) :yt = (zv/) : (xy) r which is the product in Y of z:(xy)r and w:((xy)z)r, and further z:(xy)r = z:yt while v/: ( (xy )z ) r=v,': (x( yz ) ) r=w: (yz ) t . Therefore T is a set of mappings for v/hlch the transition and output functions of definition 3.07 can be defined. Def Ini t i on 3. lU . An Egdulk Machine is a machine (T,X, ., r, Y, *) where r and (T,X,.,Y,*) are as in proposition 3.13. Proposi tion 3. 15 . Let (S, X, . , q, Y, *) be a machine. Let r be the mapping from X to the set of functions from X to Y defined by y:xr=(qx)*y for all x and y In X. Let h be defined such that qh--r and y( (nx) h) = (xy) r for all x and y in X. The mapping r is an egdulk state and (h,l,i) Is a machine homomorphi sm. Proof: For any state s in S there Is a z in X such that s=qz. y:x(sh) = y:x((qz)h) = v:Czx)r = (qzx)*y = (sx)*y for all X and y in X. (Sh^X,.,Y,*) is an autonaton and (h^l^i) Is an automaton homomorphism by proposition 3,^8. By proposition 3.1?, r is an egdu'ik state. By proposition

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3.13 and definition 5.1k, (Sh^ X, ., r, Y^ *) is an egdulk machine. Since qh=r, (h^i^i) is a machine homomorphi sm. ProDosi tion 3. 16 . Let (S^ X, ., q, Y, *) and (S'z X, .,q ', Y, *) be machines and h and h' be the respective mappings from them to egdulk machines. Let h" be a mapping from S to S'. If (h,i,i) is a machine homomorphi sm^ then h=h"h'. Proof: Proposition 3.09. Proposition 3.17 . A machine is minimal if and only if the homomorphism (h,i,i) of proposition 3.15 Is a machine isomorphism. Proof: If (h, I^i) is not a machine Isomorphism^ then IShKISj. By proposition 3.01, (S,X, ., q, Y, *) is not minimal . Assume that (S, X, ., q, Y/ *) is not minimal. Let (T,X, ./ r, Y, *) be a machine such that iT|<|S| and (qx)*y=(rx)*y for all x and y in X. Let x and z be elements of X such that qx and qz are distinct v/hlle rx = rz. y:x(qh) = {qx)*y = (rx)*y = (rz)*y = (az)*y = y:z(ah) for all y in X. The mapping h is not one-to-one.

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U2 Sect! on i: Semitnachl nes and Semi automata The definitions provide that (S,X,.) is the semi automaton, (S,X,.,q) is the semimachine, and (S,X,.,Y,*) is the automaton of the machine (S, X, . , a, Y, *) while (S.v, .) is the semi automaton of the automaton (S,X,.,Y,*). The propositions of Chapter 2 can be applied to the machines and automata of Chapter 3 with the help of proposition 3.18. Proposition 3.18 . If (h,i) is a semimachine or semiautomaton isomorphism and (sh)*y=s*y, then (h, i,i) is a machine or automaton isomorphism. Proof: (s*x) i =s*x"(sh)*x = (sh) Â• (xi ) . The rest follows from the def ini ti ons . Each machine defines an input ripht congruence F:(r) and an input congruence E(S). A machine is isomorphic to a machine with a right congruence semimachine for its semimachine by proposition 2.11 and is a homomorphic image of a machine which has as its semimachine the right congruence semimachine of the input congruence of the former machine by proposition 2.13. Proposi t i on 3.19 . The Input right congruence of any machine refines that of its egdulk machine. Proof: (h,i,i) of proposition 3.15 is a machine homomorph i sm. (h,I) is a semimachine homomorphi sm. Py

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Ij5 proposition 2.1U, the former input right congruence refines the latter. The semi automaton of each automaton is isomorphic to a semi automaton of the type described in proposition 2.32. Using proposition 3. 18, a canonical form with functions into the integers for states can be defined.

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C HAPTER it: M'JLT I PROnPAMMI MG Sect? on 1: Heneral Pi scuss i on Def ini t i on 'Â» . 1 . A Mul t i profrran^mi ng Machine is a machine (S ' .X ' , . , p, Y', *) v/ith submachine (S, X, ., p, Y, * ) such that there exist machines (V, A, ., o, ^v *) snH (V/, C, ., r, n, * ) v/here A and C are contained in X and B and D in Y and semigroup homomorphi sms j:X--A, k:Y--R/ j':A--r, and k':Y-D such that aj=a and aj'=p for all a in A^ bk=b and bk'=e for all b in B, cj=e and cj'=c for all c in C, dk=e and dk*=d for all d in D, and such that (o(x j ) ) * (yj ) =( ( px) *y)k and (r(xj '))*(yj ') = ((px)*y)k' for all x and y in X. The first machine is said to simulate the last tv/o. Def I ni t i on '1.02 . A Mul t i prop:rammi ng Automaton is an automaton (S ' ,X ' , . , Y '^ *) with subautomaton (S,X,.,Y,*) such that there exist automata (V,A, .,B,v.) and (W,C,.,n,*) where A and C are contained in X and 9 and in Y and semigroup homomorphi sms j:X--A, k:Y--B, j':X--C, and k':Y-D and relations H and M in 5;xV and <5xV7 such that aj=a and aj'=o for all e In A, bk=b and bk'=e for all b in B, cj=e and cj'-c for all c in C, dk=p and dk'=d for all d in D, there exists an s in S for each (v,w) in VxV such that (s,v) is in G, and (S/W) is in f'/ ( (sx ) *y) k = (v(xj ) )* (yj ) for all X and y in X and all (s,v) In C, and Ilk

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1*5 ((sx)*y)k'=(wCxj '))*(yj ') for all x and y in X and all (S/W) In H. The first automaton is said to simulate the last tv/o. In less precise lanpiuage, a mul ti prop;rammIng automaton is one which can simulate two other automata operating independently, starting at any given state and each receiving its inputs v/hile disregarding inputs to the other. One or both of thrse may in turn be a multiprogramming automaton so that one automaton may be capable of simulating any finite number of automata. There may be states in S which are not related by G or H to any states of V or 'Â•'. There may be inputs in X' or outputs in Y' that are not in the semigroups generated by AUC and BUD. In this way, a multiprogramming automaton may be able to do more than simulate tv/o automata operating independently. In particular, there may be control inputs which may interact v/ith the elements of A and C. This section treats the case where S=S', X=X', and Y=Y'. The addition of extra states does not change the properties of the other states. Adding elements to the input semigroup may affect several of the properties of the system. Additional output elements may be included with added input elements. The hypotheses will include any needed references to these semigroups. ProDosi ti on U . n 3 . Let ( V, A, . , q, '^, + ) ard (V.', C, . , r, n, *) be machines. Let X be the free product monoid or A and C, that is, every element of X can he

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U6 written uniquely as a product of elements of AUG no two adjacent terms in the product beins both from A or both from C where the Identity element of X is in both A and C. Let Y be the free product monoid of R and D. The machine (VxW/X, ., (q^ r), Y^*), the Internal product machine of the first twO/ Is a multiprogramming machine which simulates them. Proof: Let j and j' be the mappings from X into A and C defined such that aj=a^ cj=e, (xc) j=(cx) j=xj , (ax)j=(aj)(xj), (xa)j=(xj)(aj), aj'=e, cj'=c, (cx)j' = (cjM(xj'). (xc)j'=(xj')(cj'), and (xa) j ' = (ax) j '=xj '/ for all a In A, c in C^ and x In X. Both j and j' are clearly semigroup homomorphi sms. Since (v,w)ax=(va,w)x=(v(aj),w)x and (v,w)cx=(v,wc)x=(v,w(cj ' ) )x for all (v,w) In VxW, a I n A^ c in C, and x In X^ (q,r)x=(q(xj),r(xj')) and (v,w)x=(v(xj ),w(xj ' ) ) for all x in X and (v/W) In VxW. Let l< and l< ' be the mappings from Y Into R and D defined such that bk=b, dk=e, (yd)k=(dy)k=yk, (by)k=(bk) (yk), (yb)k=(yk)(bk), bk'=e, dk'=d, (dy)k' =(dk' ) (yk* ) . (yd)k'=(yk')(dk')/ and (by)k'=(yb)k' =yk' , for all b In B, d in D, and x In X. Both k and k' are clearly semigroup homomorphi sms. Since (v,w)*(acx) Â« ((V/W)*a)(((v^w)a)*c)(((V/W)ac)*x) ((v,w)*a)((va,w)*c) ((va,wc)*x) = (v*a) (w*c) ( (va,wc)*x) and similarly (v,w)*(cax) = (w*c) (v*a) ( ( va,wc)*x) v;hlle (v*a)k=v*a, (v*a)k' =e=(v/*c)k, and (w*c)k'=w*c for all (v,w)

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Ij7 In VxW, a in A, c in C, and x In X, it follows that (((q.r)x)*y)k = ((o(xj),r(xj'))*y)k = (a(xj))*(yj) while (((q,r)x)*y)k' = (r(xj'))*(yj '). Proposition k.Oh . Let (V, A, .,p, *) and (W,C,.,r,*) be automata. Let X be the free product monoid of A and C and Y the free product monoid of R and D. The automaton (VxVI, X, . , Y, *) , the internal product automaton of the first tv/o, is a multiprogramming automaton v/hich simulates them. Proof: Let G be the set of all ((V/W),v) and H be the set of all ((v,w),w) v/ith (v,w) in VxV/. Define the semigroup homomorph i sms j, j', k, and k' as in the proof of proposition U.03. if (s,v) Is in G, then there is a vi In V^ such that s = (v,v/). Hence, by proposition 4.03, ((sx)*y)k = ( ( (v,v/)x)*y)k = (v(xj))*(yj) in the submachine with start state 5. Similarly, there is a v in V for each (s,w) in H such that s = (v,v/) and ((sx)*y)k' = ( ( ( v,v/)*y)k' = (v;(xj ' ) )*(yj ' ) . The output function on a submachine is the restriction of the output function of the automaton to the states In the submachine. Propos I tion 't . H 5 . Let ( V, A, . , q, P, *) and (W, C, ., r, P, *) be machines such that A and C are disjoint subsemigroups of a semigroup X and R and D are disjoint subsemi groups of a semigroup Y. If there is a machine (S' ,X, . , p, Y,*) that has fewer states than Vx'' and can simulate the first tv/o machines, then at least one of the first two is not minimal.

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1(8 Proof: Let j, j\ k, and k' be as in definition 4.01. Let h be the mapping from V to egdulk states and h' be the mapping from V.' to egdulk states such that (h,i,i) and (h',i,i) are machine homomorphi sms to e?;du1k machines of the type in proposition 3.15. Since S' has fewer states than Vx''.', there is at least one pair of products ac and B^c' in X such that pac = pa'c' in S' while (oa, re) ?Â«(aa ', re' ) in VxV/. If qajSqa', then h is not one-to-one, since, for any x and y in X, (yj ) : (xj ) ( (na)h) = (yj ) : (xj ) ( (o( (ac) j )h) = (q((ac)j)(xj))*(yj) = (o( (acx) j ) )* (yj ) = ((oacx)*y)"k = ((pa'c'x)*y)k = (q( (a 'c' x) j ) ) *(yj ) = (o( (a ' c ' ) j (xj ) )* ( yj ) = (yj):(xj)((o((a'c')j)h) = (yj ) : (xj ) ( (qa ' ) h) . If rc?5rc', then h' is not one-to-one, since it can be shov;n in a similar manner that ( rc)h ' = ( re ' )H ' . Hence, one of the two homomorph i sms is not an isomorphism and, by proposition 3.16, one of the two machines is not minimal. Proposi tion k. 06 . Let (V, A, .,n,*) and (V/, C,.,D,*) be automata such that A and r are disjoint subsemi jrroups of a semigroup X and R and D are disjoint suhsemi groups of a semigroup Y. If there is an automaton (S',X,.,y,*) that has fewer states than VxV,' and can simulate the first two automata, then at least one of the first two Is not minimal . Proof: Let j, j', k, k', G, and H bo as in definition U.02. Let h be the mapping from V to egdulk states and h' be the mapping from V^ to egdulk s!:ates such that (h,i,i) and (h',I,i) are automaton homomorph i sms of the type 'n

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U9 proposition 3.08. Since S has fewer states than VxV/, there is at least one distinct pair of elements of Vx'.', (v,w) and (v',v/'), such that there is one element s in S for v/hich both (s,v) and (s,v') are in G and both (s,w) and (s,v/') are in H. If Vj^v', then h is not one-to-one, since, for any x and y in X, (yj ) : (xj ) (vh) = (v(xj))*(yj) = ((sx)*y)i' = (v'{xj))*(yj) = (yJ) :(xj)(v'h). If w^/', then h' is not one-to-one, since, for any x and y in X, (yj ' ) : (xj ' ) (v/h ' ) = (v/(xj Â•))*(yj ') = ((sx)*y)k' = (w' ( xj ' ) )^)yj ' ) . Hence, one of the two homomorphi sms is not an isomorphism and, by proposition 3.10, one of the two automata is not minimal. Propositions U.05 and 4.06 say that no multiprogramming system is more efficient than the product of efficient systems. V.'h i 1 e propositions 11.03 and U.Ofj showed the existence of a mul t i pro'^rammi ng system simulating any two given systems as an internal product when its semigroups are free products of those of the simulated systems. the following propositions examine some of the properties of systems with other semigroups. Proposi ti on tt . 07 . If the automata (S,X,.,Y,*), (V,A,.,n,*), and (W,r,.,f^,*) as defined in definition h.r^l are egdulk automata having the property that s*a is in R and s*c is in D for each s in S, a in ^^ and c in ^ where X is generated by AUC and P(v) and P(w) are, respectively, the sets of all s such that (s,v) is In G and (s,u) is in H for some u in V! and such that ('i,w) Is In H and (s,u') is in G for some u' In V, then for each w in '/ ( P(v/) , A, . , ?, * )

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50 is Isomorphic to (V,A, .,R,*) while for each v in V (P(v),C, .,D,*) Is isomorphic to (V/, C, . , D, *) . Proof: if s is in the intersection of P(v) and P(v') where V and v' are in V, then v=v' since for all a and a' in A a'-:av = va*a' = (v(aj))*a'j) = (sa*a')k = (v' (aj ) )* (a' j ) = v'a*a' a':av'. If s is in the intersection of P(w) and P(v/') where w and v/' are in VJ, then w=w'. There is at least one s in the intersection of each P(v) and each P(w). Assume that there is a (v,w) in VxW such that s and t are in the intersection of P(v) and P(w). if s and t are distinct, then there is come x and some y in X such that y:xsj^y:xt, and since y is In the semigroup generated by AUC, there are y' and y" in X and z in AUG such that y = y'zy" and z : (xy ' )5j^2 : (xy ' ) t . If z is in A, then z:(xy')s = (2:(xy')s)k = (zj ) : ( (xy ' ) j ) v = (z:(xy')t)i< = z:(xy')t. Similarly, if z is in C, then z:(xy')s = z:(xy')t. Hence, s and t cannot be distinct. For any w in V/, define the mapping h from P(v') to V by restricting ^ to P(w)xV. For any v in V, define the mapping h' by restricting H to P(v)xW. The triples of mappings (h,j,k) and (h',j',k') are automaton homomorphi sms. The triples of mappings (h,j|A,k|R) and (h ' , j ' | C, k' | D) are automaton homomorph i sms which, since they are one-to-one, are isomorphisms. Prop os 1 1 i on h . OS . If the machines (S,X,.,p,Y,*}, (V, A, . , q, P, *) , and (V.', C, . , r, D, * ) as defined In definition U.Ol are egdulk machines having the property that s*a is In B and s-*c Is In D for each s In S, a In A, and c In C where

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51 X is generated by AUC, then for each c in C the submachine (T,A, .,pc, P,*) of (S,X, .,p,Y,*) is isomorphic to (V/A, .,q, B, *) while for each a in A the submachine (T',C, .,pa,D,*) of (S,X, .,p,Y,*) is Isomorphic to (W,C, .,r,D,*). Proof: The relations G and H of definition U.02 and subsets P(v) and P(w) of proposition k.^7 can be defined. For each vj in W ( P(\7) , A, ., B, *) is automaton isomorphic to CV/A,.,B,*) and for each v in V ( P( v), C, . , P, *) is automaton isomorphic to (W,C,.,D,*) by proposition k.^7. Let c be some element of C. Since (rc(xj ' ) )*( yj ' ) = (r(cj')(xj '))*(yj') = (r((cx)j'))*(yj') = ((ncx)*y)k', (pc, re) is in H and pc is in P(rc). Since ( rc(xj ' ) )*(yj ' ) (r(cj')(aj')(xj'))*(yj ') = ( r { (cax) J ' ) ) * ( yj ' ) ( (pcax)*y)k', pea is in P(rc) for all a in A, Since (P( re). A, . ^ B, *) is automaton isomorphic to (V,A,.,t>^*) which is the automaton of a machine, any element of P(rc) can be expressed as pea. In particular, since (a(xj))*(yj) = (q(cj)(xj ))*(yj) = (n((cx)j))*(yj) (ncx*y)k, (pc)h=a and each automaton Isomorphism is a machine isomorphism. Similarly, each automaton isomorphism (h ' , j ' | C, k' | P) as defined using proposition k.f)7 is a machine isomorphism v;ith (pa)h' = r. Although the previous two propositions may seem obvious and the conditions on the output function unnecessary, there are egdulk automata v/hlch have their state spaces contained in the first projections of both of

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52 the relations G and H and yet have more states than VxV'. For example, for some s and t in S and a in A, the output mapping might be defined so that s*a=d'bd and t*a=b v;here b is in Â°> and d' and d are mutually inverse in D. Proposition U.09 . Let the automata, mappings, and relations be as defined in definition tt.02. If S is in the first projections of both '^ and H, then there exist automata (V, A, .,?,*), (W',C,.,n,*), and (V'x'a" , y, . , Z, *) and semigroup homomorphi sms f":Y--Z, f:Z---B, and f':Z--D and mappings h:V--V', h':W--W', h":S--V'xV'', g: V'x'/'' --V , and g' :V'xV.''--W' such that yf"f = yk and yf"f' = yk' for all y in Y, (hd)f"=(db)f" for all b in B and d in 0, (v',v7' )g=v 'and (v', w' )g'=v;' for all (v',w') in V'xV'', sh"=(vh,wh') for all (s,v) in G and (s,w) in H, and the triples of mappings (h, i , i ) : (V, A, . , P, *)--( V' , A, . , '*, *), (g,j/f):(V'xV'',X, .,Z,*)--(V',A, .,R,*), (h", i,f"):(S,X, .,Y,*)--(V'xV.",X, .,Z,*), (g ' , J Â• / f ' ) : ( V xVJ ',X,.,Z,*)--(V/',C, .,D,*), (h', i, i ):(K', C, .,n,*)--(W',r, .,0,*) are homomorph i sms . (S,X, .,Y,*) (V,A,.,B,*) (h",i,f") (h,i,;) ^^^(V'x'-",X,.,Z,*) (V',A,.,P,*) ^j,f) (g',j',f'r(W',C,.,n,*) Proof: iot E be the intersection of K(k) and K(k'), Z=X/p, and f" btt the natural homonorohism froni Y to Z. For all b in P and d in 0, since (bd) k=(bk) (dk)=be=eb=(dk) (bk) =(db>k and al 1 automaton (W, C, .,n,*) (h', i,i)

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53 and (bd)k'=ed=de=(db)k', it follows that (bd,db) Is in F, (bd)f"=(db)f", and F. = K(f"). Since K(f") is a subset of K(k) there is a unique horyomorph i sm f such that yk=yf"f for all y in Y (CP). There is likewise a unique homomorphism f such that yk'=yf"f' for all y in Y. Let (V, A, .,'',*) and (VJ' , r, ., D^ *) be epdulk automata and (h,i,i) and (h',i,i) the respective automaton homonorphi sms from (V,A, .,Y,*) and (W,C,.,n,*) as defined in proposition 3.08. Let g be the projection from V'x'-" to V and e' the projection to V''. if (s,v) and (s,v') are both in ^, then vh=v'h because for all a and a' in A a':a(vh)=va*a'=((sa)*a')k=v'a*a'=a':a(v'h). Similarly wh'=w'h' if (s,v/) and (s,v/') are both in !'. The mapping h" is defined by sh" = (vh, v/h ' ) v/here (s,v) is in G and (s/V/) In H. For the transition and output functions of ( V'xW , X, . , 7, *) define (vh,wh ' ) x= ( (vh) (x j ), (wh ' ) (xj ' ) ) and (vh,wh')*x=((vh)*(xj))((wh')*(xj ')). That (vh,wh' )* (xy ) = ( ( vh ) * ( ( xy ) j ) ) ( (wh ' ) * ( ( xy ) j ' ) ) ( ( vh ) *Â• ( X j ) ) ( ( vh ) ( X j ) * ( y j ) ) ( (wh ' ) * ( x j ' ) ) ( (v/h ' ) ( x j ' ) * ( y j ' ) ) = ( ( vh ) * ( X j ) ) ( ( wh ' ) * ( X j ' ) ) ( ( vh ) ( x j ) * ( y j ) ) ( (wh ' ) ( x j ' ) * ( y j ' ) ) = ( {vh,wh ' ) *x) ( (vh, v;h ' )x*y) for all x and y in X, (vh,wh')*a = ((vh)*(aj))((wh')*(aj ')) = ((vh)*a)e = (vh)*a for all a in A, and (vh,wh')*c = (v/h')*c for all c In r whenever (vh,wh') is in V'xW shows that ( V xVf' , v, . , Z, *) is the internal product automaton of (V'/A, .,^,*) and (W',C,.,n,*). Since j and g are semigroup homomorphi sns

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Sk and ((vh,wh')*x)f = ( ( ( vh)*(xj ) ) ( (wh' *(xj ' ) ) ) f (((vh)*(xj))f)(((wh')*(xj'))f) = ((vh)*(xj))e ( (vh,wh ' )g)*(x j ), (p;/j/f) is an automaton homomorDhi sm. The triple (p'/J'/f) is also an automaton homomornhi sm. Since (sh")*x= (vh,wh')*x = ( ( vh)* (xj ) ) ( (v/h ' )*(xj ' ) ) = (s*x)f" for all (s,v) in C-, (s,vi) in H, and x in X, (h",i,f") is an automaton homomorph i sm. ProDosi ti on it . 1 . Let the machines and mappings be defined as in definition 't.Ol. There exist machines (V',A, .,qh,R,*), (W',C, .,rh,C,*), and (V xV,'' , X, ., (qh^ rh)/ ?/ *) and semigroup homomorphi sms t", f, and f, and mappings h, h', h", g, and g' as in proposition k.09 such that yf"f = yk and yff =yk' for all y in v, (bd)f"=(db)f" for all b in R and d in D, (v',w)g=v' and (v',w')g'=^v/' for all (v',w') In V'xW, (px)h"=((q(xj))h,(r(xj'))h) for all x in X, and the triples of mappings (h, i , i ) : (V, A, . , q, ", *)-( V', A, . ,qh, " , *), (g/j/f ):(V'xV/',X, .,(oh,rh'),Z,*)--(V',A, .,qh,B,*), (h", i,f"):(S,X, .,p,Y,*)--(S,X, ., (ah, rh '),?,*), (g'/j ',f '):(V'xV",X, ., (ch,rh'),Z,*)--(W',C, .,rh',n,*), and (hM, i ):(W,C, .,r,D,*)--(W',C, .,rh',n,*) are all machine homomorphi sms. (S,X,.,p,Y,*) (V,A,.,q,R,*) (h'M,f") (W,C,.,r,n,*) (h,I,l) .(V'xV",X, .,(rh,ch'),D,*). (hM,i)| {V',A,.,ph,R,*) (g,j,f) (g'^j'.f) fW,r,.,qh\n,*)

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55 Proof: Let G be the set of all (px,q(xj)) and w the set of all (px,r(xj')) such that x is in X. S Is the first projection of both G and H and (S,X/.,Y,*) is a multiprogramming automaton simulating (V^A, .,Pv*) and CW, C, ., D, *) . The existence of the automata and automaton homomorphl sms follov;s from proposition '1.09. Since (p^q) is in G and (p,r) is in H, ph" = ( vh,v/h ' ) . By proposition 3.15, (h,i,i) and (h',i,l) are machine homomorphi sms. Since (ph")-=(oh, rh ' )g = ah and ( ph").'r ' = (ah, rh ' ) ' = rh ' , (h,i,i) and (h',i/i) are machine homomorphi sms. Proposition h .11 . If (S',X',.,Y',*) is a multiprogramming automaton simulating (V/A/.,B,*) and (V7,C,.,D,*) and is a subautomaton of (S", X", . , Y", * ), then (S",X", .,Y",*) simulates (V,A, .,R,*) and (W,C,.,n,*). Proof: The automaton (S,X,.,Y,*) of the definition Is also a subatomaton of (S", X", . , Y", *) . The mappings from S, X, and Y and the relations G and H serve to shov/ that any automaton with this subautomaton simulates (V/A, .,r,*) and (W,C, .,D,*). Proposition h. 12 . If (S ' , X ' , . , p, Y ' , *) is a multiprogramming machine simulating ( V, /^^ ., q, Â°>, *) and (W,C, .,r,n,*) and is a submachine of (S", X", . , p, Y", * ) , then (S",X", .,p,Y",*) simulates ( V, A, . , q, R, * ) and (W, C, . , r, P, * ) . Proof: The machine (S, X, . , p, Y, ^-) of thp definition Is also a submachine of (S", X", . .. p, Y", *) with the same start state. The mappings from S, X, and Y serve to show that any machine v/l th the same start state which has this machine as

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56 a submachine simulates (V, A, ., q, B, *) and (W, C, . , r, C^ *) . Proposi tion h. 13 . If ( V , A' , ., q, R' , *) and (V/' ,C*, . , r, r' , *) are respectively submachines of (V,A, .,q, B,*) and (W, C, . , r, D . *) and the latter pair of machines is simulated by (S ' , X ' , . , p, Y' , *), then the former pair of machines is also simulated by the same multiprogramming machine. Proof: Let X" be the subsemi ciroup of X' generated by the union of A' and C. let (S", X", . , p, Y", *) be a submachine of (S',X', .,p, Y',*) . Since X" is a subsemi p,roup of X, (S",X", ., p, Y",*) is a submachine of the (S, X, . , p, Y, *) defined in definition U.Ol. The restrictions of j and j' to X" and of k and k' to Y" are the mappineis which show that (S',X\ .,p,Y',*) simulates ( V, A' , ., q, R * , *) and (K",C',.,r,D',*). Proposition k.lk . If ( V , A' , . , R ', *) and d'J' , C , . , D' , *) are respectively subautomata of (V,A,.,R,*) and (V/,C, .,D,*) and the latter pair of automata is simulated by (S ' , X ' , . , Y ' , *) , then the former pair of automata is also simulated by the same multiprogramming automaton. * . Proof: Let X" be the subsemi proup of X' generated by the union of A' and C'. Since X" is a subsemi rroup of X, there Is a subsemlgroup Y" of Y such that (S, X", ., Y", *) is a subautoiraton of (S,X,.,Y,*) as defined in definition 'i.02. The relations H and H and the restrictions of j and j' to X" and of k and k' to Y" show that (S ' , X ' , . , Y' , *) simulates

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57 (V',A',.,B',*) and (\1\C\ . ,T}\*) . Propos i ti on tt. 15 . Any machine is trivially a mul ti programmino; machine. Proof: Let (S, X, . , p, Y, * ) = {V, A, . , q, p, *) and let (W,r,.,r,*) be a single state machine vfith input and output semigroups having only single elements. j and k are identity mappings on X and Y v/hile j' and k' map all elements of each to its identity element. Proposi t i on ti . 16 . Any automaton is trivially a multiprogramming automaton. Proof: l-et (S, X, ., Y, *) = (V, A, . , B, *) and let (W, r, .,0,^0 be a single state machine v^ith input and output semigroups having only single elements. j and k are identity mappings on X and Y while j' and k' map all elements of each to its Identity element. G is the identity relation on S while H Is the set of all (s,w) such that s is in S and w is the element of V.'. Def i ni ti on U. 17 . A Trivial f'cnoid is a monoid v/ith a single element. D ef ini ti on U . 1 8 . A Trivial Machine, Automaton, Semimachine, or Semi automaton is one with a trivial input semi group. As a result of proposition fi.l3, the investigation of the existence of a pair of nontrivial machines which a given machine (S ' , X ' , . , p, Y ' , *) cen simulate can be limited to machines v/ith cyclic subsemi groups of X' for input. This investigation can be fur':her limitfid to nontrivial

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58 cyclic monoids without proper minimal nontrivial submonold. Propos i tion JlJLB.* Hvery nontrivial monoid contains a submonold where the identity of the submonoid is that of the monoid of one of the following types: a 2element semilattice, a group of prime order, or an infinite order cyclic monoid. Each of these is nontrivial but contains no nontrivial proper submonolds. Proof: Any element other than the identity of a monoid generates a cyclic submonoid. A cyclic monoid contains an identity element and all powers of some element b. If the monoid is infinite, any submonoid Is either the trivial one or a monoid generated by some power of b. Neither type Is both minimal and nontrivial. If the monoid Is finite, some power of b is an idempotent. If this Idempotent is not the identity, the monoid consisting of the idempotent power of b and the identity Is a ^-element semllattice every submonold of which is trivial. If this Idempotent is the identity, let the n-th power of b be the identity. If n is prime, there is only one submonoid and it Is trivial. If n is not prime and n/m is prime, the m-th power of b generates a nontrivial submonold of order n/m. Proposition U.20 . If (S ' , X ', ., Y' , *) is a multiprogramming automaton simulating two nontrivial automata or If (S ' ,X ', ., p, Y' ) Is a mul t i proÂ«rrammi ng machine simulating tv/o nontrivial machines, then there are in X* two elements a and c which generate monoids of prime or

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infinite order such that the only power of a which is a power of c is the identity. Proof: If (S',X ' , .,Y ', *) simulates nontrlvial automata (V,A/./^/*) and (V/.. C, ., P, *), A and C are nontrivial and there are, by proposition '.19, an a in A and a c in C which generate submonoids of prime or infinite order. If X is a power of both a and c, then xj=x because it is a pov;er of a while x=xj=e because it is a power of c. Proposition h.^l . If (S ' , X Â» , . , y ' , *) is an automaton such that there are elements a and c in X' such that the monoid generated by eacli is either infinite or a group of prime order or a semi lattice of order two and the natural homomorphism from the monoid X generated by a and c to X/E wfiere E is the congruence on X generated by (ac,ca) is one-to-one v/hen restricted to the setv/ise product of the cyclic monoid generated by a and that generated by c, then there are nontrivial automata ^\',^,.,^,*) and (V,',C, .,D,*) such that (S',X',.,Y',*) simulates them. Proof: Let (S/X,.,Y,*) be any subautomaton such that X is generated by a and c. L'^t A be the submonoid generated by a and C that generated by c. Let f be the natural homomorphism from X to X/F. For any x in X there is a unique a' in A and a unique c' in C such that xf=(a'c')f. Define j and j' such that xj=a' and xj'=c' for each x in X. These mappings are endonorphi sms since for any x and y in X (yf)(yf) = ((xj)(;cj'))f ((yj)(yj'))f

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60 ((xj)(xj ')(yj)(yj'))f = ((xj)(yj)(xj ')(yj'>)f ((xy) j (xy) j ')f =-(xy)f. Let k and k' be the constant mappings from Y to the identity of Y. Let the machines (S,A, .,R,#) and {S,C,.,n,#) be defined with transition functions agreeing with (S,X, .,Y,*) and all output the identity. Lot and !! be the identity relation on S. For all X and y in X and all (s,s) in G ( (sx)*y)k=(s (xj ) )'Myj ) . For all K and y in X and all (s,s) in H ((sx)*y)k'=(s(xj'))#(yj'). Proposition tt.22. If (S',X',.,p,Y',*) is a machine such that there are elements a and c in X' such that the monoid generated by each is either infinite or a group of prime order or a semi lattice of order two and the natural homomorphism from the monoid X generated by a and c to X/E where E is the congruence on X generated by (ac,ca) is oneto-one when restricted to the setwise product of tlie cyclic monoid generated by a and that generated by c, then there are nontrivial machines ( V, A, . , q, B, *) and (.V},r,.,r,^^*) such that (S' ,X ' , .,p^ Y', *) simulates them. Proof: By proposition U.21/ the automaton of this machine simulates tv/o nontrivial machines. ^v proposition 'f.ltj, it simulates the automaton of any submachine of them. Any submachine of a nontrivial automaton which has the same input monoid is likev/ise nontrivial. in particular, the automaton of any submachine stcsrting with a state a such that (P/O) is in G or state r such that (o,r) is in M is simulated as automaton by the automaton (S' , X ', . , Y ' , *) .

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61 Hence, ( (Dx)*y)k=(q (xj ) )* (yj ) and ( ( px)*y)k' =( r (xj ' ) )*(yj ' ) and they are simulated as machines.

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62 Sect i on 2: Free Mono! ds of I nout and Output Def i ni t ion ii . 23 . The relation C(R) is defined for any relation R on the jrenerators of a finitely ppnerated free monoid X to be the congruence relation f^enerated by the set of all pairs (ah,ba) such that a and b are generators of X but neither (a,b) nor (b,a) is in r>. ProDos i t i on U . ?U . A necessary and sufficient condition for C(0) to be a subset of C(R) is that R be a subset of the union of 0, the inverse relation of n, and the identity relation. Proof: Assume that for every (a,b) in R either a-b, (a,b) Is in 0, or (b,a) is in n. Let x and y be tv;o elements of the finitely generated free monoid X such that (x,y) is in C(Q). Either there is a finite sequence of v/ords x',x"/ . . . , y", y' such that x differs from x', y' differs from y, and each v;ord In the sequence differs from the next only in the transposition of two distinct generators which are not related by or the inverse of or x differs from y only In the same manner or x==y. In the first case each of the pairs (x,x'), (x',x"), ..., (v*\v^), and (y',y) is In the congruence relation C{R) since thp transposed pair of generetors was not in either ^ or its Inverse and hence not In R. In the second case (x,y) is similarly In r(R).

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63 In the last case (x,y) is in C(R) because it is an equivalence relation. Assume now that there is a pair (a,b) of distinct generators which is in P but in neither <"' nor its inverse. The pair of words (ab,ba) is in r(0). Since there is no word v/hich differs from ab only in the transposition of two generators not related by P, (ab,ab) is the only pair in C(R) v/ith ab for an entry. Since ab;?ba, (ab^ba) is not in C(R). Prooos i t i on U . ? 5 . If P and '^ are equivalence relations/ then a necessary and sufficient condition for C(Q) to be a subset of C(R) is that R be a subset of 0. Proof: This is a corollary of proposition '.2H since is its own inverse and contains the identity relation. Propos i t i on 't . ? 6 . If X is the free monoid generated by the finite set X', then for any R in X'xX' C(X'xX')<.C(R)lC( l) = C(0) where I is the identity relation and the empty seii. Proof: Since I U;i= I IIKRUKX ' xV ' , this is another corollary of proposition 't.2't. Proposition 't.?7 . If P is an eauivalence relation on the generators of a finitely /generated free monoid X and P(l), P(?), . . . , P(m) are the equivalence classes of P^ then for each x in X there are elements a,b, ...,d respectively of the free monoids generated by P(1 ), ^^(2 ),..., ''(m) such that (x^ab. . .d) is in C(R).

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Proof: The proof is trivial for words of length or 1 since e=ce...e and for each generator z z=e. . . eze. . . e. Assume for any m greater than 1 that the proposition is true for ail -words of length l^ss than m. Let x be any v/ord of length m. Since x is the product of two words, z' of length m-1 and z of lenerth 1, (z',ab...d) and (x^ab...dz) are in C(R). If z is in the monoid generated by P(m), so is dz. Otherv/Ise if ab...d = y'y where z is in P(m') y' is a product of elements from ^(1) through P(rr'-l) and y a product of elements from f'Cn' + l) through P(n) (yz.zy) is in C(R) because y=e or the seouence of words v/lth z transposed v/ith each of the generators of y has each element C(R) related to yz . Finally (x, ab. . .cz. . . d) v/here cz is in the monoid generated by P(n'). Propos i tio n U. 2?^ . If the semigroups A and C of definition li.ni or 4.02 are finitely generated free monoids and X is their free product and P is the eaulvalence relation on the generators of X with the generators of A in one class and those of C in the other, r(R) is the Intersection of the kernels K(J) and K(j'). If the output semigroups B, D, and Y are similarly related and D is the analogous relation on the generators of Y, r(0) is the Intersection of the kernels K(.k) and K(!<'). Proof: Each kernel of a homomorphism Is a congruence relation. The intersection of congruence relations is a congruence relation. For each a in A and c In C (ac) j = Caj ) (cj )=ae = e5=(cj ) (aj ) = (co) j and

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55 (ac) j '=ec=ce=(ca)J ' . Since each pair (ac,ca) anH (ca,ac) where a is in A and c in C is in the intersection of the kernels K(j) and K(J') and C(R) is the congruence pienerated by such paIrS/ C(R) is a subset of the intersection of kernels. For each x in X there are a in A and c in C such that a=aj=xj and c=cj'=xj'. If (x,y) is in the intersection of the kernels, a=yj and c=yj ' . From proposition ft.l't, there are a' in A and c' in C such that (a'c',x) is in C(P). Since C(P) is In the intersection of kernels, a' = (a 'c ' ) j =xj =a and c ' = (a ' c' ) J ' =x j ' =c and (x,ac) is In C(R). Similarly (y,ac) Is in r(R). Hence the intersection of K(j) and K(j') Is in C(R). The intersection of K(k) and K(k') is shown to be equal to C(0) by a similar argument. Proposition U.?^ . If the monoids A, R, C, 0, X, and Y and the mappings j, j', k, and k' are defined as in proposition tj.28, then for all x In X and y In Y xj'=e only if xj=x, xj=e only if xj'=x, yk'=e only if yk=y, and yk=e only if yk'=y. Proof: Py proposition 'i.27, for each x in X there are a in A and c in C such that (x,ac) is in C(R) where R is defined as in proposition ti.l'K Since C(R) Is In ^(j*), if xj'=e, then e=(ac) J ' = (a j ' ) (cj ' ) ='^(cj ' ) =c. Since ac=ae is C(R) related only to itself, x-ae=a and xj=nj=x. The proof of each of the other parts of this proposition Is similar.

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66 Proposition tt.30 . If (S,X,.,Y,*) !s an automaton with Y a finitely generated free monoid, then s*p=e for all s In S. Proof: For any s in S, s*e=s* (^e) =(5*e) (se*e) =(s*e) (s*e) . Since the only idenpotent In a free monoid is tho identity, s*e=e. Proposition U.31 . If the machines (S, X, . , p,Y, *) , (V,A, .,q,B,*), and (V/, C, . , r, D, *) as defined in definition ij.Ol are egdulk machines and A, ^, C, D, X, and Y are free monoids v/here the set of .p;enerators of X I s the union of the sets of generators of A and C and theset of r:rrnerator5 of Y Is the union of the sets of generators of ^' and D, then for each c In C the submachine (T, A, . , pc, '^,* ) of (5,X,.,p, Y,*) is isomorphic to ( V, A, . , q, R, *) while for each a In A the submachine (T' ,C, . , oa, D, *) of (S, X, . , r, Y, *) Is isomorphic to ('/, C, . , r, n, *) . Proof: Since (se*a) k ' -(s(ej ' ) )* (aj ' )=se*e=e by proposition U.30, ve*3 = (se*a) !-.=se*a by proposition U.^O for all a in A. Similarly v/e*c=se*c for all c In C. Proposition I|.08 completes the proof. Proposition ti.32 . If the automata (S,X,.,Y,*), (V,A,.,B,*), and (W,C,.,n,*) as defined in definition 'f.n2 arc egdulk automata and A, 5, C, D, X, and Y are free monoids vhere the set of generators of X is the union of the sets of generators of A and C and the set of generators of Y Is the union of the sets of generators of B and D and P(v) and P(v/) are defined as in proposition ^.07, then for

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67 each w in V/ (P(w)/ A, ., B,*) Is Isomorphic to (V^A^.^R,*) while for each v In V (P(v) , C, . ,0, *) Is Isomorohic to (W,C,.,D,*). Proof: Since (se*a) k ' =se*e=e by proposition '^.30 and ve*a = (se*a)!< = se*a by proposition k.29, the result Is immediate from proposition ^.07. Proposi t Ion tÂ».33 . Any machine (S ' , X ' , .^ p, Y ' , *) with X' a noncycllc free monoid Is, nontrl vl al 1y, a multiprogramming machine. Proof: Let a and c be any two distinct generators of X*. The submonold X generated by a and c is the free product monoid of the cyclic submonoids generated by a and by c. Let E be C(l). C(l) Is the congruence relation generated by (aC/Ca). If a' and a" are pov/ers of a and c' and c" are powers of c and (a'c',a"c") Is in r(|), then a'=a" and c'=c". Hence, the natural homomorphism onto X/E Is one-toone from the setwise product of the cyclic monoids generated by a and by c In that order. By proposition I*. 22, there are non-trivial machines (V, A, . ,q, B, *) and (W,C, .,r,D,*) such that (S ' ,X ', ., p, Y ' , *) simulates them. Proposi tlon k,3k . Any automaton (S Â• , X ', . ,Y' , *) with X* a noncycllc free monoid is, nontr I vial 1 y, a multiprogramming automaton. Proof: Once more let a and c be distinct generators of X' and E be C(l) where I Is the Identity relation on the set with elements a and b. By proposition fÂ».?l there are nontrlvlal automata (V,A,.,P,*) and (W,r,.,n,*) such that

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6 8 (S' ,X', .,Y',*) simulates them. Proposi tion ti.35 . Let (S ' , X ' , . , p, Y ' , *) be a machine where X' and Y' are both noncyclic free monoids and each px*x' such that x is an element of X' and x' is a generator of X' is one of the generators of Y'. The existence of generators a and c of X' such that px*aj*px'*c for all x and x' in the submonoid generated by a and c is a necessary and sufficient condition for the existence of machines (V, A, . ,q, P,*) and (W, C, . , r, P, *) where (S' , X ' , . , p, Y' , *) simulates them and each venerator of cither A or C is a generator of X' and for each generator a of A and c of C and each element x of the free product monoid of A and C q(xj)*a and r(xj')*c am generators of v. Proof: Let (S' , X ' , . , p, Y ' , *) be a machine which simulates machines (V, A, . ,q, P, *) and (W, C, ., r, P, *) with the stated properties. For any generators a of A and c of C and element x of the free product monoid of A and C q(xj )*a=a(xj )* (aj ) = (px* j )!<' is a generator of Y'. Since (px '*c)k = p(x ' j ) *(cj )=p(x ' j )*e = e for any x' of the free product monoid of A and C, px'*c can not be equal to px*a for any x' in the free product monoid. Conversely, let a and c be generators of X' such that px*aj^px'*c for all x and x' In the submonoid X generated by a and c. Each px*a and px*c are generators of X'. Ry proposition h.Z5, there are nontrivial machines v/hlch (S ' , X ' , . , p, Y* ) simulates as a multiprogramming machine.

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69 Let (V^A^wP) and (W,C,.^r) be the semimachlnes of these machines and define new output functions on them by v*a=b=p*a for any v in V and w*c=d=p*c for any w in W. If a' is a power of a and c' Is a power of c, then q*(a'a)=(q*a')(qa'*a)=(a*a')(q*a) and r*(c'c)=( r*c' ) (rc'*c)=(r*c' ) (r*c) . Hence, if a' is the mth power of a and c' is the n-th power of C/ then q*a' is the m-th power of b and r*c' Is the n-th power of d. Since the submonoid generated by all of the elements of Y' of the form px*a and px*c is a free monoid on the subset of the generators of Y' and the 'px*a are distinct from the px*c and b and d generate free submonoids, there are homomorphi sms l< and k' such that (px*a)k=b, (px*c)k=e=(px*c)k', and (px*c)k'=d. For any x and x' in X, (q(x'j))*(xj)=a*(xj)=(p*(xj))k=(p(x' j)*(xj))k and (r(x'j'))*(xj ') = r*(xj') = (p*(xj'))k'=(p(xM')*(xj'))k' . The machine (S' /X ', .,p, Y' ,*) simulates the machines (V, A, .,q, B,*) and (W,C, .,r,*) where B is generated by b and D Is generated by d. There are machines which have px*a=px'*c for some generators a and c and x and x' In the submonoid they generate but can simulate machines with free monoids for both input and output where all outouts from generators of the Input monoid are generators of the output monoid. The following example of a machine \^/i th two generators for each of the Input and output monoids will serve to illustrate this point.

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70 Let the transition and output functions be as foHov/s . t t' t" u u' ll" 1 t ' t t u ' u u 11 t" u t' u" t u" * t t' t" u u' u" 01 n 1 1 1 II 1 1 1 The machine CS ' , X ' , . , t, Y ' , * ) has l=t*n=tl* l=t"-l, but the submachine (S,X, . , t, Y, *) where X is eienerated by 00, r^l, and 10 has the follov/Ing transition and output functions: > t U * t !! 001 t u 001 no 01 01! u t oil 01 00 101 t u 101 11 11 Since 01=-t*ni7't*10=u*10 = ll^u*01 = 00, (S, X, . , t, Y, *) simulates machines of the desired type. In particular: 2 V 00! q nil V

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71 Proposi tion U.36 . Let ( S ' , X ' , . , Y ' , * ) be an automaton where X' and Y' are both noncyclic free nonolds and each s*x such that s Is in S and x is one of the generators of X' is one of the generators of Y'. The existence of a state p in S* and of generators a and c of X' such that px*aj^px'*c for all x and x' in the subnonoid generated by a and c is a necessary and sufficient condition for the existence of automata (V/A, .,R,*) and (V;,C,.,D,*) where (S ' ^ X ' , . , Y ' , * ) simulates them and each generator of either A or C is a p:enerator of X' and for each V in V and w in 1/ v*a and w*c are j^enerators of Y'. Proof: Let (S ' , X ' , . ^ Y ' , * ) be an automaton which simulates automata (V,A, .,B,*) and (.\!,C,.,D,*) with the stated properties. Let q and r be elements of V and V! respectively. There is an element p of S' such tl^at (n^q) is in G and (.r^^r) is in H. The submachine with start state p simulates the submachines v/i th start states q and r. By proposition h.ZS, there exist generators a and c such that px*a^px'*c for all x and x' in the submonoid generated by a and c. ' Conversely, let p, a, and c be such that px*ai*px'*c for all X and x' in the submonoid generated by a and c. Ry proposition k.55, there are machines ( V, A, . , a, p, *) and (W,C, ., r,D,*) such that the submachine of (S ' , X ' , . ,Y ' , * ) with start state p simulates them. The relation H can be defined as the set of all (nx,qx) and the relation !' as the

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72 set of all (px/Tx) where x Is in the free product of A and C. The automata of the machines in the previous example^ with (t^q) and (t/V) in G: and (t,r) and (u,r) in H and the same homomorphi sms, illustrates that an automaton might simulate two automata v;i th free semigroups of input and output v;here each output from any generator of the input is a generator of the output even though it can not simulate any with these same properties where the generators of the monoids of the simulated automata are generators o^ the monoids of the simulating automaton.

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BIBLinORAPHY (A) D. N. Arden. pplaved Lop:}c and Fini tp State Machines , Quarterly Proeress Report, R.L.E., M. I .T. 62 (1961) 163-1R9. (B) T. L. Booth. Sequential Machines and Automata Theory , V'iley (1967). (BWl) A. R. Pednarek and A. D. Wallace. Fouivalences on Machine State Spaces , Mathemati ckijf Uasopis 17 (1957) 3-9. (BW2) A. R. Bednarek and A. D. Wallace. Finite Approximants of Compact , Total 1 v Disconnected Machines , Math. Systems Theory 1 (1967) 209-216. (CP) A. H. Clifford and G. B. Preston. The Algebraic Theory of Semi groups , Amer. Math. Soc. Surveys 7 (1961 and 1967). (Gl) S. Ginsburg. An. introducti on to Mathemati cal Machine Theory , Addi son-'7es ley (1962). (G2) A. Ginzburg. Six Lectures on Algebraic Theory of Automata , Carnegie Institute of Technology (1966). (G3) A. Ginzburg. Algebrai c Theory of Automata , Academic Press (196S). {Gk) Y. Give'on. On Some Propert i es of the Free Monoi ds wi th Appl i cati on to Automata Theory , Jour, of Computer and System Sciences 1 (1967) 137-15U. (G5) V. M. Glushkov. Introducti on to Cybernetics , Academic Press (1966). (HS) J. Hartmanis and R. F. Stearns. Al gebrai c Structure Theory of Spquent i al Machi nes , Prentice-Hall (1966). (KFA) R. E. Kalman, P. L. Falb, and M. A. Arbib. Topics i n f^athemat ical System Theory , McGraw-^i 1 1 (1968). (Ka) T. Kameda. P^neral i zed Transi t i on Matrix of a Sequent! al Machi ne and I ts Aool icat ions . Information and Control 12 (ige?') 259-275. (K) S. C. Kleene. Representati on of Fyents i n ^lerye Nets and Finite Automata , Automata Studies, Ann. of f'ath Studies, 3k (1955) 3-Ul.

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7it (L) YE. S. Lyapin. .S<^rr!i crrouns / Translations of f'ath. Mono?rraphs 3 (1063). ( Mc ) J . D . Mc Kn \^Mt, .1 r . K leene Quotient Theorems , Pacific Jour, of Math. Ik (196'0 13U3-1352. (Mi) M. r'insky. Computat i on ; nLnrt^ and I nf i ni te f-^achines , Prentice-Hall (1967). (Mo) E. F. Moore (Ed.). Sequentia l Machines -Â• Selected Papers, Addi son-V/esl ey (19R!+). (ij) E. M. ^ior^is. Some Structure Theorems for Toool op:i cal Machi nes . Dissertation, I'niv. of Florida, (1959). (RS) M. 0. Rabin and D. Scott. Finite Automata and thei r Peci s i on Problems , I Bfl Jour, of R^s. and Dev. 3 (1959) llti-125.

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BIOGRAPHICAL SKETCH Reverdy Edmond Wright was born 5 Aueust 1'533 In Sarasota, Florida. He v/as graduated from Sarasota High School in June 1951. From September 1951 until January 1956, he attended the Massachusetts Institute of Technology. From June 1956 through flay 1958, he served v/ith the U. S. Army. After attending the University of Florida from September 1958 through January 1960, he received with honors the degree of ^.achelor of Science in mathematics. From January 1959 through June 1950, he worked as a prograrMmer for tlie University of Florida Statistical Laboratory. From July I960 to January 1961, he served with the U. S. Army. From January 1961 to Aotil 1S65, he vies a programmer for the Statistical Sf^ction of the Univi'^rsity of Florida's APiri cul tural flxpsriment Station. From May 1953 through August 1965, he was the systems suporv'sor for the University o'^ Florida Computing Center. From September 1965 through August ]9B9, he vvas a graduate student ', n the deportment of mathematics. During the 1969-70 and 1970-71 academic years, he has been an assistant pro^ossor of computer science at the Virginia Polyteclinic Institute. Reverdy Fdmcpd '/'right is maTied to the former Lydia Rc^iike. They have three chi!di-on, Tamara, f'M-anwen, and [inid. He is a niemher o*" the Association ^or Computing Machinery, the Ar^erlcan Association of i.'ni v?. rs i ty Professors, and the America;) Association ''or the Advancerifint of Sci'-^nce.

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I certify that I have read this study and that in my opinion it confcrns to acceptable standards ol' scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Piiilosophy. Rail*! G. Professor i f (fi dge, Chai rman f athenat i cs I certify that i have read this study and that in !i y <^ip 1 I ! I Up II. t-i.M 1 I '> I Hiu i.i^ d c v.. c i^ L c! D I t; i.. L a 1 n. c4 I I i :> k.i i :Â•> <, I \ O i ci i i y presentation and is fully adequate, in scope and quality, as a dissertation for the de^^ree of Doctor of Philosophy. /[.UvJI^^d^ ^. l^jJ^^lJLA 1 1 r> /-^ r% 1 R. Professor of !"athenatics I certify that I have read this study arid that i il rny opinion it conforms to accepta'ale standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the dG?;refof [doctor 0/7 Ph i 1 osopiv/. JYX^^ I c h ae 1 L' . Professor of Industrial Systens engineering I crrriify that I Iiave read tins study '"ru; f".hat in r.y opinion it conforms to acceptable j.tandards of scholarly presentation and is fully adequate, in scope and quality, as a d i sser tat i OTi for the degree or noctcr of F!-! i I030 uhy , _i.v ^ryyu?^^.,.. Assistant Profossor of Ma Lhem.at i cs

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This ci 1 ssertat 1 on v.'as submitted to the Dean of the Coller^e of Arts and Sciences and to the Graduate Council/ and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. June/ 1971 Dean, Col 1 ege o s and Sciences Dean, Graduate School

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