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
helpful suggestions.
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
TABLE OF CONTENTS
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 Free Semi groups 62
Bibliography 73
10DL 111 llotation
f: AB
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:STf, j:XW
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
wiith union for addition
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 ntuples
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~histhtat 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
given multiprogramming task.
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
"blackbox" 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 nonempty 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:ST, j:XZ, j is a
semi group homomorphism, and (sx)h=(sh)(x)). 'f both h and
j are onetoone 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:Yl 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 isomorphism 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 semiautomaton
(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 product of the Si, W' is the subsemigroup2
of X generated by the union of the Xi, and 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
elementIs other than t~he identity in their inerein th r
eFffct of an! Inout Fro' 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 rir.
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 rxs,
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 srx=t. x and y are
elements of sh if and only if rxs=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 rxry.
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, eaaba, 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
property 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 ndimensional vector space
V', and any onetoone mapipiling h from 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 onetoone andi
onto, {hI)s a semriautomaton isonlorphism.
ain the vectcor space is the set of ntuples 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 ntuple by a matrix amounts to
selecting a row of the matrix since each ntuple 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 ntuples
over any semi ring with zero and one which coriesponds 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 ntuple 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 onesided units if X has
onesided 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 pge 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 twPosided 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 onesided
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 ntuples 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 nbyn 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 nth power of M(1) is M~n).
Proof: The case when n=13 is trivial. Assume th~e case! for
n1 and calculate the product M~(n1)M(1). If x in X is of
length n or less, then there is a y of length n1 or less
and a z of length I or less such that x=yz. xm'AM(n) only
if ym'iM(n1) 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 shth row of x) will be th if
and only if sxet. In other words the thth entry of the
ghth 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 nonemipty
entry in this position.
PLQposition 2.32. Let k' now be defined to be any
onetoone 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 onetoone, 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") .
Hence 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 scsc'.
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)is~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 onetoone
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 plhr 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 rxrz.
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 onetoone.
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)*yssy, 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 Ja1XA2, k:8 ':XC, 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:XA, kY,Jl:XC, 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, dkr 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 onetoone, 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')
onetoone, 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 onetoone, 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
onetoone, 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 onetoone, 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 are 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 teab 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":YZ, f:Z.B, and f':ZD
and m~appings h:VV', h:W, h":SV'xW!', 8Vx'
and e':V'xW' W' such that yf"'Fyk 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 th~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') wrhere (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"fyk 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 set: 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~hisms.
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 proposition4.13, 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 2element Semilattice every
submonoid of which is trivial. if this idempotent is the
identity, let the nth 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 mth 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 generat~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 onetoone w~hen restricted 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
toone 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 ab, (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 m1 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. Otherwise 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, xae=a and x)=ajx. 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)kse~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 oneto
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 nontrivi 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 nth power of c, then o*a' Is
the mth power of b and rec' Is the nth 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)ke=(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
11 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=11ju;* 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. Let (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 pgenerators 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) 39.
(BW12) A. R. Bednarek and A. D. Wlallace. Finite
Approximants of Compact, Totally Di~sconneczted
Irachines, M~ath. Systems Theory 1 (1967)
209216.
(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
Machine Theory, AddisonWesley (1962).
(G2) A. Ginzburg. Six Lectures on Alpebraic Theory of
Automata, Carnegie Institute of Technology
(1966).
(G3) A. GinzburR. Alprhraic Theory of Automata,
Academic Press (1968).
(G4) Y. Give'on. On Some Propertiees of the Free rMonoids
with Application to Automata Theory, Jour.
of Computer and System Sciences 1 (1967)
137154.
(05) V. M. Glushk~ov. Introduction in Cybernetics,
Academic Press (1966).
(HS) J. Hartmanis and R. E. Stearns. Aleebraic
Structure Thqory of Seauential Machines,
PrenticeHall (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) 259275.
(K) S. C. Kleene. Representation of Fvents In r! rve
Engs and Finite Automata, Auitomata Studies,
Ann. of IPath Studies, '34 (1956) 341.
( 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) 131131352.
(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) 1114125.
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
rxperime~nt
he was the
Computinff
Center. Fom September 1965 through August 1000, he was a
pgraduate students ;n the department of m~athemratics. Drn
the 19 89701 and 1.97071 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
Adv~anceme:nti of Scienre.
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
preent;io ad Is; fully adequate,, in scopec 3nii quallit::,
as ;; dissortati;onl fr~ the degrec 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;itation 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 ;'cathemaitic
Dean, Graduate School
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
