REMARKS ON ALGEBRAIC DECOMPOSITION OF AUTOMATA

by

A. R. Meyer and C. Thompson

Carnegie-Mellon University

Pittsburgh, Pennsylvania

August, ]968

This work was supported by the Advanced Research Projects Agency

of the Office of the Secretary of Defense (F 44620-67-C-0058)and is monitored by the Air Force Office of Scientific Research.

Dhis document has been approved for public release and sale; itsdistribution is unlimited.

REMARKS ON ALGEBRAIC DECOMPOSITION OF AUTOMATA

by

A. R. Meyer and C. Thompson

ABSTRACT

A version of the Krohn-Rhodes decomposition theorem for finite automata

is proved in which capabilities as well as semigroups are preserved. Another

elementary proof of the usual Krohn-Rhodes theorem is also presented.

]. INTRODUCTION

The constructive half of Krohn and Rhodes' decomposition theorem for

finite automata states that any finite automaton can be simulated by a

cascade of reset and permutation automata. Moreover, the groups of the

permutation automata in the cascade need only be simple groups which divide

the semigroup of the original automaton. Assorted proofs of this theorem

appear in [], 2, 3, 4, 5, 7] and we include our own elementary proof in

Section 5.

Our object in this paper is to supply the few extra steps necessary

to prove a corrected version of a slightly stronger decomposition theorem

stated by Hartmanis and Stearns [4]. This theorem appears in Section 3.

In Section 4 we exhibit a counter-example to the theorem as originally

stated by Hartmanis and Stearns_ and briefly consider cascades of '_alf-

reset" automata.

-2-

2. PREL IMINARIE S

Our notation follows Ginzburg [3]. In particular, function arguments

appear on the left (so that xf is the value of function f at argument x).

Composition of functions is designated by concatenation, with the leftmost

function understood to apply first (so that xfg = (xf)g). Fo= a function f

and set S, the restriction of f to S is f_S. The cardinality of S is IsI.

we use "_' to mean improper inclusion• For a set S and family _ of func-

• IIs_tltions with domains including S, S_ is {sfJsE s, f ] As usual,

means {s]_, and "Sf" means S[f].

A semiautomaton (or state machine) A consists of a finite set QA (of

states), a finite set A (of inputs), and a set of (transition) functions

from QA into QA indexed by A. The function from QA into QA indexed by

2A AE is _ . When the context is unambiguous, we shall frequently omit

Asuperscripts and identify _ with _ .

Let A and B be semiautomata. B is a subautomaton of A iff _B c_ ,A

QB c QA and _B = A_QB for each _ EE B. A subautomaton B of A is non-

B A QA QBtrivial if _ = _ and I I > I I > ]" B is an image of A if there are

functions _]- QA QB B A = )A. -+ and _: _ -+E such that _ is onto and _ B (_

for each _ E E B. The function _ is then called a homomorphism from A

(on)to B. A covers B, in symbols "A _ B", iff B is an image of a subautomaton

of A. Covering is transitive. A and B are equivalent iff A _ B and B >_A.

A partition _ of QA is an admissible partition of A iff for every

X E _ and _ E _A there is a Y E _ such that X_ c Y. The _uotient semi-

automaton A/_ (defined for admissible _)has state set _, inputs _A, and

transitions given by: X_A/_ = Y where Y is the (necessarily unique)

element of _ such that X_ c Y. The semiautomaton A/_ is an image of A.

-3-

Given a (cjonnecting) function _: QA _A _BX -+ , the cascade rod_

(AOB) is the semiautomaton with state set QA QB× , inputs _A and transi-u0

tions given by: (p,q)_ = (p_,q((p,_)_)) for (p,q) E QA X QB and _ E EA.

We usually suppress mention of the connecting function and simply write

"AOB ''. Cascade product is associative in the sense that given (AOB)OC,

there is an equivalent semiautomaton AO(BOC). A cascade product of a

sequence of three or more semiautomata is any parenthesization of the

sequence into a cascade product of pairs of semiautomata.

If B _ D_ then for every connecting function _ there is a connecting

function _' such that (A°B)w' _ (A°D)w" Similarly, if A _ C, there is an

A' equivalent to A such that A'OB _ COB.

A is a permutation semiautomaton iff every _ E _A is a permutation of

QA. A is a reset iff every _ E _A is a constant or identity function on

QA; constant functions are also called resets. A is an identity semiautomaton

iff every _ E A is the identity on QA.

We assume the reader is familiar with the elementary facts about

groups and semigroups. Let S and T be semigroups. S is a subgroup of T

iff S is a subsemigroup of T_ and S is (abstractly isomorphic to) a group.

S divides T, in symbols "SIT" , iff S is a homomorphic image of a subsemi-

group of T. Division is transitive. "T -_ S" means S is a homomorphic

image of T_ and "S=T" means S and T are isomorphic. Most of the semi-

groups in this paper are transformation semigroups, but we use "-+ " and "="

to mean homomorphism and isomorphism of abstract semigroups (though it will

usually be clear when an abstract homomorphism is actually a transformation

homomorphism). When T is a group, "S4T" means S is a normal subgroup.

-4-

The semisroup of a semiautomaton A is the transformation semigroup

generated by [_AI_ C A] under composition. The monoid GA is the semi-

group of A with AA the identity on QA added if it is not already in the

semigroup. If A _ B, then GBIG A. The converse is not true. A is an

identity semiautomaton iff IGAI = ]. A is a permutation semiau_omaton iff

GA is a group. Corresponding statements with the semigroup of A in place

of GA are not true.

3. THE DECOMPOSITION THEOREM

The following version of Krohn and Rhodes decomposition theorem is

proved in [2, 3, 4, 7].

Theorem ]. For any semiautomaton A, there is a cascade product of semi-

automata A],A2,...,A which covers A such that for all i (] _ i _ n)n

either

A. A.

(1) G I . group*, i GAis a simple and G I

or (2) A. is a two-state reset.l

Moreover, if GA is a group, those A. which are resets will actually be1

identity semiaUtomata.

The components A° of the cascade covering A are no more complicatedi

than A, insofar as semigroups reflect the complexity of semiautomata. On

the other hand, Theorem ] does not prohibit the A. from being larger thani

A, and in fact the usual decomposition techniques applied to a five state

machine whose semigroup consists of resets and the alternating group of

degree five yields an A. with sixty states The following theorem eliminatesl

this flaw.

* GAiWe remind the reader that A. is a permutation semiautomaton iff is a group.l

-5-

Definition• Let A be a semiautomaton. The completion of A is the semi-

Q_ = QA _ GA GA _ df.automaton _ such that , = , and for g E , g = g.

Theorem 2. Theorem ] is true when (4) is replaced by

A.

(4') G i is a simple group , and _ > A..l

GA

N

Clearly = GA, and since A _ B implies GBI GA, we observe that

Theorem 2 implies Theorem 1.

We take Theorem ] as our starting point, and prove Theorem 2 from the

following lemmas.

Len_a 1. Let C be a semiautomaton such that GC is a group and N,G C. Let

= [qNiq E QC}. _ is an admissible partition and GC/N -+G C/_.

Proof: The elements of _ are the orbits of QC under the group of trans-

Cformations N, and so _ is clearly a partition of Q . Moreover, _ is

admissible: Ng = gN for all g E GC since N is normal, and so for all

qN E _ it follows that (qN)g = q(Ng) = q(gN) = (qg)N E _. Observe that

the elements of GC/_ are simply the elements of GC acting on _. Hence,

GC GC/_-+ and N is trivially included in the kernel of the homomorphism;

therefore, GC/N -_G C/_. Q.E.D

Len_na 2. Let A be a semiautomaton such that H is a simple group and HIGA.

there is a semiautomaton B such that _ z B and GB = H.

Proof: It is easy to show (cf. Ginzburg [3], section ].46) that if a

group divides a semigroup, then the group is actually a homomorphic image

of a subr_ of the semigroup. Let K be a subgroup of GA of minimum

size such that K -+H, and let NqK be the kernel of the homomorphism. Let

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C be the subautomaton of _ such that QC = Q_ and _C = K. Then GC = K

(as the reader may verify)* QCand by Lemma 1, _ = {qN lq E ] is an admis-

sible partition of C. Finally, let B = C/_.

Clearly, _ _ B.

Lemma ] also implies that GC/N -+G B. But GC/N = K/N = H,and H is

simple, so that if IGBI _ I, it must be that GB = H as required.

On the other hand, suppose IGBI = I. Then every element of K = GC

acts as an identity on _, viz., (qN)k = qN for every k E K and qN E _.

For q E QC, let K = {k E KIqk=q} Since q E qN = qNk for q E Qc itq

follows that K intersects every coset of N in K, and so the restrictionq

to K of the canonical homomorphism from K onto K/N is also onto. There-q

fore, K -_K/N = H (obviously K is a group), and since K is of minimumq q

size, K = K for all q E QC. But this implies K = [AC], which is absurd,q

since H is a non-trivial image of K. Q.E.D.

Lemma 3. If A and B are semiautomata such that GA = GB, then there is

cascade product of copies of B and an identity semiau_omaton which covers A.

Proof: For convenience assume QB = [],2,...,n}. The cascade covering A

will consist of an identity machine with state set QA and n copies of B,

all acting in parallel For q0 E QA QB GB" ' qi E ] <_ i <_n, and g E , the

dr.

transitions in the cascade are defined by (q0,ql,...,qn)g = (q0,qlg,...,qng).

QB QBThe states qi C QB uniquely determine a function f: -+ by the condi-

tion f(i) = qi' ] _ i _ n. If it happens that f E GB, the state q0f E QA

GBis also uniquely determined by the isomorphism between and GA.

The states of the cascade which determine functions f C GB obviously

form a subautomaton of the cascade, and the mapping of <q0,q],...,qn > to*

This is not qui_e immediate, s_nce one must argue that the identity of Krestricted to QV is actually A-.

-7-

m

q0f defines a homomorphism from this subautomaton onto A (and hence onto

A), as is easily verified. Q.E.D.

Lemma 3 emphasizes the difficulty in interpreting the Krohn-Rhodes

theorem as a "prime" decomposition theorem fo_.._rmachines (as opposed to

semigroups). We might tentatively define A to be prime if (1) GA is simple,

GB Aand (2) _ _ B implies either B _ A or _ G . There

will then be prime machines for the same simple group which are incomparable

under covering. Lemma 3 then leads to the unsatisfactory situation that

one prime divides (is covered by) a power (cascade product of copies) of

another prime.A.l

The proof of Theorem 2 is now straightforward. Each A. such that Gl

is a simple group can be covered according to Lemma 3 by a cascade ofA.

copies of B and an identity semiau_omaton_ for any B such that GB = G i.A.

Since G I IGA_ Lemma 2 implies that such an automaton B can be found for

which _ _ B (and hence _ _ B). The identity semiautomata which are intro-

duced can trivially be replaced by cascades of two-state identity semiautomata_

and the proof is complete.

Hartmanis and Stearns' notion "A has the capability of B" is equivalent

to "A _ B". Theorem 2 above is thus a restatement of Theorem 7.]0 of

Hartmanis and Stearns [4],except that their Theorem 7.]0 makes the additional

m

assertion that A >_ A. even when A. is a reset This is false_ as we showl l "

in the next section.

-8-

4. HALF-RESETS

Let R0 be the semiautomaton whose state set and input set equals

[0,1}, and whose transitions are given by ordinary multiplication. Any

semiautomaton covered by R0 will be called a half-reset. Except for

permutation semiautomata, every semiautomaton has the capability of R0.

Definition. Let A be a semiautomaton, p,q E QA. Then q is accessible

from p iff q = pg for some g E GA. A is partially ordered (p.o_ iff acces-

sibility is a partial order on QA.

R0 is trivially p.o., and it is easy to show that if A is p.o. and

A _ B, then B is p.o. Likewise, if A and B are p.o., so is AOB. Con-

versely, if A is p.o. (and not already a half-reset), then A has a non-

trivial subautomaton which is a half-reset. We let the reader convince

himself that A can then be covered by a p.o. semiautomaton with one fewer

state followed by a half-reset (cf., Method I of Section 5). In short,

we have

Theorem 4. A semiautomaton is covered by a cascade of half-resets if and

only if it is partially ordered.

The regular events associated with p.o. semiautomata are obviously

finite unions of events of the form "F]_]F2_ 2....F " such that F. is an l

finite set of input symbols and _i _ Fi (] _ i <_n). These events form a

Boolean algebra, and can also be characterized by an inductive definition

resembling that of the star-free events [6]. One can also define partially

ordered semigroups in the obvious way, and conclude that A is p.o. iff

GA is p.o.

-9-

Consider a semiautomaton A with state set {1,2,3} and inputs x and y

such that ]x=2, 2y=I and the remaining transitions lead to 3. No non-

trivial groups divide GA, so in the decomposition of A satisfying Theorem

2, only two-state resets appear. By Theorem 4, not all of these two-state

resets can be half-resets (because states ] and 2 are mutually accessible,

viz., A is not p.o.). But the only two-state resets covered by _ are

half-resets (as can be verified by exhaustion), and so A cannot have the

capability of all the components in its decomposition.

5. PROOF OF THEOREM ]

There are at least three elementary proofs of Theorem ] in the literature:

Ginzburg's [3] corrected version of Zeiger's proof using set systems or

covers, Arbib's [2] version of Krohn-Rhodes' proof, and the elegant proof

of Zeiger [7]. Nevertheless, no=e of these proofs are very simple, and

so we feel another proof may still be of interest. Readers familiar with

the other proofs will note that our method I is essentially dual to that

of Zeiger [7], and our Method III is almost the same as that of Arbib [2].

The following lemma appears in [2, 3, 4] and we shall not repeat the

proof.

,The proof of Zeiger [7] is given in only two and a half pages, and separ-

ates non-permutation semiautomata into only two cases. Unfortunately,

Zeiger's remark that his method applies to permutation-reset semiautomata

is false, as can be seen by applying it to any permutation-reset semi-

automaton. Moreover, a semiautomaton with state set {:1,233], reset in-puts to each state, and an additional input leaving states 1 and 2 fixedand sending state 2 to state 3 is a counter-example to Zeiger's assertion

that his second method reduces the number of non-permutation, non-reset

elements. This counter-example invalidates the proof that his method

terminates. When these errors are corrected, Zeiger's proof turns outto be no simpler than ours.

-10-

Lemma 4. Let A be a permutation semiautomaton. A can be covered by a

cascade of two-state identity semiautomata and permutation semiautomata

whose monoids are the factor groups in a composition series for GA (and

hence are simple groups dividing GA).

We refer to permutation semiautomata and two state resets as basic.

Theorem ] follows immediately from Lemma 4 and

Theorem 5. For any semiautomaton A, there is a cascade product of basic

semiautomata A],A2,...,A which covers A such that for all i (] < i _ n),nA.

if G i is a group, then GAilG A.

A natural way to prove Theorem 5 is to show that any semiautomaton

can be covered by a product of two "smaller" semiautomata_ and then use

induction. (A disadvantage of the proof using set systems [3,4] is that

it does not conform to this description.) The proper interpretation of

"smaller" is necessarily a little devious.

Definition. For any transformation monoid S, N(S) is the submonoid gen-

erated by the nonconstant (i.e._ non-reset) elements of S. For any semi-

automaton A_ the measure of A is the triple of positive integers

_(A) df. (IN(GA) I, IQAI, IGAI).

Measures will be well-ordered lexicographically in the usual manner:

Definition. If x = (x],x2,x3) and y = (y],y2,Y3) are triples of integers,

then X > Y iff x] > y], or, x]=y] and x2 > Y2' or, x]=y] and x2=Y 2 and

x3 > Y3"

,

By convention, N(S) = the identity when S acts on a singleton.

-11-

Lemma 5. For any semiautomaton A, which is not basic, there are semiautomata

B and C such that

(I) Boc _ A,

(2) N(GB) IGA, and either _(B) < _(A) or B is basic,

(3) N(GC) IGA and _(C) < _(A).

Proof of Theorem 5: Let A be a semiautomaton. If A is basic (and in

particular if _(A) = (1,1,1) is minimum), then Theorem 5 is true trivially.

Proceeding by (transfinite) induction, suppose Theorem 5 is true for all

semiautomata with measures smaller than _(A). Theorem 5 is then true by

hypothesis for the semiautomata B and C produced by Lemma 5. Let Bi, ] _ i _ n,

be the basic semiautomata in the cascade covering of B, and likewise for

Ci, ] _ i _ m. Since Boc m A, a cascade of the B. (or semiautomata equiva-lB.

lentB to the Bi) followed by the Ci covers A. Suppose G I is"a group; then

G ilGB. But if a group G divides a transformation monoid S, then it mustB

be that GIN(S ). Hence, G ilN(GB), by Lemma 5 N(GB) IGA, and by transitivityB

G i lGA. The same reasoning applies to the Ci, and it follows that Theorem 5

is true for A. Q.E.D.

Proof of Lemma 5: We describe three decomposition methods, one of which

will yield appropriate B and C for any semiautomaton A which is not basic.

Definition. For any semiautomaton A, let _be the subautomaton of A

obtained by eliminating all reset inputs from A.

Method I. N(A) has a non-trivial subautomaton.

Let QC equal the states of the non-trivial subautomaton of N(A), and

let QB = (QA . QC) U {d] for d _ QA. Transitions in Boc are given by:

-12-

(b_,c) if b_d and b_ _ QC,

(d,b_) if b_d and bo E QC,

(b,c)_ = QA . QC(r,c) if cy is a reset to r E •

(b,cc_) otherwise

Since QC is the state set of a subautomaton of N(A), it is closed

under non-reset inputs. Hence the fourth clause only applies when b=d

and c_ E QC so the transitions of Boc are well defined.

When b_d map (b,c) to b, and when b=d map (b,c) to c. This mapping

defines a homomorphism from Boc onto A (as is immediately verified by

checking the four types of transitions in Boc), so part (I) of the lemma

is satisfied.

Note that the singletons in QA . QC together with QC form an admissible

partition _ of A 3 and that A/_ is isomorphic to B. We conclude that GA -+GB

and that N(GB) IGA. Moreover, IQBI = IQA - qC I + I < IQAI since the sub-

QCautomaton on is non-trivial. This guarantees that part (2) is satisfied.

The only non-identity, non-reset transitions in GC arise from the

fourth clause in the definition of transitions of Boc. It follows that

N(G C) = {g_QClg E N(GA)_. Hence N(G A) -+N(GC), and since IQCI < IQAI, part

(3) is satisfied.

Method II. GA contains a non-identity permutation.

Let P be the subgroup of GA generated by the permutations, and T the

subsemigroup generated by GA - P. Note that T_ (otherwise GA is a group

and A is basic), and that T is trivially a (two-sided) ideal. Let QB = p,

and QC = QA. Transitions in Boc are given by:

-13-

(p_,q) if _ E P(p,q)_ =

_(p, qp_p- I) _ Eif T

Since GA is the disjoint union of P and T, the transitions of Boc

are well defined. Mapping (p,q) to qp defines a homomorphism from Boc

onto A.

Clearly GB = P, so N(G B)[G A and B is basic. Likewise GC = T U _AC}, so

N(G C) IGA and N(G C) does not contain the non-identity permutation in N(GA).

Therefore, _(C) < _(A).

Method II!. GA = V U T where V is a subsemigroup such that IN(V)[ < IN(GA)[,

and T is a proper left ideal of GA - [A}.

Let QB = V and QC = QA. Transitions in BoC are given by:

_(v_,q) if _ E V-T,

(v,q)_ = <_(A,qv_) if _ E T.

Note that f__ T (and hence A E V) since T is proper, and so the transi-

tions in Boc are well defined. Nmpping (v,q) to qv defines a homomorphism

from Boc onto A.

Clearly GC = T U {A}, so GCIG A. Moreover, QC = QA and IGC[ < [GAI

since T is proper. Hence, _(C) < _(A).

Note that N(G B) is a submonoid (generated by V-T) of V acting on itself

by right multiplication,and so N(G B) Iv. Moreover, any r E V which is a

reset (on QA) is certainly a reset when V acts on itself. Therefore, N(G B)

is isomorphic to a submonoid of N(V), and we actually have N(GB) IN(V). In

particualr, [N(GB)[ _< IN(V)[. By hypothesis, IN(V)[ < [N(GA)[, so

_(B) < _(A).

-14-

Let A be a semiautomaton such that neither method I nor method II

applies to A and A is not basic. We claim that method III applies to A_

which completes the proof of Lemma 5.

To verify the claim, let S = GA - [A}. S is a subsemigroup because

GA contains no non-identity permutations. There is a non-reset element

s E S (otherwise A is a reset and method I applies). If GAs U {resets} = S,

then N(A) has a non-trivial subautomaton on the states in the range of s2

and method I applies. Therefore GAs U [resets] _ S, and in particular S

has proper left ideals (e.g., GAs).

Let T be a maximal left ideal of S, and let V = GAx U {A_ for any

x C S-T. Then (V - [A]) U T is a left ideal of S properly containing T,

which implies (V - [A}) U T = S and V U T = GA. If x=s, we have observed

that V U [resets} _ GA, and so IN(V) I < IN(GA) I. Alternatively, if T

contains every non-reset s E S_ then x is a reset_ hence GAx contains

only resets and IN(V) I = ] < IN(GA) I. Q.E.D.

There are usually many ways to decompose a semiautomaton into two

semiautomata with smaller measures_ and it is far from clear which

choices ultimately yield the most satisfactory decomposition into basic

semiau_omata. It may even be desirable at times to cover a semiautomaton

with semiautomata which have larger measures (but which presumably are

"smaller" in some more general sense).

-15-

Re ference s

1. Arbib• M., Algebraic .Theory of Machines• (1968), to appear.

2. Arbib, M. • Theorie s of Abstract Automata• Prentice-Hall (I968) • toappear.

3. Ginzburg_ A.• Algebraic Theor_ of Automata, ACM Monograph Series, toappear •

4. Hartmanis• J. and Stearns, R. E.• Algebraic Structure Theory of

Sequentia! Machines , Prentice-Hall, Englewood Cliffs, N. J. Yi966).

"Algebraic Theory of Machines• I•" Trans5. Krohn• K. and Rhodes• J.•

Amer. Math• Soc., Vol. 116 (1965)• 450-464•

• " JACM (1968) to aa_."A Note on Star-free Events• •6 Meyer, A.R.• ___- ---

7. Zeiger_ P._ "Yet Another Proof of the Cascade Decomposition Theorem

for Finite Automata"• Mathematical Systems Theory, Vol. 1, No. 3(1966), 225-288•

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REMARKS ON ALGEBRAIC DECOMPOSITION OF AUTOMATA

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Scientific Interim6. AUTHOR(S)(PJtet Mme, midge In|tlal,'_ae# nnm_

A. R. Meyer and C. Thompson

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A version of the Krohn-Rhodes decomposition theorem forfinite automata is proved in which capabilities as well as semigroupsare preserved. Another elementary proof of the usual Krohn-Rhodestheorem is also presented.

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