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Labelled Cayley graphs and minimal automata

A.V. Kelarev

School of Computing University of Tasmania Private Bag 100, Hobart

Tasmania 7001 Australia

Abstract

Cayley graphs considered as language recognisers are as powerful as the more general finite state automata. This paper applies Cayley graphs to define a class of automata and describe minimal automata of this type, all their congruences and the Nerode equivalence of states.

Throughout, the word graph means a finite directed graph without multiple edges but possibly with loops, and D= (V, E) is a graph. Alanguage is a set of words over a finite alphabetX. For standard concepts of automata and languages theory the reader is referred to [5], [7], [14] and [16].

LetGbe a groupoid, i.e., a set with a binary operation, and letSbe a nonempty subset ofG. TheCayley graph Cay(G, S) ofGrelative toSis defined as the graph with vertex setGand edge setE(S) consisting of all ordered pairs (x, y) such that

xs= y for some s S. Cayley graphs of groups have received serious attention in the literature (see, in particular, [1], [2], [4]). They are significant both in group theory and in constructions of interesting graphs with nice properties.

If we are interested in language recognition, then the concept of a Cayley graph turns out to be as powerful as the more general notion of a finite state automaton (FSA). Indeed, ifLis recognised by an FSA, then it is well known and easily verified thatLis also recognised by the finite labelled Cayley graph of

Syn(L) =X∗/µL,

whereµLis the theMyhill congruence on the free monoidX∗of all words overX:

µL={(w1, w2)|ContL(w1) = ContL(w2)},

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ContL(w) ={(a, b)|awb∈L}.

Cayley graphs of groupoids have been used in [8] and [9] to define two-sided automata of labelled graphs and investigate properties of languages recognized by them. The aim of this paper is to give necessary and sufficient conditions for a two-sided automaton of this type to be minimal. To this end we describe all congruences on these FSA and, in particular, Nerode equivalences on the sets of states.

Let:X→ {+,−}andf :X→V be any mappings, and letT be a subset of

V. Thetwo-sided automaton Atm(D) = Atm(D, T) = Atm(D, T, f, ) of the graph

Dis the (possibly incomplete) finite state acceptor with

(DA1) the set of statesV ∪ {1}; (DA2) the initial state 1;

(DA3) the set of terminal statesT;

(DA4) the next-state function given, for a stateuand a letterx∈X, by the rule

u·x=

f(x) if(x) = + and (u, f(x))∈E, or ifu= 1,

u if(x) =and (f(x), u)∈E.

If a vertex v ∈V does not belong to f(X), then this state is inaccessible in Atm(D, T), and so without loss of generality we may assume that all vertices are images of letters of the alphabetX.

LetY ⊆V, and letbe an equivalence relation onY. The class ofcontaining

x is denoted byx/. If there is no need to indicate the set Y explicitly, we may call the equivalence relation anincompleteequivalence relation onV. Often we omit the word ‘incomplete’ when there is no ambiguity. The setY is called theground set of , and is denoted by G. An incomplete equivalence relation on the set of states of Atm(D, T) is called an incompletecongruence if it defines the quotient automaton recognizing the same language. Defining the quotient automaton modulo an incomplete congruence, as usual, one has to drop all states which do not belong to the ground set of the congruence, and then introduce a new transition function on the set of all equivalence classes of the relation. Henceis an incomplete congruence if and only if the following conditions hold, for alla, b∈V ∪ {1},x∈X,

(C1) if (a, b)anda·xis defined, thenb·xis defined too, and (a·x, b·x);

(C2) if (a, b)anda∈T, thenb∈T;

(C3) (1,1)and the class containing 1 is a singleton;

(C4) if 1·x1· · ·xn∈T for somex1, . . . , xn∈X, then

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Let1 and2 be incomplete relations with ground setsS1 andS2, respectively.

Then we write1 2if S1 ⊇S2 and1(S2×S2)2. The largest incomplete

congruence on Atm(D, T) is theNerode equivalence ηT described by

ηT = {(1,1)} ∪ {(a, b)|a·u∈T iff b·u∈T for allu∈X∗;

and (a·X∗)∩T =∅} (1)

(see, e.g., [5] or [16]). Denote the equality relation on Atm(D, T) byι. A congruence is said to beproper if it is distinct fromιandηT. The following sets of letters and vertices are used in our main theorem and proofs:

X(+) = {xX|(x) = +},

X() = {xX|(x) =−},

V(+) = {vV | ∃xX+, f(x) =v},

V() = {vV | ∃xX(), f(x) =v}.

Note that the intersectionV(+)∩V()may be nonempty in general. If v∈V and

S⊆V, then put

In(v) = {w∈V()|(w, v)∈E}, Out+(v) = {w∈V(+)|(v, w)∈E},

In(S) = ∪s∈SIn(s), Out+(S) = ∪s∈SOut+(s). For a subsetSofV, define new equivalence relations

αS = {(1,1)} ∪ {(a, b)|a, b∈V \S,In(a) = In(b)}, (2) Θ(T) = {(1,1)} ∪(T ×T)((V \T)×(V \T)), (3)

and consider auxiliary sets

βT

S = {(a, b)| Out+(a)\S= Out+(b)\Sanda, b∈T},

βVS\T = {(a, b)| Out+(a)\S= Out+(b)\Sanda, b∈V \T}. We introduce new relationβS as the following disjoint union

βS={(1,1)} ∪βST∪βSV\T. (4) Clearly,βS is an equivalence relation on the set of states of Atm(D, T). Our main theorem describes all incomplete congruences on the automaton Atm(G, T, f, ).

Apathin the graphD= (V, E) means a directed path, i.e., a sequence of vertices

v0, v1, . . . , vn such that (vi, vi+1)∈E fori= 0,1, . . . , n−1. Denote byT+the set of

all elementsv∈V such that eitherv∈T or there exist a vertext∈T∩V(+)and a pathv=v0, v1, . . . , vn=t, fromvtotwithn≥1 and all verticesv1, . . . , vninV(+). LetC =CD be the set of all vertices c∈V such thatc /∈T+and ifc∈V()then

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THEOREM 1 The automaton Atm(D, T, f, )is minimal if and only if α∅∩β∅∩Θ(T) =ι,

and for eachc∈V()eitherc∈T+ or there existsv∈T+ such that(v, c)∈E.

THEOREM 2 Letbe an incomplete equivalence relation on Atm(D, T, f, ), and letS=V\G. Thenis a congruence of this automaton if and only ifSis a subset

ofCD and

αS∩βS∩Θ(T). (5)

COROLLARY 3 The Nerode equivalence on Atm(D, T, f, ) is equal to

η= (αS∩βS∩Θ(T))\(CD×CD). (6)

Proofof Theorem 2. The ‘only if’ part. Take any incomplete congruence of the automaton Atm(D, T, f, ).

Let us begin by showing that condition (C4) implies thatS=V\Gis a subset ofC =CD. To this end consider any vertexvwhich does not belong toC. All we have to verify is thatvlies inG. In view of the definitions ofC the following cases may occur.

Case 1. v∈T. Then 1·v=v∈T, and so 1·v∈Gby condition (C4). Case 2. v /∈T andv∈T+. Then the definition ofT+ means that there exist a

vertext∈T∩V(+)and a pathv=v0, v1, . . . , vn=t, fromvtotwithn≥1 and all verticesv1, . . . , vn inV(+). By the definition of Atm(D, T, f, ) we get

1·v0v1. . . vn=t∈T.

Hence condition (C4) yieldsv= 1·v0∈G.

Case 3. c ∈V() and (v, c) ∈E for some v∈T. By (DA4), 1·vc= v ∈T. Thereforev= 1·v∈Gin view of (C4).

Case 4. c ∈V() and (v, c) E for some v T+, v /∈ T. Then there exist

t∈T∩V(+)and a pathv=v

0, v1, . . . , vn=t, fromvtotwithn≥1 and all vertices

v1, . . . , vn inV(+)such that (v, c)∈E. It follows from (DA4) that

1·vcv1v2· · ·vn=t∈T.

Hence (C4) impliesv∈Gagain.

Thus in all cases it follows from (C4) and the definition of Atm(D, T, f, ) that

v∈G. This means thatS⊆C.

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Suppose to the contrary that (a, b)∈/αS. SinceS=V\Gand⊆G×G, we havea, b /∈S. Therefore it follows from the definition of αS that In(a)= In(b). We may assume that there existsu∈ In(a)\ In(b). ChoosexinX()such that

f(x) =u. Then ax =a andbxis undefined. This contradicts condition (C1) and shows that (a, b)∈αS.

If there exists an elementuin Out+(a)∩S\Out+(b)∩S, thenu=f(x) for some

x∈X(+); whenceax =x andbxis undefined, a contradiction to (C1). Therefore

Out+(a)∩S Out+(b)∩S. The reversed inclusion is proven in exactly the same way; whence

Out+(a) = Out+(b). (7)

In proving that (a, b) ∈β first note that if a, b∈T, then Out+(a) = Out+(b) implies (a, b) ∈βT. If, however, aor b is not in T, thena, b∈V \T as indicated above. Hence (a, b)∈βV\T again, and we get (a, b)∈β. Therefore (a, b)∈βin both cases. Thus (5) is satisfied.

The ‘if’ part. Letbe an incomplete equivalence relation such thatS=V \G is a subset ofCand the inclusion (5) holds. We claim thatis a congruence.

Indeed, sinceΘ(T), conditions (C2) and (C3) are obvious. In order to verify (C1), choose an arbitrary pair (a, b) andx∈X such thatax is defined. Note thata, b /∈Sby the definition of G. The following cases are possible.

Case 1: (x) = . Since (a, b) αS, we get In(a) = In(b). If f(x) ∈/ In(a), thenaxandbxare undefined. This contradiction shows thatf(x) In(a). Thereforeax=a,bx=b, and so (ax, bx).

Case 2: (x) = +. Since (a, b)∈β, we get Out+(a)\S= Out+(b)\S. Iff(x)∈/ Out+(a), thenaxandbxare undefined, a contradiction. Thereforef(x) Out+(a)\

S⊆ Out+(b), and so bxis defined too. It follows that (ax, bx) = (f(x), f(x)), because(x) = +.

Thus, ifaxis defined, then (ax, bx) always belongs to, i.e., (C1) holds. It remains to prove condition (C4). Choose any elementsx1, . . . , xn ∈X such that 1·x1· · ·xn ∈T. All we have to verify is thatx1, . . ., xn, 1·x1, 1·x1x2, . . .,

1·x1· · ·xn are not inS.

Denote byi1,i2, . . .,imall integers such thatxi1, . . .,xim∈X(+)andi1≤i2

· · · ≤im. Clearly, allxi1, . . .,xim are inT+. Consider anyksuch that 1≤k≤n. First, suppose that(xk) = +. Thenk=q for someq, andxq= 1·x1· · ·xn. If q = m, then it follows from (DA4) that xq = 1·x1· · ·xn T. Hence 1·x1· · ·xq=xq∈/C.

Ifq < m, then (DA4) implies

1·xq· · ·xn= 1·x1· · ·xn∈T

Hencexq∈T+, and so 1·x1· · ·xq=xq∈/C again. Second, consider the case where(xk) =.

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1·x1· · ·xq∈/C. Besides, the same equality together with the definition ofCviaT+

yield thatxq∈/C.

If i1 k, then denote by r the maximum integer with ir k. We get 1·

x1· · ·xq=xir∈T+. This equality immediately shows that 1·x1· · ·xq∈/C. Besides, the same equality together with the definition ofCalso yield thatxq∈/C.

Thus we see thatxqand 1·x1· · ·xqare not inC. Therefore they do not belong toS⊆C. This means that (C4) is satisfied, which completes the proof. 2

Proof of Corollary 3 follows immediately from Theorem 2, because the Nerode equivalence is the largest congruence and the intersection in the right hand side of (5) is an equivalence relation. 2

Proof of Theorem 1. An automaton is minimal if and only if its Nerode equivalence is the identity relation. Hence the proof follows from the definition of the setCD and Theorem 2 or Corollary 3.2

References

[1] L. Babai, Automorphism groups, isomorphism, reconstruction, inHandbook of Combinatorics, Elsevier Sci., 1995, 1447–1540.

[2] N. Biggs,Algebraic Graph Theory, Cambridge University Press, 1994.

[3] D. Gusfield,Algorithms on Strings, Trees, and Sequences, Cambridge University Press, 1997.

[4] M.-C. Heydemann, Cayley graphs and interconnection networks, inGraph Sym-metry: Algebraic Methods and Applications(Montreal, Canada, July 1–12, 1996), Kluwer, Dordrecht, 1997, 167–224.

[5] M. Ito, Algebraic Theory of Automata and Languages, World Scientific, New York, 2001.

[6] A. V. Kelarev, On undirected Cayley graphs,Australas. J. Combin.25 (2002), 73–78.

[7] A. V. Kelarev,Graph Algebras and Automata, Marcel Dekker, 2003.

[8] A. V. Kelarev, M. Miller and O. V. Sokratova, Directed graphs and closure prop-erties for languages, 12th Australasian Workshop on Combinatorial Algorithms (Ed. E.T. Baskoro), 2001, 118–125.

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[10] A. V. Kelarev and C. E. Praeger, On transitive Cayley graphs of groups and semigroups,European J. Combinatorics24(2003), 59–72.

[11] A. V. Kelarev and O. V. Sokratova, Languages recognized by a class of finite automata, Acta Cybernetica15(2001), 45–52.

[12] A. V. Kelarev and O. V. Sokratova,On congruences of automata defined by di-rected graphs, Theoretical Computer Science, in press.

[13] G. Lallement, Semigroups and Combinatorial Applications, Wiley, New York, 1979.

[14] G. P˘aun and A. Salomaa (Eds.), New Trends in Formal Languages, Springer-Verlag, Berlin, 1997.

[15] J. E. Pin (Ed.), Formal Properties of Finite Automata and Applications, Lec. Notes Comp. Science 386, Springer, New York, 1989.

[16] G. Rozenberg, A. Salomaa (Eds.),Handbook of Formal Languages, Vol. 1, 2, 3, Springer, New York, 1997.

References

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