Labelled Cayley graphs and minimal automata

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. A language is a set of words over a finite alphabet X . For standard concepts of automata and languages theory the reader is referred to [5], [7], [14] and [16]. Let G be a groupoid, i.e., a set with a binary operation, and let S be a nonempty subset of G. The Cayley graph Cay(G,S) of G relative to S is defined as the graph with vertex set G and edge set E(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, if L is recognised by an FSA, then it is well known and easily verified that L is also recognised by the finite labelled Cayley graph of Syn(L) = X∗/μL, where μL is the the Myhill congruence on the free monoid X ∗ of all words over X : μL = {(w1, w2) | ContL(w1) = ContL(w2)}, ∗ The author was supported by IRGS grant K12511 and by ARC Discovery grant DP0449469.