Limitedness Theorem on Finite Automata with Distance Functions: An Algebraic Proof

A finite automaton .d with a distance function d is a sextuple, (C, Q, M, S, F, d), such that C is the input alphabet, Q is the finite set of states, M: Q x C -+ 2Q is the transition function, S c Q and F c Q are the sets of initial and final states, respectively, d: Q x C x Q -+ (0, 1, co } is the distance function, where co denotes infinity, and d satisfies the following: for any (q,a,q’)E Q XCX Q, d(q,a,q’= co iff q’ eM(q,a). M is extended to Q XC* and d is extended to Q x C* x Q in the usual way. & is said to be limited in distance if there exists a nonnegative integer k such that for any w accepted by .d, d(q, W, q’) < k for some q E S and q’ E F. This paper shows that there exists an algorithm for deciding whether or not an arbitrary finite automaton with a distance function is limited in distance.