Finite-Valued Distance Automata

Abstract Distance automata are a model of finite-state machines which charge to an input word the expense of the cheapest successful computation consuming that word, called its distance. The distance of such a machine is the maximal distance of an input word recognized by it or is infinite, depending on whether or not a maximum exists. The valuedness of a distance automaton is introduced by considering this machine to be a transducer with unary output alphabet. The distance of a finite-valued distance automaton with n states is either infinite or at most 6 n ·(2 n ) n 2 − 1. In the former case its growth is linear in the input length. The problem of deciding whether a given finite-valued (2-valued, respectively) distance automaton has infinite distance is PSPACE-complete. It is decidable in deterministic double exponential time whether two given finite-valued distance automata are equivalent, i.e., every input word has the same distance in both machines. There is an inherently infinite-valued distance automaton and, for each k , an inherently k -valued distance automaton such that the growth of the distance in all these machines is linear in the input length. It is decidable in deterministic polynomial time whether a 2-valued distance automaton given as the disjoint union of two single-valued distance automata is inherently 2-valued.