Approximate comparison of functions computed by distance automata

Distance automata are automata weighted over the semiring (Formula presented.) (the tropical semiring). Such automata compute functions from words to (Formula presented.). It is known from Krob that the problems of deciding ‘ f≤g’ or ‘ f=g’ for f and g computed by distance automata is an undecidable problem. The main contribution of this paper is to show that an approximation of this problem is decidable. We present an algorithm which, given ε>0 and two functions f,g computed by distance automata, answers “yes” if f≤(1−ε)g, “no” if f≦̸g, and may answer “yes” or “no” in all other cases. The core argument behind this quasi-decision procedure is an algorithm which is able to provide an approximated finite presentation of the closure under products of sets of matrices over the tropical semiring. Lastly, our theorem of affine domination gives better bounds on the precision of known decision procedures for cost automata, when restricted to distance automata.