Approximate comparison of distance automata

Distance automata are automata weighted over the semiring (N [ {∞}, min, +) (the tropical semiring). Such automata compute functions from words to N [ {∞} such as the number of occurrences of a given letter. It is known that testing f 6 g is an undecidable problem for f, g computed by distance automata. The main contribution of this paper is to show that an approximation of this problem becomes decidable. We present an algorithm which, given ≤ 0 and two functions f, g computed by distance automata, answers yes if f 6 (1-ε)g, no if f ≰ g, and may answer yes or no in all other cases. This result highly refines previously known decidability results of the same type. The core argument behind this quasi-decision procedure is an algorithm which is able to provide an approximated finite presentation to the closure under products of sets of matrices over the tropical semiring. We also provide another theorem, of affine domination, which shows that previously known decision procedures for cost-automata have an improved precision when used over distance automata.