Approximate Comparison of Functions Computed by Distance Automata

Distance automata are automata weighted over the semiring (N∪{∞},min,+) (the tropical semiring). Such automata compute functions from words to N∪{∞}. 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

Green's Relations and Their Use in Automata Theory

International audienceThe objective of this survey is to present the ideal theory of monoids, the so-called Green's relations, and to illustrate the usefulness of this tool for solving automata related questions. We use Green's relations for proving four classical results related to automata theory: The result of Schützenberger characterizing star-free languages, the theorem of factorization forests of Simon, the characterization of infinite words of decidable monadic theory due to Semenov, and the result of determinization of automata over infinite words of McNaughton