On syntactic congruences for ω-languages

Abstract

In this paper we investigate several questions related to syntactic congruences and to minimal automata associated with ω-languages. In particular we investigate relationships between the so-called simple (because it is a simple translation from the usual definition in the case of finitary languages) syntactic congruence and its infinitary refinement (the iteration congruence) investigated by Arnold (Theoret. Comput. Sci. 39 (1985) 333–335). We show that in both cases not every ω-language having a finite syntactic monoid is regular and we give a characterization of those ω-languages having finite syntactic monoids. Among the main results we derive a condition which guarantees that the simple syntactic congruence and Arnold's syntactic congruence coincide and show that all (including infinitestate) ω-languages in the Borel class Fσ∩ G δ satisfy this condition. We also show that all ω-languages in this class are accepted by their minimal-state automaton — provided they are accepted by any Muller automaton. Finally we develop an alternative theory of recognizability of ω-languages by families of right-congruence relations, and define a canonical object (much smaller then Arnold's monoid) associated with every ω-language. Using this notion of recognizability we give a necessary and sufficient condition for a regular ω-language to be accepted by its minimal-state automaton.