On Syntactic Congruences for ω−languages∗

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 [Ar85]. 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 iteration congruence coincide and show that all (including infinite-state) ω-languages in the Borel class Fσ ∩ Gδ 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 than the iteration 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.