For ω-languages several notions of syntactic congruence were defined. The present paper investigates 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 refinements 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. As the main result we derive a condition which guarantees that the simple syntactic congruence and Arnold’s syntactic congruence coincide and show that all ω-languages in the Borel class Fσ ∩ Gδ satisfy this condition. Finally we define an alternative canonical object for ω-languages, namely a family of right-congruence relations. Using this object we give a necessary and sufficient condition for a regular ω-language to be accepted by its minimal-state automaton.
CITATION STYLE
Maler, O., & Staiger, L. (1993). On syntactic congruences for ω-languages. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 665 LNCS, pp. 586–594). Springer Verlag. https://doi.org/10.1007/3-540-56503-5_58
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