Regular omega-Languages with an Informative Right Congruence

TL;DR

This paper investigates the properties of right congruences in regular omega-languages, showing that while weak regular omega-languages have an informative right congruence, more expressive classes do not, and explores the characterization of languages with trivial right congruence.

Contribution

The paper extends understanding of right congruences in regular omega-languages, identifying classes with informative right congruences and exploring their expressiveness hierarchy.

Findings

Weak regular omega-languages have an informative right congruence.

Many regular omega-languages have a trivial right congruence.

The paper discusses the expressiveness hierarchy of omega-languages.

Abstract

A regular language is almost fully characterized by its right congruence relation. Indeed, a regular language can always be recognized by a DFA isomorphic to the automaton corresponding to its right congruence, henceforth the Rightcon automaton. The same does not hold for regular omega-languages. The right congruence of a regular omega-language is not informative enough; many regular omega-languages have a trivial right congruence, and in general it is not always possible to define an omega-automaton recognizing a given language that is isomorphic to the rightcon automaton. The class of weak regular omega-languages does have an informative right congruence. That is, any weak regular omega-language can always be recognized by a deterministic B\"uchi automaton that is isomorphic to the rightcon automaton. Weak regular omega-languages reside in the lower levels of the expressiveness hierarchy of regular omega-languages. Are there more expressive sub-classes of regular omega languages that have an informative right congruence? Can we fully characterize the class of languages with a trivial right congruence? In this paper we try to place some additional pieces of this big puzzle.