Acceptance conditions for omega-languages and the Borel hierarchy

TL;DR

This paper explores the expressive power of various acceptance conditions for omega-automata, establishing their relation to the Borel hierarchy and introducing a new acceptance condition with full characterization.

Contribution

It classifies acceptance conditions based on their expressive power and introduces a novel acceptance condition with a complete characterization.

Findings

Acceptance conditions definable in MSO logic recognize at most omega-regular languages.

A new acceptance condition is introduced and fully characterized.

Progress is made in understanding the expressive power of (fin, =) acceptance conditions.

Abstract

This paper investigates acceptance conditions for finite automata recognizing omega-regular languages. As a first result, we show that, under any acceptance condition that can be defined in the MSO logic, a finite automaton can recognize at most omega-regular languages. Starting from this, the paper aims at classifying acceptance conditions according to their expressive power and at finding the exact position of the classes of omega-languages they induced according to the Borel hierarchy. A new interesting acceptance condition is introduced and fully characterized. A step forward is also made in the understanding of the expressive power of (fin, =).