Index problems for game automata

TL;DR

This paper proves the decidability of index problems for a broad class of regular languages recognized by game automata, extending previous results limited to deterministic automata.

Contribution

It demonstrates that all three index problems are decidable for languages recognized by game automata, and also provides a method to determine if a language can be recognized by such automata.

Findings

All three index problems are decidable for game automata-recognized languages.

It is decidable whether a regular language can be recognized by a game automaton.

Game automata extend the class of deterministic automata and handle higher complexity in the index hierarchy.

Abstract

For a given regular language of infinite trees, one can ask about the minimal number of priorities needed to recognize this language with a non-deterministic, alternating, or weak alternating parity automaton. These questions are known as, respectively, the non-deterministic, alternating, and weak Rabin-Mostowski index problems. Whether they can be answered effectively is a long-standing open problem, solved so far only for languages recognizable by deterministic automata (the alternating variant trivializes). We investigate a wider class of regular languages, recognizable by so-called game automata, which can be seen as the closure of deterministic ones under complementation and composition. Game automata are known to recognize languages arbitrarily high in the alternating Rabin-Mostowski index hierarchy; that is, the alternating index problem does not trivialize any more. Our main contribution is that all three index problems are decidable for languages recognizable by game automata. Additionally, we show that it is decidable whether a given regular language can be recognized by a game automaton.