Rabin-Mostowski Index Problem: A Step beyond Deterministic Automata

Abstract

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

On the Weak Index Problem for Game Automata

“…Despite structural simplicity, they have enough expressive power to inhabit all levels of the non-deterministic index hierarchy and the alternating index hierarchy. In [FMS13] it was shown that these two hierarchies are decidable when the input language is recognized by a game automata.…”

“…In [FMS13], it was shown that for the class of game automata (the closure of the class of deterministic automata under complementation and substitution) the non-deterministic and alternating index problems are solvable. The deterministic index problem being already solved by [NW98], the only case left is the weak index, known to coincide with the quantifier alternation depth for the weak monadic second order logic [Mos91b].…”

Disjunctive form and the modal μ alternation hierarchy

“…Deciding the modal µ alternation hierarchy is exactly equivalent to deciding the Rabin-Mostowski index of alternating parity automata. The corresponding problem has also been studied for automata operating on words [3] and automata which are deterministic [13,12], or non-deterministic [4,7] rather than alternating. As will be highlighted throughout this paper, many of the methods used here are similar to methods applied to different types of automata.…”

Index Problems for Game Automata

“…As the characterization effectively yields an equivalent game automaton, we obtain procedures computing the alternating, weak alternating, and non-deterministic index for a given alternating automaton equivalent to some game automaton. This paper collects results from two conference papers: [Facchini et al 2013] and [Facchini et al 2015]. Additionally, it contains a discussion of the maximality of the class of game automata, which adapts a reasoning from [Duparc et al 2011] to the index problem.…”