Half-Positional Objectives Recognized by Deterministic B\"uchi Automata

TL;DR

This paper characterizes which omega-regular objectives recognized by deterministic B"uchi automata are half-positional, enabling the protagonist to always play optimally with memoryless strategies, and provides a polynomial-time decision algorithm.

Contribution

It offers a novel characterization of half-positional objectives recognized by deterministic B"uchi automata and a polynomial-time algorithm to decide this property.

Findings

Characterization of half-positional objectives via language-theoretic conditions

Polynomial-time algorithm for deciding half-positionality

Applicable to objectives recognized by deterministic B"uchi automata

Abstract

In two-player games on graphs, the simplest possible strategies are those that can be implemented without any memory. These are called positional strategies. In this paper, we characterize objectives recognizable by deterministic B\"uchi automata (a subclass of omega-regular objectives) that are half-positional, that is, for which the protagonist can always play optimally using positional strategies (both over finite and infinite graphs). Our characterization consists of three natural conditions linked to the language-theoretic notion of right congruence. Furthermore, this characterization yields a polynomial-time algorithm to decide half-positionality of an objective recognized by a given deterministic B\"uchi automaton.