Semiring Provenance for B\"uchi Games: Strategy Analysis with Absorptive Polynomials

TL;DR

This paper applies semiring semantics to B"uchi games, enabling analysis of strategies and winning conditions through polynomial evaluations, which offers insights into minimal effort strategies and potential game modifications.

Contribution

It introduces a semiring-based fixed-point approach for B"uchi games, revealing absorption-dominant strategies and their relation to positional and persistent strategies.

Findings

Semiring semantics generalize Boolean evaluation for B"uchi games.

Evaluation of fixed-point formulas yields strategies with minimal effort.

Provides a framework for game synthesis and minimal modifications.

Abstract

This paper presents a case study for the application of semiring semantics for fixed-point formulae to the analysis of strategies in B\"uchi games. Semiring semantics generalizes the classical Boolean semantics by permitting multiple truth values from certain semirings. Evaluating the fixed-point formula that defines the winning region in a given game in an appropriate semiring of polynomials provides not only the Boolean information on who wins, but also tells us how they win and which strategies they might use. This is well-understood for reachability games, where the winning region is definable as a least fixed point. The case of B\"chi games is of special interest, not only due to their practical importance, but also because it is the simplest case where the fixed-point definition involves a genuine alternation of a greatest and a least fixed point. We show that, in a precise sense, semiring semantics provide information about all absorption-dominant strategies - strategies that win with minimal effort, and we discuss how these relate to positional and the more general persistent strategies. This information enables applications such as game synthesis or determining minimal modifications to the game needed to change its outcome.