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 winning strategies and minimal effort strategies through fixed-point evaluations in polynomial semirings.

Contribution

It introduces a novel semiring-based approach to analyze strategies in B"uchi games, capturing not only winners but also how they win and the effort involved.

Findings

Semiring semantics provide detailed information about winning strategies.

Absorption-dominant strategies can be characterized using this approach.

The method enables applications like 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\"uchi 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. Lastly, we discuss limitations of our approach and present questions that cannot be immediately answered by semiring semantics.