2 research outputs found
Semiring Provenance for B\"uchi Games: Strategy Analysis with Absorptive Polynomials
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 further 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.Comment: Full version of a paper submitted to GandALF 202