Synthesizing Permissive Winning Strategy Templates for Parity Games

We present a novel method to compute \emph{permissive winning strategies} in two-player games over finite graphs with $\omega$-regular winning conditions. Given a game graph $G$ and a parity winning condition $\Phi$, we compute a \emph{winning strategy template} $\Psi$ that collects an infinite number of winning strategies for objective $\Phi$ in a concise data structure. We use this new representation of sets of winning strategies to tackle two problems arising from applications of two-player games in the context of cyber-physical system design -- (i) \emph{incremental synthesis}, i.e., adapting strategies to newly arriving, \emph{additional} $\omega$-regular objectives $\Phi'$, and (ii) \emph{fault-tolerant control}, i.e., adapting strategies to the occasional or persistent unavailability of actuators. The main features of our strategy templates -- which we utilize for solving these challenges -- are their easy computability, adaptability, and compositionality. For \emph{incremental synthesis}, we empirically show on a large set of benchmarks that our technique vastly outperforms existing approaches if the number of added specifications increases. While our method is not complete, our prototype implementation returns the full winning region in all 1400 benchmark instances, i.e., handling a large problem class efficiently in practice.Comment: CAV'2

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

Computing Adequately Permissive Assumptions for Synthesis

We solve the problem of automatically computing a new class of environment assumptions in two-player turn-based finite graph games which characterize an ``adequate cooperation'' needed from the environment to allow the system player to win. Given an $\omega$-regular winning condition $\Phi$ for the system player, we compute an $\omega$-regular assumption $\Psi$ for the environment player, such that (i) every environment strategy compliant with $\Psi$ allows the system to fulfill $\Phi$ (sufficiency), (ii) $\Psi$ can be fulfilled by the environment for every strategy of the system (implementability), and (iii) $\Psi$ does not prevent any cooperative strategy choice (permissiveness). For parity games, which are canonical representations of $\omega$-regular games, we present a polynomial-time algorithm for the symbolic computation of adequately permissive assumptions and show that our algorithm runs faster and produces better assumptions than existing approaches -- both theoretically and empirically. To the best of our knowledge, for $\omega$-regular games, we provide the first algorithm to compute sufficient and implementable environment assumptions that are also permissive.Comment: TACAS 202

Optimal Strategies in Weighted Limit Games

We prove the existence and computability of optimal strategies in weighted limit games, zero-sum infinite-duration games with a B\"uchi-style winning condition requiring to produce infinitely many play prefixes that satisfy a given regular specification. Quality of plays is measured in the maximal weight of infixes between successive play prefixes that satisfy the specification.Comment: In Proceedings GandALF 2020, arXiv:2009.09360. Full version at arXiv:2008.1156

Algorithms for Büchi Games

The classical algorithm for solving Büchi games requires time O(n · m) for game graphs with n states and m edges. For game graphs with constant outdegree, the best known algorithm has running time O(n 2 / log n). We present two new algorithms for Büchi games. First, we give an algorithm that performs at most O(m) more work than the classical algorithm, but runs in time O(n) on infinitely many graphs of constant outdegree on which the classical algorithm requires time O(n 2). Second, we give an algorithm with running time O(n · m · log δ(n) / log n), where 1 ≤ δ(n) ≤ n is the outdegree of the game graph. Note that this algorithm performs asymptotically better than the classical algorithm if δ(n) = O(log n)