Fatal attractors in parity games

We study a new form of attractor in parity games and use it to define solvers that run in PTIME and are partial in that they do not solve all games completely. Technically, for color c this new attractor determines whether player c%2 can reach a set of nodes X of color c whilst avoiding any nodes of color less than c. Such an attractor is fatal if player c%2 can attract all nodes in X back to X in this manner. Our partial solvers detect fixed-points of nodes based on fatal attractors and correctly classify such nodes as won by player c%2. Experimental results show that our partial solvers completely solve benchmarks that were constructed to challenge existing full solvers. Our partial solvers also have encouraging run times. For one partial solver we prove that its runtime is in O(|V |3), that its output game is independent of the order in which attractors are computed, and that it solves all B¨uchi games

Fatal Attractors in Parity Games: Building Blocks for Partial Solvers

Attractors in parity games are a technical device for solving "alternating" reachability of given node sets. A well known solver of parity games - Zielonka's algorithm - uses such attractor computations recursively. We here propose new forms of attractors that are monotone in that they are aware of specific static patterns of colors encountered in reaching a given node set in alternating fashion. Then we demonstrate how these new forms of attractors can be embedded within greatest fixed-point computations to design solvers of parity games that run in polynomial time but are partial in that they may not decide the winning status of all nodes in the input game. Experimental results show that our partial solvers completely solve benchmarks that were constructed to challenge existing full solvers. Our partial solvers also have encouraging run times in practice. For one partial solver we prove that its run-time is at most cubic in the number of nodes in the parity game, that its output game is independent of the order in which monotone attractors are computed, and that it solves all Buechi games and weak games. We then define and study a transformation that converts partial solvers into more precise partial solvers, and we prove that this transformation is sound under very reasonable conditions on the input partial solvers. Noting that one of our partial solvers meets these conditions, we apply its transformation on 1.6 million randomly generated games and so experimentally validate that the transformation can be very effective in increasing the precision of partial solvers

Winning Cores in Parity Games

We introduce the novel notion of winning cores in parity games and develop a deterministic polynomial-time under-approximation algorithm for solving parity games based on winning core approximation. Underlying this algorithm are a number properties about winning cores which are interesting in their own right. In particular, we show that the winning core and the winning region for a player in a parity game are equivalently empty. Moreover, the winning core contains all fatal attractors but is not necessarily a dominion itself. Experimental results are very positive both with respect to quality of approximation and running time. It outperforms existing state-of-the-art algorithms significantly on most benchmarks

Static Analysis of Parity Games: Alternating Reachability Under Parity

It is well understood that solving parity games is equivalent, up to polynomial time, to model checking of the modal mu-calculus. It is a long-standing open problem whether solving parity games (or model checking modal mu-calculus formulas) can be done in polynomial time. A recent approach to studying this problem has been the design of partial solvers, algorithms that run in polynomial time and that may only solve parts of a parity game. Although it was shown that such partial solvers can completely solve many practical benchmarks, the design of such partial solvers was somewhat ad hoc, limiting a deeper understanding of the potential of that approach. We here mean to provide such robust foundations for deeper analysis through a new form of game, alternating reachability under parity. We prove the determinacy of these games and use this determinacy to define, for each player, a monotone fixed point over an ordered domain of height linear in the size of the parity game such that all nodes in its greatest fixed point are won by said player in the parity game. We show, through theoretical and experimental work, that such greatest fixed points and their computation leads to partial solvers that run in polynomial time. These partial solvers are based on established principles of static analysis and are more effective than partial solvers studied in extant work

Static analysis of parity games: alternating reachability under parity

It is well understood that solving parity games is equivalent, up to polynomial time, to model checking of the modal mu-calculus. It is a long-standing open problem whether solving parity games (or model checking modal mu-calculus formulas) can be done in polynomial time. A recent approach to studying this problem has been the design of partial solvers, algorithms that run in polynomial time and that may only solve parts of a parity game. Although it was shown that such partial solvers can completely solve many practical benchmarks, the design of such partial solvers was somewhat ad hoc, limiting a deeper understanding of the potential of that approach. We here mean to provide such robust foundations for deeper analysis through a new form of game, alternating reachability under parity. We prove the determinacy of these games and use this determinacy to define, for each player, a monotone fixed point over an ordered domain of height linear in the size of the parity game such that all nodes in its greatest fixed point are won by said player in the parity game. We show, through theoretical and experimental work, that such greatest fixed points and their computation leads to partial solvers that run in polynomial time. These partial solvers are based on established principles of static analysis and are more effective than partial solvers studied in extant work

Effective partial solvers for parity games

Partial methods play an important role in formal methods and beyond. Recently such methods were developed for parity games, where polynomial-time partial solvers decide the winners of a subset of nodes. We investigate here how effective polynomial-time partial solvers can be in principle by studying polynomial-time interactions of partial solvers. Concretely, we propose simple, generic composition patterns for partial solvers that preserve polynomialtime computability. We show that an implementation of this semantic framework manually discovers new partial solvers – including those that merge node sets that have the same but unknown winner – by studying games that composed partial solvers can neither solve nor simplify. We experimentally validate that this data-driven approach to refinement leads to polynomial-time partial solvers that can solve all standard benchmarks of structured games. For one of these polynomial-time partial solvers, we were unable to find even a sole random game that it won’t solve completely, although we generated a few billion random games of varying configurations to that end. However, the work presented here does not yet offer any deeper characterisations of which games are completely solved by such partial solvers