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Register a new userSynthesising Strategy Improvement and Recursive Algorithms for Solving 2.5 Player Parity Games
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Added by
Ernst Moritz Hahn
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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2.5 player parity games combine the challenges posed by 2.5 player reachability games and the qualitative analysis of parity games. These two types of problems are best approached with different types of algorithms: strategy improvement algorithms for 2.5 player reachability games and recursive algorithms for the qualitative analysis of parity games. We present a method that - in contrast to existing techniques - tackles both aspects with the best suited approach and works exclusively on the 2.5 player game itself. The resulting technique is powerful enough to handle games with several million states.
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Zielonkas classic recursive algorithm for solving parity games is perhaps the simplest among the many existing parity game algorithms. However, its complexity is exponential, while currently the state-of-the-art algorithms have quasipolynomial complexity. Here, we present a modification of Zielonkas classic algorithm that brings its complexity down to $n^{mathcal{O}left(logleft(1+frac{d}{log n}right)right)}$, for parity games of size $n$ with $d$ priorities, in line with previous quasipolynomial-time solutions.
The recent breakthrough paper by Calude et al. has given the first algorithm for solving parity games in quasi-polynomial time, where previously the best algorithms were mildly subexponential. We devise an alternative quasi-polynomial time algorithm based on progress measures, which allows us to reduce the space required from quasi-polynomial to nearly linear. Our key technical tools are a novel concept of ordered tree coding, and a succinct tree coding result that we prove using bounded adaptive multi-counters, both of which are interesting in their own right.
We consider parity games on infinite graphs where configurations are represented by control-states and integer vectors. This framework subsumes two classic game problems: parity games on vector addition systems with states (vass) and multidimensional energy parity games. We show that the multidimensional energy parity game problem is inter-reducible with a subclass of single-sided parity games on vass where just one player can modify the integer counters and the opponent can only change control-states. Our main result is that the minimal elements of the upward-closed winning set of these single-sided parity games on vass are computable. This implies that the Pareto frontier of the minimal initial credit needed to win multidimensional energy parity games is also computable, solving an open question from the literature. Moreover, our main result implies the decidability of weak simulation preorder/equivalence between finite-state systems and vass, and the decidability of model checking vass with a large fragment of the modal mu-calculus.
This article extends the idea of solving parity games by strategy iteration to non-deterministic strategies: In a non-deterministic strategy a player restricts himself to some non-empty subset of possible actions at a given node, instead of limiting himself to exactly one action. We show that a strategy-improvement algorithm by by Bjoerklund, Sandberg, and Vorobyov can easily be adapted to the more general setting of non-deterministic strategies. Further, we show that applying the heuristic of all profitable switches leads to choosing a locally optimal successor strategy in the setting of non-deterministic strategies, thereby obtaining an easy proof of an algorithm by Schewe. In contrast to the algorithm by Bjoerklund et al., we present our algorithm directly for parity games which allows us to compare it to the algorithm by Jurdzinski and Voege: We show that the valuations used in both algorithm coincide on parity game arenas in which one player can surrender. Thus, our algorithm can also be seen as a generalization of the one by Jurdzinski and Voege to non-deterministic strategies. Finally, using non-deterministic strategies allows us to show that the number of improvement steps is bound from above by O(1.724^n). For strategy-improvement algorithms, this bound was previously only known to be attainable by using randomization.
Parity games are infinite two-player games played on directed graphs. Parity game solvers are used in the domain of formal verification. This paper defines parametrized parity games and introduces an operation, Justify, that determines a winning strategy for a single node. By carefully ordering Justify steps, we reconstruct three algorithms well known from the literature.
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