7,266 research outputs found
A Comparison of BDD-Based Parity Game Solvers
Parity games are two player games with omega-winning conditions, played on
finite graphs. Such games play an important role in verification,
satisfiability and synthesis. It is therefore important to identify algorithms
that can efficiently deal with large games that arise from such applications.
In this paper, we describe our experiments with BDD-based implementations of
four parity game solving algorithms, viz. Zielonka's recursive algorithm, the
more recent Priority Promotion algorithm, the Fixpoint-Iteration algorithm and
the automata based APT algorithm. We compare their performance on several types
of random games and on a number of cases taken from the Keiren benchmark set.Comment: In Proceedings GandALF 2018, arXiv:1809.0241
A Delayed Promotion Policy for Parity Games
Parity games are two-player infinite-duration games on graphs that play a
crucial role in various fields of theoretical computer science. Finding
efficient algorithms to solve these games in practice is widely acknowledged as
a core problem in formal verification, as it leads to efficient solutions of
the model-checking and satisfiability problems of expressive temporal logics,
e.g., the modal muCalculus. Their solution can be reduced to the problem of
identifying sets of positions of the game, called dominions, in each of which a
player can force a win by remaining in the set forever. Recently, a novel
technique to compute dominions, called priority promotion, has been proposed,
which is based on the notions of quasi dominion, a relaxed form of dominion,
and dominion space. The underlying framework is general enough to accommodate
different instantiations of the solution procedure, whose correctness is
ensured by the nature of the space itself. In this paper we propose a new such
instantiation, called delayed promotion, that tries to reduce the possible
exponential behaviours exhibited by the original method in the worst case. The
resulting procedure not only often outperforms the original priority promotion
approach, but so far no exponential worst case is known.Comment: In Proceedings GandALF 2016, arXiv:1609.0364
Robust Exponential Worst Cases for Divide-et-Impera Algorithms for Parity Games
The McNaughton-Zielonka divide et impera algorithm is the simplest and most
flexible approach available in the literature for determining the winner in a
parity game. Despite its theoretical worst-case complexity and the negative
reputation as a poorly effective algorithm in practice, it has been shown to
rank among the best techniques for the solution of such games. Also, it proved
to be resistant to a lower bound attack, even more than the strategy
improvements approaches, and only recently a family of games on which the
algorithm requires exponential time has been provided by Friedmann. An easy
analysis of this family shows that a simple memoization technique can help the
algorithm solve the family in polynomial time. The same result can also be
achieved by exploiting an approach based on the dominion-decomposition
techniques proposed in the literature. These observations raise the question
whether a suitable combination of dynamic programming and game-decomposition
techniques can improve on the exponential worst case of the original algorithm.
In this paper we answer this question negatively, by providing a robustly
exponential worst case, showing that no intertwining of the above mentioned
techniques can help mitigating the exponential nature of the divide et impera
approaches.Comment: In Proceedings GandALF 2017, arXiv:1709.0176
A Parity Game Tale of Two Counters
Parity games are simple infinite games played on finite graphs with a winning
condition that is expressive enough to capture nested least and greatest
fixpoints. Through their tight relationship to the modal mu-calculus, they are
used in practice for the model-checking and synthesis problems of the
mu-calculus and related temporal logics like LTL and CTL. Solving parity games
is a compelling complexity theoretic problem, as the problem lies in the
intersection of UP and co-UP and is believed to admit a polynomial-time
solution, motivating researchers to either find such a solution or to find
superpolynomial lower bounds for existing algorithms to improve the
understanding of parity games. We present a parameterized parity game called
the Two Counters game, which provides an exponential lower bound for a wide
range of attractor-based parity game solving algorithms. We are the first to
provide an exponential lower bound to priority promotion with the delayed
promotion policy, and the first to provide such a lower bound to tangle
learning.Comment: In Proceedings GandALF 2019, arXiv:1909.0597
Oink: an Implementation and Evaluation of Modern Parity Game Solvers
Parity games have important practical applications in formal verification and
synthesis, especially to solve the model-checking problem of the modal
mu-calculus. They are also interesting from the theory perspective, as they are
widely believed to admit a polynomial solution, but so far no such algorithm is
known. In recent years, a number of new algorithms and improvements to existing
algorithms have been proposed. We implement a new and easy to extend tool Oink,
which is a high-performance implementation of modern parity game algorithms. We
further present a comprehensive empirical evaluation of modern parity game
algorithms and solvers, both on real world benchmarks and randomly generated
games. Our experiments show that our new tool Oink outperforms the current
state-of-the-art.Comment: Accepted at TACAS 201
Simple Fixpoint Iteration To Solve Parity Games
A naive way to solve the model-checking problem of the mu-calculus uses
fixpoint iteration. Traditionally however mu-calculus model-checking is solved
by a reduction in linear time to a parity game, which is then solved using one
of the many algorithms for parity games. We now consider a method of solving
parity games by means of a naive fixpoint iteration. Several fixpoint
algorithms for parity games have been proposed in the literature. In this work,
we introduce an algorithm that relies on the notion of a distraction. The idea
is that this offers a novel perspective for understanding parity games. We then
show that this algorithm is in fact identical to two earlier published fixpoint
algorithms for parity games and thus that these earlier algorithms are the
same. Furthermore, we modify our algorithm to only partially recompute deeper
fixpoints after updating a higher set and show that this modification enables a
simple method to obtain winning strategies. We show that the resulting
algorithm is simple to implement and offers good performance on practical
parity games. We empirically demonstrate this using games derived from
model-checking, equivalence checking and reactive synthesis and show that our
fixpoint algorithm is the fastest solution for model-checking games.Comment: In Proceedings GandALF 2019, arXiv:1909.0597
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