34 research outputs found
An Ordered Approach to Solving Parity Games in Quasi Polynomial Time and Quasi Linear Space
Parity games play an important role in model checking and synthesis. In their
paper, Calude et al. have shown that these games can be solved in
quasi-polynomial time. We show that their algorithm can be implemented
efficiently: we use their data structure as a progress measure, allowing for a
backward implementation instead of a complete unravelling of the game. To
achieve this, a number of changes have to be made to their techniques, where
the main one is to add power to the antagonistic player that allows for
determining her rational move without changing the outcome of the game. We
provide a first implementation for a quasi-polynomial algorithm, test it on
small examples, and provide a number of side results, including minor
algorithmic improvements, a quasi bi-linear complexity in the number of states
and edges for a fixed number of colours, and matching lower bounds for the
algorithm of Calude et al
An Ordered Approach to Solving Parity Games in Quasi-Polynomial Time and Quasi-Linear Space
Parity games play an important role in model checking and synthesis. In their paper, Calude et al. have recently shown that these games can be solved in quasi-polynomial time. We show that their algorithm can be implemented efficiently: we use their data structure as a progress measure, allowing for a backward implementation instead of a complete unravelling of the game. To achieve this, a number of changes have to be made to their techniques, where the main one is to add power to the antagonistic player that allows for determining her rational move without changing the outcome of the game. We provide a first implementation for a quasi-polynomial algorithm, test it on small examples, and provide a number of side results, including minor algorithmic improvements, a quasi-bi-linear complexity in the number of states and edges for a fixed number of colours, matching lower bounds for the algorithm of Calude et al., and a complexity index associated to our approach, which we compare to the recently proposed register index
Succinct progress measures for solving parity games
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
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