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

REGISTER GAMES

The complexity of parity games is a long standing open problem that saw a major breakthrough in 2017 when two quasi-polynomial algorithms were published. This article presents a third, independent approach to solving parity games in quasi-polynomial time, based on the notion of register game, a parameterised variant of a parity game. The analysis of register games leads to a quasi-polynomial algorithm for parity games, a polynomial algorithm for restricted classes of parity games and a novel measure of complexity, the register index, which aims to capture the combined complexity of the priority assignement and the underlying game graph. We further present a translation of alternating parity word automata into alternating weak automata with only a quasi-polynomial increase in size, based on register games; this improves on the previous exponential translation. We also use register games to investigate the parity index hierarchy: while for words the index hierarchy of alternating parity automata collapses to the weak level, and for trees it is strict, for structures between trees and words, it collapses logarithmically, in the sense that any parity tree automaton of size n is equivalent, on these particular classes of structures, to an automaton with a number of priorities logarithmic in n

The Strahler number of a parity game

The Strahler number of a rooted tree is the largest height of a perfect binary tree that is its minor. The Strahler number of a parity game is proposed to be defined as the smallest Strahler number of the tree of any of its attractor decompositions. It is proved that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices~n and linear in (d/2k)k, where d is the number of priorities and k is the Strahler number. This complexity is quasi-polynomial because the Strahler number is at most logarithmic in the number of vertices. The proof is based on a new construction of small Strahler-universal trees. It is shown that the Strahler number of a parity game is a robust parameter: it coincides with its alternative version based on trees of progress measures and with the register number defined by Lehtinen~(2018). It follows that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices and linear in (d/2k)k, where k is the register number. This significantly improves the running times and space achieved for parity games of bounded register number by Lehtinen (2018) and by Parys (2020). The running time of the algorithm based on small Strahler-universal trees yields a novel trade-off k⋅lg(d/k)=O(logn) between the two natural parameters that measure the structural complexity of a parity game, which allows solving parity games in polynomial time. This includes as special cases the asymptotic settings of those parameters covered by the results of Calude, Jain Khoussainov, Li, and Stephan (2017), of Jurdziński and Lazić (2017), and of Lehtinen (2018), and it significantly extends the range of such settings, for example to d=2O(lgn√) and k=O(lgn−√)

Quasipolynomial Set-Based Symbolic Algorithms for Parity Games

Solving parity games, which are equivalent to modal $\mu$-calculus model checking, is a central algorithmic problem in formal methods. Besides the standard computation model with the explicit representation of games, another important theoretical model of computation is that of set-based symbolic algorithms. Set-based symbolic algorithms use basic set operations and one-step predecessor operations on the implicit description of games, rather than the explicit representation. The significance of symbolic algorithms is that they provide scalable algorithms for large finite-state systems, as well as for infinite-state systems with finite quotient. Consider parity games on graphs with $n$ vertices and parity conditions with $d$ priorities. While there is a rich literature of explicit algorithms for parity games, the main results for set-based symbolic algorithms are as follows: (a) an algorithm that requires $O(n^d)$ symbolic operations and $O(d)$ symbolic space; and (b) an improved algorithm that requires $O(n^{d/3+1})$ symbolic operations and $O(n)$ symbolic space. Our contributions are as follows: (1) We present a black-box set-based symbolic algorithm based on the explicit progress measure algorithm. Two important consequences of our algorithm are as follows: (a) a set-based symbolic algorithm for parity games that requires quasi-polynomially many symbolic operations and $O(n)$ symbolic space; and (b) any future improvement in progress measure based explicit algorithms imply an efficiency improvement in our set-based symbolic algorithm for parity games. (2) We present a set-based symbolic algorithm that requires quasi-polynomially many symbolic operations and $O(d \cdot \log n)$ symbolic space. Moreover, for the important special case of $d \leq \log n$, our algorithm requires only polynomially many symbolic operations and poly-logarithmic symbolic space.Comment: Published at LPAR-22 in 201

Improving the complexity of Parys' recursive algorithm

Parys has recently proposed a quasi-polynomial version of Zielonka's recursive algorithm for solving parity games. In this brief note we suggest a variation of his algorithm that improves the complexity to meet the state-of-the-art complexity of broadly $2^{O((\log n)(\log c))}$, while providing polynomial bounds when the number of colours is logarithmic

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