An ordered approach to solving parity games in quasi polynomial time and quasi linear space

Abstract

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, and matching lower bounds for the algorithm of Calude et al.

Quasipolynomial Set-Based Symbolic Algorithms for Parity Games

“…By the results in [29] the order (W, ) and lift meet the requirements of our algorithms. Theorem 1 together with Lemma 5 imply the following theorem.…”

“…In this section, we plug the ordered approach to progress measure (OPM) described by Fearnley et al [29] into Algorithm 1. To do this, we recall the witnesses they use in their algorithm and encode it with a specially-tailored technique to obtain an algorithm with a sublinear amount of symbolic space.…”

“…To do this, we recall the witnesses they use in their algorithm and encode it with a specially-tailored technique to obtain an algorithm with a sublinear amount of symbolic space. Finally, we argue that the function lift : W × C → W and the total order (W, ) described in [29] can be used to fully implement Algorithm 1.…”

“…For the details of the lift function we refer the reader to the work of Fearnley et al [29]. An implementation of the lift operation can be found at the GitHub repository of the Oink system [56].…”

“…We now discuss the bounds on |W|. The breakthrough result of [12] shows that |W| is quasi-polynomial (n O(log d) ) in general and polynomial when d ≤ log n. Using the refined analysis of [29], we obtain the following bound on |W|: in general, min(n · log(n) d−1 , h · n c1.45+log 2 (h) ), where c 1.45 = log 2 (e) < 1.45 and h = ⌈1 + d/ log(n)⌉; and if d ≤ log n, then |W| is polynomial due to [12,Theorem 2.8] and [29,Corollary 8.8].…”