Quasipolynomial Set-Based Symbolic Algorithms for Parity Games

Abstract

Solving parity games, which are equivalent to modal µ-calculus model checking, is a central algorithmic problem in formal methods, with applications in reactive synthesis, program repair, verification of branching-time properties, etc. 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) the basic 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. In this work, 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 immediately 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 · log n) symbolic space. Moreover, for the important special case of d ≤ log n, our algorithm requires only polynomially many symbolic operations and poly-logarithmic symbolic space.

Oink: An Implementation and Evaluation of Modern Parity Game Solvers

“…A number of parallel implementations have been proposed for the Playstation 3 [6], for multi-core architectures [25,37] and for GPUs [8,24]. Furthermore, Chatterjee et al proposed an implementation using BDDs [10]. Different types of progress measures were introduced after the recent breakthrough of a quasi-polynomial time algorithm due to Calude et al [9], which resulted in the progress measures algorithms by Jurdziński et al [28] and by Fearnley et al [18].…”

A Comparison of BDD-Based Parity Game Solvers

“…Assuming that the maximal priority is of parity α, the outcome of the first recursion yields a dominion forᾱ in the subgame. In case this dominion isᾱ-maximal in the entire game, α wins all vertices outside the dominion of playerᾱ (lines [11][12]. This can be seen as follows: since the game restricted to V \ B (where B is computed in line 10) isᾱ-closed, the opponent can choose to stay within the subgame W α ; choosing to do so means she will lose.…”

“…In case the region is open in the current state (line [11][12][13][14], the priority function of the current state is updated to set all vertices in A to the currently dominating priority m g . This is achieved by the update…”

“…Another line of research has concentrated on giving set-based versions of classical algorithms such as the small progress measures algorithm, the bigstep algorithm and the subexponential dominion decomposition algorithm, see e.g. [9,11]. Symbolic algorithms studied for more restricted types of games include [1,7].…”

Symbolic Parity Game Solvers that Yield Winning Strategies

“…Their online implementation in LTSmin [22] appears to be capable of generating the winning strategy, but this is not reported in the paper and the solver is embedded within the LTSmin model checker. A symbolic implementation of the quasi-polynomial "ordered progress measures" algorithm [17] has been proposed [8] which requires only quasi-polynomially many symbolic operations. In [32], Di Stasio et al implement Zielonka's recursive algorithm, the fixpoint algorithm APT [24], and a symbolic small progress measures [21] algorithm.…”