Simple Fixpoint Iteration To Solve Parity Games

Abstract

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.

Symbolic Parity Game Solvers that Yield Winning Strategies

“…This estimation is then iteratively refined by considering the direct successors of each vertex. By updating the estimation in a strict order, starting with the lowest-priority vertices, and resetting the estimation of lower-priority vertices whenever a higherpriority vertex is updated, the solution of parity games can be computed [13]. We consider two fixpoint iteration algorithms proposed in recent literature.…”

“…We consider two fixpoint iteration algorithms proposed in recent literature. The DFI algorithm proposed by Van Dijk and Rubbens [13] is a fixpoint iteration algorithm based on the concept of distractions. The freezing extension of DFI enables straightforward strategy computation.…”

“…We call these simply the fixpoint iteration algorithms. These are the APT algorithm [32], a fixpoint iteration algorithm we refer to as BFL [4], as well as the recent DFI [13] and FPJ [25] algorithms.…”

“…In recent work, Van Dijk and Rubbens [13] and Lapauw et al [25] propose fixpoint algorithms that yield strategies in a straightforward way. Previous fixpoint algorithms either do not produce a winning strategy [24] or require a secondary, complicated algorithm [4].…”

“…The contributions of this paper are as follows. We propose and implement two novel symbolic parity game algorithms based on [13] and [25]. We evaluate these algorithms empirically using benchmarks from the 2020 edition of SYNTCOMP [19].…”