Simple Fixpoint Iteration To Solve Parity Games

TL;DR

This paper introduces a simple fixpoint iteration method for solving parity games, demonstrating its equivalence to existing algorithms, and shows it is effective and fast in practical model-checking scenarios.

Contribution

The paper presents a novel fixpoint iteration algorithm for parity games based on the concept of distraction, revealing its equivalence to prior algorithms and enhancing practical performance.

Findings

The proposed algorithm is equivalent to two earlier fixpoint algorithms.

Modifying the algorithm allows for efficient partial recomputation of fixpoints.

Empirical results show the algorithm is the fastest for model-checking 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.