# Simple Fixpoint Iteration To Solve Parity Games

Tom van Dijk, Bob Rubbens

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

2 Citations (Scopus)

## 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.
Original language English Proceedings Tenth International Symposium on Games, Automata, Logics, and Formal Verification, GandALF 2019 Bordeaux, France, 2-3rd September 2019 Jérôme Leroux, Jean-François Raskin ArXiv.org 123-139 17 https://doi.org/10.4204/EPTCS.305.9 Published - 2019 10th International Symposium on Games, Automata, Logics, and Formal Verification, GandALF 2019 - Bordeaux, FranceDuration: 2 Sept 2019 → 3 Sept 2019Conference number: 10

### Publication series

Name Electronic Proceedings in Theoretical Computer Science (EPTCS) arXiv 305 2075-2180

### Conference

Conference 10th International Symposium on Games, Automata, Logics, and Formal Verification, GandALF 2019 GandALF France Bordeaux 2/09/19 → 3/09/19

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