Down the Borel Hierarchy: Solving Muller Games via Safety Games

TL;DR

This paper introduces a novel reduction from Muller games to safety games, enabling efficient computation of winning strategies and leading to new memory structures and strategies for infinite games.

Contribution

It presents a new reduction technique transforming Muller games into safety games, facilitating strategy computation and generalizing to other winning conditions.

Findings

Reduction from Muller to safety games with (n!)^3 vertices

Development of an antichain-based memory structure

Applicability to various infinite game conditions

Abstract

We transform a Muller game with n vertices into a safety game with (n!)^3 vertices whose solution allows to determine the winning regions of the Muller game and to compute a finite-state winning strategy for one player. This yields a novel antichain-based memory structure and a natural notion of permissive strategies for Muller games. Moreover, we generalize our construction by presenting a new type of game reduction from infinite games to safety games and show its applicability to several other winning conditions.