How Much Lookahead is Needed to Win Infinite Games?

TL;DR

This paper improves the understanding of delay games with infinite duration by providing exponential time algorithms and tight bounds on lookahead requirements for winning strategies under various conditions.

Contribution

It introduces an exponential time algorithm and tight exponential bounds on lookahead, improving previous doubly-exponential results for solving delay games with $$-regular conditions.

Findings

Exponential time algorithm for solving delay games.

Exponential upper bound on necessary lookahead.

PSPACE-completeness for reachability conditions.

Abstract

Delay games are two-player games of infinite duration in which one player may delay her moves to obtain a lookahead on her opponent's moves. For $\omega$-regular winning conditions it is known that such games can be solved in doubly-exponential time and that doubly-exponential lookahead is sufficient. We improve upon both results by giving an exponential time algorithm and an exponential upper bound on the necessary lookahead. This is complemented by showing EXPTIME-hardness of the solution problem and tight exponential lower bounds on the lookahead. Both lower bounds already hold for safety conditions. Furthermore, solving delay games with reachability conditions is shown to be PSPACE-complete. This is a corrected version of the paper https://arxiv.org/abs/1412.3701v4 published originally on August 26, 2016.