Degrees of Lookahead in Regular Infinite Games

TL;DR

This paper investigates the impact of lookahead strategies in regular infinite games, demonstrating decidability of continuous strategies and bounding lookahead size, with implications for distributed systems and automata theory.

Contribution

It introduces and analyzes different degrees of lookahead in regular infinite games, establishing decidability results and bounds on lookahead size for winning strategies.

Findings

Decidability of continuous strategies in regular infinite games.

Reduction of continuous strategies to bounded lookahead strategies.

Bound on lookahead size is at most doubly exponential in automaton size.

Abstract

We study variants of regular infinite games where the strict alternation of moves between the two players is subject to modifications. The second player may postpone a move for a finite number of steps, or, in other words, exploit in his strategy some lookahead on the moves of the opponent. This captures situations in distributed systems, e.g. when buffers are present in communication or when signal transmission between components is deferred. We distinguish strategies with different degrees of lookahead, among them being the continuous and the bounded lookahead strategies. In the first case the lookahead is of finite possibly unbounded size, whereas in the second case it is of bounded size. We show that for regular infinite games the solvability by continuous strategies is decidable, and that a continuous strategy can always be reduced to one of bounded lookahead. Moreover, this lookahead is at most doubly exponential in the size of a given parity automaton recognizing the winning condition. We also show that the result fails for non-regular gamesxwhere the winning condition is given by a context-free omega-language.