Playing Safe, Ten Years Later

TL;DR

This paper establishes tight bounds on the memory size needed for strategies to ensure safety objectives in two-player graph games, revealing fundamental limits and applications in game theory.

Contribution

It provides a general characterization of minimal memory requirements for safety strategies, applicable without regularity assumptions, and explores various game-theoretic applications.

Findings

Memory size equals the size of the maximal antichain of left quotients.

Characterization of opponent memory requirements in reachability games.

Existence of positional strategies in counter-based games.

Abstract

We consider two-player games over graphs and give tight bounds on the memory size of strategies ensuring safety objectives. More specifically, we show that the minimal number of memory states of a strategy ensuring a safety objective is given by the size of the maximal antichain of left quotients with respect to language inclusion. This result holds for all safety objectives without any regularity assumptions. We give several applications of this general principle. In particular, we characterize the exact memory requirements for the opponent in generalized reachability games, and we prove the existence of positional strategies in games with counters.