Reasoning under uncertainty in the game of Cops and Robbers

TL;DR

This paper introduces a formal logical framework, ELCR, for analyzing the game of Cops and Robbers with imperfect information, enabling automated reasoning about players' knowledge and strategies.

Contribution

It presents the first formal logical model for Cops and Robbers considering partial information, including a novel dynamic operator for information updates.

Findings

ELCR effectively models players' knowledge and inference during the game.

The framework supports axiomatization and decidability analysis.

Comparison with existing paradigms highlights its novelty in handling imperfect information.

Abstract

The game of Cops and Robbers is an important model for studying computational queries in pursuit-evasion environments, among others. As recent logical explorations have shown, its structure exhibits appealing analogies with modal logic. In this paper, we enrich the game with a setting in which players may have imperfect information. We propose a new formal framework, Epistemic Logic of Cops and Robbers (ELCR), to make the core notions of the game precise, for instance, players' positions, observational power and inference. Applying ELCR to analyze the game, we obtain an automated way to track interactions between players and characterize their information updates during the game. The update mechanism is defined by a novel dynamic operator, and we compare it with some relevant paradigms from the game and logic perspectives. We study various properties of ELCR including axiomatization and decidability. To our knowledge, this is the first attempt to explore these games from a formal point of view where (partial) information available to players is taken into account.