Detecting danger in gridworlds using Gromov's Link Condition

TL;DR

This paper uses topological and geometric methods to analyze gridworlds in AI, revealing safety limitations by detecting failures in Gromov's Link Condition that correspond to dangerous states.

Contribution

It introduces a modified state complex framework capturing agent braiding, linking topological defects to unsafe states in gridworlds, and applies geometric group theory tools to AI safety analysis.

Findings

Failures in Gromov's Link Condition indicate dangerous states.

Modified state complexes effectively identify safety limitations.

Topological methods provide new insights into gridworld safety analysis.

Abstract

Gridworlds have been long-utilised in AI research, particularly in reinforcement learning, as they provide simple yet scalable models for many real-world applications such as robot navigation, emergent behaviour, and operations research. We initiate a study of gridworlds using the mathematical framework of reconfigurable systems and state complexes due to Abrams, Ghrist & Peterson. State complexes represent all possible configurations of a system as a single geometric space, thus making them conducive to study using geometric, topological, or combinatorial methods. The main contribution of this work is a modification to the original Abrams, Ghrist & Peterson setup which we introduce to capture agent braiding and thereby more naturally represent the topology of gridworlds. With this modification, the state complexes may exhibit geometric defects (failure of Gromov's Link Condition). Serendipitously, we discover these failures occur exactly where undesirable or dangerous states appear in the gridworld. Our results therefore provide a novel method for seeking guaranteed safety limitations in discrete task environments with single or multiple agents, and offer useful safety information (in geometric and topological forms) for incorporation in or analysis of machine learning systems. More broadly, our work introduces tools from geometric group theory and combinatorics to the AI community and demonstrates a proof-of-concept for this geometric viewpoint of the task domain through the example of simple gridworld environments.