  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.
