Two-Robot Computational Landscape: A Complete Characterization of Model Power in Minimal Mobile Robot Systems

TL;DR

This paper provides the first complete characterization of the computational capabilities of two autonomous mobile robots across various models and schedulers, revealing fundamental differences from the general multi-robot case.

Contribution

It offers the first full classification of two-robot computational power across key models and schedulers, using a novel simulation-free approach.

Findings

FSTA^F and LUMI^F coincide under full synchrony.

FSTA and FCOM are orthogonal, with problems solvable in one but not the other.

The results reveal a landscape that differs significantly from the general n-robot case.

Abstract

The computational power of autonomous mobile robots under the Look-Compute-Move (LCM) model has been widely studied through an extensive hierarchy of robot models defined by the presence of memory, communication, and synchrony assumptions. While the general n-robot landscape has been largely established, the exact structure for two robots has remained unresolved. This paper presents the first complete characterization of the computational power of two autonomous robots across all major models, namely OBLOT, FSTA, FCOM, and LUMI, under the full spectrum of schedulers (FSYNCH, SSYNCH, ASYNCH, and their atomic variants). Our results reveal a landscape that fundamentally differs from the general case. Most notably, we prove that FSTA^F and LUMI^F coincide under full synchrony, a surprising collapse indicating that perfect synchrony can substitute both memory and communication when only two robots exist. We also show that FSTA and FCOM are orthogonal: there exists a problem solvable in the weakest communication model but impossible even in the strongest finite-state model, completing the bidirectional incomparability. All equivalence and separation results are derived through a novel simulation-free method, providing a unified and constructive view of the two-robot hierarchy. This yields the first complete and exact computational landscape for two robots, highlighting the intrinsic challenges of coordination at the minimal scale.