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

TL;DR

The paper resolves the complete two-robot computational landscape under the Look–Compute–Move model across OBLOT, FSTA, FCOM, and LUMI, for the major schedulers and their atomic variants. It provides a simulation-free, constructive classification of equivalences and separations, revealing a surprising collapse $FSTA^F \equiv LUMI^F$ under full synchrony and establishing a true orthogonality between $FCOM$ and $FSTA$ in the minimal two-robot setting. Novel two-robot specific problems, such as DMSD and RDAM, separate CM-atomic from general Asynch and illustrate the limits of symmetry-breaking under different models. The work also presents a more efficient two-robot simulator framework and explicit reduction techniques that refine prior simulation-based results. Overall, it delivers the first complete, exact computability landscape for two autonomous robots, highlighting intrinsic coordination challenges at minimal scale.

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.

Figures (3)

• Figure 1: Computational landscape of two-robot models under major scheduler classes. Models enclosed by the same ellipse are equivalent in computational power. Each connecting line represents a separating problem: the task above the line is solvable in the stronger model, whereas the one below it is not.
• Figure 2: Global phase pairs. Left: Fsynch; right: RR.
• Figure 3: Transition Diagram of SIM$(\mathcal{A})$. Each node encodes the colored configuration of the two robots, where the upper (resp. lower) semicircle and the symbol above (resp. below) it represent the state of robot $a$ (resp. $b$). The symbols e, c and r denote exc, cpy, and rst, respectively. Directed edges correspond to possible moves between nodes. An edge labeled $r \in \{q,b\}$ indicates that robot performs an action at that transition, whereas a label $r(\mathcal{A}) (r \in \{q,b\})$ means that robot $r$ executes one step of the simulated algorithm $\mathcal{A}$. These labeled edges specify how the pair of robots progresses through the state space during the simulation.

Theorems & Definitions (36)

• Theorem 1
• Lemma 1
• proof
• Theorem 2
• Lemma 2
• Theorem 3
• Lemma 3
• Theorem 4
• Theorem 5
• Theorem 6: Separation via CGE$^*$; even for $n=2$