Finite-State Robots in the Land of Rationalia

Abstract

Advancing technologies have enabled simple mobile robots that collaborate to perform complex tasks. Understanding how to achieve such collaboration with simpler robots leverages these advances, potentially allowing more robots for a given cost and/or decreasing the cost of deploying a fixed number of robots. This paper is a step toward understanding the algorithmic strengths and weaknesses of robots that are identical mobile finite-state machines (FSMs)—FSMs being the avatar of “simple” digital computers. We study the ability of (teams of) FSMs to identify and search within varied-size quadrants of square $n \times n$) meshes of tiles—such meshes being the avatars of tesselated geographically constrained environments. Each team must accomplish its assigned tasks scalably—i.e., in arbitrarily large meshes (equivalently, for arbitrarily large values of $n$). Each subdivision of a mesh into quadrants is specified via a pair of fractions $\langle \varphi , \psi \rangle$, where $0 < \varphi , \psi < 1$, chosen from a fixed, finite repertoire of such pairs. The quadrants specified by the pair $\langle \varphi , \psi \rangle$ are delimited by a horizontal line and a vertical line that cross at anchor mesh-tile $v^{(\varphi , \psi )} = \langle \lfloor \varphi (n-1) \rfloor , \lfloor \psi (n-1) \rfloor \rangle$. The current results: $\bullet$ A single FSM cannot identify tile $v^{(\varphi , \psi )}$ in meshes of arbitrary sizes, even for a single pair $\langle \varphi , \psi \rangle$—except when $v^{(\varphi , \psi )}$ resides on a mesh-edge. $\bullet$ A pair of identical FSMs can identify tiles $v^{(\varphi _i, \psi _i)}$ in meshes of arbitrary sizes, for arbitrary fixed finite sets of $k$ pairs $\{\langle \varphi _i, \psi _i \rangle \}_{i=1}^k$. The pair can sweep each of the resulting quadrants in turn. $\bullet$ Single FSMs can always verify (for all pairs and meshes) that all of the tiles of each quadrant are labeled in a way that is unique to that quadrant. This process parallelizes linearly for teams of FSMs.