Region Management by Finite-State Robots1

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 avatars of simple, yet non-trivial, discrete control structures. We study the ability of (teams of) FSMs to identify and search varied-size quadrants of square (i.e. n×n) meshes of tiles—such meshes being the avatars of simple tesselated geographically constrained environments. Each team must be able to accomplish its assigned tasks in arbitrarily large meshes—i.e. for arbitrarily large values of n. Partitions of a mesh into quadrants are specified via pairs of rational numbers ˂ φ, ψ ˃, where 0 < φ, ψ < 1, chosen from a fixed, finite repertoire of such pairs. The quadrants specified by a pair ˂ φ, ψ ˃ are delimited by a horizontal line and a vertical line that that cross at the anchor mesh-tile v = ˂ ⌊ φ (n−1) ⌋, ⌊ ψ (n−1) ⌋ ˃. The following results are established. A single FSM cannot identify anchor tiles in meshes of arbitrary sizes, even for a single pair ˂ φ, ψ ˃—except when the anchor tile resides on an edge of the mesh. A pair of identical FSMs can identify anchor tiles in meshes of arbitrary sizes, for arbitrary fixed finite sets of k pairs {˂ φi, ψi ˃}^i=1k. After identifying an anchor tile, the pair can sweep each of the resulting quadrants in turn. Single FSMs can verify—for arbitrary fixed finite sets of pairs and for meshes of arbitrary sizes—that all of the tiles of each quadrant are labeled in a way that is unique to that quadrant. Deploying multiple FSMs for this task achieves linear parallel speedup.