Shape recognition by a finite automaton robot

Motivated by the problem of shape recognition by nanoscale computing agents, we investigate the problem of detecting the geometric shape of a structure composed of hexagonal tiles by a finite-state automaton robot. In particular, in this paper we consider the question of recognizing whether the tiles are assembled into a parallelogram whose longer side has length ` = f(h), for a given function f(·), where h is the length of the shorter side. To determine the computational power of the finite-state automaton robot, we identify functions that can or cannot be decided when the robot is given a certain number of pebbles. We show that the robot can decide whether ` = ah+ b for constant integers a and b without any pebbles, but cannot detect whether ` = f(h) for any function f(x) = ω(x). For a robot with a single pebble, we present an algorithm to decide whether ` = p(h) for a given polynomial p(·) of constant degree. We contrast this result by showing that, for any constant k, any function f(x) = ω(x ⁶k ⁺²) cannot be decided by a robot with k states and a single pebble. We further present exponential functions that can be decided using two pebbles. Finally, we present a family of functions f _n(·) such that the robot needs more than n pebbles to decide whether ` = f _n(h).