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Shape Recognition by a Finite Automaton Robot

Authors Robert Gmyr, Kristian Hinnenthal, Irina Kostitsyna, Fabian Kuhn, Dorian Rudolph, Christian Scheideler

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Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Design and analysis of algorithms
Keywords
  • finite automata
  • shape recognition
  • computational geometry

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Abstract

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 l = 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 l = ah+b for constant integers a and b without any pebbles, but cannot detect whether l = f(h) for any function f(x) = omega(x). For a robot with a single pebble, we present an algorithm to decide whether l = 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) = omega(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 l = f_n(h).

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Robert Gmyr, Kristian Hinnenthal, Irina Kostitsyna, Fabian Kuhn, Dorian Rudolph, and Christian Scheideler. Shape Recognition by a Finite Automaton Robot. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.MFCS.2018.52

Author Details

Robert Gmyr
  • Paderborn University, Germany
Kristian Hinnenthal
  • Paderborn University, Germany
Irina Kostitsyna
  • TU Eindhoven, the Netherlands
Fabian Kuhn
  • University of Freiburg, Germany
Dorian Rudolph
  • Paderborn University, Germany
Christian Scheideler
  • Paderborn University, Germany

Funding

This work is partly supported by DFG grant SCHE 1592/3-1. Fabian Kuhn is supported by ERC Grant 336495 (ACDC).

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