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Complexity of Motion Planning of Arbitrarily Many Robots: Gadgets, Petri Nets, and Counter Machines

Authors Joshua Ani, Michael Coulombe, Erik D. Demaine, Yevhenii Diomidov, Timothy Gomez, Dylan Hendrickson, Jayson Lynch

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Author Details

Joshua Ani
  • Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA
Michael Coulombe
  • Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA
Erik D. Demaine
  • Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA
Yevhenii Diomidov
  • Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA
Timothy Gomez
  • Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA
Dylan Hendrickson
  • Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA
Jayson Lynch
  • Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA

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Joshua Ani, Michael Coulombe, Erik D. Demaine, Yevhenii Diomidov, Timothy Gomez, Dylan Hendrickson, and Jayson Lynch. Complexity of Motion Planning of Arbitrarily Many Robots: Gadgets, Petri Nets, and Counter Machines. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SAND.2023.5

Abstract

We extend the motion-planning-through-gadgets framework to several new scenarios involving various numbers of robots/agents, and analyze the complexity of the resulting motion-planning problems. While past work considers just one robot or one robot per player, most of our models allow for one or more locations to spawn new robots in each time step, leading to arbitrarily many robots. In the 0-player context, where all motion is deterministically forced, we prove that deciding whether any robot ever reaches a specified location is undecidable, by representing a counter machine. In the 1-player context, where the player can choose how to move the robots, we prove equivalence to Petri nets, EXPSPACE-completeness for reaching a specified location, PSPACE-completeness for reconfiguration, and ACKERMANN-completeness for reconfiguration when robots can be destroyed in addition to spawned. Finally, we consider a variation on the standard 2-player context where, instead of one robot per player, we have one robot shared by the players, along with a ko rule to prevent immediately undoing the previous move. We prove this impartial 2-player game EXPTIME-complete.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Gadgets
  • robots
  • undecidability
  • Petri nets

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References

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