Impossibility of Gathering, a Certification

TL;DR

This paper provides a formal certification of the impossibility of gathering autonomous robots in certain conditions, extending previous results and ensuring correctness of these fundamental limitations in distributed robotics.

Contribution

It introduces a formal framework to certify impossibility results in autonomous robot gathering, including a proof of the Suzuki and Yamashita impossibility theorem and its generalization.

Findings

Certified proof of the impossibility of gathering for two robots.

Extended impossibility result for an even number of robots with potential initial overlaps.

Formal verification of fundamental limitations in autonomous robot coordination.

Abstract

Recent advances in Distributed Computing highlight models and algorithms for autonomous swarms of mobile robots that self-organise and cooperate to solve global objectives. The overwhelming majority of works so far considers handmade algorithms and proofs of correctness. This paper builds upon a previously proposed formal framework to certify the correctness of impossibility results regarding distributed algorithms that are dedicated to autonomous mobile robots evolving in a continuous space. As a case study, we consider the problem of gathering all robots at a particular location, not known beforehand. A fundamental (but not yet formally certified) result, due to Suzuki and Yamashita, states that this simple task is impossible for two robots executing deterministic code and initially located at distinct positions. Not only do we obtain a certified proof of the original impossibility result, we also get the more general impossibility of gathering with an even number of robots, when any two robots are possibly initially at the same exact location.