Impossibility of gathering, a certification

Abstract

International audienceRecent 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

Optimally Gathering Two Robots

“…However, we expect the complexity of the proof to go beyond what is tractable by a human, and would like to consider the possibility of using formal methods. Currently, modelchecking [15,4,17,21] and program synthesis [6,19] cannot scale to an arbitrary number of robots, and proof assistant techniques [2,11,10,3] do not yet permit to reason about the ASYNC model. Most likely, solving self-stabilizing gathering with n robots in ASYNC will require significant advances in mobile robot formalization.…”

“…The gathering tasks consist in all robots (each considered as a dimensionless point in a 2-dimensional Euclidean space) reaching a single point, not known beforehand, in finite time. A foundational result [22,9] shows that in the SSYNC model, no deterministic algorithm can solve gathering for two robots without additional assumptions. This impossibility result naturally extends to the ASYNC model.…”

Parameterized verification of algorithms for oblivious robots on a ring

“…Formal methods encompass a long-lasting path of research that is meant to overcome errors of human origin. Unsurprisingly, this mechanized approach to protocol correctness was successively used in the context of mobile robots [7,13,5,2,20,9,4,22,3]. When robots are not constrained to evolve on a particular topology (but instead are allowed to move freely in a bidimensional Euclidian space), the Pactole (http://pactole.lri.fr) framework has been proven useful.…”

“…Developed for the Coq proof assistant, Pactole enabled the use of high-order logic to certify impossibility results [2] for the problem of convergence: for any positive ε, robots are required to reach locations that are at most ε apart. Another classical impossibility result that was certified with Pactole is the impossibility of gathering starting from a bivalent configuration [9]. Recently, positive certified results for SSYNC gathering with multiplicity detection [10], and for FSYNC gathering without multiplicity detection [3] were provided.…”

StarL

“…Formal verification for distributed systems There is a large body of work on formal models and proof techniques for general classes of distributed systems that include robotic swarms [6,10,16,31,32,37,38]. Less expressive models that have been used for automatic verification (see, for example, [2,20,23]).…”