Impossibility of Gathering, a Certification

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.Comment: 10

A Certified Universal Gathering Algorithm for Oblivious Mobile Robots

We present a new algorithm for the problem of universal gathering mobile oblivious robots (that is, starting from any initial configuration that is not bivalent, using any number of robots, the robots reach in a finite number of steps the same position, not known beforehand) without relying on a common chirality. We give very strong guaranties on the correctness of our algorithm by proving formally that it is correct, using the COQ proof assistant. To our knowledge, this is the first certified positive (and constructive) result in the context of oblivious mobile robots. It demonstrates both the effectiveness of the approach to obtain new algorithms that are truly generic, and its managability since the amount of developped code remains human readable

Certified Universal Gathering in R2R^2 for Oblivious Mobile Robots

We present a unified formal framework for expressing mobile robots models, protocols, and proofs, and devise a protocol design/proof methodology dedicated to mobile robots that takes advantage of this formal framework. As a case study, we present the first formally certified protocol for oblivious mobile robots evolving in a two-dimensional Euclidean space. In more details, we provide a new algorithm for the problem of universal gathering mobile oblivious robots (that is, starting from any initial configuration that is not bivalent, using any number of robots, the robots reach in a finite number of steps the same position, not known beforehand) without relying on a common orientation nor chirality. We give very strong guaranties on the correctness of our algorithm by proving formally that it is correct, using the COQ proof assistant. This result demonstrates both the effectiveness of the approach to obtain new algorithms that use as few assumptions as necessary, and its manageability since the amount of developed code remains human readable.Comment: arXiv admin note: substantial text overlap with arXiv:1506.0160

Parameterized Verification of Algorithms for Oblivious Robots on a Ring

We study verification problems for autonomous swarms of mobile robots that self-organize and cooperate to solve global objectives. In particular, we focus in this paper on the model proposed by Suzuki and Yamashita of anonymous robots evolving in a discrete space with a finite number of locations (here, a ring). A large number of algorithms have been proposed working for rings whose size is not a priori fixed and can be hence considered as a parameter. Handmade correctness proofs of these algorithms have been shown to be error-prone, and recent attention had been given to the application of formal methods to automatically prove those. Our work is the first to study the verification problem of such algorithms in the parameter-ized case. We show that safety and reachability problems are undecidable for robots evolving asynchronously. On the positive side, we show that safety properties are decidable in the synchronous case, as well as in the asynchronous case for a particular class of algorithms. Several properties on the protocol can be decided as well. Decision procedures rely on an encoding in Presburger arithmetics formulae that can be verified by an SMT-solver. Feasibility of our approach is demonstrated by the encoding of several case studies

Optimal torus exploration by oblivious robots

International audienceWe deal with a team of autonomous robots that are endowed with motion actuators and visibility sensors. Those robots are weak and evolve in a discrete environment. By weak, we mean that they are anonymous, uniform, unable to explicitly communicate, and oblivious. We first show that it is impossible to solve the terminating exploration of a simple torus of arbitrary size with less than 4 or 5 such robots, respectively depending on whether the algorithm is probabilistic or deterministic. Next, we propose in the SSYNC model a probabilistic solution for the terminating exploration of torus-shaped networks of size ℓ×L, where 7≤ℓ≤L, by a team of 4 such weak robots. So, this algorithm is optimal w.r.t. the number of robots

On the Synthesis of Mobile Robots Algorithms: the Case of Ring Gathering

International audienceRecent advances in Distributed Computing highlight models and algorithms for autonomous swarms of mobile robots that self-organize and cooperate to solve global objectives. The overwhelming majority of works so far considers handmade algorithms and correctness proofs.This paper is the first to propose a formal framework to automatically design distributed algorithms that are dedicated to autonomous mobile robots evolving in a discrete space. As a case study, we consider the problem of gathering all robots at a particular location, not known beforehand. Our contribution is threefold. First, we propose an encoding of the gathering problem as a reachability game. Then, we automatically generate an optimal distributed algorithm for three robots evolving on a fixed size uniform ring. Finally, we prove by induction that the generated algorithm is also correct for any ring size except when an impossibility result holds (that is, when the number of robots divides the ring size)

Optimal Torus Exploration by Oblivious Robots

International audienceWe consider autonomous robots that are endowed with motion actuators and visibility sensors. The robots we consider are weak, i.e., they are anonymous, uniform, unable to explicitly communicate, and oblivious (they do not remember any of their past actions). In this paper, we propose an optimal (w.r.t. the number of robots) solution for the terminating exploration of a torus-shaped network by a team of $k$ such robots. In more details, we first show that it is impossible to explore a simple torus of arbitrary size with (strictly) less than four robots, even if the algorithm is probabilistic. If the algorithm is required to be deterministic, four robots are also insufficient. This negative result implies that the only way to obtain an optimal algorithm (w.r.t. the number of robots participating to the algorithm) is to make use of probabilities. Then, we propose a probabilistic algorithm that uses four robots to explore all simple tori of size $\ell \times L$, where $7 \leq \ell \leq L$. Hence, in such tori, four robots are necessary and sufficient to solve the (probabilistic) terminating exploration. As a torus can be seen as a 2-dimensional ring, our result shows, perhaps surprisingly, that increasing the number of possible symmetries in the network (due to increasing dimensions) does not come at an extra cost w.r.t. the number of robots that are necessary to solve the problem

Formal verification of Mobile Robot Protocols

Mobile robot networks emerged in the past few years as a promising distributed computing model. Existing work in the literature typically ensures the correctness of mobile robot protocols via \emph{ad hoc} handwritten proofs, which, in the case of asynchronous execution models, are both cumbersome and error-prone. In this paper, we propose the first formal model and general verification (by model-checking) methodology for mobile robot protocols operating in a discrete space (that is, the set of possible robot positions is finite). Our contribution is threefold. First, we formally model using synchronized automata a network of mobile robots operating under various synchrony (or asynchrony) assumptions. Then, we use this formal model as input model for the DiVinE model-checker and prove the equivalence of the two models. Third, we verify using DiVinE two known protocols for variants of the ring exploration in an asynchronous setting (exploration with stop and perpetual exclusive exploration). The exploration with stop we verify was manually proved correct only when the number of robots is $k>17$, and $n$ (the ring size) and $k$ are co-prime. As the necessity of this bound was not proved in the original paper, our methodology demonstrates that for several instances of $k$ and $n$ \emph{not covered} in the original paper, the algorithm remains correct. In the case of the perpetual exclusive exploration protocol, our methodology exhibits a counter-example in the completely asynchronous setting where safety is violated, which is used to correct the original protocol

Discovering and Assessing Fine-Grained Metrics in Robot Networks Protocols

International audienceThe model of autonomous oblivious and anonymous mobile robots recently emerged as an attractive distributed computingabstraction that permits to assess the intrinsic difficulties of many fundamentals tasks, such as exploring orgathering in a discrete space. We present and implement a generic method for obtaining all possible protocols for aswarm of mobile robots operating in a particular discrete space. We use the exclusive perpetual exploration ofanonymous rings as a case study. Our method permits to discover new protocols that solve the problem, and to assessspecific optimization criteria (such as individual coverage, visits frequency, etc.) that are met by those protocols.To our knowledge, this is the first attempt to mechanize the discovery and fine-grained property testing ofdistributed mobile robot protocols

Brief Announcement: Discovering and Assessing Fine-Grained Metrics in Robot Networks Protocols

International audienceIn discrete anonymous environments, robot algorithms consist in a list of rules, where each rule takes a configuration of the system as input and outputs the set of robots that are required to move when the system is in this configuration. Based on these rules and on the robots' activations by the scheduler, the system globally evolves and, when the algorithm is correct, it solves the targeted problem