Certified Impossibility Results for Byzantine-Tolerant Mobile Robots

Abstract

Abstract. We propose a framework to build formal developments for robot networks using the COQ proof assistant, to state and to prove formally various properties. We focus in this paper on impossibility proofs, as it is natural to take advantage of the COQ higher order calculus to reason about algorithms as abstract objects. We present in particular formal proofs of two impossibility results for convergence of oblivious mobile robots if respectively more than one half and more than one third of the robots exhibit Byzantine failures, starting from the original theorems by Bouzid et al.. Thanks to our formalization, the corresponding COQ developments are quite compact. To our knowledge, these are the first certified (in the sense of formally proved) impossibility results for robot networks.

StarL

“…(loc(j,s) = move wait IMPLIES subset? (list2set(myseq(j,s)),crit set(j,s))) AND 6 (loc(j,s) = mutex wait IMPLIES list2set(myseq(j,s))= req set(j,s) AND myseq(j,s) = Path(sd(j))) AND (loc(j,s) = S0 IMPLIES myseq(j,s) = Path(sd(j))) 8 …”

“…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]).…”

Verification of Asynchronous Mobile-Robots in Partially-Known Environments

“…On the other hand, we prove that parameterised verification is decidable for scenarios in which the number of times that the robots take turns is bounded. 4 This decidability result is very robust:…”

“…This literature is mostly mathematical, and theorems from this literature are parameterised, i.e., they may involve graph-parameters (e.g., the maximum degree, the number of vertices, the diameter), memory parameters (e.g., the number of internal states of the robot protocol), and the number of robots may be a parameter. Only recently has there been emphasis on formal analysis of correctness of robots in a parameterised setting [22,4,27,23,30,29]. In these formal analyses, typically it is the number of agents that is treated as the parameter [22,4,27,23].…”

“…Only recently has there been emphasis on formal analysis of correctness of robots in a parameterised setting [22,4,27,23,30,29]. In these formal analyses, typically it is the number of agents that is treated as the parameter [22,4,27,23]. In contrast, in this paper (as in [30]) we apply formal methods to the parameterised verification problem in which it is the environment that is parameterised.…”