Topological Characterization of Consensus under General Message Adversaries

Abstract

In this paper, we provide a rigorous characterization of consensus solvability in synchronous directed dynamic networks controlled by an arbitrary message adversary using point-set topology: We extend the approach introduced by Alpern and Schneider in 1985 by introducing two novel topologies on the space of infinite executions: the process-view topology, induced by a distance function that relies on the local view of a given process in an execution, and the minimum topology, which is induced by a distance function that focuses on the local view of the process that is the last to distinguish two executions. We establish some simple but powerful topological results, which not only lead to a topological explanation of bivalence arguments, but also provide necessary and sufficient topological conditions on the admissible graph sequences of a message adversary for solving consensus. In particular, we characterize consensus solvability in terms of connectivity of the set of admissible graph sequences. For non-compact message adversaries, which are not limit-closed in the sense that there is a convergent sequence of graph sequences whose limit is not permitted, this requires the exclusion of all "fair" and "unfair" limit sequences that coincide with the forever bivalent runs constructed in bivalence proofs. For both compact and non-compact message adversaries, we also provide tailored characterizations of consensus solvability, i.e., tight conditions for impossibility and existence of algorithms, based on the broadcastability of the connected components of the set of admissible graph sequences.

Derivation of Heard-of Predicates from Elementary Behavioral Patterns

“…This model was put to use in many ways. Obviously computability and complexity results were proven: new algorithms for consensus in the original paper by Charron-Bost and Schiper [4]; characterizations for consensus solvability by Coulouma et al [5] and Nowak et al [10]; a characterization for approximate consensus solvability by Charron-Bost et al [3]; a study of k set-agreement by Biely et al [1]; and more.…”

K-set agreement bounds in round-based models through combinatorial topology

“…Combination of Topology and Round-based models. Two papers at least applied topology (combinatorial or not) to general roundbased models in order to study agreement problems: Godard and Perdereau [14] used combinatorial topology to study -set agreement in models with omission failures; and Nowak et al [24] characterization of consensus for general round-based models (not necessarily oblivious) using point-set topology.…”

“…Actually, there is at least one example of a general mathematical technique used in this context: the characterization of consensus solvability through point-set topology by Nowak et al [24]. We propose what might be seen as an extension to higher dimension of this intuition, by applying combinatorial topology (instead of point-set topology) to bear on -set agreement (instead of just consensus).…”

Back to the Coordinated Attack Problem

“…We are aware of Nowak et al (2019) that appeared between revisions of this paper. We underline that Theorem 33 cannot be obtained in a straightforward way from Nowak et al (2019).…”