A categorical and logical framework for iterated protocols
Eric Goubault, Bernardo Hummes Flores, Roman Kniazev, Jeremy Ledent, Sergio Rajsbaum
TL;DR
The paper addresses the challenge of giving a unified, category-theoretic semantics to the protocol-complex approach for round-based distributed systems. It models protocol states and one-round transitions as functors between chromatic augmented semi-simplicial sets, shows that these protocol complex functors arise as Yoneda extensions, and obtains natural $F$-algebras and free $F$-algebras to support iterated protocols. It then develops a temporal-epistemic logic whose semantics live on free algebras over chromatic simplicial sets, provides axioms and task-specification examples, and discusses extensions to dynamic networks and robotics. The framework yields a modular, compositional avenue for specifying and verifying distributed computations, with potential applications to robotics and broader topological invariants.
Abstract
In this article, we show that the now classical protocol complex approach to distributed task solvability of Herlihy et al. can be understood in standard categorical terms. First, protocol complexes are functors, from chromatic (semi-) simplicial sets to chromatic simplicial sets, that naturally give rise to algebras. These algebras describe the next state operator for the corresponding distributed systems. This is constructed for semi-synchronous distributed systems with general patterns of communication for which we show that these functors are always Yoneda extensions of simpler functors, implying a number of interesting properties. Furthermore, for these protocol complex functors, we prove the existence of a free algebra on any initial chromatic simplicial complex, modeling iterated protocol complexes. Under this categorical formalization, protocol complexes are seen as transition systems, where states are structured as chromatic simplicial sets. We exploit the epistemic interpretation of chromatic simplicial sets and the underlying transition system (or algebra) structure to introduce a temporal-epistemic logic and its semantics on all free algebras on chromatic simplicial sets. We end up by giving hints on how to extend this framework to more general dynamic network graphs and state-dependent protocols, and give example in fault-tolerant distributed systems and mobile robotics.