Concurrencies in Reversible Concurrent Calculi

The algebraic specification and representation of networks of agents have been greatly impacted by the study of reversible phenomena: reversible declensions of the calculus of communicating systems (CCSK and RCCS) offer new semantic models, finer congruence relations, original properties, and revisits existing theories and results in a finer light. But much remains to be done: concurrency, a central notion in establishing causal consistency–a crucial property for reversible systems–, was never given a syntactical definition in CCSK. We remedy this gap by leveraging a definition of concurrency developed for forward-only calculi using proved transition systems, and prove that CCSK still enjoys causal consistency for this elegant and syntactical notion of reversible concurrency. We also compare it to a definition of concurrency inspired by reversible π -calculus, discuss its relation with structural congruence, and prove that it can be adapted to any CCS-inspired reversible system and is equivalent—or refines—existing definitions of concurrency for those systems.