Up-to Techniques for Branching Bisimilarity

Abstract

Ever since the introduction of behavioral equivalences on processes one has been searching for efficient proof techniques that accompany those equivalences. Both strong bisimilarity and weak bisimilarity are accompanied by an arsenal of up-to techniques: enhancements of their proof methods. For branching bisimilarity, these results have not been established yet. We show that a powerful proof technique is sound for branching bisimilarity by combining the three techniques of up to union, up to expansion and up to context for Bloom's BB cool format. We then make an initial proposal for casting the correctness proof of the up to context technique in an abstract coalgebraic setting, covering branching but also η, delay and weak bisimilarity. ⋆ The research of the second author was supported by a Marie Curie Fellowship (grant code 795119). An extended abstract has been published in the proceedings of SOF-SEM 2020, see https://doi.

Divergence-Preserving Branching Bisimilarity

“…Ad hoc up-to techniques for divergence-preserving branching bisimilarity have already been used, e.g., in the congruence proof in [12] and in proof that the π-calculus is behaviourally complete [20]. Recently, several more generic up-to techniques for branching bisimilarity were proved sound [9]. An interesting direction for future work would be to consider extending those up-to techniques for divergencepreserving branching bisimilarity too.…”

Combining Semilattices and Semimodules

“…Remark 25. By comparing (14) and (13) with (4) and (5) in [25], it is immediate to see that our monadP coincides with a slight variation of Jacobs's convex powerset monad C, the only difference being that we do allow for ∅ to be in P a c X. Jacobs insisted on the necessity of C(X) to be the set of non-empty convex subsets of X, because otherwise he was not able to define a semimodule structure on C(X) such that 0 • ∅ = {0 a }. However, we do manage to do so, since by (13), 0 • A = 0 a for all A and in particular for A = ∅.…”

“…Consider for instance automata theory: deterministic automata can be conveniently regarded as certain kind of coalgebras on Set [33], nondeterministic automata as the same kind of coalgebras but on EM(P f ) [35], and weighted automata on EM(S) [4]. Here, P f is the finite powerset monad, modelling nondeterministic computations, while S is the monad of semimodules over a semiring S, modelling various sorts of quantitative aspects when varying the underlying semiring S. It is worth mentioning two facts: first, rather than taking coalgebras over EM(T ), the category of algebras for the monad T , one can also consider coalgebras over Kl(T ), the Kleisli category induced by T [20]; second, these two approaches based on monads have lead not only to a deeper understanding of the subject, but also to effective proof techniques [6,7,14], algorithms [1,8,22,36,39] and logics [19,21,27].…”

Combining Semilattices and Semimodules

“…Consider for instance automata theory: deterministic automata can be conveniently regarded as certain kind of coalgebras on Set [33], nondeterministic automata as the same kind of coalgebras but on EM(P f ) [35], and weighted automata on EM(S) [4]. Here, P f is the finite powerset monad, modelling nondeterministic computations, while S is the monad of semimodules over a semiring S, modelling various sorts of quantitative aspects when varying the underlying semiring S. It is worth mentioning two facts: first, rather than taking coalgebras over EM(T ), the category of algebras for the monad T , one can also consider coalgebras over Kl(T ), the Kleisli category induced by T [20]; second, these two approaches based on monads have lead not only to a deeper understanding of the subject, but also to effective proof techniques [6,7,14], algorithms [1,8,22,36,39] and logics [19,21,27].…”