Coalgebraic Weak Bisimulation from Recursive Equations over Monads

Abstract

Abstract. Strong bisimulation for labelled transition systems is one of the most fundamental equivalences in process algebra, and has been generalised to numerous classes of systems that exhibit richer transition behaviour. Nearly all of the ensuing notions are instances of the more general notion of coalgebraic bisimulation. Weak bisimulation, however, has so far been much less amenable to a coalgebraic treatment.Here we attempt to close this gap by giving a coalgebraic treatment of (parametrized) weak equivalences, including weak bisimulation. Our analysis requires that the functor defining the transition type of the system is based on a suitable order-enriched monad, which allows us to capture weak equivalences by least fixpoints of recursive equations. Our notion is in agreement with existing notions of weak bisimulations for labelled transition systems, probabilistic and weighted systems, and simple Segala systems.

Behavioural equivalences for coalgebras with unobservable moves

“…weak) behavioural morphisms on single systems only. The restriction to single systems aligns us to other recent works on weak bisimulations [10,18]. We do this without loss of generality as all results presented in this paper readily extend to the case of multiple systems.…”

“…To this end, we provide an equivalent characterisation based on least solutions to (PS). In fact, as we illustrate in Section 4, weak bisimulations (e.g., [5,31,18]) are based on recursive equations subsumed 3 by (PS).…”

“…Convex combinations monad(s) The convex combinations monad was first introduced in its full generality by Jacobs in [22] to study trace semantics for combined possibilistic and probabilistic systems. Independently, Brengos [10] and Goncharov and Pattison [18] have tweaked Jacobs' construction slightly, so that the resulting monads are more suitable to model Segala systems and their weak bisimulations. Jacobs' monad, Brengos' monad and Goncharov-Pattison's monad form Kleisli categories which are ω-Cpoenriched and whose hom-sets admit binary joins.…”

“…Consider the endofunctor F W (A τ × Id) ∼ = F W (A × Id + Id) modelling Weighted LTSs with unobservable actions [18,31]. The lifting of A τ × Id is given, on any f ∈ Kl(F W )(X, Y ), as:…”

A general account of coinduction up-to

“…We leave as future work to integrate in our framework the coalgebraic treatment of weak bisimilarity, developed for example in [13,14,21] for systems modelled as colagebras in an order-enriched setting. Thus, we expect to extend our results to encompass fully probabilistic and Segala models [49,50].…”