Multiset Bisimulations as a Common Framework for Ordinary and Probabilistic Bisimulations
David de Frutos-Escrig, Miguel Palomino, Ignacio Fábregas
TL;DR
The paper addresses unifying ordinary and probabilistic bisimulations within a single coalgebraic framework based on multisets. It develops a general theory where natural transformations between functors map F-bisimulations to G-bisimulations, and, when the transformation is epi, allows G-bisimulations to be reflected as simulations of F-representations, via kernel-based quotients. The authors extend the framework to arbitrary orders on functors and to alternating probabilistic systems, providing a cohesive treatment of nondeterminism and probability within multiset coalgebras. This approach yields a concise, general theory that subsumes traditional bisimulations as a special case of categorical simulations and enables near-trivial proofs and broad applicability to diverse transition-system models.
Abstract
Our concrete objective is to present both ordinary bisimulations and probabilistic bisimulations in a common coalgebraic framework based on multiset bisimulations. For that we show how to relate the underlying powerset and probabilistic distributions functors with the multiset functor by means of adequate natural transformations. This leads us to the general topic that we investigate in the paper: a natural transformation from a functor F to another G transforms F-bisimulations into G-bisimulations but, in general, it is not possible to express G-bisimulations in terms of F-bisimulations. However, they can be characterized by considering Hughes and Jacobs' notion of simulation, taking as the order on the functor F the equivalence induced by the epi-mono decomposition of the natural transformation relating F and G. We also consider the case of alternating probabilistic systems where non-deterministic and probabilistic choices are mixed, although only in a partial way, and extend all these results to categorical simulations.