Essential transitions to bisimulation equivalences

Abstract

Minimization of a labelled transition system (Its) is useful e.g. while condensing the global state space of a concurrent system compositionally for verification. In this paper new minimality results for both weak and branching bisimilarities are proven. It is well known that an equivalent Its with the minimal number of states can be, in the case of bisimilarities, found by identifying all equivalent states of an Its. However, the question has been partially open whether an equivalent Its with the minimal number of states and transitions can be found. We give a proof that for every weak-image-finite Its there is a unique bisimilar Its that contains the minimal number of states and transitions. We study divergence preserving bisimilarities, since divergence — i.e. the possibility to use system resources infinitely without any output — should not be ignored when liveness properties of systems have to be checked. Our results are shown to be valid also for commonly used divergence ignoring bisimilarities, weak bisimilarity and branching bisimilarity.