Computation Tree Logic with Deadlock Detection

We study the equivalence relation on states of labelled transition systems of satisfying the same formulas in Computation Tree Logic without the next state modality (CTL-X). This relation is obtained by De Nicola & Vaandrager by translating labelled transition systems to Kripke structures, while lifting the totality restriction on the latter. They characterised it as divergence sensitive branching bisimulation equivalence. We find that this equivalence fails to be a congruence for interleaving parallel composition. The reason is that the proposed application of CTL-X to non-total Kripke structures lacks the expressiveness to cope with deadlock properties that are important in the context of parallel composition. We propose an extension of CTL-X, or an alternative treatment of non-totality, that fills this hiatus. The equivalence induced by our extension is characterised as branching bisimulation equivalence with explicit divergence, which is, moreover, shown to be the coarsest congruence contained in divergence sensitive branching bisimulation equivalence

Branching Bisimilarity with Explicit Divergence

We consider the relational characterisation of branching bisimilarity with explicit divergence. We prove that it is an equivalence and that it coincides with the original definition of branching bisimilarity with explicit divergence in terms of coloured traces. We also establish a correspondence with several variants of an action-based modal logic with until- and divergence modalities