Computation Tree Logic with Deadlock Detection

TL;DR

This paper investigates the limitations of CTL-X in detecting deadlocks within non-total Kripke structures and proposes an extension to improve expressiveness and congruence properties.

Contribution

It introduces an extension of CTL-X that captures deadlock properties and ensures congruence with parallel composition, addressing previous limitations.

Findings

The existing equivalence is not a congruence for parallel composition.

CTL-X lacks expressiveness for deadlock detection in non-total structures.

The proposed extension achieves the coarsest congruence with divergence sensitivity.

Abstract

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.