A Logic with Reverse Modalities for History-preserving Bisimulations

We introduce event identifier logic (EIL) which extends Henn essy-Milner logic by the addition of (1) reverse as well as forward modalities, and (2) identifier s to keep track of events. We show that this logic corresponds to hereditary history-preserv ing (HH) bisimulation equivalence within a particular true-concurrency model, namely stable configu ration structures. We furthermore show how natural sublogics of EIL correspond to coarser equivalences. In particular we provide logical characterisations of weak history-preserving (WH) and history-preserving (H) bisimulation. Logics corresponding to HH and H bisimulation have been given previously, but not to WH bisimulation (when autoconcurrency is allowed), as far as we are aware. We also present characteristic formulas which characterise individual structures with respect to h istory-preserving equivalences. The paper presents a modal logic that can express simple properties of computation in the true concurrency setting of stable configuration structures. We aim, li ke Hennessy-Milner logic (HML) [19] in the interleaving setting, to characterise the main true concur rency equivalences and to develop characteristic formulas for them. We focus in this paper on history-preserving bisimulation equivalences. HML has a “diamond” modality haiφ which says that an event labelled a can be performed, taking us to a new state which satisfies φ . The logic also contains negation (¬), conjunction (∧) and a base formula which always holds (tt). HML is strong enough to distinguish any two processes which are not bisimilar. We are interested in making true concurrency distinctions between processes. These processes will be event structures, where the current state is represented by the set of events w hich have occurred so far. Such sets are called configurations. Events have labels (ranged over by a, b,...), and different events may have the same label. We shall refer to example event structures using a CCS-like notation, with a| b denoting an event labelled with a in parallel with another labelled with b, a.b denoting two events labelled a and b where the first causes the second, and a+ b denoting two events labelled a and b which conflict.