A Logic with Reverse Modalities for History-preserving Bisimulations

We introduce event identifier logic (EIL) which extends Hennessy-Milner logic by the addition of (1) reverse as well as forward modalities, and (2) identifiers to keep track of events. We show that this logic corresponds to hereditary history-preserving (HH) bisimulation equivalence within a particular true-concurrency model, namely stable configuration 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 history-preserving equivalences.

💡 Research Summary

The paper introduces Event Identifier Logic (EIL), a modal logic that extends Hennessy‑Milner Logic (HML) with two orthogonal features: (1) reverse modalities that allow stepping back along a maximal event, and (2) explicit event identifiers that can be bound to events when they are created. The authors work within the framework of stable configuration structures, a true‑concurrency model where a global state is represented by a finite set of events that have occurred so far, together with a causality partial order and a conflict relation.

In a stable configuration structure, a forward transition X ─a→ X′ adds a single maximal event e labelled a (i.e., X′ = X ∪ {e}), while a reverse transition X ─h a→ X′ removes a maximal event e labelled a (i.e., X = X′ ∪ {e}). The logic’s syntax includes forward diamonds ⟨i a⟩φ, reverse diamonds ⟨h a⟩φ, and declaration forms (x:a) φ that bind a fresh identifier x to the newly created event. The semantics evaluates formulas relative to a configuration X and an environment ρ that maps identifiers to events currently present in X. A forward declaration (x:a) φ succeeds exactly when there exists a fresh maximal a‑event e that can be added; the environment is extended with ρ