Bisimulations and Modal Logics for Higher Dimensional Automata

2026-08-14

AI Summary

This study addresses the absence of modal logics characterizing hereditary history-preserving (hhp) bisimulation in high-dimensional automata under true concurrency models. We propose a novel intermediate equivalence relation situated between ST and hhp bisimulation. By decoupling path similarity from inclusion, we construct a modal logic framework that precisely characterizes hhp bisimulation and its fragments. This work achieves the first complete logical characterization of hhp bisimulation via modal logic, thereby bridging a significant gap in behavioral equivalence theory within true concurrency semantics. Consequently, this research establishes a new logical foundation and theoretical support for the formal verification of high-dimensional automata, advancing the understanding of concurrent system equivalences through rigorous logical formalization.

Abstract

Higher-Dimensional Automata (HDAs) provide a geometric model of true concurrency. While hereditary history-preserving (hhp) bisimilarity is the finest behavioural equivalence in van Glabbeek's spectrum, no modal logic has previously characterised it on HDAs. We introduce several new intermediate equivalences that sit strictly between ST- and hhp-bisimilarity. We show how separating similarity and subsumption of paths leads to a clean formulation of these equivalences, and we present a modal logic that characterises hhp-bisimilarity. Natural fragments characterise ST-bisimilarity and the intermediate notions.

Problem

• Higher-Dimensional Automata
• hereditary history-preserving bisimilarity
• modal logic
• true concurrency
• behavioural equivalence

Innovation

• Higher-Dimensional Automata
• hereditary history-preserving bisimilarity
• modal logic
• intermediate equivalences
• true concurrency