The Glory of the Past and Geometrical Concurrency

This paper contributes to the general understanding of the "geometrical model of concurrency" that was named higher dimensional automata (HDAs) by Pratt and van Glabbeek. In particular we provide some understanding of the modal logics for such models and their expressive power in terms of the bisimulation that can be captured.The geometric model of concurrency is interesting from two main reasons: its generality and expressiveness, and the natural way in which autoconcurrency and action refinement are captured.Logics for this model, though, are not well investigated, where a simple, yet adequate, modal logic over HDAs was only recently introduced.As this modal logic, with two existential modalities, "during" and "after", captures only split bisimulation, which is rather low in the spectrum of van Glabbeek and Vaandrager, the immediate question was what small extension of this logic could capture the more fine-grained hereditary history preserving bisimulation (hh)?In response, the work in this paper provides several insights. One is the fact that the geometrical aspect of HDAs makes it possible to use for capturing the hh-bisimulation, a standard modal logic that does not employ event variables, opposed to the two logics (over less expressive models) that we compare with. The logic that we investigate here uses standard backward-looking modalities (i.e., past modalities) and extends the previously introduced logic (called HDML) that had only forward, action-labelled, modalities.Since the direct proofs are rather intricate, we try to understand better the above issues by introducing a related model that we call ST-configuration structures, which extend the configuration structures of van Glabbeek and Plotkin. We relate this model to HDAs, and redefine and prove the earlier results in the light of this new model. These offer a different view on why the past modalities and geometrical concurrency capture the hereditary history preserving bisimulation.Additional correlating insights are also gained.