In this paper we revisit some pioneering eorts to equip Petri nets with compact operational models for expressing causality. The models we propose have a bisimilarity relation and a minimal representative for each equivalence class, and they can be fully explained as coalgebras on a presheaf category on an index category of partial orders. First, we provide a set-theoretic model in the form of a a causal case graph, that is a labeled transition system where states and transitions represent markings and rings of the net, respectively, and are equipped with causal information. Most importantly, each state has a poset representing causal dependencies among past events. Our rst result shows the correspondence with behavior structure semantics as proposed by Trakhtenbrot and Rabinovich. Causal case graphs may be innitely-branchi ng and have innitely many states, but we show how they can be rened to get an equivalent nitely-branching model. In it, states only keep the most recent causes for each token, are up to isomorphism, and are equipped with a symmetry, i.e., a group of poset isomorphisms. Symmetries are essential for the existence of a minimal, often nite-state, model. This rst part requires no knowledge of category theory. The next step is constructing a coalgebraic model. We exploit the fact that events can be represented as names, and event generation as name generation. Thus we can apply the Fiore-Turi framework, where the semantics of nominal calculi are modeled as coalgebras over presheaves. We model causal relations as a suitable category of posets with action labels, and generation of new events with causal dependencies as an endofunctor on this category. Presheaves indexed by labeled posets represent the functorial association between states and their causal information. Then we dene a well-behaved category of coalgebras. Our coalgebraic model is still innite-state, but we exploit the equivalence between coalgebras over a class of presheaves and History Dependent automata to derive a compact representation, which is equivalent to our set-theoretical compact model. Remarkably, state reduction is automatically performed along the equivalence.
The category of event structures is known to embed fully and faithfully in the category of presheaves over pomsets. Here a characterisation of the presheaves represented by event structures is presented. The proof goes via a characterisation of the presheaves represented by event structures when the morphisms on event structures are "strict" in that they preserve the partial order of causal dependency. This paper establishes a bridge between presheaf models for concurrency and the more operationally-informative world of event structures. It concentrates on a particular presheaf category, consisting of presheaves over finite partial orders of events; such presheaves form a model of nondeterministic processes in which the computation paths have the shape of partial orders. It is shown how with the introduction of symmetry event structures represent all presheaves over finite partial orders. This is in contrast with plain event structures which only represent certain separated presheaves. Specifically a coreflection from the category of presheaves to the category of event structures with symmetry is exhibited. It is shown how the coreflection can be cut down to an equivalence between the presheaf category and the subcategory of graded event structures with symmetry. Event structures with strong symmetries are shown to represent precisely all the separated presheaves. The broader context and specific applications to the unfolding of higher-dimensional automata and Petri nets, and weak bisimulation on event structures are sketched.