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.
In this paper we recast the classical Darondeau---Degano's causal semantics of concurrency in a coalgebraic setting, where we derive a compact model. Our construction is inspired by the one of Montanari and Pistore yielding causal automata, but we show that it is instance of an existing categorical framework for modeling the semantics of nominal calculi, whose relevance is further demonstrated. The key idea is to represent events as names, and the occurrence of a new event as name generation. We model causal semantics as a coalgebra over a presheaf, along the lines of the Fiore---Turi approach to the semantics of nominal calculi. More specifically, we take a suitable category of finite posets, representing causal relations over events, and we equip it with an endofunctor that allocates new events and relates them to their causes. Presheaves over this category express the relationship between processes and causal relations among the processes' events. We use the allocation operator to define a category of well-behaved coalgebras: it models the occurrence of a new event along each transition. Then we turn the causal transition relation into a coalgebra in this category, where labels only exhibit maximal events with respect to the source states' poset, and we show that its bisimilarity is essentially Darondeau---Degano's strong causal bisimilarity. This coalgebra is still infinite-state, but we exploit the equivalence between coalgebras over a class of presheaves and History Dependent automata to derive a compact representation, where states only retain the poset of the most recent events for each atomic subprocess, and are isomorphic up to order-preserving permutations. Remarkably, this reduction of states is automatically performed along the equivalence.