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 coalgebraic framework developed for the classical process algebras, and in particular its advantages concerning minimal realizations, does not fully apply to the π-calculus, due to the constraints on the freshly generated names that appear in the bisimulation.In this paper we propose to model the transition system of the π-calculus as a coalgebra on a category of name permutation algebras and to define its abstract semantics as the final coalgebra of such a category. We show that permutations are sufficient to represent in an explicit way fresh name generation, thus allowing for the definition of minimal realizations.We also link the coalgebraic semantics with a slightly improved version of history dependent (HD) automata, a model developed for verification purposes, where states have local names and transitions are decorated with names and name relations. HD-automata associated with agents with a bounded number of threads in their derivatives are finite and can be actually minimized. We show that the bisimulation relation in the coalgebraic context corresponds to the minimal HD-antomaton. In this paper we propose a new approach to check history-preserving equivalence for Petri nets. Exploiting this approach, history-preserving bisimulation is proved decidable for the class of finite nets which are n-safe for some n (the approaches of [17] and of [8] work just for 1-safe nets). Moreover, since we map nets on ordinary transition systems, standard results and algorithms can be re-used, yielding for instance the possibility of deriving minimal realizations. The proposed approach can be applied also to other concurrent formalisms based on partial order semantics, like CCS with causality [4].  The semantics of name-passing calculi is often defined employing coalgebraic models over presheaf categories. This elegant theory lacks finiteness properties, hence it is not apt to implementation. Coalgebras over named sets, called history-dependent automata, are better suited for the purpose due to locality of names. A theory of behavioural functors for named sets is still lacking: the semantics of each language has been given in an ad-hoc way, and algorithms were implemented only for the @p-calculus. Existence of the final coalgebra for the @p-calculus was never proved. We introduce a language of accessible functors to specify history-dependent automata in a modular way, leading to a clean formulation and a generalisation of previous results, and to the proof of existence of a final coalgebra in a wide range of cases.