An Axiomatization of the Category of Petri Net Computations

We introduce the notion of strongly concatenable process as a refinement of concatenable processes which can be expressed axiomatically via a functor Qn(_) from the category of Petri nets to an appropriate category of symmetric strict monoidal categories, in the precise sense that, for each net N, the strongly concatenable processes of N are isomorphic to the arrows of Qn(N). In addition, we identify a coreflection right adjoint to Qn(_) and characterize its replete image, thus yielding an axiomatization of the category of net computations

Algebraic Models for Contextual Nets

We extend the algebraic approach of Meseguer and Montanari from ordinary place/transition Petri nets to contextual nets, covering both the collective and the individual token philosophy uniformly along the two interpretations of net behaviors

Functorial Semantics for Petri Nets under the Individual Token Philosophy

Although the algebraic semantics of place/transition Petri nets under the collective token philosophy has been fully explained in terms of (strictly) symmetric (strict) monoidal categories, the analogous construction under the individual token philosophy is not completely satisfactory because it lacks universality and also functoriality. We introduce the notion of pre-net to recover these aspects, obtaining a fully satisfactory categorical treatment centered on the notion of adjunction. This allows us to present a purely logical description of net behaviours under the individual token philosophy in terms of theories and theory morphisms in partial membership equational logic, yielding a complete match with the theory developed by the authors for the collective token view of net

Two Algebraic Process Semantics for Contextual Nets

We show that the so-called 'Petri nets are monoids' approach initiated by Meseguer and Montanari can be extended from ordinary place/transition Petri nets to contextual nets by considering suitable non-free monoids of places. The algebraic characterizations of net concurrent computations we provide cover both the collective and the individual token philosophy, uniformly along the two interpretations, and coincide with the classical proposals for place/transition Petri nets in the absence of read-arcs

Petri Nets and Other Models of Concurrency

This paper retraces, collects, and summarises contributions of the authors --- in collaboration with others --- on the theme of Petri nets and their categorical relationships to other models of concurrency

Elements of Petri nets and processes

We present a formalism for Petri nets based on polynomial-style finite-set configurations and etale maps. The formalism supports both a geometric semantics in the style of Goltz and Reisig (processes are etale maps from graphs) and an algebraic semantics in terms of free coloured props: the Segal space of P-processes is shown to be the free coloured prop-in-groupoids on P. There is also an unfolding semantics \`a la Winskel, which bypasses the classical symmetry problems. Since everything is encoded with explicit sets, Petri nets and their processes have elements. In particular, individual-token semantics is native, and the benefits of pre-nets in this respect can be obtained without the need of numberings. (Collective-token semantics emerges from rather drastic quotient constructions \`a la Best--Devillers, involving taking $\pi_0$ of the groupoids of states.)Comment: 44 pages. The math is intended to be in reasonably final form, but the paper may well contain some misconceptions regarding the place of this material in the theory of Petri nets. All feedback and help will be greatly appreciated. v2: fixed a mistake in Section