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

A Comparison of Petri Net Semantics under the Collective Token Philosophy

In recent years, several semantics for place/transition Petri nets have been proposed that adopt the collective token philosophy. We investigate distinctions and similarities between three such models, namely configuration structures, concurrent transition systems, and (strictly) symmetric (strict) monoidal categories. We use the notion of adjunction to express each connection. We also present a purely logical description of the collective token interpretation of net behaviours in terms of theories and theory morphisms in partial membership equational logic

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

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

Zero-Safe Nets, or Transition Synchronization Made Simple

Abstract In addition to ordinary places, called stable, zero-safe nets are equipped with zero places, which in a stable marking cannot contain any token. An evolution between two stable markings, instead, can be a complex computation called stable transaction, which may use zero places, but which is atomic when seen from stable places: no stable token generated in a transaction can be reused in the same transaction. Every zero-safe net has an ordinary Place-Transition net as its abstract counterpart, where only stable places are maintained, and where every transaction becomes a transition. The two nets allow us to look at the same system from both an abstract and a refined viewpoint. To achieve this result no new interaction mechanism is used, besides the ordinary token-pushing rules of nets. The refined zero-safe nets can be much smaller than their corresponding abstract P/T nets, since they take advantage of a transition synchronization mechanism. For instance, when transactions of unlimited length are possible in a zero safe net, the abstract net becomes infinite, even if the refined net is finite. In the second part of the paper two universal constructions - both following the Petri nets are monoids approach and the collective token philosophy - are used to give evidence of the naturality of our definitions. More precisely, the operational semantics of zero-safe nets is characterized as an adjunction, and the derivation of abstract P/T nets as a coreflection

Formalization of Petri Nets with Individual Tokens as Basis for DPO Net Transformations

Reconfigurable place/transition systems are Petri nets with initial markings and a set of rules which allow the modification of the net structure during runtime. They have been successfully used in different areas like mobile ad-hoc networks. In most of these applications the modification of net markings during runtime is an important issue. This requires the analysis of the interaction between firing and rule-based modification. For place/transition systems this analysis has been started explicitly without using the general theory of M-adhesive transformation systems, because firing cannot be expressed by rule-based transformations for P/T systems in this framework. This problem is solved in this paper using the new approach of P/T nets with individual tokens. In our main results we show that on one hand this new approach allows to express firing by transformation via suitable transition rules. On the other hand transformations of P/T nets with individual tokens can be shown to be an instance ofM-adhesive transformation systems, such that several well-known results, like the local Church-Rosser theorem, can be applied. This avoids a separate conflict analysis of token firing and transformations. Moreover, we compare the behavior of P/T nets with individual tokens with that of classical P/T nets. Our new approach is also motivated and demonstrated by a network scenario modeling a distributed communication system

Configuration structures, event structures and Petri nets

In this paper the correspondence between safe Petri nets and event structures, due to Nielsen, Plotkin and Winskel, is extended to arbitrary nets without self-loops, under the collective token interpretation. To this end we propose a more general form of event structure, matching the expressive power of such nets. These new event structures and nets are connected by relating both notions with configuration structures, which can be regarded as representations of either event structures or nets that capture their behaviour in terms of action occurrences and the causal relationships between them, but abstract from any auxiliary structure. A configuration structure can also be considered logically, as a class of propositional models, or—equivalently— as a propositional theory in disjunctive normal from. Converting this theory to conjunctive normal form is the ke