Recently there has been a growing interest towards algebraic structures that are able to express formalisms different from the standard, tree-like presentation of terms. Many of these approaches reveal a specific interest towards their application in the "distributed and concurrent systems" field, but an exhaustive comparison between them is difficult because their presentations can be quite dissimilar. This work is a first step towards a unified view, which is able to recast all those formalisms into a more general one, where they can be easily compared. We introduce a general schema for describing a characteristic normal form for many algebraic formalisms, and show that those normal forms can be thought of as arrows of suitable concrete monoidal categories.

When employing Petri nets to model distributed systems, one must be aware that the basic activities of each component can vary in duration and can involve smaller internal activities, i.e., that transitions are conceptually refined into transactions. We present an approach to the modeling of transactions based on zero-safe nets. They extend ordinary pt nets with a simple mechanism for transition synchronization. We show that the net theory developed under the two most diffused semantic interpretations (collective token and individual token philosophies) can be uniformly adapted to zero-safe nets. In particular, we show that each zero-safe net has associated two pt nets which represent the abstract counterparts of the modeled system according to the two philosophies. We show several applications of the framework, a distributed interpreter for zs nets based on classical net unfolding (here extended with a commit rule) and discuss some extensions to other net flavours.

We propose rewriting logic as a unifying framework for a wide range of Petri nets models. We treat in detail place/transition nets and important extensions of the basic model by individual tokens, test arcs, and time. Based on the idea that "Petri nets are monoids" suggested by Meseguer and Montanari we define a rewriting semantics that maps place/transition nets into rewriting logic specifications. We furthermore generalize this result to a general form of algebraic net specifications subsuming also colored Petri nets as a special case. The soundness and completeness results we state relate the commutative process semantics of Petri nets proposed by Best and Devillers to the model-theoretic semantics of rewriting logic in the sense of natural isomorphisms between suitable functors. In addition we show how place/transition nets with test arcs and timed Petri nets can be equipped with a rewriting semantics and discuss how other extensions can be treated along similar lines.

Distributed systems are often composed by many heterogeneous agents that can work concurrently and exchange information. Therefore, in their modeling via pt nets we must be aware that the basic activities of each system can vary in duration and can be constituted by smaller internal activities, i.e., transitions are conceptually refined into transactions. We address the issue of modeling transactions in distributed systems by using zero-safe nets, which extend pt nets with a simple mechanism for transition synchronization. In particular, starting from the zero-safe net that represents a certain system, we give a distributed algorithm for executing the transactions of the system as transitions of a more abstract pt net. Among the advantages of our approach, we emphasize that the zero-safe net can be much smaller than its abstract counterpart, due to the synchronization mechanism.

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.

Zero-safe nets have been introduced to extend classical Petri nets with a primitive notion of transition synchronization. To this aim, besides ordinary places, called stable, zero-safe nets are equipped with zero places, which cannot contain any token in a stable marking . An evolution between two stable markings is called transaction and can be a complex computation that involves zero places, with the restriction that no stable token generated in a transaction can be reused in the same transaction. The abstract counterpart of a generic zerosafe net B consists of an ordinary pt net whose places are the stable places of B, and whose transitions are the transactions of B. The two nets offer the refined and the abstract model of the same system, where the former can be much smaller than the latter, because of the transition synchronization mechanism. Depending on the chosen approach --- collective vs individual token philosophy --- two notions of transaction may be defined, each leading ...

This paper retraces, collects, and summarises the contributions of the author — both individually and in collaboration with others — on the theme of algebraic, compositional approaches to the semantics of Petri nets.

This paper is concerned with the asymptotic properties of a restricted class of Petri nets equipped with stochastic mass action semantics. We establish a simple algebraic criterion for the existence of an equilibrium, that is to say an invariant probability that satisfies the detailed balance condition familiar from the thermodynamics of reaction networks. We also find that when such a probability exists, it can be described by a free energy function which combines an internal energy term and an entropy one. Under strong additional conditions, we show how the entropy term can be deconstructed using the finer-grained individual-token semantics of Petri nets. 1.