Process versus Unfolding Semantics for Place/Transition Petri Nets

In the last few years, the semantics of Petri nets has been investigated in several different ways. Apart from the classical "token game," one can model the behaviour of Petri nets via non-sequential processes, via unfolding constructions, which provide formal relationships between nets and domains, and via algebraic models, which view Petri nets as essentially algebraic theories whose models are monoidal categories. In this paper we show that these three points of view can be reconciled. In our formal development a relevant role is played by DecOcc, a category of occurrence nets appropriately decorated to take into account the history of tokens. The structure of decorated occurrence nets at the same time provides natural unfoldings for Place/Transition (PT) nets and suggests a new notion of processes, the decorated processes, which induce on Petri nets the same semantics as that of unfolding. In addition, we prove that the decorated processes of a net can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification

On the Semantics of Petri Nets

Petri Place/Transition (PT) nets are one of the most widely used models of concurrency. However, they still lack, in our view, a satisfactory semantics: on the one hand the "token game"' is too intensional, even in its more abstract interpretations in term of nonsequential processes and monoidal categories; on the other hand, Winskel's basic unfolding construction, which provides a coreflection between nets and finitary prime algebraic domains, works only for safe nets. In this paper we extend Winskel's result to PT nets. We start with a rather general category {PTNets} of PT nets, we introduce a category {DecOcc} of decorated (nondeterministic) occurrence nets and we define adjunctions between {PTNets} and {DecOcc} and between {DecOcc} and {Occ}, the category of occurrence nets. The role of {DecOcc} is to provide natural unfoldings for PT nets, i.e. acyclic safe nets where a notion of family is used for relating multiple instances of the same place. The unfolding functor from {PTNets} to {Occ} reduces to Winskel's when restricted to safe nets, while the standard coreflection between {Occ} and {Dom}, the category of finitary prime algebraic domains, when composed with the unfolding functor above, determines a chain of adjunctions between {PTNets} and {Dom}

On the Semantics of Place/Transition Petri Nets

Place/Transition (PT) Petri nets are one of the most widely used models of concurrency. However, they still lack, in our view, a satisfactory semantics: on the one hand the 'token game' is too intensional, even in its more abstract interpretations in terms of nonsequential processes and monoidal categories; on the other hand, Winskel's basic unfolding construction, which provides a coreflection between nets and finitary prime algebraic domains, works only for safe nets. In this paper we extend Winskel's result to PT nets. We start with a rather general category PTNets of PT nets, we introduce a category DecOcc of decorated (nondeterministic) occurrence nets and we define adjunctions between PTNets and DecOcc and between DecOcc and Occ, the category of occurrence nets. The role of DecOcc is to provide natural unfoldings for PT nets, i.e., acyclic safe nets where a notion of family is used for relating multiple instances of the same place. The unfolding functor from PTNets to Occ reduces to Winskel's when restricted to safe nets; moreover, the standard coreflection between Occ and Dom, the category of finitary prime algebraic domains, when composed with the unfolding functor above, determines a chain of adjunctions between PTNets and Dom

On the Semantics of Place/Transition Petri Nets

Place/Transition (PT) Petri nets are one of the most widely used models of concurrency. However, they still lack, in our view, a satisfactory semantics: on the one hand the 'token game' is too intensional, even in its more abstract interpretations in terms of nonsequential processes and monoidal categories; on the other hand, Winskel's basic unfolding construction, which provides a coreflection between nets and finitary prime algebraic domains, works only for safe nets. In this paper we extend Winskel's result to PT nets. We start with a rather general category PTNets of PT nets, we introduce a category DecOcc of decorated (nondeterministic) occurrence nets and we define adjunctions between PTNets and DecOcc and between DecOcc and Occ, the category of occurrence nets. The role of DecOcc is to provide natural unfoldings for PT nets, i.e., acyclic safe nets where a notion of family is used for relating multiple instances of the same place. The unfolding functor from PTNets to Occ reduces to Winskel's when restricted to safe nets; moreover, the standard coreflection between Occ and Dom, the category of finitary prime algebraic domains, when composed with the unfolding functor above, determines a chain of adjunctions between PTNets and Dom

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

Distributed Non-Interference

Information flow security properties were defined some years ago (see, e.g., the surveys \cite{FG01,Ry01}) in terms of suitable equivalence checking problems. These definitions were provided by using sequential models of computations (e.g., labeled transition systems \cite{GV15}), and interleaving behavioral equivalences (e.g., bisimulation equivalence \cite{Mil89}). More recently, the distributed model of Petri nets has been used to study non-interference in \cite{BG03,BG09,BC15}, but also in these papers an interleaving semantics was used. We argue that in order to capture all the relevant information flows, truly-concurrent behavioral equivalences must be used. In particular, we propose for Petri nets the distributed non-interference property, called DNI, based on {\em branching place bisimilarity} \cite{Gor21b}, which is a sensible, decidable equivalence for finite Petri nets with silent moves. Then we focus our attention on the subclass of Petri nets called {\em finite-state machines}, which can be represented (up to isomorphism) by the simple process algebra CFM \cite{Gor17}. DNI is very easily checkable on CFM processes, as it is compositional, so that it does does not suffer from the state-space explosion problem. Moreover, we show that DNI can be characterized syntactically on CFM by means of a type system

A Process Calculus for Expressing Finite Place/Transition Petri Nets

We introduce the process calculus Multi-CCS, which extends conservatively CCS with an operator of strong prefixing able to model atomic sequences of actions as well as multiparty synchronization. Multi-CCS is equipped with a labeled transition system semantics, which makes use of a minimal structural congruence. Multi-CCS is also equipped with an unsafe P/T Petri net semantics by means of a novel technique. This is the first rich process calculus, including CCS as a subcalculus, which receives a semantics in terms of unsafe, labeled P/T nets. The main result of the paper is that a class of Multi-CCS processes, called finite-net processes, is able to represent all finite (reduced) P/T nets.Comment: In Proceedings EXPRESS'10, arXiv:1011.601