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 categoryPTNetsof PT nets, we introduce a categoryDccOccof decorated (nondelerministic) occurrence nets and we define adjunctions betweenPTNetsandDecOccand betweenDecOccandOcc, the category of occurrence nets. The role ofDecOccis 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 fromPTNetstoOccreduces to Winskel's when restricted to safe nets, while the standard coreflection betweenOccandDom, the category of finitary prime algebraic domains, when composed with the unfolding functor above, determines a chain of adjunctions betweenPTNetsandDom.
KeywordsFull Subcategory Monoidal Category Left Adjoint Forgetful Functor Place Component
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