Contextual Petri Nets, Asymmetric Event Structures, and Processes

Abstract

We present an event structure semantics for contextual nets, an extension of P/T Petri nets where transitions can check for the presence of tokens without consuming them (read-only operations). A basic rôle is played by asymmetric event structures, a generalization of Winskel's prime event structures where symmetric conflict is replaced by a relation modelling asymmetric conflict or weak causality, used to represent a new kind of dependency between events arising in contextual nets. Extending Winskel's seminal work on safe nets, the truly concurrent event based semantics of contextual nets is given at categorical level via a chain of coreflections leading from the category SW-CN of semi-weighted contextual nets to the category Dom of finitary prime algebraic domains. First an unfolding construction generates from a contextual net a corresponding occurrence contextual net, from where an asymmetric event structure is extracted. Then the configurations of the asymmetric event structure, endowed with a suitable order, are shown to form a finitary prime algebraic domain. We also investigate the relation between the proposed unfolding semantics and several deterministic process semantics for contextual nets in the literature. In particular, the domain obtained via the unfolding is characterized as the collection of the deterministic processes of the net endowed with a kind of prefix ordering.

Models of Computation: A Tribute to Ugo Montanari’s Vision

“…The price to pay was the introduction of more complex event structures [3,28,2,4,5]. Significantly related is also [1].…”

Domain and event structure semantics for Petri nets with read and inhibitor arcs*1

“…The proof of Condition (i) in the definition of enabling (see Definition 2), i.e., • µh T (A) + µh T (A) ≤ µh S (M), is essentially the same as for ordinary nets, adapted to take into account also the read arcs (see [6] for details). As for Condition (ii), which involves the inhibiting places, notice that…”

“…Then, disregarding the inhibitor arcs, we apply to N c the unfolding construction for contextual nets defined in [6], which produces an occurrence contextual net U a (N c ). Finally, if a place s and a transition t were originally connected by an inhibitor arc in the net N, then we insert an inhibitor arc between each occurrence of s and each occurrence of t in U a (N c ), thus obtaining the unfolding U i (N) of the net N. Then the characterisation of the unfolding as a universal construction can be lifted from contextual to inhibitor nets.…”

“…Furthermore we give some intuition on how such results have been extended in [6] to structures with asymmetric conflict. These notions and results will be useful later in the treatment of inhibitor event structures.…”

“…Asymmetric event structures have been introduced in [6] as a generalisation of prime event structures where the conflict relation is allowed to be non-symmetric. Formally, an asymmetric event structure (AES) is a triple G = E, ≤, ր , where E is a set of events, ≤ is the causality relation and ր is a binary relation on E called asymmetric conflict.…”