Building Occurrence Nets from Reveals Relations

Abstract

Occurrence nets are a well known partial order model for the concurrent behavior of Petri nets. The causality and conflict relations between events, which are explicitly represented in occurrence nets, induce logical dependencies between event occurrences: the occurrence of an event e in a run implies that all its causal predecessors also occur, and that no event in conflict with e occurs. But these structural relations do not express all the logical dependencies between event occurrences in maximal runs: in particular, the occurrence of e in any maximal run may imply the occurrence of another event that is not a causal predecessor of e, in that run. The reveals relation has been introduced to express this dependency between two events. Here we generalize the reveals relation to express more general dependencies, involving more than two events, and we introduce ERL logic to express them as boolean formulas. Finally we answer the synthesis problem that arises: given an ERL formula ϕ, is there an occurrence net N such that ϕ describes exactly the dependencies between the events of N ? S. Balaguer et al. / Building Occurrence Nets from Reveals Relationsfor the partial order semantics referred to as unfoldings; they are nets in which all transitions, called events, are executable and the flow relation induced by the arcs is acyclic. Paths between events represent causality.The representation of all runs of a Petri net as an unfolding [15,23] allows one to avoid the statespace explosion due to interleavings when exploring the runs of a Petri net. Unfoldings are infinite in general, but can be represented efficiently by a finite complete prefix [16,21], for instance to check LTL formulas [19].The structure of an occurrence net induces three relations over its events, causality, concurrency and conflict, thus generating a prime event structure [23]. Causality represents the partial ordering of events due to the progress of the run. When two events may occur in the same run, but are not related by causality, they are concurrent. The last possibility is that two events never occur in the same run; then they are in conflict. The causality and conflict relations induce logical dependencies between event occurrences: the occurrence of an event e in a run implies that all its causal predecessors also occur, and that no event in conflict with e ever occurs.Here, we focus on a particular setting where weak fairness [25] is assumed, i.e. any enabled event has to occur or to be disabled, and when we consider these maximal runs, the structural relations do not express all the logical dependencies between event occurrences. Indeed, in this context, concurrency does not necessarily mean logical independency: it is possible that the occurrence of an event implies the eventual occurrence of another one, which is structurally concurrent. This happens with events a and c in Fig. 2(a): we have to observe that a is in conflict with b and that any maximal run contains either b or c. Therefore, if a occurs in a maximal run, then b does not ...

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