Building Tight Occurrence Nets from Reveals Relations

Abstract

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 in [1] 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 ?

A Canonical Contraction for Safe Petri Nets

“…We will identify macro-transitions in safe Petri nets that allow contraction with preservation of maximal semantics, and thus to give a contracted normal form for any given Petri net. If the definition is applied to occurrence nets, we obtain exactly the facets according to [6,7,1,2]. At the same time, the reduced net has never more, and generally much fewer, transitions than the original net.…”

“…Definition 11 (Reveals relation [6,7,1,2]). We say that event e reveals event f , and write e f , iff ∀ω ∈ Ω max , (e ∈ ω ⇒ f ∈ ω).…”

“…The structures obtained can be embedded into non-branching occurrence nets, allowing the approach to be compared with ours. However, while the construction of facets appears geometrically similar, the approach of [6,7,1,2] focuses on the question of logical occurrence regardless of the order in which events occur. The theory of the reveals relation and of reduced occurrence nets is given in [6,7,1,2].…”

“…However, while the construction of facets appears geometrically similar, the approach of [6,7,1,2] focuses on the question of logical occurrence regardless of the order in which events occur. The theory of the reveals relation and of reduced occurrence nets is given in [6,7,1,2]. Figure 3(a) (whose formal discussion is postponed to Section 2) illustrates the facets of an occurrence net; the contraction of its facets yields the reduced occurrence net in Figure 3(b).…”

“…The paper is organized as follows: We begin by recalling the basic definitions on unfoldings, and results from [6,7,1,2] concerning facets in occurrence nets, based on the reveals-relation, in Section 2. Section 3 contains the core of the present work, with the study of macro-transitions that generalize facets from occurrence nets to safe Petri nets.…”