Partial Order Semantics of Box Expressions

We develop a partial order semantics for the process expressions underlying the Petri Box Calculus. We aim at a semantics which would be equivalent to the standard partial order semantics of the Petri nets (Boxes) corresponding to such expressions. The solution we present is a variant of step sequence semantics in which actions are annotated with an additional information about the relative position of the parts of the expression from which they were derived, as first proposed by Degano, De Nicola and Montanari. This information is then used to capture all essential causal dependencies among actions, leading to the definition of a partial order of action occurrences. To represent Petri net markings within process expressions we employ an overbarring and underbarring technique which is related to that used in the event systems due to Boudol and Castellani. The partial order operational model turns out to be consistent with that defined in the Petri net theory. More precisely, if an expression can execute a partial order then the same holds for the corresponding Petri Box. The converse holds for all guarded expressions.