Preserving Liveness with Rule - Based Refinement of Place / Transition Systems

The focus of this paper is to endow step-wise modification of nets with refinement that preserves liveness. Modification of nets uses rules that allow substituting subnets of the left hand side by subnets of the right hand side of the rule. Such rules are then restricted so that the liveness is preserved. Rule-based modification is expressed in terms of spans of morphisms and two pushout constructions, thus being in line with the so-called double pushout approach to graph transformations. We then extend the span of morphisms with collapsing morphisms from the right-hand side to the left-hand side of the rule. As these morphisms respect live-ness the extended rules preserve liveness from left to right. Under adequate matches of the rule to a net the corresponding transformation preserves liveness as well.