Multi-Amalgamation in Adhesive Categories

Abstract

Abstract. Amalgamation is a well-known concept for graph transformations in order to model synchronized parallelism of rules with shared subrules and corresponding transformations. This concept is especially important for an adequate formalization of the operational semantics of statecharts and other visual modeling languages, where typed attributed graphs are used for multiple rules with general application conditions. However, the theory of amalgamation for the double pushout approach has been developed up to now only on a set-theoretical basis for pairs of standard graph rules without any application conditions. For this reason, we present the theory of amalgamation in this paper in the framework of adhesive categories for a bundle of rules with (nested) application conditions. In fact, it is also valid for weak adhesive HLR categories. The main result is the Multi-Amalgamation Theorem, which generalizes the well-known Parallelism and Amalgamation Theorems to the case of multiple synchronized parallelism. The constructions are illustrated by a small running example. A more complex case study for the operational semantics of statecharts based on multi-amalgamation is presented in a separate paper.

Multi-amalgamation of rules with application conditions in -adhesive categories

“…The work in the current paper extends Golas et al (2010) in several ways. First, we consider amalgamated transformations in any M-adhesive category, while Golas et al (2010) only used adhesive categories. Second, we present the firing semantics of Petri nets as a new case study.…”

Generalised Compositionality in Graph Transformation

“…We do not require that the monos are strict (i.e., pullbacks). This makes the definition quite a bit more general than similar notions in [19,9], a fact which is at the core of this paper's contribution.…”

“…Though ordinary cospan rules and span rules have been shown to be equally expressive, cospan rules turn out to be advantageous in the presence of interfaces and composition. Another important difference with the usual concept of rule interface (called kernel in [9]) is that the relation between rule and interface is less strict (we do not insist on pullbacks).…”

Canonical Derivations with Negative Application Conditions

“…A similar construction is also presented in [10], but for two rules only. Paper [8] considers amalgamation of several rules with nested application conditions, which generalize negative ones, in arbitrary (M-)adhesive categories. However, rules are amalgamated along a single common subrule, thus this approach is not applicable to our setting.…”