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

Abstract

Amalgamation is a well-known concept for graph transformations that is used to model synchronised parallelism of rules with shared subrules and corresponding transformations. This concept is especially important for an adequate formalisation of the operational semantics of statecharts and other visual modelling languages, where typed attributed graphs are used for multiple rules with nested application conditions. However, the theory of amalgamation for the double-pushout approach has so far only been developed on a set-theoretical basis for pairs of standard graph rules without any application conditions.For this reason, in the current paper we present the theory of amalgamation for$\mathcal{M}$-adhesive categories, which form a slightly more general framework than (weak) adhesive HLR categories, for a bundle of rules with (nested) application conditions. The two main results are the Complement Rule Theorem, which shows how to construct a minimal complement rule for each subrule, and the Multi-Amalgamation Theorem, which generalises the well-known Parallelism and Amalgamation Theorems to the case of multiple synchronised parallelism. In order to apply the largest amalgamated rule, we use maximal matchings, which are computed according to the actual instance graph. The constructions are illustrated by a small but meaningful running example, while a more complex case study concerning the firing semantics of Petri nets is presented as an introductory example and to provide motivation.

Compositionality of Rewriting Rules with Conditions

“…Lemma 5 (Compositionality of Shift; compare [45], Fact 3.14) Let X ∈ obj(C) be an object, c X an application condition over X, and let f : X → Y and g : Y → Z be two morphisms of C. Then the following equivalence of conditions holds:…”

“…Concretely (cf. Section 2), it has proved essential to suitably adapt the requirements on the host categories for both DPO-and SqPO-type rewriting with application conditions on objects and morphisms [42,45,46,51,60], concluding that these categories should be M-adhesive and possess certain additional properties such as the existence of epi-M-factorizations and M-effective unions as described in Assumptions 1 (DPO-case) and 2 (SqPO-case).…”

A Generalized Concurrent Rule Construction for Double-Pushout Rewriting

“…Parallel and amalgamated rules are further kinds of rules arising by composition. Whereas concurrent rules combine the sequential application of two rules, an amalgamated rule combines the application of two (or more) rules to the same object into the application of a single rule [4,17]. In categories with coproducts, the parallel rule is just the sum of two rules; for plain rules (i.e., without application conditions) it is a special case of the concurrent as well as of the amalgamated rule construction.…”

“…Classical kinds of rule composition are the ones of parallel and concurrent [11] as well as of amalgamated rules [4]. Considering the double-pushout approach to graph transformation, these rule constructions have been lifted from ordinary graphs to the general framework of M-adhesive categories and from plain rules to such with application conditions [5,9,17,7]. These central forms of rule composition have also been developed for other variants of transformation, like single-or sesqui-pushout rewriting [21,22,1].…”

Amalgamation of domain specific languages with behaviour

“…Notice that this construction does not coincide with multi-amalgamation, as defined in [26], where n rules are amalgamated given a single kernel rule -the construction in [26] is also provided for HLR categories and for rules with (nested) application conditions. It is not difficult to show, based on the HLR-adhesiveness of our cate- R i → R , and f R : R → R , defined by the corresponding colimits, are monomorphisms.…”