Parallelism in AGREE Transformations

Abstract

The AGREE approach to graph transformation allows to specify rules that clone items of the host graph, controlling in a finegrained way how to deal with the edges that are incident, but not matched, to the rewritten part of the graph. Here, we investigate in which ways cloning (with controlled embedding) may a↵ect the dependencies between two rules applied to the same graph. We extend to AGREE the classical notion of parallel independence between the matches of two rules to the same graph, identifying su cient conditions that guarantee that two rules can be applied in any order leading to the same result.? This work has been partly funded by projects CLIMT (ANR/(ANR-11-BS02-016), VeriTeS (CNPq 485048/2012-4 and 309981/2014-0), PEPSégalité (CNRS).

On the Essence of Parallel Independence for the Double-Pushout and Sesqui-Pushout Approaches

“…We present here explicit proofs of the equivalence of the three conditions introduced in the previous section for dpo and sqpo rewriting. The equivalence between the Standard and the Pullback Conditions was proved in an indirect way in [4], by exploiting some results of [5]. The proofs that follow are complete and in a way simpler than those in [4], by reducing both conditions to the new one.…”

“…G is uniquely determined (up to isomorphism) if it exists, and thus also object H is unique up to iso by the universal property of the right-hand side pushout. 5 Sesqui-Pushout transformations were proposed in [7] as a little variation of dpo ones, able to guarantee the above form of determinism by the very definition. The only di↵erence with respect to dpo is the property that the left square of (1) has to satisfy.…”

On the Definition of Parallel Independence in the Algebraic Approaches to Graph Transformation

“…In category Graph, (⇢ 1 , m 1 ) and (⇢ 2 , m 2 ) are two redexes in a graph G, as in diagram (3). The intersection of matches m 1 and m 2 in G is preserved along the interfaces, that is,…”

“…In fact, we show that in the case of non-left-linear productions, Condition 3 is equivalent to the conjunction of Condition 2 and Condition 4. The last part of the proof exploits results to appear in [3], and requires an additional condition on category C, namely that it has a partial maps classifier (or equivalently, since C is assumed to have final pullback complements, a subobject classifier ). We refer to [1] for the definition of partial maps classifiers and the relationship with subobject classifiers, and to [2] for their use in the AGREE approach to graph transformation.…”

“…For Condition 4, the fact that the squares in (8) are pullbacks is proved in Lemma 1 of [3] by exploiting Condition 3 and an additional condition involving the partial maps classifier, and formulated for the more general framework of AGREE transformation. The latter condition, instantiated to SqPO transformation, requires that the left square of (12) is a pullback.…”