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

Abstract

Parallel independence between transformation steps is a basic and well-understood notion of the algebraic approaches to graph transformation, and typically guarantees that the two steps can be applied in any order obtaining the same resulting graph, up to isomorphism. The concept has been redefined for several algebraic approaches as variations of a classical "algebraic" condition, requiring that each matching morphism factorizes through the context graphs of the other transformation step. However, looking at some classical papers on the doublepushout approach, one finds that the original definition of parallel independence was formulated in set-theoretical terms, requiring that the intersection of the images of the two left-hand sides in the host graph is contained in the intersection of the two interface graphs. The relationship between this definition and the standard algebraic one is discussed in this position paper, both in the case of left-linear and non-left-linear rules.

Confluence of Graph Rewriting with Interfaces

“…In a critical pair, L 1 and L 2 must additionally overlap in S , so that the two rewriting steps are not parallel independent (see e.g. [17]). For the purposes of this paper, this restriction is immaterial.…”

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

“…That is, there must exist the two dotted arrows of Diagram (4) so that the resulting triangles commute, i.e., g 1 m 2d = m 2 and g 2 m 1d = m 1 . However, as shown explicitly with a counterexample in [4], this condition only works for dpo and sqpo with leftlinear rules. For sqpo with non-left-linear rules the commutativity of the two triangles is not su cient, but it is necessary to require the stronger condition that m 1 is reflected along g 2 by m 1d , and symmetrically for m 2 .…”

“…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.…”

“…As discussed in [4], the classical condition of Diagram (4) is not a direct translation of this set-theoretical one, as the categorical counterpart of intersections are pullbacks. Diagram (7) shows the two pullbacks corresponding to the left side ( 1 ) and to the right side ( 2 ) of Equation (6).…”

“…, by the universal property of pullback 1 . By exploiting these pullbacks, the following condition of parallel independence was proposed in [4] as a direct categorical translation of Equation (6).…”