Reversible Sesqui-Pushout Rewriting

Abstract

Abstract. The paper proposes a variant of sesqui-pushout rewriting (SqPO) that allows one to develop the theory of nested application conditions (NACs) for arbitrary rule spans; this is a considerable generalisation compared with existing results for NACs, which only hold for linear rules (w.r.t. a suitable class of monos). Besides this main contribution, namely an adapted shifting construction for NACs, the paper presents a uniform commutativity result for a revised notion of independence that applies to arbitrary rules; these theorems hold in any category with (enough) stable pushouts and a class of monos rendering it weak adhesive HLR. To illustrate results and concepts, we use simple graphs, i.e. the category of binary endorelations and relation preserving functions, as it is a paradigmatic example of a category with stable pushouts; moreover, using regular monos to give semantics to NACs, we can shift NACs over arbitrary rule spans.

Knowledge Representation and Update in Hierarchies of Graphs

“…This establishes that the two definitions of G − coincide. The above argument amounts to a proof of the abstract property that PBCs are stable under PBs: this requires a commutative cube whose front face is a PBC, whose left and bottom faces are PBs and where any one of the other faces is also a PB (in our case, the top face); see also Proposition 6 of [13] or Lemma 1 of [6].…”

“…, we define L Gi and L Gj as in (6) and apply the UP of L Gj to obtain the unique arrowĥ ij : L Gi → L Gj satisfying:…”

Parallelism in AGREE Transformations

“…Then G ) ⇢1,m1 H 1 and G ) ⇢2,m2 H 2 are parallel independent if the following are satisfied: 1. In the left diagram of (7) where the inner and the outer squares are built as pullbacks, the mediating morphism K 1 K 2 ! L 1 L 2 is an isomorphism.…”

“…Proof. In the left cube, the front-left face is a pullback by construction of step G ) ⇢2,m2 H 2 , the bottom face is a pullback by hypothesis (see (7)), and the back-right face is trivially a pullback. In addition the front-right face commutes: in fact on one hand we have T (⇡ L 2 ) ⇡ L 1 = '(⇡ L 1 , ⇡ L 2 ) by property (4) of partial maps classifiers, on the other hand the right diagram of (9) proves that m 2 m 1 = '(⇡ L 1 , ⇡ L 2 ).…”

“…Those problems have been studied in many GT approaches: double pushout (DPO) [5], single pushout [12], sesqui-pushout (SqPO) [4], reversible sesquipushout [7], with negative application conditions [13], borrowed contexts [1] and nested application conditions [11]. To our knowledge, in all these approaches (but for [7]) rules are required to be linear, i.e. both the left-and the right-hand side have to be monomorphisms.…”