-adhesive transformation systems with nested application conditions. Part 1: parallelism, concurrency and amalgamation

Abstract

Nested application conditions generalise the well-known negative application conditions and are important for several application domains. In this paper, we present Local Church–Rosser, Parallelism, Concurrency and Amalgamation Theorems for rules with nested application conditions in the framework of$\mathcal{M}$-adhesive categories, where$\mathcal{M}$-adhesive categories are slightly more general than weak adhesive high-level replacement categories. Most of the proofs are based on the corresponding statements for rules without application conditions and two shift lemmas stating that nested application conditions can be shifted over morphisms and rules.

Attributed Graph Transformation via Rule Schemata: Church-Rosser Theorem

“…Among other results, the ChurchRosser Theorem is proved in this setting. The approach is further studied in [4] by adding nested application conditions and proving the previous results for this more expressive approach. Both are a generalized version of the Church-Rosser Theorem of [9].…”

k-Inductive Invariant Checking for Graph Transformation Systems

“…Additionally, we require (nested) application conditions [8] and (nested) graph constraints [10] to describe more complex conditions over morphisms and graphs, respectively. Here, an application condition (graph constraint) can also be interpreted as describing the set of morphisms (graphs) that satisfy it.…”

“…Application conditions (or nested application conditions) are inductively defined as in [8]: (1) for every graph P, true is an application condition over P; (2) for every morphism a ∶ P ↪ C and every application condition ac over C, ∃(a, ac) is an application condition over P. Application conditions can also be extended over boolean combinations: (3) for application conditions ac, ac i over P (for all index sets I), ¬ac and ⋀ i∈I ac i are application conditions over P.…”

Obscuring Provenance Confidential Information via Graph Transformation

“…To address the problem of false (in)dependency, we have established that, when a node needs to be deleted, we need to consider not only the edges incident to that node, but also the edges between its adjacent nodes. To do so, graph transformation rules need to be equipped with a variety of graph rewriting capabilities, such as negative application conditions (NAC) [11] and nested constraints [12].…”