Well-Structured Graph Transformation Systems with Negative Application Conditions

Abstract

Given a transition system and a partial order on its states, the coverability problem is the question to decide whether a state can be reached that is larger than some given state. For graphs, a typical such partial order is the minor ordering, which allows to specify "bad graphs" as those graphs having a given graph as a minor. Well-structuredness of the transition system enables a finite representation of upward-closed sets and gives rise to a backward search algorithm for deciding coverability. It is known that graph tranformation systems without negative application conditions form well-structured transition systems (WSTS) if the minor ordering is used and certain condition on the rules are satisfied. We study graph transformation systems with negative application conditions and show under which conditions they are well-structured and are hence amenable to a backwards search decision procedure for checking coverability.

Well-structured graph transformation systems

“…More extensions are possible (possibly introducing over-approximations) and we especially plan to further investigate the integration of rules with negative application conditions as for the induced subgraph ordering. In [18] we introduced an extension with negative application conditions for the minor ordering, but still, the interplay of the well-quasi-order and conditions has to be better understood. Naturally, we plan to look for additional orders, for instance the induced minor and topological minor orderings [20] in order to see whether they can be integrated into this framework and to study application scenarios.…”

“…the subgraph ordering, since rules may become inapplicable to a graph by adding nodes or edges. Restricted GTSs may still be well-structured even with negative application conditions, as we have shown in [18] for the minor ordering. However, since the induced subgraph ordering is finer, there are graph transformation systems with negative application conditions, which satisfy the compatibility condition naturally (wrt.…”

A General Framework for Well-Structured Graph Transformation Systems

“…More extensions are possible (possibly introducing over-approximations) and we especially plan to further investigate the integration of rules with negative application conditions as for the induced subgraph ordering. In [15] we introduced an extension with negative application conditions for the minor ordering, but still, the interplay of the well-quasi-order and conditions has to be better understood. Naturally, we plan to look for additional orders, for instance the induced minor and topological minor orderings [8] in order to see whether they can be integrated into this framework and to study application scenarios.…”

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].…”