Termination Analysis for Graph Transformation Systems

Abstract

We introduce two techniques for proving termination of graph transformation systems. We do not fix a single initial graph, but consider arbitrary initial graphs (uniform termination), but also certain sets of initial graphs (non-uniform termination). The first technique, which can also be used to show non-uniform termination, uses a weighted type graph to assign weights to graphs. The second technique reduces uniform termination of graph transformation systems of a specific form to uniform termination of cycle rewriting, a variant of string rewriting. 2 Preliminaries We first introduce graphs, morphisms, and graph transformation, in particular the double pushout approach [5]. We use edge-labeled, directed graphs.

Proving Termination of Graph Transformation Systems Using Weighted Type Graphs over Semirings

“…We have shown how to generalize the tropical and arctic weighted type graphs of [5] to weighted type graphs over general semirings and their application to the termination analysis of graph transformation systems. This enables us to work in the arithmetic semiring and to prove termination of systems that could not be handled with previous approaches.…”

“…In [5] we have shown how to adapt methods from string rewriting [17,12] and to develop a technique based on weighted type graphs, which was implemented in the tool Grez. Despite its simplicity the method is quite powerful and finds termination arguments also in cases which are difficult for human intuition.…”

Kruskal's Tree Theorem for Acyclic Term Graphs

“…[21]) these works require a term representation of the heap, which would be much more naturally be encoded as term dags (see the definition below). However, complexity and termination analysis of term graph rewrite systems have only recently be conceived attention in the literature [8,11,10,4]. In particular, at the moment there are no automated tools, which would allow an application for program analysis and could be compared to existing approaches using either AProVE [13] or T C T [5].…”

“…Bonfante et al [8]. Also Bruggink et al [11,10] use an interpretation method, where they use type graphs to assign weights to graphs to prove termination. Finally, in [4] complexity of acyclic term graph rewriting is investigated, based on the use of interpretations and suitable adaptions of the dependency pair framework.…”

Termination of Cycle Rewriting

“…Cycle rewriting can be seen as a special instance of graph transformation. In a separate paper [1] we investigate how the techniques of this paper extend to the general setting of graph transformation. In particular there we show that for a graph transformation system in which all rules are string rewrite rules, termination on all cycles coincides with termination on all graphs.…”