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

We introduce techniques for proving uniform termination of graph transformation systems, based on matrix interpretations for string rewriting. We generalize this technique by adapting it to graph rewriting instead of string rewriting and by generalizing to ordered semirings. In this way we obtain a framework which includes the tropical and arctic type graphs introduced in a previous paper and a new variant of arithmetic type graphs. These type graphs can be used to assign weights to graphs and to show that these weights decrease in every rewriting step in order to prove termination. We present an example involving counters and discuss the implementation in the tool Grez

Non-simplifying Graph Rewriting Termination

So far, a very large amount of work in Natural Language Processing (NLP) rely on trees as the core mathematical structure to represent linguistic informations (e.g. in Chomsky's work). However, some linguistic phenomena do not cope properly with trees. In a former paper, we showed the benefit of encoding linguistic structures by graphs and of using graph rewriting rules to compute on those structures. Justified by some linguistic considerations, graph rewriting is characterized by two features: first, there is no node creation along computations and second, there are non-local edge modifications. Under these hypotheses, we show that uniform termination is undecidable and that non-uniform termination is decidable. We describe two termination techniques based on weights and we give complexity bound on the derivation length for these rewriting system.Comment: In Proceedings TERMGRAPH 2013, arXiv:1302.599

Non-size increasing Graph Rewriting for Natural Language Processing

International audienceA very large amount of work in Natural Language Processing use tree structure as the first class citizen mathematical structures to represent linguistic structures such as parsed sentences or feature structures. However, some linguistic phenomena do not cope properly with trees: for instance, in the sentence "Max decides to leave", "Max" is the subject of the both predicates "to decide" and "to leave". Tree-based linguistic formalisms generally use some encoding to manage sentences like the previous example. In former papers, we discussed the interest to use graphs rather than trees to deal with linguistic structures and we have shown how Graph Rewriting could be used for their processing, for instance in the transformation of the sentence syntax into its semantics. Our experiments have shown that Graph Rewriting applications to Natural Language Processing do not require the full computational power of the general Graph Rewriting setting. The most important observation is that all graph vertices in the final structures are in some sense "predictable" from the input data and so, we can consider the framework of Non-size increasing Graph Rewriting. In our previous papers, we have formally described the Graph Rewriting calculus we used and our purpose here is to study the theoretical aspect of termination with respect to this calculus. In our framework, we show that uniform termination is undecidable and that non-uniform termination is decidable. We define termination techniques based on weight, we prove the termination of weighted rewriting systems and we give complexity bounds on derivation lengths for these rewriting systems

Preliminary Version

Abstract Visual rewriting techniques are increasingly used to model transformations of systems specified through diagrammatic sentences. Graph transformations, in particular, are a widespread formalism with several applications, from parsing to model animation or transformation. Although a wealth of rewriting models have been proposed, differing in the expressivity of the types of rules and in the complexity of the rewriting mechanism, basic results concerning the formal properties of these models are still missing for many of them. In this paper, we propose a contribution towards solving the termination problem for rewriting systems with external control mechanisms for rule application. In particular, we obtain results of more general validity by extending the concept of transformation unit to high-level replacement systems, a generalization of graph transformation systems. For the resulting highlevel replacement units, we state and prove several abstract properties based on termination criteria. Then, we instantiate the high-level replacement systems by attributed graph transformation systems and present concrete termination criteria. These are used to show the termination of some replacement units needed to express model transformations as a consequence of software refactoring

Kruskal's Tree Theorem for Acyclic Term Graphs

In this paper we study termination of term graph rewriting, where we restrict our attention to acyclic term graphs. Motivated by earlier work by Plump we aim at a definition of the notion of simplification order for acyclic term graphs. For this we adapt the homeomorphic embedding relation to term graphs. In contrast to earlier extensions, our notion is inspired by morphisms. Based on this, we establish a variant of Kruskal's Tree Theorem formulated for acyclic term graphs. In proof, we rely on the new notion of embedding and follow Nash-Williams' minimal bad sequence argument. Finally, we propose a variant of the lexicographic path order for acyclic term graphs.Comment: In Proceedings TERMGRAPH 2016, arXiv:1609.0301

On termination of Graph Rewriting Systems through language theory

The termination issue we tackle is rooted in natural language processing where graph rewriting systems (GRS) may contain a large number of rules, often in the order of thousands. Decidable concepts thus become mandatory to verify the termination of such systems. The notion of graph rewriting consider does not make any assumption on the structure of graphs (they are not “term graphs”, “port graphs” nor drags). The lack of algebraic structure in our setting led us to proposing two orders on graphs inspired from language theory: the matrix multiset-path order and the rational embedding order. We show that both are stable by context, which we then use to obtain the main contribution of the paper: under a suitable notion of “interpretation”, a GRS is terminating if and only if it is compatible with an interpretation

12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto

Termination of graph rewriting systems through language theory

International audienceThe termination issue that we tackle is rooted in Natural Language Processing where computations are performed by graph rewriting systems (GRS) that may contain a large number of rules, often in the order of thousands. This asks for algorithmic procedures to verify the termination of such systems. The notion of graph rewriting that we consider does not make any assumption on the structure of graphs (they are not "term graphs", "port graphs" nor "drags"). This lack of algebraic structure led us to proposing two orders on graphs inspired from language theory: the matrix multiset-path order and the rational embedding order. We show that both are stable by context, which we then use to obtain the main contribution of the paper: under a suitable notion of "interpretation", a GRS is terminating if and only if it is compatible with an interpretation

On graph rewriting systems termination through language theory

The termination issue that we tackle is rooted in Natural Language Processing where computations are performed by graph rewriting systems (GRS) that may contain a large number of rules, often in the order of thousands. Thus algorithms become mandatory to verify the termination of such systems. The notion of graph rewriting that we consider does not make any assumption on the structure of graphs (they are not "term graphs", "port graphs" nor "drags"). This lack of algebraic structure led us to proposing two orders on graphs inspired from language theory: the matrix multiset-path order and the rational embedding order. We show that both are stable by context, which we then use to obtain the main contribution of the paper: under a suitable notion of "interpretation", a GRS is terminating if and only if it is compatible with an interpretation