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

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

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

A Graph Transformation Approach to Software Architecture Reconfiguration

The ability of reconfiguring software architectures in order to adapt them to new requirements or a changing environment has been of growing interest. We propose a uniform algebraic approach that improves on previous formal work in the area due to the following characteristics. First, components are written in a high-level program design language with the usual notion of state. Second, the approach deals with typical problems such as guaranteeing that new components are introduced in the correct state (possibly transferred from the old components they replace) and that the resulting architecture conforms to certain structural constraints. Third, reconfigurations and computations are explicitly related by keeping them separate. This is because the approach provides a semantics to a given architecture through the algebraic construction of an equivalent program, whose computations can be mirrored at the architectural level

Termination of Graph Transformation Systems Using Weighted Subgraph Counting

We introduce a termination method for the algebraic graph transformation framework PBPO+, in which we weigh objects by summing a class of weighted morphisms targeting them. The method is well-defined in rm-adhesive quasitoposes (which include toposes and therefore many graph categories of interest), and is applicable to non-linear rules. The method is also defined for other frameworks, including SqPO and left-linear DPO, because we have previously shown that they are naturally encodable into PBPO+ in the quasitopos setting. We have implemented our method, and the implementation includes a REPL that can be used for guiding relative termination proofs.Comment: 36 pages. Preprint submitted to LMCS. Extends the conference version published at the 16th International Conference on Graph Transformation (ICGT 2023

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

Specification of Software Architecture Reconfiguration

In the past years, Software Architecture has attracted increased attention by academia and industry as the unifying concept to structure the design of complex systems. One particular research area deals with the possibility of reconfiguring architectures to adapt the systems they describe to new requirements. Reconfiguration amounts to adding and removing components and connections, and may have to occur without stopping the execution of the system being reconfigured. This work contributes to the formal description of such a process. Taking as a premise that a single formalism hardly ever satisfies all requirements in every situation, we present three approaches, each one with its own assumptions about the systems it can be applied to and with different advantages and disadvantages. Each approach is based on work of other researchers and has the aesthetic concern of changing as little as possible the original formalism, keeping its spirit. The first approach shows how a given reconfiguration can be specified in the same manner as the system it is applied to and in a way to be efficiently executed. The second approach explores the Chemical Abstract Machine, a formalism for rewriting multisets of terms, to describe architectures, computations, and reconfigurations in a uniform way. The last approach uses a UNITY-like parallel programming design language to describe computations, represents architectures by diagrams in the sense of Category Theory, and specifies reconfigurations by graph transformation rules