Infinite hypergraphs I. Basic properties

AbstractBasic properties of the category of infinite directed hyperedge-labelled hypergraphs are studied. An algebraic structure is given which enables us to describe such hypergraphs by means of infinite expressions. It is then shown that two expressions define the same hypergraphs if and only if they are congruent with respect to some rewriting system. These results will be used in the second part of this paper to solve systems of recursive equations on hypergraphs and characterize their solutions

Graph Relabelling Systems A Tool for Encoding, Proving, Studying and Visualizing Distributed Algorithms

International audienc

Visualization of Distributed Algorithms Based on Graph Relabelling Systems1 1This work has been supported by the European TMR research network GETGRATS, and by the “Conseil Régional d' Aquitane”.

AbstractIn this paper, we present a uniform approach to simulate and visualize distributed algorithms encoded by graph relabelling systems. In particular, we use the distributed applications of local relabelling rules to automatically display the execution of the whole distributed algorithm. We have developed a Java prototype tool for implementing and visualizing distributed algorithms. We illustrate the different aspects of our framework using various distributed algorithms including election and spanning trees

Characterizing Compressibility of Disjoint Subgraphs with NLC Grammars

We consider compression of a given set S of isomorphic and disjoint subgraphs of a graph G using node label controlled (NLC) graph grammars. Given S and G, we characterize whether or not there exists a NLC graph grammar consisting of exactly one rule such that (1) each of the subgraphs S in G are compressed (i.e., replaced by a nonterminal) in the (unique) initial graph I , and (2) the set of generated terminal graphs is the singleton {G}.acceptance rate: 39%status: publishe

Representing First-Order Logic Using Graphs

Abstract. We show how edge-labelled graphs can be used to represent first-order logic formulae. This gives rise to recursively nested structures, in which each level of nesting corresponds to the negation of a set of existentials. The model is a direct generalisation of the negative application conditions used in graph rewriting, which count a single level of nesting and are thereby shown to correspond to the fragment ∃¬∃ of first-order logic. Vice versa, this generalisation may be used to strengthen the notion of application conditions. We then proceed to show how these nested models may be flattened to (sets of) plain graphs, by allowing some structure on the labels. The resulting formulae-as-graphs may form the basis of a unification of the theories of graph transformation and predicate transformation

Parallel Rewriting of Graphs through the Pullback Approach

We continue here the development of our description of the pullback approach to graph rewriting - already shown to encompass both the NCE and the doublepushout approach, by describing parallel application of rewriting rules. We show that this new framework provides a genuine definition of parallel rewriting (parallel application of several rewriting rules at several different places in the graph is actually expressed through the application of one single mathematical operation) and even further that a deterministic graph grammar can be described by a single rule which we call P-grammar. 1 Introduction In two earlier papers [1,2], we have presented a new categorical approach to graph rewriting where the traditional pushout is replaced by a pullback. As shown in [1] by the coding of (B)NLC [5] and HR [3,4] rules, this new approach provides a unified framework to both node and edge rewriting. The definitions used there were simplified and tailored to the problem we were adressing : codin..