Michel Bauderon

Graph expressions and graph rewritings

We define an algebraic structure for the set of finite graphs, a notion of graph expression for defining them, and a complete set of equational rules for manipulating graph expressions. (By agraph we mean an oriented hypergraph, the hyperedges of which are labeled with symbols from a fixed finite ranked alphabet and that is equipped with a finite sequence of distinguished vertices). The notion of a context-free graph grammar is introduced (based on the substitution of a graph for a hyperedge in a graph). The notion of an equational set of graphs follows in a standard way from the algebraic structure. As in the case of context-free languages, a set of graphs is contextfree iff it is equational. By working at the level of expressions, we derive from the algebraic formalism a notion of graph rewriting which is as powerful as the usual one (based on a categorical approach) introduced by Ehrig, Pfender, and Schneider.

A Uniform Approach to Graph Rewriting: The Pullback Approach

Most of the works in the theory of graph rewriting can be put into two main categories: edge (or hyperedge) rewriting and node rewriting. Each has been described by a specific formalism, both have given rise to many significant developments and many works have been devoted to the comparison of both approaches. In this paper, we describe a new categorical formalism, which provides a common framework to both approaches and makes their comparison much clearer.

A Unified Framework for Designing, Implementing and Visualizing Distributed Algorithms

We present a general method and a toolkit for designing, implementing and visualizing distributed algorithms. We make use of the high level encoding of distributed algorithms as graph rewriting systems. The result is a unified and simple framework for describing, implementing and visualizing a large family of distributed algorithms.

Pullback as a Generic Graph Rewriting Mechanism

Rewriting usually relies on a notion of substitution which can be understood as the succession of three basic operations: deletion of the part to be rewritten to provide a context, union of this context with the right-hand side of a rule, liaison of those two parts, most often by identification of some corresponding items.In the field of graph rewriting, this has led to the elegant, productive and therefore popular method known as the double push-out approach to graph rewriting. Yet this method has met its descriptive limits when trying to deal with the various notions of node replacement.In this paper we show how – when set in a proper framework – products (or pullbacks) can provide a very generic and uniform rewriting mechanism which extends uniformly to arbitrary complicated graph-like structures.

Node rewriting in graphs and hypergraphs: a categorical framework

Vertex rewriting in graphs is a very powerful mechanism which has been studied for quite a long time. In this paper we eventually provide a categorical theory of vertex rewriting and show how it can extend in a uniform way to node and pattern rewriting mechanisms in hypergraphs. Copyright 2001 Elsevier Science B.V.

Infinite hypergraphs: ii. systems of recursive equations

Abstract The results and tools which were developed in Part I of this paper [4] are used to solve systems of recursive equations on hypergraphs and to characterize their solutions completely. A hypergraphs initial solutions are called equational hypergraphs. It is then shown that the context-free graphs of Muller and Schupp [15] are equational.

On Systems of Equations Defining Infinite Graphs

A framework is described in which we can solve equations and systems of equations on oriented edge labelled hypergraphs with a finite sequence of distinguished sources. We show that this cannot be done with the standard order-theoretic methods, but implies the use of some category-theoretic tools and results.

Searching for weakly autocorrelated binary sequences

Exhaustive search for minimally autocorrelated binary sequences has not met any conclusive answer yet. We present here a new algorithm which allows us to obtain in a reasonable time results for heretofore unreachable values.

Visualisation of distributed algorithms based on graph relabelling systems

Abstract In 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.

Node Rewriting in Hypergraphs

Pullback rewriting has recently been introduced as a new and unifying paradigm for vertex rewriting in graphs. In this paper we show how to extend it to describe in a uniform way more rewriting mechanisms such as node and handle rewriting in hypergraphs.