Publisher Summary This chapter presents three mathematical tools that can be used to describe graph-grammars and the sets generated by them. The chapter describes graph properties by logical formulas and presents a comparison of the powers of several logical languages. The chapter discusses first-order logic, second-order logic, and monadic second-order logic, together with a few variants and restrictions of these three languages. It presents algebraic techniques that are useful for defining and studying graph rewriting rules and context-free graph-grammars. The chapter also presents the use of category theory for specifying graph rewriting rules in a precise and concise way, and for properly defining the least or rather the initial solution of a system of graph equations. It presents some links between the context-free graph-grammars and monadic second-order logic. Some applications to the definition of sets of graphs by forbidden configurations and to the theory of NP-completeness are also presented in the chapter.

Graph Rewriting: An Algebraic and Logic Approach.

Handle-rewriting hypergraph grammars

Graph rewriting: an algebraic and logic approach

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.

https://link.springer.com/content/pdf/10.1007/BF01692060.pdf

Graph expressions and graph rewritings

Formulas of monadic second-order logic can be used to specify graph transductions, i.e., multi-valued functions from graphs to graphs. We obtain in this way classes of graph transductions, called monadic second-order definable graph transductions (or, more simply, definable transductions ) that are closed under composition and preserve the two known classes of context-free sets of graphs, namely the class of hyperedge replacement (HR) and the class of vertex replacement (VR) sets. These two classes can be characterized in terms of definable transductions and recognizable sets of finite trees, independently of the rewriting mechanisms used to define the HR and VR grammars. When restricted to words, the definable transductions are strictly more powerful than the rational transductions such that the image of every finite word is finite; they do not preserve context-free languages. We also describe the sets of discrete (edgeless) labelled graphs that are the images of HR and VR sets under definable transductions: this gives a version of Parikh's theorem (i.e., the characterization of the commutative images of context-free languages) which extends the classical one and applies to HR and VR sets of graphs

Monadic second-order definable graph transductions: a survey

Graph grammars with node-label controlled rewriting and embedding

Abstract The degree of a graph H is the maximum among the degrees of its nodes. A set of graphs L is of bounded degree if there exists a positive integer n such that the degree of each graph in L does not exceed n . We demonstrate that it is decidable whether or not the (graph) language of an arbitrary node label controlled (NLC) grammar is of bounded degree. Moreover, it is shown that, given an arbitrary NLC grammar G generating the language L(G) of bounded degree, one can effectively compute the maximum integer which appears as the degree of a graph in L(G) .

The Bounded Degree Problem for NLC Grammars Is Decidable ; CU-CS-267-84

The bounded degree problem for NLC grammars is decidable

In the las t years the theory of graph grammars has developed in to a chal lenging and mathematically in te res t ing area in computer science that is wel l -mot ivated by i t s many potent ia l f i e l ds of app l ica t ions, such as data bases, data f low analys is , concurrency, pattern recogni t ion and developmental biology (one can f ind more deta i l s in , e .g . , ~ER] and ~ I ) . Although there ex is ts already a considerable amount of l i t e rature on th is top ic , the mathematical theory of graph grammars is s t i i l much less developed than ti le theory of s t r ing grammars. In pa r t i cu la r there is a need for a un i fy ing framework for graph grammars w i th in which various issues can be discussed and compared, because in the ex is t ing theory one often encounters the undesirable s i tuat ion that various issues are invest igated w i th in various models, which makes i t d i f f i cu l t to compare the resu l ts . In ~R1] and~R2] NLC grammars were introduced as a f i r s t step towards such a un i fy ing framework. We would l i ke to bui ld up a sol id theory of NLC grammars f i r s t and then consider var#ous extensions and var ia t ions of the basic model in order to obtain a qui te general theory of graph grammars based on the NLC model. We hope that th is approach w i l l also y ie ld a better ins igh t in what are the central notions of the theory and what are the main problems to be solved yet . In th is paper we present an overview of (a part of) the resul ts obtained for NLC grammars in the las t few years. A more complete overview, together with proofs of the resul ts mentioned can be found in the Ph.D. thesis by D. Janssens (to appear).

Dirk Janssens

Papers

Graph grammars with node-label controlled rewriting and embedding

The Bounded Degree Problem for NLC Grammars Is Decidable ; CU-CS-267-84

The bounded degree problem for NLC grammars is decidable

A Survey of NLC Grammars

Computation graphs for actor grammars