Syntactic Characterization of the Two-Dimensional Grid Graphs

An operation of concatenation is defined for graphs. This allows strings to be viewed as expressions denoting graphs, and string languages to be interpreted as graph languages. For a class \(K\) of string languages, \({\rm Int}(K)\) is the class of all graph languages that are interpretations of languages from \(K\). For the classes REG and LIN of regular and linear context-free languages, respectively, \({\rm Int}({\rm REG}) = {\rm Int}({\rm LIN})\). \({\rm Int}({\rm REG})\) is the smallest class of graph languages containing all singletons and closed under union, concatenation and star (of graph languages). \({\rm Int}({\rm REG})\) equals the class of graph languages generated by linear HR (= Hyperedge Replacement) grammars, and \({\rm Int}(K)\) is generated by the corresponding \(K\)-controlled grammars. Two characterizations are given of the largest class \(K'\) such that \({\rm Int}(K') = {\rm Int}(K)\). For the class CF of context-free languages, \({\rm Int}({\rm CF})\) lies properly inbetween \({\rm Int}({\rm REG})\) and the class of graph languages generated by HR grammars. The concatenation operation on graphs combines nicely with the sum operation on graphs. The class of context-free (or equational) graph languages, with respect to these two operations, is the class of graph languages generated by HR grammars.

Graph Operations, Graph Transformations and Monadic Second-Order Logic A survey

Graph Operations, Graph Transformations and Monadic Second-Order Logic:: A survey

Adaptive star grammars and their languages

In this paper we discuss the use of views to address the problem of providing useful answers to provenance queries while ensuring that privacy concerns are met. In particular, we propose a hierarchical workflow model, based on context-free graph grammars, in which fine-grained dependencies between the inputs and outputs of a module are explicitly specified. Using this model, we examine how privacy concerns surrounding data, module function, and workflow structure can be addressed.

Recursive queries and context-free graph grammars

On the Complexity of Pattern Matching for Highly Compressed Two-Dimensional Texts

The aim of this paper is to prove that the system of simple graph grammars (Einfache Graph-Grammatiken) in[1]is complete. A graph grammar is a generalization of a Chomsky-grammar which defines a graph language, i.e。a language composed of a set of directed graphs with labelled nodes and arcs。 A system of graph grammars is said to be complete if it satisfies the following property: For each finite set T of labels, the class of all graph languages over T defined by graph grammars of the system is identical with the class of ail recussively enumerable sets of labelled graphs over T. Where a set of labelled graphs over T is recursively enumerable if the associated set of integers by a Godel numbering from the labelled graphs over T into the integers is recursively enumerable。 Several authors have introduced their system of graph grammars (e・g.[11,[21, [31,[41,[51,[6j,[7])。However they did not refer to the completeness of their systems。 The system of simple graph grammars was introduced in[1]as a subsystem of the system in [2]。In this paper we modify the definition of the system in the following manner: A production of our system is an ordered triple(K,F,K’)which consists of two labelled graphs 瓦A7 and a one・to・one function F from a set of nodes of 瓦to a set of nodes of K'. If iK, F, m is a production, and if K occurs in a labelled graph G, then the production is applicable to G, and the effect of the application is to replace an occurrence of K in G by 瓦へ The function F specifies the embedding of 瓦7into the remainder of removing K from G; if a node n' of K' is the image of a node n of K, thenがis put upon the trace of n in the remainder. This is illustrated by Figure O。 The formal definitions of labelled graphs and several concepts with labelled graphs are given in the first section of this paper. The definition of simple graph grammars is given in the second section. In the third section it is proved that the system of simple graph grammars is complete. The final section gives a generalization of the system of simple graph grammars.

The Complexity of the Kn,n-Problem for Node Replacement Graph Languages

In each graph-grammar approach it is defined how and under which conditions graph productions can be applied to a given graph in order to obtain a derived graph. The conditions under which productions can be applied are called application conditions. Although the generative power of most of the known general graph-grammar approaches is sufficient to generate any recursively enumerable set of graphs, it is often convenient to have specific application conditions for each production. Such application conditions, on the one hand, include context conditions like the existence or non-existence of nodes, edges, or certain subgraphs in the given graph as well as embedding restrictions concerning the morphisms from the left-hand side of the production to the given graph. In this paper, the concept of application conditions introduced by Ehrig and Habel is restricted to contextual conditions, especially negative ones. In addition to the general concept, we state local confluence and the Parallelism Theorem for derivations with application conditions. Finally we study context-free graph grammars with application conditions with respect to their generative power.

Given a sample from an unknown probability distribution over strings, there exist algorithms for inferring the structure and parameters of stochastic grammatical representations of the unknown distribution, i.e. string grammars. Despite the fact that research on grammatical representations of sets of graphs has been conducted since the late 1960’s, almost no work has considered the possibility of stochastic graph grammars and no algorithms exist for inferring stochastic graph grammars from data. This paper presents PEGG, an algorithm for estimating the parameters of stochastic context-free graph grammars given a sample from an unknown probability distribution over graphs. It is established formally that for a certain class of graph grammars PEGG finds parameter estimates in polynomial time that maximize the likelihood of the data, and preliminary empirical results demonstrate that the algorithm performs well in practice.KeywordsGraph TransformationGraph GrammarMaximum Likelihood ParameterStart SymbolEstimate Maximum Likelihood ParameterThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Layout graph grammars are extensions of context-free graph grammars and are introduced as a tool for syntax directed constructions of graph layouts. The constructions are based on a layout specification of the productions, which are consistently transferred to the derivations. The layout specification consists of rules for a placement of the vertices and a partial routing of the edges. It specifies minimal distances between the vertices in X- or Y-dimension. These distances can be optimized according to some formal cost measures.

Apex graph grammars are a particular type of directed node-label controlled (DNLC) graph grammars: the embedding edges are established between terminal nodes only. Apex graph grammars, slightly generalized, can generate the sets of dependency graphs of attribute grammars. The other way around, every apex graph language can be obtained from such a dependency graph language by a graph replacement (which is an operation analogous to a string homomorphism).

Many industries are shifting from mass production to mass customization, which demands quick response to the needs of individual customers with high quality and low costs. The development of product families has received an increasing interest in recent years because, by sharing components across products, a family of products can be derived to cater variety while maintaining the economy of scale. Aiming at the computerization, and eventual automation, of product family design, this paper tackles the formal representation issue surrounding this economically important class of engineering design problem. Breaking free from conventional understanding of product families, which is limited as shared components, the paper defines a product family as a structured system to create variety of products with shared core product technologies. It not only involves the shared base product, but also encompasses customization modules, standard designs, and primary patterns of variety to generate custom designs. The paper introduces graph grammar formalisms to the modeling of such a product family. Based on Programmed Attributed Graph Grammars (PAGG), the graph language is developed to specify the design space of the product family. The process of customizing the base product through manipulating particular modules is modeled by rewriting the starting graph using a series of productions according to the control diagram. Configuration constraints are dealt with by defining application conditions for production rules. Control diagrams are constructed to capture complex relationships among modules and used to control the application sequence of production rules. A case study of power supplies is presented to demonstrate the potential of the graph grammar based modeling approach.

We present a heuristic algorithm for k-broadcasting in an arbitrary network. This heuristic generates optimal k-broadcast time in grid graph when k/spl ges/2. In two-dimensional torus graph, it also generates optimal k-broadcast time when k/spl ges/3, while giving a bound of /spl lfloor/m/2/spl rfloor/+/spl lfloor/n/2/spl rfloor/+1 when k=2, where m and n are the number of rows and columns in the graph. In practice, the new heuristic outperforms best known 1-broadcast algorithm for three different network design models. The new algorithm runs fast. The time complexity of the algorithm is O(R /spl middot/ m), where R represents the rounds of broadcasting, and m stands for the total number of edges in the graph.