Ehrig, H.: Introduction to the algebraic theory of graph grammars. In Claus, V., Ehrig, H., Rozenberg, G., eds.: Graph Grammars and Their Application to Computer Science and Biology. Volume 73 of Lecture Notes in Computer Science., Springer-Verlag (1979) 1--69  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The definition of parallel independence demands that the occurrences of two independent transformations do only share items which are preserved by both steps. Under this assumption, the local ChurchRosser theorem states that the two steps can be executed in any order with the same overall result [4]. 7 Given two transformations G =# H 1 and G =# H 2 , G =# H 1 is (weakly) parallel independent of G =# H 2 if the occurrence o 1 (L 1 ) of the left hand side of p 1 is preserved by the application of p 2 . This is the case if o 1 (L 1 ) o 2 (L 2 g 2 (D 2 ) #, that is, o 1 ....

....Church Rosser theorem) Given two parallel independent transformations H 1 =# H 2 with H 1 H 2 , there are transformations H 1 =# X with o # 1 = h 2 #g 1 2 #o and o # 2 = h 1 g 1 1 o 2 . The local Church Rosser theorem has been shown for colored graphs in [4]. Simply rephrasing its proof for typed graphs would lead to the proof of proposition 1. In the case of attributed graph transformation, where attribute values are modelled as vertices and attribute links are edges, any modification of an attribute corresponds to a deletion of an attribute link ....

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H. Ehrig. Introduction to the Algebraic Theory of Graph Grammars (A Survey). In Graph Grammars and their Application to Computer Science and Biology. Springer LNCS 73, 1979.

....and in data base languages like Hyperlog [36] In this paper we introduce hierarchical hypergraphs in which certain hyperedges, called frames, contain hypergraphs that can be hierarchical again, with an arbitrary depth of nesting. We show that the double pushout approach to graph transformation [11, 15] extends smoothly to these hierarchical hypergraphs, by giving recursive constructions for pushouts and pushout complements in the category of hierarchical graphs. Hierarchical transformation rules consist of hierarchical graphs and can be applied at all levels of the hierarchy, where the ....

....there is a unique morphism n: OOE P satisfying n p n 1 =n 1 and n p n 2 =n 2 . Depicted as a diagram, this looks as follows: OL OOE n , n We recall the following well known facts about pushouts and pushout complements in the category of graphs and graph morphisms (see [15]) Let m 1 :GQH 1 and m 2 :GQH 2 be morphisms. Then there is a graph H and there are morphisms such that (m 1 ,m 2 ,n 1 ,n 2 ) is a pushout. Furthermore, H and the n i are determined as follows. Let HOE be the disjoint union of H 1 and H 2 , and let be the equivalence relation on A HOE ....

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H. Ehrig, Introduction to the algebraic theory of graph grammars, in Claus et al. [10, pp. 1--69].

....several different areas yielded similar results based on similar proofs. The most prominent area was the algebraic approach to graph grammars. This algebraic approach to graph grammars was introduced in the early seventies by H. Ehrig, M. Pfender and H.J. Schneider, summarized in 1979 by H. Ehrig [Ehr79] and was investigated intensively by research groups in Berlin, Bremen, Erlangen and L Aquila. It is based on double pushouts, that is to say productions are represented as two pushouts in the category of graphs with total graph morphisms: the first represents a kind of deleting, the second a kind ....

H. Ehrig, Introduction to the algebraic theory of graph grammars, 1st Graph Grammar Workshop, Lecture Notes in Computer Science 73, Springer, 1979, pp. 1--69.

....and in data base languages like Hyperlog [36] In this paper we introduce hierarchical hypergraphs in which certain hyperedges, called frames, contain hypergraphs that can be hierarchical again, with an arbitrary depth of nesting. We show that the double pushout approach to graph transformation [15, 11] extends smoothly to these hierarchical hypergraphs, by giving recursive constructions for pushouts and pushout complements in the category of hierarchical graphs. Hierarchical transformation rules consist of hierarchical graphs and can be applied at all levels of the hierarchy, where the ....

....morphism n : O P satisfying n n 1 = n 1 and n n 2 = n 2 . Depicted as a diagram, this looks as follows: O O 1 O 2 O P m 1 m 2 n 1 n 2 n 9 n We recall the following well known facts about pushouts and pushout complements in the category of graphs and graph morphisms (see [15]) Let m 1 : G H 1 and m 2 : G H 2 be morphisms. Then there is a graph H and there are morphisms n 1 : H 1 H and n 2 : H 2 H such that (m 1 ; m 2 ; n 1 ; n 2 ) is a pushout. Furthermore, H and the n i are determined as follows. Let H be the disjoint union of H 1 and H 2 , and let be ....

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Hartmut Ehrig. Introduction to the algebraic theory of graph grammars. In Claus et al. [10], pages 1-69.

....hand, it lifts the substitutive transformation of at graphs [20] to nested graphs; on the other hand, it extends hierarchical graph transformation [5] with respect to the use of variables. Hierarchical graph transformation has in turn been de ned by lifting double pushout graph transformation [6] to hierarchical graphs (for injective occurrences, as studied in [12] The paper [8] de nes double pushout transformation of hierarchical graphs where edges may cross the border of components (called packages) yet without investigating un 117 der which conditions the hierarchy stays intact. A ....

H. Ehrig. Introduction to the algebraic theory of graph grammars. In V. Claus, H. Ehrig, and G. Rozenberg, editors, Proc. Graph Grammars and Their Application to Computer Science and Biology, number 73 in Lecture Notes in Computer Science, pages 1-69. Springer, 1979.

....is terminating, that is, whether every computation of a system eventually halts [4] Moreover, their proof implies that termination of string rewriting systems is undecidable. In the present paper, it is shown that termination of graph rewriting systems in the so called double pushout approach [2, 1] is undecidable, too. Huet and Lankford simulated Turing machines by term rewriting systems such that a given machine halts on all inputs if and only if the corresponding term rewriting system terminates for all terms. Thus, they obtained a reduction of the uniform halting problem for Turing ....

....in Section 3 the main result is proved, and Section 4 concludes by stating an undecidability result following from the proof of the main result. 2. Graph rewriting Below the double pushout approach to graph rewriting is briefly reviewed. For a comprehensive survey, the reader may consult [1] or [2]. A label alphabet Sigma = h Sigma V ; Sigma E i is a pair of finite sets of vertex labels and edge labels. A graph over Sigma is a system G = hVG ; EG ; s G ; t G ; l G ; mG i consisting of two finite sets VG and EG of vertices (or nodes) and edges, two source and target functions s G ; t G : ....

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Hartmut Ehrig. Introduction to the algebraic theory of graph grammars. In Proc. GraphGrammars and Their Application to Computer Science and Biology, volume 73 of Lecture Notes in Computer Science, pages 1--69. Springer-Verlag, 1979.

....over that whole. Graph rewriting There is a well developed algebraic theory of graph rewriting, especially in relation to concurrency. The third volume of a recent Handbook [7] is devoted to this topic. A prominent part is played by the double pushout (DPO) construction introduced by Ehrig [6] for representing rewriting rules, called here reaction rules. This construction works in what we may call an embedding category, with graphs as objects and graph morphisms or embeddings as arrows. In our terminology, a reaction rule consists of a 999 9999 9999 of embeddings, where is the ....

Ehrig, H. (1979), Introduction to the algebraic theory of graph grammars. Proc. first International Workshop on Graph Grammars and their Application to Computer Science and Biology, LNCS Vol 73, pp1--69.

.... Gamma . If the subscript (m; n) is not important we omit it. If G 2 N Gamma , the order of the nodes in bottom(G) and top(G) is represented by their left to right order in pictures. The following definition of net rewriting is a special case of the algebraic double pushout approach, see e.g. [Ehr79]. Definition 5. A net rewrite rule j is a pair j = L; R) of elements of AN Gamma . A direct derivation D from G 2 N Gamma to H 2 N Gamma (by j) is a tuple (G; j; g; H) where g : L G is an injective morphism and H is constructed from G, g and j as follows. In a first step delete ....

H. Ehrig. Introduction to the algebraic theory of graph grammars. In Proceedings of the 1st International Workshop on Graph-Grammars and Their Application to Computer Science and Biology, Lect. Notes Comput. Sci. 73, pages 1--69. Springer--Verlag, 1979.

....of the edges of L. Then the graph G is extended by adding the new nodes in R (i.e. the nodes in VR VL ) and the edges of R. Observe that the (images of) the nodes in L are preserved , i.e. not a ected by the rewriting step. The reader which is familiar with the double pushout (DPO) approach [4] to graph rewriting would have recognized that our rules (L; R; can be seen as DPO rules (L VL , R) and that our notion of rewriting is equivalent to a DPO construction. Hence compared to general DPO rules L L K R R we have 4 that (i) K is discrete, i.e. it contains no edges, ....

H. Ehrig. Introduction to the algebraic theory of graph grammars. In V. Claus, H. Ehrig, and G. Rozenberg, editors, Proceedings of the 1st International Workshop on Graph-Grammars and Their Application to Computer Science and Biology, volume 73 of LNCS, pages 1-69. Springer Verlag, 1979.

....transition and of his statement of dissection, it is di#cult to see how these could be generalised to embrace wiring without the benefit of the notion of RPOs or other universal constructions. In this dissertation, I have made no use of the double pushout techniques developed in graph rewriting [Ehr79] These are a way to describe the occurrence of a subgraph especially a redex in a graph. To avoid confusion, I should emphasise that relative pushouts play quite a di#erent role. In my work, subgraph occurrences are handled by embeddings and contexts; the nature of the graphs (with forked ....

H. Ehrig. Introduction to the algebraic theory of graph grammar. In Proc. first international Workshop on Graph Grammars and their application to Computer Science and Biology, volume 73 of Lecture Notes in Computer Science, pages 1--69. Springer-Verlag, 1979. {15}

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H. Ehrig. Introduction to the Algebraic Theory of Graph Grammars (A Survey). In Graph Grammars and their Application to Computer Science and Biology, pages 1-69. Springer LNCS 73, 1979.

No context found.

H. Ehrig. Introduction to the algebraic theory of graph grammars. In V. Claus, H. Ehrig, and G. Rozenberg, editors, 1st Graph Grammar Workshop, Lecture Notes in Computer Science 73, pages 1-69. Springer Verlag, 1979.

No context found.

Ehrig, H.: Introduction to the algebraic theory of graph grammars. In Claus, V., Ehrig, H., Rozenberg, G., eds.: Graph Grammars and Their Application to Computer Science and Biology. Volume 73 of Lecture Notes in Computer Science., Springer-Verlag (1979) 1--69

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H. Ehrig. Introduction to the Algebraic Theory of Graph Grammars. In V. Claus, H. Ehrig, and G. Rozenberg, editors, Proc. Int. Workshop Graph-Grammars and Their Application to Computer Science and Biology, LNCS 73. Springer Verlag, 1979.

No context found.

H. Ehrig. Introduction to the algebraic theory of graph grammar. In Proc. first international Workshop on Graph Grammars and their application to Computer Science and Biology, volume 73 of Lecture Notes in Computer Science, pages 1--69. Springer-Verlag, 1979.

No context found.

H. Ehrig. Introduction to the algebraic theory of graph grammars. In V. Claus, H. Ehrig, and G. Rozenberg, editors, Proc. of the 1st International Workshop on Graph-Grammars and Their Application to Computer Science and Biology, volume 73 of LNCS, pages 1-69. Springer Verlag, 1979. 15

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H. Ehrig. Introduction to the algebraic theory of graph grammars. In V. Claus, H. Ehrig, and G. Rozenberg, editors, Proceedings of the 1st International Workshop on Graph-Grammars and Their Application to Computer Science and Biology, volume 73 of LNCS, pages 1-69. Springer Verlag, 1979. 18

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H. Ehrig. Introduction to the algebraic theory of graph grammars. In V. Claus, H. Ehrig, and G. Rozenberg, editors, Proceedings of the 1st International Workshop on Graph-Grammars and Their Application to Computer Science and Biology, volume 73 of LNCS, pages 1--69. Springer Verlag, 1979.

No context found.

H. Ehrig. Introduction to the Algebraic Theory of Graph Grammars. In V. Claus, H. Ehrig, and G. Rozenberg, editors, Proc. Int. Workshop GraphGrammars and Their Application to Computer Science and Biology, LNCS 73. Springer Verlag, 1979.

No context found.

H. Ehrig, Introduction to the Algebraic Theory of Graph Grammars, 1st Graph Grammar Workshop (V. Claus, H. Ehrig, G. Rozenberg, Eds.), pp. 1-69, LNCS 93, Spring-Verlag, 1979.

No context found.

Hartmut Ehrig. Introduction to the algebraic theory of graph grammars. In Proc. Graph-Grammars and Their Application to Computer Science and Biology, volume 73 of Lecture Notes in Computer Science, pages 1-69. Springer-Verlag, 1979.

No context found.

H. Ehrig. Introduction to the Algebraic Theory of Graph Grammars (A Survey). In Graph Grammars and their Application to Computer Science and Biology. Springer LNCS 73, 1979.

No context found.

H. Ehrig. Introduction to the Algebraic Theory of Graph Grammars. In Proc. 1st Graph Grammar Workshop, Springer LNCS 73, pages 1-69. Springer Verlag, 1979.

No context found.

Hartmut Ehrig. Introduction to the algebraic theory of graph grammars. In Proc. Graph-Grammars and Their Application to Computer Science and Biology, pages 1--69. Springer Lecture Notes in Computer Science 73, 1979.

No context found.

Ehrig, 11., "Introduction to the Algebraic Theory of Graph Grammars," Plot. of lnternahonal Workshop on Graph (;rammars, LNC' 73, 1-69, 1979.

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