Habel, A.: Hyperedge replacement: grammars and languages. Volume 643 of Lecture Notes in Computer Science. Springer-Verlag Inc., New York, NY, USA (1992)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....G ) t , where G , H , and t are the attened versions of G, H , and t, respectively. Obviously, the converse cannot hold as structural information gets lost in the attening process. For the de nition of attening, we extend the well known concept of hyperedge replacement (see [25, 13]) to hierarchical graphs. Let H be a hierarchical graph. For every set E EH of edges, H E denotes the hierarchical graph obtained by deleting all the edges in E from H , i.e. H E = hG; FH n E; ctsi where G = hVH ; EH nE; att ; labi, att and lab being the restrictions of att H and lab H to EH ....

Annegret Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, 1992.

....unification of context free and non context free approaches for graph grammars have been obtained already by the groups in Bordeaux, Bremen and Leiden in the first phase of COMPUGRAPH. An excellent survey of hyperedge replacement (HR) is given in the PhD thesis by A. Habel (Bremen Hildesheim) Hab92a] In the second phase the relationship between hyperedge and vertex replacement has been studied in more detail by B. Courcelle (Bordeaux) Cou93a] and an interesting extension of HR grammars, called collage grammars, was studied in the Bremen group, where important decidability results and ....

....Replacement, and C edNCE (or NR) grammars, which are based on Node Replacement (with Neighbourhood Controlled Embedding) Graph transductions can be defined in different ways, by monadic second order formulas, by tree transducers, by hyperedge replacement grammars. Surveys In the monograph [Hab92a] the theory of HR grammars and languages is developed in detail. Among the most important topics are: context freeness, characterizations of HR languages, a pumping lemma for HR languages, generative power of HR grammars, predicates and numeric functions on hypergraphs compatible with the ....

A. Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer, 1992.

....essential aspects are context free grammars and transformations, called transductions in language theory. There are many types of context free graph grammars, some based on node replacement and others on (hyper)edge replacement. For the historical developments, see, e.g. Nag79, Nag80, ER96] and [Hab92a, DHK96] respectively. General requirements for a (graph) grammar to be called context free are formulated in [Cou87] the replacement mechanism should be confluent and associative. During the last ten years two main types of context free graph grammars have emerged: HR grammars, which are based ....

.... CER93] by formulas of Monadic Second Order Logic (MSOL) Eng91, CE95] and by handle replacement grammars [CER93] Graph transductions can also be defined in different ways, e.g. by HR grammars [Rao93] by tree transducers [Dre93c] or by MSOL formulas [Cou91, Eng91] Surveys In the monograph [Hab92a] the theory of HR grammars and languages is developed in detail. Among the most important topics are: context freeness, characterizations of HR languages, a pumping lemma for HR languages, generative power of HR grammars, predicates and numerical functions on hypergraphs compatible with hyperedge ....

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A. Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer, 1992.

....and each node has those hyperedges as children which replace the node s hyperedge in the derivation sequence. A formal definition of derivation trees for context free hypergraph grammars and a detailed discussion of their properties and relationships to derivation sequences can be found in [31,53]. The derivation tree of a terminal hypergraph starting at the axiom represents the set of all equivalent derivation sequences and thus the syntactic structure of the hypergraph. Therefore, this tree is the preferred representation of the derivation history as shown in Figure 3.1. If nodes and ....

A. Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1992. 176 CHAPTER 3. APPLICATION TO VISUAL LANGUAGES

....0 H 0 , where G 0 , H 0 , and t 0 are the attened versions of G, H , and t, respectively. Obviously, the converse cannot hold as structural information gets lost in the attening process. For the de nition of attening, we extend the well known concept of hyperedge replacement (see [Hab92, DHK97] to hierarchical graphs. Let H be a hierarchical graph. For every set E EH of edges, H E denotes the hierarchical graph obtained by deleting all the edges in E from H , i.e. H E = hG; FH n E; ctsi where G = hV H ; EH n E; att ; labi, att and lab being the restrictions of att H and lab ....

Annegret Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. SpringerVerlag, 1992.

....H 0 , where G 0 , H 0 , and t 0 are the flattened versions of G, H , and t, respectively. Obviously, the converse cannot hold as structural information gets lost in the flattening process. For the definition of flattening, we extend the well known concept of hyperedge replacement (see [Hab92, DHK97] to hierarchical graphs. Let H be a hierarchical graph. For every set E EH of edges, H Gamma E denotes the hierarchical graph obtained by deleting all the edges in E from H , i.e. H Gamma E = hG; FH n E; ctsi where G = hV H ; EH n E; att ; labi, att and lab being the restrictions of ....

Annegret Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. SpringerVerlag, 1992.

....and is therefore skipped in this short version. 5 Flattening A natural operation on hierarchical graphs is the flattening operation which removes the hierarchy by recursively replacing every frame with its contents. For this, we use the well known concept of hyperedge replacement (see [9, 4]) in a slightly generalized form. Flattening is similar to (a recursive version of) the operation considered in Section 3, but it removes all frames and identifies their attached nodes with the corresponding points of their contents. If the numbers of attached nodes and points differ, the ....

A. Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1992.

....to those of finite state transductions on strings. In particular, they are closed under composition and they preserve the class of context free graph languages. In fact, there are two large classes of context free graph languages, namely, those generated by hyperedge replacement (HR, see, e.g. [Hab92, DKH97, Eng97]) and those generated by node replacement (NR, see, e.g. ER97] Both of them are preserved by (two different types of) MSO transductions. Moreover, both of them can be characterized in terms of MSO transductions: they are obtained by applying MSO transductions to regular tree languages [EvO97, ....

A. Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, 1992.

....forward and is therefore skipped in this short version. 5 Flattening A natural operation on hierarchical graphs is the attening operation which removes the hierarchy by recursively replacing every frame with its contents. For this, we use the well known concept of hyperedge replacement (see [8, 4]) in a slightly generalized form. Flattening is similar to (a recursive version of) the operation considered in Section 3, but it removes all frames and identi es their attached nodes with the corresponding points of their contents. If the numbers of attached nodes and points di er, the ....

A. Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1992.

....areas of circles. 3 Hypergraph representation of diagrams Hypergraphs have proved to be an intuitive means for internally representing diagrams [12, 13, 15, 22] A hypergraph is a generalization of a graph, in which edges are hyperedges which can be connected to any (fixed) number of nodes [7]. Each hyperedge has a type and a number of connection points ( tentacles ) that attach to nodes. We say the hyperedge visits these nodes. The familiar directed graph can be seen as a hypergraph in which all hyperedges visit exactly two nodes. A specification of a diagram language based on such a ....

.... grammars: Similar to context free string grammars, context free hypergraph grammars are defined in terms of their productions: A context free production consists of a single nonterminal hyperedge as lhs, a discrete interface containing each node of the lhs, and injective morphisms p l ; p r [7]. In contrast, restricted contextsensitive hypergraph grammars consist of context free as well as embedding productions that add hyperedges to a certain context. Figure 9 shows the productions of a restricted context sensitive grammar for MSC. For VEX and the hypergraph model described in Section ....

A. Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1992.

....(e) this hyperedge is not present in H 1 j V 1 , but nevertheless it may be in Hj V 1 [V 2 because port H 2 (p) 2 VH . Therefore, we only get H 1 j V 1 = H 2 j V 2 Hj V 1 [V 2 in this case. Another type of hypergraph operations is provided by the well known notion of hyperedge replacement (cf. [BC87, HK87, Hab92]) For this, we consider hypergraphs whose sets of port labels are initial segments of N . Call a hypergraphs H with PH = f1; kg for some k 2 N a k hypergraph. In particular, a 0 hypergraph is a portless hypergraph. Then, a hyperedge e in a k hypergraph H can be replaced with a type H ....

....operation on hypergraphs. The result of its application to n argument hypergraphs is obtained by replacing the n hyperedges by the arguments. To the author s knowledge the first to take this view of hyperedge replacement was Courcelle in [Cou91] 3. 4 Definition (hyperedge replacement, cf. [BC87, HK87, Cou91, Hab92]) Let H be a k hypergraph for some k 2 N and let e 1 ; e n 2 EH be pairwise distinct, so called virtual hyperedges. For i = 1; n let H i be a type H (e i ) hypergraph. 1. Let H 0 = HtF (H 1 )t Delta Delta Delta tF (H n ) The (simultaneous) replacement of H 1 ; H n ....

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A. Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer, 1992.

.... [6,7] efficient evaluation of functional expressions [8,9] and logic programs [10] synchronization mechanisms [11] distributed systems [11,12,13,5] object oriented systems [14] applied software systems [15] implementation of abstract data types [16] and context free hyperedge replacement [17] (see also [18] in this book) Historically, the first of the algebraic approaches to graph transformation is the so called double pushout (DPO) approach introduced in [1] which owes its name to the basic algebraic construction used to define a direct derivation step: This is modelled indeed by ....

A. Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer Verlag, 1992.

....which replace the node s hyperedge in the derivation sequence. A formal definition of derivation trees for context free hypergraph grammars and a detailed discussion of their 1.3. DEFINING THE SYNTAX OF VISUAL LANGUAGES 21 properties and relationships to derivation sequences can be found in [62,63]. The derivation tree of a terminal hypergraph starting at the axiom represents the set of all equivalent derivation sequences and thus the syntactic structure of the hypergraph. Therefore, this tree is the preferred representation of the derivation history as shown in Figure 1.1. If nodes and ....

A. Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1992.

.... study of relations which can be computed by tree transductions and in particular our second main result, which is explained next was inspired by the work of Habel, Kreowski, and Vogler [HKV91] on the decidability of boundedness problems for so called compatible functions on graphs (see also [Hab92, Chapter VII]) Further work in this direction was presented by Courcelle and Mosbah [CM93] by Seidl [Sei94b] and by Wanke [Wan94] For a detailed discussion of the relationship of computability by tree transductions with the notions studied in these papers see [Dre96a, Sections 3.3, 3.4, and 4.2] We give a ....

....of tree transductions that transform k input trees into one output tree) As discussed in [Dre96a, Sections 3.3 and 3. 4] the case C = dtTD (of deterministic and total top down tree transductions) yields a notion which is roughly equivalent to the notions of compatibility known from [HKV91] and [Hab92, Chapter VII], and of inductive computability studied in [CM93] The tree automata with cost functions introduced in [Sei94b] correspond to C = BU, and those with multi dimensional cost functions lead to the more general case that C = dtTD ffi BU. The second remark concerns the fact that only effective TBY ....

Annegret Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, 1992.

....In fact, the results of this paper also yield a new proof for the characterization by Engelfriet and Heyker. 1 Introduction Hyperedge replacement graph grammars, also called context free hypergraph grammars, are well studied devices for the generation of graph and hypergraph languages (see, e.g. [Hab92, Eng97, DHK97]) Their basic operation is the replacement of a non terminal hyperedge by a hypergraph. This works by removing the replaced hyperedge e and adding the replacing hypergraph H in such a way that, if e has k incident nodes, these nodes are identified with k distinguished nodes of H called its ....

....that the power of hyperedgereplacement graph grammars depends on the maximum type of right hand side hypergraphs (called the order of the grammar) where the type of a hypergraph is the number of points. In fact, it is well known that the generation of the set of all k trees requires order k (see [Hab92]) But even if the very special case of string graphs (i.e. the natural representation of strings by edge labelled graphs) is considered, the generating power of hyperedge replacement graph grammars increases properly whenever the order is increased by 2 (see [EH91, Hab92] There is an important ....

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Annegret Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, 1992.

....in a euclidean space of some dimension) where the overlay of the parts yields the picture. The generation of collages is based on the replacement of nonterminals and is closely related to hyperedge replacement which is known as a context free type of graph and hypergraph generation (see, e.g. [Hab92,DHK97]) The replacement of a nonterminal by other nonterminals or by parts involves the application of affine transformations. As replacements of different nonterminals are independent of each other, one cannot expect that generated languages can consist of collages with a non linear growth concerning ....

Annegret Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, 1992.

....results of this paper also yield a new proof for the characterization by Engelfriet and Heyker. 1 Introduction Hyperedge replacement graph grammars (hr grammars) also called contextfree hypergraph grammars, are well studied devices for the generation of graph and hypergraph languages (see, e.g. [Hab92,Eng97,DHK97]) Their basic operation is the replacement of a non terminal hyperedge by a hypergraph. This works by removing the replaced hyperedge e and adding the replacing hypergraph R in such a way that, if e has k incident nodes, these nodes are identified with k distinguished nodes of R called its ....

....side hypergraphs (called the order of the grammar) where the type of a hypergraph is the number of points. In fact, even if only string graphs (i.e. edge labelled, chain like graphs of type 2) are generated, the power of hr grammars increases properly whenever the order is increased by 2 (see [EH91,Hab92]) The string case is interesting because even the use of string graphs as right hand sides suffices to generate the set of all string graphs over a fixed alphabet. However, making use of larger and larger types, one can specify more and more sophisticated subsets of this basic set. For example, ....

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Annegret Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, 1992.

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Annegret Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer, 1992.

....the set FULLTREE of all full binary trees of height n (for some n 2 IN 0 ) can be generated by a context free graph grammar with positive and negative application conditions. On the other hand, FULLTREE is a non semilinear language. Thus, by Parikh s theorem for context free graph languages (see [14]) FULLTREE cannot be generated by a context free graph grammar without application conditions. Theorem 6.5 L(CF ) 0 L(CF; nc) 4 Proof: As shown in example 6.2.2, the set FULLTREE of all full binary trees of height n (for some n 2 IN 0 ) can be generated by a context free graph grammar with ....

....example 6.2.2, the set FULLTREE of all full binary trees of height n (for some n 2 IN 0 ) can be generated by a context free graph grammar with negative application conditions. On the other hand, FULLTREE is a non semilinear language. Thus, by Parikh s theorem for context free graph languages (see [14]) FULLTREE cannot be generated by a context free graph grammar without application conditions. Theorem 6.6 L(CF ) 0 L(CF; pc) 4 Proof: The set L = fa 2 n jn 2 IN 0 g of all strings over fag of length 2 n (for some n 2 IN 0 ) is a context sensitive string language. The corresponding ....

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Annegret Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1992.

....this may no longer be true for generated fractals because the generated languages may be infinite. The idea of hyperedge replacement stems from the field of graph grammars, where hyperedge replacement is used very successfully as a context free mechanism to generate graph languages (see, e. g, [Hab92]) 1 2 Preliminaries In this section the basic notions and notations concerning collages are recalled (cf. HK91, HKT93] For a set X, X) denotes the powerset of X and X denotes the set of all finite sequences over X, including the empty sequence . Given sets X and Y , Y Gamma X denotes ....

....to the literature. In [HK91, HKT93] collage grammars (invented by Habel and Kreowski in [HK88] are studied with main emphasis on context freeness and fixed point results. It is shown that collage grammars are context free in much the same sense as hyperedge replacement graph grammars (cf. [Hab92]) are. Due to context freeness, every derivation in a collage grammar that starts with a single hyperedge can be divided into the first step (yielding the right hand side of the applied production) and the subderivations originating from the hyperedges in the right hand side. Then the replacement ....

A. Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer, 1992.

....programming languages, compiler description, database systems, specification of abstract data types, specification of distributed systems, etc. This development is documented in proceedings and other collections of selected papers as well as on some monographs ( CER79, CM95, EKR91, ENR83, ENRR87, Hab92, PvE93, SE94, SPvE93] The aim of this survey paper is to point out some recent trends in applied graph transformation as a rule based framework for the specification and development of systems, languages, and tools. The central part of the paper (Section 3) describes four case studies that ....

....to offer a variety of graph grammar approaches. At least there will be variants for a PROGRES like approach [Sch90, Sch91b] and the algebraic one [Ehr79, Low93] Further variants will be restricted classes of graph grammars like node, edge or hyperedge replacement grammars (see e.g. JR80, HK87, Hab92] A final goal of GRACE is to allow user defined approaches. Operational semantics. The operational semantics of a graph grammar is usually defined as the set of all derivations that begin with a distinguished initial graph. If one is not interested in the intermediate graphs of a derivation or ....

Annegret Habel. Hyperedge Replacement: Grammars and Languages, volume 643 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1992.

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Habel, A.: Hyperedge replacement: grammars and languages. Volume 643 of Lecture Notes in Computer Science. Springer-Verlag Inc., New York, NY, USA (1992)

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