A. Habel, H.-J. Kreowski, May we introduce to you : hyperedge replacement, Lecture Notes in Computer Science 291 (1987), 15-26.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to the context according to some embedding mechanism. Due to the very structure of graphs, this embedding mechanism is often given in an algorithmic way, leading to graph rewriting systems which are not context preserving in our sense (e.g. the hyperedge replacement approach of Habel and Kreowski [10, 15], the Node Label Controlled approach of Janssens and Rozenberg [17] or its extensions [18, 25, 26] Hence, expanding graph relabeling systems can be viewed as graph rewriting systems without any embedding mechanism. This notion of context preservation can be related to the complexity of the ....

A. Habel, H.-J. Kreowski, May we introduce to you : hyperedge replacement, Lecture Notes in Computer Science 291 (1987), 15-26.

....languages generated by HR grammars. 1 Introduction Context free graph languages are generated by context free graph grammars, which are usually graph replacement systems. One of the most popular types of context free graph grammar is the Hyperedge Replacement System, or HR grammar (see, e.g. [Hab, HabKre, HabKV]) A completely di#erent way of generating graphs is to select a number of graph operations, to generate a set of expressions (built from these operations) and to interpret the expressions as graphs. The set of expressions is generated by a classical context free grammar generating strings (or ....

A.Habel, H.-J.Kreowski; May we introduce to you: hyperedge replacement, in Graph-Grammars and their Application to Computer Science (H.Ehrig, M.Nagl, G.Rozenberg, A.Rosenfeld, eds.), Lecture Notes in Computer Science 291, Springer-Verlag, Berlin, 1987, pp.15-26

....graphs by mapping nodes to graphs (and includes NLC and NCE) and so edges in the original graph are only manipulated as a consequence of node transformations, the latter rewrite (hyper)edges 6 to (hyper)graphs. Hyper)edge rewriting grammars have been primarily investigated by Kreowski [45,23], Lautemann [71,70] and Courcelle [13,15] where they have been successfully used to establish many decidability properties for graph languages. The extension to handle rewriting hypergraph grammars is through the inclusion of the vertices to which the hyperedge is attached in the rewrite; such a ....

Annegret Habel and Hans-Jorg Kreowksi. May we introduce to you: Hyperedge replacement. In H. Ehrig, M. Nagl, G. Rozenberg, and A. Rosenfeld, editors, Proceedings of the 3rd International Workshop on Graph Grammars and Their Application to Computer Science, number 291 in Lecture Notes in Computer Science, pages 15--26, Warrenton, Virginia, December 2--6, 1986. SpringerVerlag.

....bounded transformation h k; Then, the graph problem can be solved in a reasonable amount of time and space in many practical applications. Another interesting way to de ne classes of graphs is by graph grammars. For instance, for each positive integer k there is a hyperedge replacement system [HK87] whose language is the set of all partial k trees. However, we do not know any already de ned rewriting system whose language is exactly the set of all k NLC graphs or k NLC trees. Rewriting systems whose languages are similar to k NLC graphs are certain versions of node label controlled (NLC) ....

A. Habel and H.J. Kreowski. May we introduce to you: Hyperedge replacement. In H. Ehrig, M. Nagl, A. Rosenfeld, and G. Rozenberg, editors, Graph-Grammars and Their Application to Computer Science, volume 291 of Lecture Notes in Computer Science, pages 15-26. Springer-Verlag, Berlin/New York, 1987.

....vertices) and edges (or arcs) Were a graph is composed of two kinds of objects two schools can be recognised in graph rewriting theory: one concentrating on replacements of nodes and one concentrating on the replacement of edges. Here only edge replacement, as originating from the Berlin school[Habel87,Ehrig91] is discussed. f f f TRANSFORMATIONAL DESIGN OF DIGITAL SYSTEMS RELATED TO GRAPH REWRITING 4 Definition A directed graph is a 4 tuple G = N, E, s, t in which N is the set of nodes and E the set of directed edges s : E N gives the source of the edges Figure 3: a) usual representation ....

....between the inputs and outputs, external ports, is meant. For example in the graph of figure 6 the nodes labelled with x, y are wanted to be distinguished as inputs and the node labelled z as output. External ports need to be defined. For this purpose the definition of multi ported graphs[Habel87 5 ], also called graphs with distinguished nodes can be used. Definition: A multi ported directed hypergraph is a 4 tuple G = N, E, s, t, P in , P out in which N is the set of nodes and E the set of hyperedges s : E N gives the sources of the hyperedges Figure 8: A multi ported graph of ....

A. Habel H.-J. Kreowski, May we introduce to you: Hyperedge replacement, in H. Ehrig, M. Nagl, G. Rozenberg, A. Rosenfeld(Eds) Graph-Grammars and their Application to Computer Science , LNCS 291, (Springer-Verlag, Berlin Heidelberg,1987) pp.15-26.

....vertices) and edges (or arcs) Where a graph is composed of two kinds of objects two schools can be recognised in graph rewriting theory: one concentrating on replacements of nodes and one concentrating on the replacement of edges. Here only edge replacement, as originating from the Berlin school [17,18] is discussed. Hypergraphs are graphs in which the edges, called hyperedges, are allowed to have more than one source and more than one target. Sources and targets of a hyperedge are given by lists of nodes. The ordering of sources and targets is explicitly part of the structure of hypergraphs. ....

A. Habel H.-J. Kreowski, May we introduce to you: Hyperedge replacement, in H. Ehrig, M. Nagl, G. Rozenberg, A. Rosenfeld(Eds) Graph-Grammars and their Application to Computer Science , LNCS 291, (Springer-Verlag, Berlin Heidelberg,1987) pp.15-26.

....can specify the top down design, or syntactic generation, of such a model. Three main types of hypergraph grammars which are context free in the sense of [Cou87] are considered in the literature, depending on the kind of subgraph which is rewritten in one derivation step: hyperedge rewriting [BC87,HK87], separated handle rewriting [CER93] see also [KJ99] and con uent node rewriting [Kle96] With respect to their generative power, hyperedge rewriting and separated handle rewriting grammars are known to be incomparable [CER93] and the union of their generated languages is properly contained in ....

Annegret Habel and Hans-Jorg Kreowski. May we introduce to you: Hyperedge replacement. In H. Ehrig, M. Nagl, G. Rozenberg, and A. Rosenfeld, editors, Proc. Third Intl. Workshop on Graph Grammars and Their Application to Comp. Sci., volume 291 of Lecture Notes in Computer Science, pages 15-26, 1987.

....transformation h k; Pi . Then, the graph problem can be solved in a reasonable amount of time and space in many practical applications. Another interesting way to define classes of graphs is by graph grammars. For instance, for each positive integer k there is a hyperedge replacement system [HK87] whose language is the set of all partial k trees. However, we do not know any already defined rewriting system whose language is exactly the set of all k NLC graphs or k NLC trees. Rewriting systems whose languages are similar to k NLC graphs are certain versions of node label controlled (NLC) ....

A. Habel and H.J. Kreowski. May we introduce to you: Hyperedge replacement. In H. Ehrig, M. Nagl, A. Rosenfeld, and G. Rozenberg, editors, Graph-Grammars and Their Application to Computer Science, volume 291 of Lecture Notes in Computer Science, pages 15--26. Springer-Verlag, Berlin/New York, 1987.

....graph languages (see [9, 12, 14] The equivalence of HR with these sets of operations will be shown in Proposition 8. Thus, they all define the HR context free graph languages. It was proved in [1] that these are exactly the graph languages generated by the Hyperedge Replacement graph grammars of [33, 32, 18, 1]. The set of operations VR was defined in [15] To be precise, it was defined for a more general type of graphs (in which one node can have several port labels) the equivalence of the two sets of operations was shown in Lemma 1.1 of [8] using a tree transducer and applying Proposition A, as in ....

A.Habel, H.-J.Kreowski; May we introduce to you: hyperedge replacement, in [20], pp. 15-26

.... which was first investigated in, e.g. Kau, Bra, Schu, Oos, Eng2] One advantage of the class C edNCE is that it is the largest known class of context free graph languages (where context free is taken in the sense of [Cou1] It includes, e.g. the HR (i.e. Hyperedge Replacement) languages of [BC, HK, Hab], the B NLC languages of [RW1, RW2] and the B edNCE languages of [ELW, EL] Thus, results on C edNCE apply to a quite large class of recursive praph properties. A second advantage of C edNCE is that it seems to be robust in the sense that it can be characterized in several different ways. It is ....

....of cliques is bounded for the graphs in L. ut Note that, of course, all decidability results of this section hold for classes of graph grammars that are effectively contained in RPD. Thus, apart from the C edNCE grammars, they also hold, e.g. for the Hyperedge Replacement grammars (see, e.g. [BC, HK, ER1]) the B NLC graph grammars (see [RW1] and the C NLC graph grammars (see [Cou1] Acknowledgement: The first author wishes to thank Bruno Courcelle, George Leih, and Roderick Bloem for stimulating discussions. ....

A.Habel, H.-J.Kreowski; May we introduce to you: hyperedge replacement, in

....to the context according to some embedding mechanism. Due to the very structure of graphs, this embedding mechanism is often given in an algorithmic way, leading to graph rewriting systems which are not context preserving in our sense (e.g. the hyperedge replacement approach of Habel and Kreowski [10, 15], the Node Label Controlled approach of Janssens and Rozenberg [17] or its extensions [18, 25, 26] Hence, expanding graph relabeling systems can be viewed as graph rewriting systems without any embedding mechanism. This notion of context preservation can be related to the complexity of the ....

A. Habel, H.-J. Kreowski, May we introduce to you : hyperedge replacement, Lecture Notes in Computer Science 291 (1987), 15-26.

....(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 ....

A. Habel, H.-J. Kreowski. May we introduce to you: Hyperedge replacement. Lecture Notes in Computer Science 291, 15--26, 1987.

....It turns out that the two formalisms generate the same class of hypergraph languages. twentysix Hypergraphs, hypergraph languages and hypergraph expressions 4.1 Hypergraphs A variety of different definitions of hypergraphs and hypergraph grammars is in use. Examples can be found in [EngHey91] [HabKre87], Hab92] and [BauCou87] We chose for the definitions in [Hab92] since this book was also the source for the predicates and numerical functions on hypergraphs and hypergraph languages. All definitions and theorems in this section and the sections 4.2 and 4.3 are from [Hab92] sometimes with a ....

Annegret Habel, H. J. Kreowski, May We Introduce to You: Hyperedge Replacement , in: Graph Grammars and Their Applications to Computer Science, Lecture Notes in Computer Science 291, 1987, pp. 15--26.

....and recursive families of graphs In this section, we consider some types of graph grammars, the recursive families of graphs and their relationships to treewidth. We will limit the presentation here to a few notions and results, and direct the readers for further reading to other sources, e.g. [30, 36, 46, 45]. 8.1 Hyperedge replacement grammars First, we consider the notion of hyperedge replacement grammar, introduced by Habel and Kreowski. We only give an informal description here: for a good introduction to this topic, see e.g. 45, 46] The framework of context free graph grammars of Bauderon ....

....readers for further reading to other sources, e.g. 30, 36, 46, 45] 8.1 Hyperedge replacement grammars First, we consider the notion of hyperedge replacement grammar, introduced by Habel and Kreowski. We only give an informal description here: for a good introduction to this topic, see e.g. [45, 46]. The framework of context free graph grammars of Bauderon and Courcelle is essentially similar [8] See also [66] Hyperedge replacement grammars work with hypergraphs, where each hyperedge is represented as a sequence of vertices. A hyperedge also has a label, which is either a terminal label, ....

A. Habel and H. J. Kreowski, May we introduce to you: hyperedge replacement, in Proc. Graph-Grammars and their Applications to Computer Science '86, H. Ehrig, M. Nagl, and A. Rosenberg, eds., Springer Verlag, Lecture Notes in Computer Science, vol. 291, 1987, pp. 15--26.

....: ae G [ H if K = G H, undefined otherwise : Definition 14 (Interface graph) An interface graph A = G; K) consists of a graph G and a subgraph K G. We call G the body and K the interface of A. Interface graphs are similar to hypergraphs with sources [BC87] multipointed hypergraphs [HK87] or partially abstract graphs [CM91] 3.1 Concrete Graph Rewriting Definition15 (Graph production and graph grammar) A graph production p = L; K;R) consists of graphs L, K, R such that K ae L and K R. The nodes of K are called glueing nodes. A graph grammar GG = P; S) consists of a set ....

A. Habel and H.-J. Kreowski. May we introduce to you: Hyperedge replacement. Lecture Notes in Computer Science 291, Springer, pages 15--25, 1987.

....languages generated by HR grammars. 1 Introduction Context free graph languages are generated by context free graph grammars, which are usually graph replacement systems. One of the most popular types of context free graph grammar is the Hyperedge Replacement System, or HR grammar (see, e.g. [Hab, HabKre, HabKV]) A completely different way of generating graphs is to select a number of graph operations, to generate a set of expressions (built from these operations) and to interpret the expressions as graphs. The set of expressions is generated by a classical context free grammar generating strings (or ....

A.Habel, H.-J.Kreowski; May we introduce to you: hyperedge replacement, in Graph-Grammars and their Application to Computer Science (H.Ehrig, M.Nagl, G.Rozenberg, A.Rosenfeld, eds.), Lecture Notes in Computer Science 291, Springer-Verlag, Berlin, 1987, pp.15-26

....one nonterminal node. It is easy to see that LIN edNCE B edNCE. It is shown in [EL1] that A edNCE and LIN edNCE are incomparable subclasses of B edNCE. The class B edNCE also contains the B NLC languages of [RW] and (as shown, e.g. in [ER1] the hyperedge replacement (HR) graph languages of [BC, HK, Hab]. Grammar G 1 from Example 1 is both linear and apex. Grammar G 0 2 is boundary, but not linear or apex. Grammars G 2 and G 3 are not boundary. The class A edNCE can be characterized within the class C edNCE: there is a simple condition on a graph language L 2 C edNCE that expresses membership ....

A.Habel, H.-J.Kreowski; May we introduce to you: hyperedge replacement, in

....grammars have two characteristics in common. Namely, A) they are all based on node replacement graph grammars and (B) none of them is based on a context free type of graph grammar. In [34] grammars based on edge replacement were introduced and later generalized to hyperedge replacement grammars [35]. These grammars, also called context free hypergraph grammar (for short, cfhg grammars) turned out to have a number of attractive features, such as an easy and uniform graph embedding mechanism and the straightforward representation of strings and terms (with and without sharing of common ....

.... (c) b) a) fl fl 1 2 1 3 1 Figure 3: The jungle ga , the parjungle gb , and the tree hypergraph gc tree(g b ) ffi (oe(y 1 ; y 2 ) ff; fl(y 2 ) and tree(g c ) tree(g a ) Let us now formulate the notion of context free hypergraph grammars (for short, cfhg grammars) which was introduced in [4, 35]. The concept of cfhg grammars is based on the replacement of hyperedges. A hyperedge e labeled by a nonterminal A will be replaced by the right hand side of a production which has the nonterminal A as left hand side. The left hand side of each production contains only a single nonterminal symbol. ....

A. Habel, H.-J. Kreowski, May we introduce to you: hyperedge replacement. In: [20], 15--26.

....rewriting rule. Shaded nodes in (a) represent the injective mapping of the distinguished gluing points. Application of the rule shown in (a) to the graph shown in (b) results in the graph shown in (c) A.5. 3 Hyperedge Replacement Systems The hyperedge replacement systems described in [HaKr87b] [HaKV89] DrKr91] manipulate hypergraphs, which consist of hyperedges. Each hyperedge has a label, m ordered incoming tentacles, and n ordered outgoing tentacles (Figure#A.9a) A hypergraph consists of a set of hyperedges joined at sockets (Figure#A.9b) Each rewrite rule specifies the ....

A. Habel, J.-J. Kreowski, "May We Introduce to You: Hyperedge Replacement," in [IWGG87], pp. 15-26.

....show that the problem of query answering can be reduced to checking the query on all expansions of a database. We then show that binary intensional queries are undecidable, even when the program contains only linear rules. 37 Hypergraph edge replacement grammars, introduced by Habel and Kreowski [47, 48], are generalizations of context free grammars, generating sets of hypergraphs instead of sets of strings. We present Courcelle s [22] slightly more general version of these grammars. Let A be an alphabet with a rank function : A N. A hypergraph on A is a tuple hV; E; lab; vert; srci where V ....

A. Habel and H.-J. Kreowski. May we introduce to you: Hyperedge replacement. In H. Ehrig, M. Nagl, G. Rozenberg, and A. Rosenfeld, editors, Graph-Grammars and their Applications to Computer Science 3rd International Workshop, pages 15--26. Springer LNCS No. 291, 1987.

....We show that the problem of query answering can be reduced to checking the query on all expansions of a database. We then show that binary intensional queries are undecidable, even when the program contains only linear rules. Hypergraph edge replacement grammars, introduced by Habel and Kreowski [10, 11], are generalizations of context free grammars, generating sets of hypergraphs instead of sets of strings. We present Courcelle s [5] slightly more general version of these grammars. Let A be an alphabet with a rank function : A N. A hypergraph on A is a tuple hV; E; lab; vert; srci where V ....

A. Habel and H.-J. Kreowski. May we introduce to you: Hyperedge replacement. In H. Ehrig, M. Nagl, G. Rozenberg, and A. Rosenfeld, editors, Graph-Grammars and their Applications to Computer Science 3rd International Workshop, pages 15--26. Springer LNCS No. 291, 1987.

....classes. Then g=R denotes the hypergraph (V 0 ; E g ; lab g ; nod 0 ; ext 0 ) where V 0 = V g =jR ) for every e 2 E g and i 2 [rk(e) nod 0 (e) i) OE(nod g (e) i) and for every i 2 [rk(g) ext 0 (i) OE(ext g (i) Hypergraph substitution is defined as in [BC87] see also [HK87a, HK87b, Hab92]. Roughly speaking, in a hypergraph g, an edge e of rank m is replaced by a hypergraph h of rank m by pairwise identifying the nodes that are incident with e, with corresponding external nodes (cf. Fig. 1) Formally, for a ranked set Gamma, let g 2 n HGR( Gamma) with n 0. Let e 2 E g with rank ....

A. Habel and H.-J. Kreowski. May we introduce to you: hyperedge replacement. In H. Ehrig, M. Nagl, G. Rozenberg, and A. Rosenfeld, editors, Graph grammars and their application to computer science, LNCS 291, pages 15--26. Springer-Verlag, 1987.

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