Joost Engelfriet. Graph grammars and tree transducers. In S. Tison, editor, Proc. CAAP 94, volume 787 of Lecture Notes in Computer Science, pages 15-37. Springer, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....interactions [1] and the theory of optimal reductions [2] The present paper is independent of these calculus related theories. We develop several tools for the MS logic of higher level trees. Our rst tool is a signature of graph operations studied by Courcelle, Olariu, Engelfriet and others [10,16,6,9,13]. We use operations which allow to add edges to a vertex labelled graph, from every vertex labelled by, say, a to every vertex labelled by, say, b. Other operations rename labels and a binary operation builds the disjoint union of two labelled graphs. Using these operations, graphs are generated ....

J. Engelfriet. Graph grammars and tree transducers. In S. Tison, editor, Trees in Algebra and Programming { CAAP'94, pages 15-36, Edinburgh, Apr. 1994. LNCS 787.

....the theory of optimal reductions [AG98] The present paper is independent of these calculus related theories. We develop several tools for the MS logic of higher level trees. Our tools are based on graph operations extensively studied during the past decade by Courcelle, Engelfriet and others [CER93,Eng94,Cou92,Cou97]. We use operations which allow to add edges from the vertices of a given label to the vertices of another given label, to rename labels of vertices and to make disjoint union of two graphs. Using these operations, graphs are generated from constants, each of which represents a single labeled ....

Joost Engelfriet. Graph grammars and tree transducers. In Sophie Tison, editor, Trees in Algebra and Programming { CAAP'94, pages 15-36, Edinburgh, April 1994. LNCS 787.

.... adds edges between every vertex labelled by p and every vertex labelled by q (this last operation together with disjoint union is at the base of the de nition of clique width) See (Bauderon, Courcelle 1987, Courcelle 1992, Courcelle 1997, Courcelle, Makowsky, Rotics 2000, Courcelle, Olariu 2000, Engelfriet 1994) about these graph operations. In general, for every choice of a nite set F of operations on nite structures Fusion and MS Properties 3 (including constant symbols denoting basic structures) one gets a mapping val F from the set T (F) of nite well formed terms over F to the considered class ....

....applies to hypergraphs and relational structures. The so called Hyperedge Replacement (HR) context free) sets of hypergraphs, and the so called Vertex Replacement (VR) context free) sets of graphs can be de ned in this way in terms of appropriate operations. See (Bauderon, Courcelle 1987, Engelfriet 1994, or Courcelle 97) We say that a set of structures is F context free , if it is a component of the least solution of a system of recursive set equations over the operations in F . If a nite set F of operations on structures is such that val F is MS compatible, then, for every F context free set ....

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J. Engelfriet (1994) Graph grammars and tree transducers, Proceedings CAAP'94, Lec. Notes Comp. Sci. 787 (1994), pp.5-36.

....to graphs are immediate for any set of graph operations. For appropriate graph operations, one obtains the notions of HR and NR context free grammars in terms of systems of algebraic equations. A survey on the use of tree transducer theory, in this Universal Algebra setting, is presented in [Eng94] Hyperedge Replacement In the theory of HR graph grammars and languages essentially three different types of graph properties with nice decidability and structural results have been found: the compatible, recognizable (or finite) and inductive graph properties. In [Cou90b, HKL93] it is shown ....

....EHL94] and linear [EL89b] NR grammars. Boundary NR grammars are more powerful than HR grammars [Vog93, ER90c] Many types of NLC (Node Label Controlled) graph grammars [JR80, ER91] are also special cases of the C edNCE grammars. The results of [HKV91] are generalized to NR grammars in [Eng91, Eng94] In [Cou95c] a decidable condition is given for a set of graphs generated by an NR graph grammar to be generable by an HR grammar. The condition implies that the NR and HR grammars generate the same class of graph languages of bounded degree. It is shown in [EHL94] that this is exactly the ....

J. Engelfriet. Graph grammars and tree transducers. In Proc. CAAP'94, Lecture Notes in Computer Science 787, pp.15--36, 1994.

....the ESPRIT Working Group APPLIGRAPH through the University of Bremen. 1 The idea to associate a tree grammar or tree transducer with an algebra that maps trees into a semantic domain was already mentioned in [14] For graphs generated by node or hyperedge replacement this has been worked out in [15] (see also [3, 11, 9] For the field of picture generation the idea seems to be new, however. In the main part of the paper four of the well known classes of picture generating devices found in the literature, namely collage grammars [22] mutually recursive function systems (a generalised type ....

Joost Engelfriet. Graph grammars and tree transducers. In S. Tison, editor, Proc. CAAP 94, volume 787 of Lecture Notes in Computer Science, pages 15--37. Springer, 1994.

....(and all its adjacent hyperedges) is replaced with another hypergraph. The latter is embedded by adding new hyperedges to the former neighbours of the vertex. These hyperedges and their labels further depend on the labels of the vertices involved and the removed hyperedges. We refer the reader to [21, 22, 16] for a comparison between both approaches of hypergraph rewriting, in particular on the algebraic level. Key words and phrases. Monadic second order logic, simple hypergraphs, systems of regular equations. 1 The article [7] is not very precise in this respect. Actually, the equational ....

....would be slightly more convenient to write (and the copies L i could be abandoned) if add l 1 ;l 2 ;l 3 and create l 1 ;l 2 ;i 1 : i k operations were combined into add 0 l 1 ;l 2 ;l 3 ;i 1 : i k operations and these in turn would be merged into parallel compositions. Like the operation C in [21], they would only introduce edges between their arguments. 18 KLAUS BARTHELMANN (Th(val(t) v 1 : vn ) v 1 : vn ) is always an edge in val(t 0 ) If k dioeerent leaves are needed, the third item in the above construction is used at least k Gamma 1 times. On the other hand, for ....

J. Engelfriet, Graph grammars and tree transducers, in: S. Tison, ed., Trees in Algebra and Programming CAAP '94, Lecture Notes in Computer Science, Vol. 787 (Springer, 1994) 1536.

....it has a least upper bound B i . It is obvious from the de nitions that B i = occ n i (G i ) Lemma 3.7. It is decidable whether jsat(G i ; OE)j = 1 for i 2 [k] and OE 2 QF n i ( x) Proof. Since all expressions in Proposition 3. 6 are written with addition and multiplication, the results of [25, 23] apply. The sets of nite graphs fH ij j j 2 Ng are context free for every i 2 [k] as already indicated. We conclude that B i = occ n i (G i ) can be computed eoeectively. After all formulas OE with jsat(G i ; OE)j = 1, i 2 [k] have been identi ed the other values can be computed iteratively ....

J. Engelfriet, Graph grammars and tree transducers, in: S. Tison, ed., Trees in Algebra and Programming CAAP '94, Lecture Notes in Computer Science, Vol. 787 (Springer, 1994) 1536.

....does not necessarily need an HR algebra on the left, and the functions may also be partial, or even arbitrary relations. This use of tree transducers has already been discussed by Engelfriet in [Eng80] For more recent work on transducibility see also the papers by Engelfriet and Drewes [Eng94, Dre94]. For many classes of graphs definable by hyperedge replacement (such as, for example, series parallel graphs, k trees, etc. there are single natural HR algebras whose operations allow to generate just these graphs. For such a standard algebra it is often quite easy to see whether a given ....

....loops are not allowed, that is, the nodes a hyperedge is incident with are pairwise distinct. Some nodes can be distinguished from the others by pairwise distinct labels. These so called ports are needed to defined certain operations on hypergraphs. 3. 1 Definition (hypergraph with ports, cf. [Cou92, Eng94]) Let L be a set of hyperedge labels and let P be a set of port labels. A hypergraph with hyperedge labels in L and port labels in P is a tuple H = V; E; att ; lab; port) where ffl V is the finite set of nodes, ffl E is the finite set of hyperedges, ffl att : E V is the attachment of ....

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J. Engelfriet. Graph grammars and tree transducers. In Proc. CAAP 94, Lecture Notes in Computer Science 787, 15--37, 1994.

....We now will show that it is decidable for a realizable function f 0 , whether or not a regular tree grammar, generating hypergraph expressions, generates expressions that evaluate to hypergraphs whose function values under f 0 grow beyond any bound. These results were presented in another form in [Eng94]. We use numerical functions that can be realized by a top down finite tree transducer, which transforms the hypergraph expressions into expressions in the Gamma N algebra N = N; ff) for the arithmetic caclulus, which was defined in Definition 5.1. For the simplicity of the proofs, we will ....

....(boundedness theorem) Let Delta SUB and f 0 : HG N a ( Delta; N) realizable function. Then it is decidable for a regular tree grammar G over Delta, whether or not there is a v 2 N, such that for all t 2 L(G) f 0 (val(t) v. Proof The proof is similar to the proof of Proposition 15 in [Eng94]. Let G = N; Sigma; R; S) be an arbitrary regular tree grammar over Delta. We have to show that it is decidable for G, whether or not f 0 (val(L(G) is bounded or, equivalently, whether or not f 0 (val(L(G) is finite. Clearly, Sigma is a finite subset of Delta, so there exists a total ....

Joost Engelfriet, Graph Grammars and Tree Transducers, proceedings CAAP'94, Sophie Tison (ed.), Lecture Notes in Computer Science, Vol. 787, Springer Verlag, Berlin, Germany, pp. 15--36.

....a term t that denotes an object O may be transformed by a tree transduction into a term (t) that denotes another object O 0 possibly with respect to a different algebra. Thus, a transduction of terms induces a transformation of objects between possibly different types of data (see also [Eng80, Eng94, Dre96a]) From a theoretical point of view maybe the most interesting aspect is that many properties of the generated (sets of) objects and of the induced transformations turn out to be decidable, provided that the chosen grammars, Partially supported by the EC TMR Network GETGRATS (General Theory of ....

....and of the induced transformations turn out to be decidable, provided that the chosen grammars, Partially supported by the EC TMR Network GETGRATS (General Theory of Graph Transformation Systems) through the University of Bremen. transducers, and algebras are taken from suitable classes [Eng94, Dre96a, DE98]. In this paper, a software system called Treebag (Tree Based Generator) is presented in which these facts are exploited in order to allow for the generation, manipulation, and visualization of objects of various kinds. From the user s point of view (which in many ways just reflects the internal ....

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Joost Engelfriet. Graph grammars and tree transducers. In S. Tison, editor, Proc. CAAP 94, volume 787 of Lecture Notes in Computer Science, pages 15--37. Springer, 1994.

....[BC87] and Habel and Kreowski [HK87] This work was continued in diverse directions. For references see, e.g. Hab92] Eng97] and [Roz97, Chapters 1, 2, and 5] Results that establish in different ways connections between tree transductions and hyperedge replacement can be found in, e.g. [EH91, EH92, Eng94, EV94, EV96, Dre96]. The notion of hypergraphs to be used in this paper is defined below. 3.1 Definition (pointed hypergraph) A pointed hypergraph is a tuple G = V; E; att ; lab; point) where ffl V is the finite set of nodes, ffl E with E V = is the finite set of hyperedges, ffl att : E V is the ....

Joost Engelfriet. Graph grammars and tree transducers. In S. Tison, editor, Proc. CAAP 94, volume 787 of Lecture Notes in Computer Science, pages 15--37. Springer, 1994.

....: Hn yields the hypergraph obtained from the disjoint union of H and F (H 1 ) F (Hn ) by removing e 1 ; e n and identifying att H (e i ; j) with port H i (j) for i = 1; n and j = 1; type H (e i ) The resulting hypergraph keeps the port labelling of H. 1 See [8] for this terminology. Using hyperedge replacement, every hypergraph H as above can be seen as a (partial) n ary operation on (isomorphism classes of) hypergraphs (see [3] by defining H(H 1 ; Hn ) H[e 1 =H 1 Delta Delta Delta e n =Hn ] For simplicity, we assume that an arbitrary, ....

....bu: T(A) T(B) bu(t) is defined to be the set of all terms t 0 2 T(B) such that t fl(t 0 ) for some final state fl. By TB we denote the set of all transductions 1 ffi Delta Delta Delta ffi n composed of finitely many top down and bottom up tree transductions (see also [6, 8]) To define tree to graph to tree transductions (tgt transductions, see [9] we 1 Delta Delta Delta Delta Delta Delta f 1 n n 1 1 e 1 1 e n 1 f 1 2 3 g 1 2 f 1 2 3 a 1 c 1 a 1 H f hyp(f(g(a) f(c; a) Figure 1: The HR operation H f and the term hypergraph for f(g(a) ....

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Joost Engelfriet. Graph grammars and tree transducers. In Proc. CAAP 94, Lecture Notes in Computer Science 787, 15--37, 1994.

....regular tree language R over Sigma , the set Val Sigma (R) fval Sigma (t) j t 2 Rg is a cf HR language. The class fval Sigma j Sigma is a ranked alphabet of hypergraph operationsg is denoted by VALHR . Then HR = VALHR (RGT ) where RGT denotes the class of regular tree languages (see, e.g. [BC87, Eng94, Dre96]) In the same way we can consider derivation trees of CDHR grammars in tmode. Now the question arises, for which class X of tree languages is VALHR (X) equal to CDHR(t) In [Ful96] CDRGT grammars have been introduced. It will turn out that the class of CDRGT languages in t mode is precisely ....

J. Engelfriet. Graph grammars and tree transducers. In S. Tison, editor, Proc. CAAP'94, volume 787 of LNCS, pages 15--36. Springer-Verlag, 1994.

....of Bremen. with every derivation tree the corresponding picture. The idea to associate a tree grammar or tree transducer with an algebra that maps trees into a semantic domain was already mentioned in [Eng80] For graphs generated by node or hyperedge replacement this has been worked out in [Eng94] (see also [CE95, DE96, Dre97] For the field of picture generation the idea seems to be new, however. In the main part of the paper four of the well known classes of picture generating devices found in the literature, namely collage grammars [HK91] mutually recursive function systems (or ....

Joost Engelfriet. Graph grammars and tree transducers. In S. Tison, editor, Proc. CAAP 94, volume 787 of Lecture Notes in Computer Science, pages 15--37. Springer, 1994.

....Section 3.3] our second main result can be viewed as a generalization of the main result of [HKV91] which corresponds to the case of regular tree languages S and top down tree transductions r. A formulation of the proof of [HKV91] in terms of mappings computed by tree transductions was given in [Eng94]. There, the tree transductions considered are compositions of top down and bottom up tree transductions (which makes it already a little more general than the result in [HKV91] and the output algebra is N with operations 0, 1, addition, multiplication, and maximum (as in [HKV91] Here we ....

.... transductions considered are compositions of top down and bottom up tree transductions (which makes it already a little more general than the result in [HKV91] and the output algebra is N with operations 0, 1, addition, multiplication, and maximum (as in [HKV91] Here we generalize the proof of [Eng94] in two ways. Firstly, thanks to the decision algorithm that checks finiteness of tree languages in R TBY it is now possible to consider the much larger class TBY of tree transductions. Secondly, and this is the part that requires additional work, we abstract from the concrete domain of ....

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Joost Engelfriet. Graph grammars and tree transducers. In S. Tison, editor, Proc. CAAP 94, volume 787 of Lecture Notes in Computer Science, pages 15--37. Springer, 1994.

....q (s) M q (s) Hence M 0 2 T fc and M = M 00 ffi M 0 . By a straightforward product construction. ut 4 Tree Graph Operations To state the characterization of TR(HR) by Drewes in [Dre97] we have to recall the characterization of HR in terms of operations on hypergraphs (cf. [BC87, Eng94, Dre96]) A hypergraph that contains variables z 1 ; z k can be seen as a k ary operation on hypergraphs. Let Z k = fz j i j 1 i k; j 0g with rank(z j i ) j. If g 2 HGR( Delta [ Z k ) and for every 1 i k there is exactly one j i such that z j i i appears in g, then g is a k ary ....

J. Engelfriet. Graph grammars and tree transducers. In S. Tison, editor, Proc. CAAP'94, volume 787 of LNCS, pages 15--36. Springer-Verlag, 1994.

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Joost Engelfriet. Graph grammars and tree transducers. In S. Tison, editor, Proc. CAAP 94, volume 787 of Lecture Notes in Computer Science, pages 15-37. Springer, 1994.

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