Plump, D., Term graph rewriting, in: H. Ehrig and G. Engels et alii,editors, Handbook of Graph Grammars and Computing by Graph Transformation, II: Applications, Languages and Tools, Theoretical Computer Science 2,World Scientific, 1999 pp. 3--61.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....analyzed statically, represent potential conflicts in a minimal context. If the rewrite system is terminating, confluence follows if all critical pairs can be joined [15] This theory of critical pairs and confluence has been transferred to transformation systems on term graphs and hyper graphs [17, 18]. However, in most applications of graph transformation to visual modelling techniques, attributed graphs are used to represent diagrams with textual, numerical, or layout information, semantic annotations, etc. To develop the theory of critical pairs and confluence in this case is the aim of this ....

D. Plump. Term graph rewriting. In G. Engels, H.-J. Kreowski, and G. Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, Volume 2: Applications, Languages, and Tools, pages 3 -- 62. World Scientific, 1999.

.... investigated since about 1970 and has been applied in several areas of Computer Science see the recent handbook volumes [17, 18, 40] Graph transformation has been used, for example, as an efficient computational model for term rewriting systems and functional programming languages [5, 34] and for specifying visual languages and generating associated editors [4, 31] Besides specific applications of graph transformation, several programming languages have been developed that are based on graph transformation rules. Examples of such languages are Progres [43] Agg [22] Gamma [3] ....

D. Plump, Term graph rewriting, in "Handbook of Graph Grammars and Computing by Graph Transformation," Vol. 2, Chap. 1, pp. 3--61, World Scientific, Singapore, 1999.

....graphs consist of nite trees of components. In certain applications, components with equal contents might be shared, e.g. control ow graphs of the same procedure. The discussion of the adequacy of collapsed representations could draw from the results concerning collapsed representations of terms [18]. Even cyclic sharing of components makes sense, e.g. for representing control ow graphs with recursive calls. Then results concerning the cyclic representations of in nite terms could be employed [14] Shapes are just a structural way of classifying values according to their (graph ical) ....

D. Plump. Term graph rewriting. In Engels et al. [6], chapter 1, pages 3-102.

....consist of finite trees of components. In certain applications, components with equal contents might be shared, e.g. control flow graphs of the same procedure. The discussion of the adequacy of collapsed representations could draw from the results concerning collapsed representations of terms [18]. Even cyclic sharing of components makes sense, e.g. for represent ing control flow graphs with recursive calls. Then results concerning the cyclic representations of infinite terms could be employed [14] Shapes are just a structural way of classifying values according to their (graph ical) ....

D. Plump. Term graph rewriting. In Engels et al. [6], chapter 1, pages 3-102.

.... the presence of cycles, a nite number of axioms cannot capture interesting notions of equivalence (see e.g. Sew94,BE00] This implies that traditional rewriting cannot produce normal forms of interesting cyclic structures (note, though, that there is a literature on cyclic rewriting e.g. AK96,Plu98] although we are unsure how this relates. 2.2 Induction and Coinduction We can better understand the di erence between the pessimistic and optimistic approaches in terms of inductive and coinductive de nitions of a set in some universe X : Informally, to de ne a set inductively we start ....

....equalities via rewrite proofs. Both approaches are based on an inductively de ned equality relation across theories, and both approaches require that we iteratively apply decision procedures (improvement passes) to decide a combined theory (combined improvement) Term graphs (e.g. Plu98,AK96] provide a useful view of this problem. Figure 7 represents the expression f(z 1) f(1 car(cons(z; w) as a term graph, and we have grouped vertices into subgraphs corresponding to their appropriate theory. Note that this term graph is acyclic; neither Nelson Oppen nor iterated ....

D. Plump. Term graph rewriting. In H. Ehrig, G. Engels, H.-J. Kreowski, , and G. Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, volume 2. World Scienti c, 1998.

.... the presence of cycles, a finite number of axioms cannot capture interesting notions of equivalence (see e.g. Sew94,BE00] This implies that traditional rewriting cannot produce normal forms of interesting cyclic structures (note, though, that there is a literature on cyclic rewriting e.g. AK96,Plu98] although we are unsure how this relates. 2.2 Induction and Coinduction We can better understand the di#erence between the pessimistic and optimistic approaches in terms of inductive and coinductive definitions of a set in some universe X: Informally, to define a set inductively we start ....

....equalities via rewrite proofs. Both approaches are based on an inductively defined equality relation across theories, and both approaches require that we iteratively apply decision procedures (improvement passes) to decide a combined theory (combined improvement) Term graphs (e.g. Plu98,AK96] provide a useful view of this problem. Figure 7 represents the expression f(z 1) f(1 car(cons(z, w) as a term graph, and we have grouped vertices into subgraphs corresponding to their appropriate theory. Note that this term graph is acyclic; neither Nelson Oppen nor iterated ....

D. Plump. Term graph rewriting. In H. Ehrig, G. Engels, H.-J. Kreowski, , and G. Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, volume 2. World Scientific, 1998.

....= root(x) and arg (x; j) arg(x; j) for all x 2 X and all j. Then unsh(X ; x 0 ; root ; arg ) is well de ned and equals t. which is straightforwardly proved by induction on the structure of t. 2 Maximal sharing is essentially the same as what is called the fully collapsed tree in [10]. In [8] it is shown that the maximally shared representation is unique, and that the original term can be reconstructed from it. In implementations some care has to be taken in order to keep terms maximally shared. In essence, when constructing a term, a hash table is used to nd out whether a ....

Plump, D. Term graph rewriting. In Handbook of Graph Grammars and Computing by Graph Transformation, volume 2: Applications, Languages (1999), H.-J. K. H. Ehrig, G. Engels and G. Rozenberg, Eds., World Scienti c, pp. 3-61.

....of its properties. Section 6 concludes the paper. Due to lack of space, most of the proofs have been omitted. However, we reported in the appendices the sketches of the main results. 2 Definitions and notations Many different notations are used in the literature to investigate graph rewriting [8, 19, 16]. The aim of this section is to recall briefly some key definitions in order to make easier the understanding of the paper. We are mostly consistent with [5] Some precise definitions which are omitted can be found in [6] A many sorted signature Sigma = hS; Omega i consists of a set S of sorts ....

D. Plump. Term graph rewriting. In H. Ehrig, G. Engels, H.-J. Kreowski, and G. Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, volume 2. World Scientific, to appear.

....like to perform narrowing steps over (possibly maximally) collapsed graphs. The motivation of this work comes from the fact that non tractable (exponential) computations over terms may turn to tractable (polynomial) ones over graphs. Consider, for instance, the case of the Fibonacci function [21] (see Fig. 1) It is clear that the processing of collapsed graphs improves drastically the operational semantics of functional logic programming languages. fib(5) fib(4) fib(3) fib(2) fib(1) fib(3) fib(2) fib(1) fib(0) Figure1. In Fig. 2, we show three di erent narrowing ....

....De nition 4, G [n2;L R] G 0 . We can prove that the set of atgs is closed under rewriting with admissible rules, i.e. if g 1 is an atg and g 1 [p; R] g 2 is a rewriting step, then g 2 is an atg. We can also prove that the rewrite relation is con uent (and con uent modulo the bisimilarity [6, 21]) with respect to atgs [9] i.e. for all atgs g 1 , g 2 , g 0 1 and g 0 2 such that g 1 g 2 (resp. g 1 : g 2 ) g 1 g 0 1 and g 2 g 0 2 , there exist two atgs g 00 1 and g 00 2 such that g 0 1 g 00 1 , g 0 2 g 00 2 and g 00 1 g 00 2 (resp. ....

D. Plump. Term graph rewriting. In H. Ehrig, G. Engels, H. J. Kreowski, and G. Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, volume 2. World Scientic, 1998.

....graphs consist of nite trees of components. In certain applications, components with equal contents might be shared, e.g. control ow graphs of the same procedure. The discussion of the adequacy of collapsed representations could draw from the results concerning collapsed representations of terms [18]. Even cyclic sharing of components makes sense, e.g. for representing control ow graphs with recursive calls. Then results concerning the cyclic representations of in nite terms could be employed [14] Shapes are just a structural way of classifying values according to their (graph ical) ....

D. Plump. Term graph rewriting. In Engels et al. [6], chapter 1, pages 3-102. 14

....the state of the object as what has been computed so far [ASS85] The semantics of sharing. Ecient implementations of lazy functional languages (or of computer algebras) require some sharing mechanism to avoid multiple computations of a single argument. Term graphs [Wad71, Tur79, BVEG 87, Plu99] have been studied as a representation of program expressions intermediate between abstract syntax trees and concrete representations in memory, and term graph rewriting provides a formal operational semantics of functional programming sensitive to sharing. However, compared with thinking with ....

....rewriting systems can be found in the Appendix A. 3. 1 Sharing Sharing has been extensively studied in the context of obtaining implementations of lazy functional programming languages [PJ87, PvE93] and the initial studies of sharing in the notations of term graph rewriting systems [BVEG 87, Plu99] were indeed motivated by this application. Sharing of computation. Consider the function square de ned by square(x) times(x; x) It is clear that an implementation of this function should not duplicate its input x in the expression times(x; x) but optimize this by only copying a pointer to ....

D. Plump. Term graph rewriting. In H. Ehrig, H.-J. Kreowski, and G. Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, volume 2. World Scientic, 1999. To appear.

....and Un = pn (Un Gamma1 ; Tn Gamma1 ) Considering Tn as a term its size #(Tn ) is exponential in n. However, the only subterms of Tn are true, false, and T i and U i for i n, hence # sh (Tn ) is linear in n. ut Maximal sharing is essentially the same as what is called the fully collapsed tree in [10]. In implementations some care has to be taken in order to keep terms maximally shared. In essence, when constructing or modifying a term, a hash table is used to find out whether a node representing this term exists already. If so, this node is reused; otherwise a new node is created. In order to ....

Plump, D. Term graph rewriting. In Handbook of Graph Grammars and Computing by Graph Transformation, volume 2: Applications, Languages (1999), H.-J. K. H. Ehrig, G. Engels and G. Rozenberg, Eds., World Scientific, pp. 3--61.

....and arg 0 (x; j) arg(x; j) for all x 2 X and all j. Then unsh(X 0 ; x 0 ; root 0 ; arg 0 ) is well defined and equals t. which is straightforwardly proved by induction on the structure of t. 2 Maximal sharing is essentially the same as what is called the fully collapsed tree in [10]. In [8] it is shown that the maximally shared representation is unique, and that the original term can be reconstructed from it. In implementations some care has to be taken in order to keep terms maximally shared. In essence, when constructing a term, a hash table is used to find out whether a ....

Plump, D. Term graph rewriting. In Handbook of Graph Grammars and Computing by Graph Transformation, volume 2: Applications, Languages (1999), H.-J. K. H. Ehrig, G. Engels and G. Rozenberg, Eds., World Scientific, pp. 3--61.

....on a set of terms, graph transformation rules could be used for specifying the equivalence of graphs. This raises the classical questions of the theory of rewriting like termination, confluence, orthogonality, etc. which have so far been studied in depth only for term graph rewriting systems (see [35] for a survey) The relevance of the general case is demonstrated in [22, 23] see also the invited presentation of M. Gogolla at this workshop) where reduction and equivalence rules on UML diagrams are used. Summarizing, there is some evidence that the general idea of using graphs instead of ....

PLUMP, D. Term graph rewriting. In Engels et al. [15], pp. 3 -- 62. 10 Proceedings in Informatics

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Plump, D., Term graph rewriting, in: Handbook of Graph Grammars and Computing by Graph Transformation, 2: Applications, Languages and Tools, World Scienti c, 1999 pp. 3-61.

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Detlef Plump. Term graph rewriting. In H. Ehrig, G. Engels, H.-J. Kreowski, and G. Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, volume 2, chapter 1, pages 3-61. World Scienti c, 1999.

No context found.

Detlef Plump. Term graph rewriting. In H. Ehrig, G. Engels, H.-J. Kreowski, and G. Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, volume 2, chapter 1, pages 3-61. World Scienti c, 1999.

No context found.

Detlef Plump. Term graph rewriting. In Ehrig et al. [EEKR99], chapter 1, pages 3-62.

....Representing expressions as graphs allows to share common subexpressions, improving the efficiency of term rewriting in space and time. Besides efficiency, term graph rewriting differs from term rewriting in properties like termination and confluence. A survey of term graph rewriting is given in [Plu99] where emphasis is given to the relations between term and term graph rewriting. The paper focusses on soundness of term graph rewriting with respect to term rewriting, on completeness for proving validity of equations and for computing term normal forms, on termination and confluence, and on ....

Detlef Plump. Term graph rewriting. In Hartmut Ehrig, Gregor Engels, Hans-Jorg Kreowski, and Grzegorz Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, Volume 2: Applications, Languages, and Tools, chapter 1, pages 3--62. World Scientific,

....Representing expressions as graphs allows to share common subexpressions, improving the eciency of term rewriting in space and time. Besides eciency, term graph rewriting di ers from term rewriting in properties like termination and con uence. A survey of term graph rewriting is given in [Plu99b] where emphasis is given to the relations between term and term graph rewriting. The paper focusses on soundness of term graph rewriting with respect to term rewriting, on completeness for proving validity of equations and for computing term normal forms, on termination and con uence, and on ....

Detlef Plump. Term graph rewriting. In H. Ehrig, G. Engels, H.-J. Kreowski, and G. Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, volume 2, chapter 1. World Scientic, 1999.

....related to term rewriting, the two models di er with respect to important properties like termination and con uence. The reason is that sharing common subexpressions excludes certain rewrite sequences. In this paper, we consider acyclic term graph rewriting according to the approach of [Plu93b, Plu99] The de nition of rewrite steps in this setting is as far as acyclic term graphs are concerned equivalent to the corresponding de nitions in [BvEG 87, KKSdV94, AK96] We remark, however, that this equivalence fails for cyclic graphs. In particular, a collapsing term rewrite rule like id(x) ....

....we select from every isomorphism class of term graphs a unique standard term graph and tacitly assume that we are dealing with standard term graphs only. Standard term graphs can be constructed by a canonical numbering of nodes, similar to the numbering of positions in terms. Details are in [Plu99] De nition 2.3 (Collapsing and copying) Given two term graphs G and H, G collapses to H if there is a graph morphism G H mapping root G to root H . This is denoted by G H or, if the morphism is non injective, by G H. The latter kind of collapsing is said to be proper. The inverse relation ....

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Detlef Plump. Term graph rewriting. In H. Ehrig, G. Engels, H.-J. Kreowski, and G. Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, volume 2, chapter 1. World Scientic, 1999. To appear. 30

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Plump, D., Term graph rewriting, in: H. Ehrig and G. Engels et alii,editors, Handbook of Graph Grammars and Computing by Graph Transformation, II: Applications, Languages and Tools, Theoretical Computer Science 2,World Scientific, 1999 pp. 3--61.

No context found.

PLUMP, D. Term graph rewriting. In Engels et al. [15], pp. 3 -- 62.

No context found.

D. Plump. Term graph rewriting. In H. Ehrig, G. Engels, H.-J. Kreowski, and G. Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, volume 2, pages 3-61. World Scienti c, 1999.

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D. Plump. Term graph rewriting. In H. Ehrig, G. Engels, H.-J. Kreowski, , and G. Rozenberg, editors, Handbook of Graph Grammars and Computing by Graph Transformation, volume 2. World Scientific, 1998. URL citeseer.nj.nec.com/plump98term.html.

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