J. R. Kennaway, J. K. Klop, M. R. Sleep, and F. J. De Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM Transactions on Programming Languages and Systems, 16(3):493--523, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....B(T,F,x) ones f(x,1) cons(x,n:cons(1,n) B(F,x,T) ones f(1,x) n:cons(1,n) B(x,T,F) ones It is well known that weakly orthogonal term rewriting systems are confluent. However, this property does not hold for graphs even if the considered graph rewriting system is orthogonal [10]. Indeed, the orthogonal system which consists of the rules (R1) A(x) x and (R2) B(x) x induces a confluent rewrite relation on terms and a non confluent rewrite relation on graphs as it is shown by the following counter example [10] the graph n:A(B(n) may be rewritten into two different ....

.... even if the considered graph rewriting system is orthogonal [10] Indeed, the orthogonal system which consists of the rules (R1) A(x) x and (R2) B(x) x induces a confluent rewrite relation on terms and a non confluent rewrite relation on graphs as it is shown by the following counter example [10] : the graph n:A(B(n) may be rewritten into two different normal forms, namely, u:A(u) and v:B(v) The source of the non confluence of the graph rewriting system above comes from the use of the so called collapsing rules. A rule is collapsing if its right hand side is a variable. However, ....

J. R. Kennaway, J. K. Klop, M. R. Sleep, and F. J. De Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM Transactions on Programming Languages and Systems, 16(3):493--523, 1994.

....to the appendix. 2 Preliminaries Many di erent notations are used in the literature to investigate graphs (see [11, 22] for a compilation) The aim of this section is to give brie y some key de nitions in order to make easier the understanding of the paper. Our notations are similar to those of [7, 16]. We are consistent with [9, 8] A many sorted signature = hS; i consists of a set S of sorts and an S indexed family of sets of operation symbols = s2S s with s = w;s)2S S w;s . We shall write f : s 1 : s n s whenever f 2 s 1 : s n ;s and say that f is of sort s ....

J. R. Kennaway, J. K. Klop, M. R. Sleep, and F. J. De Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM Transactions on Programming Languages and Systems, 16(3):493-523, 1994.

....we refer to [75, 43, 51, 61, 23, 89] In the literature there exists a variety of definitions of term graphs. Besides hypergraphs, directed graphs, terms with labels, and recursion equations have been used as underlying structures. Acyclic graphs have been dealt with in [34, 95, 96, 97] while [83, 92, 59, 15, 37, 63, 32] also consider cyclic graphs. By equipping function symbols with additional labels, sharing of different occurrences of a subterm in a term can be expressed through identical labels. Such labelled terms correspond to acyclic term graphs and have been studied in [76, 74, 82] In [36, 4, 2, 67] ....

....Term graph rewriting was first studied in [95] where it was shown that nonoverlapping rules give rise to a subcommutative rewrite relation. The name term graph rewriting was introduced in [15] This paper focusses on normalizing strategies and states the soundness of ) for left linear rules. In [15, 63], term graph rewrite rules are considered which operate on possibly cyclic term graphs. The application of such a rule involves the redirection of all edges pointing to the root of the left hand side, to the root of the right hand side. An alternative, transitive version of redirection is ....

[Article contains additional citation context not shown here]

Richard Kennaway, Jan Willem Klop, Ronan Sleep, and Fer-Jan de Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM Transactions on Programming Languages and Systems, 16(3):493--523, 1994.

....or analyzers can often be done in a modular fashion, simply by adding or removing sets of equations. ffl Equational systems are particularly amenable to efficient mechanical implementation, using such techniques as Knuth Bendix completion [Dershowitz and Jouannaud 1990] term graph rewriting [Kennaway et al. 1994], and compilation to C [Borovansky et al. 1996; Didrich et al. 1994; Kamperman and Walters 1996] ffl Equational semantics is very well understood. A rich body of notions and results can be brought to bear to prove useful properties of equational systems, such as completeness, soundness, and ....

KENNAWAY, J. R., KLOP, J. W., SLEEP, M. R., AND DE VRIES, F. J. 1994. On the adequacy of graph rewriting for simulating term rewriting. ACM Trans. Program. Lang. Syst. 16, 3, 493--523.

....di Informatica of Pisa and the Technical University of Berlin. a very interesting subclass of infinite terms, because they have a finitary representation; usually, however, this is not unique. Infinitary extensions of Term Rewriting have been considered by various authors during the last decade [12, 11, 15, 16, 7, 20, 21, 22, 9, 8]. Most of those contributions are concerned with the study of the rewriting relation induced by a set of finite term rules on infinite terms, presenting results about the existence of normal forms (possibly reachable after steps) confluence and so on. Only a few of them, namely [20, 21, 8] ....

....20, 21, 22, 9, 8] Most of those contributions are concerned with the study of the rewriting relation induced by a set of finite term rules on infinite terms, presenting results about the existence of normal forms (possibly reachable after steps) confluence and so on. Only a few of them, namely [20, 21, 8], focus on the subclass of rational terms, regarded essentially as the semantics of some finite but possibly cyclic structures (term graphs or terms) The goal of this paper is to provide a solid mathematical basis for the theory of rational term rewriting. One main requisite for us is that such ....

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J.R. Kennaway, J.W. Klop, M.R. Sleep, and F.J. de Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM Trans. Program. Lang. Syst., 16:493--523, 1994.

.... that, given the operational interpretation of the program scheme, fixed point itself enter the picture only as the specific side effect of suitable implementation strategies of the graph rewriting, as shown by [30] and since then cyclic term graph rewriting has been considered by various authors [1, 12, 29, 41]. To a certain extent, we consider our work at the borderline of both areas. It fits into the term graph rewriting community, since it discusses an alternative solution for the collapsing rules problem. But it also provides a linear syntax presentation for both term graphs (the data structure) ....

J.R. Kennaway, J.W. Klop, M.R. Sleep, and F.J. de Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM Trans. Program. Lang. Syst., 16:493--523, 1994.

.... term rewriting systems (while reduction of sharing graphs correspond to complete family reductions in the calculus only up to book keeping steps) And restriction to complete family reductions is inevitable when one studies an adequate simulation of term rewriting with graph rewriting [KKSV94] (although the adequacy concept in [KKSV94] does not use redex families explicitly) or simulation of a duplicating system with a non duplicating one in general [KG97c, KG98] In order to make the introduced concepts computationally more meaningful, we relativize them w.r.t. the semantics one ....

.... of sharing graphs correspond to complete family reductions in the calculus only up to book keeping steps) And restriction to complete family reductions is inevitable when one studies an adequate simulation of term rewriting with graph rewriting [KKSV94] although the adequacy concept in [KKSV94] does not use redex families explicitly) or simulation of a duplicating system with a non duplicating one in general [KG97c, KG98] In order to make the introduced concepts computationally more meaningful, we relativize them w.r.t. the semantics one may be interested in. For example, in the ....

[Article contains additional citation context not shown here]

Kennaway J. R., Klop J. W., Sleep M. R, de Vries F.-J. On the adequacy of Graph Rewriting for simulating Term Rewriting. ACM Transactions on Programming Languages and Systems, 16(3):493-523, 1994.

....In this case, the correspondence with terms has to be reconsidered, because in general a cyclic term graph unravels to a rational term, i.e. a possibly infinite term with a finite number of distinct sub terms. The relationship between rational terms and cyclic term graphs is addressed in [9, 38], while in [11] rational terms are related to terms (which are essentially a subclass of term graphs) None of these papers, however, phrases this relationship categorically as a functor between suitable theories, as we did for algebraic and gsmonoidal theories in Section 5. This is certainly an ....

J.R. Kennaway, J.W. Klop, M.R. Sleep, and F.J. de Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM Trans. Program. Lang. Syst., 16:493--523, 1994.

....jungles, we refer to [76,43,51,62,23,91] In the literature there exists a variety of definitions of term graphs. Besides hypergraphs, directed graphs, terms with labels, and recursion equations have been used as underlying structures. Acyclic graphs have been dealt with in [34,97,98,99] while [85,94,60,15,37,64, 32] also consider cyclic graphs. By equipping function symbols with additional labels, sharing of different occurrences of a subterm in a term can be expressed through identical labels. Such labelled terms correspond to acyclic term graphs and have been studied in [77,75,83,84] In [36,4,2,68] ....

....Term graph rewriting was first studied in [97] where it was shown that nonoverlapping rules give rise to a subcommutative rewrite relation. The name term graph rewriting was introduced in [15] This paper focusses on normalizing strategies and states the soundness of ) for left linear rules. In [15,64], term graph rewrite rules are considered which operate on possibly cyclic term graphs. The application of such a rule involves the redirection of all edges pointing to the root of the left hand side, to the root of the right hand side. An alternative, transitive version of redirection is ....

[Article contains additional citation context not shown here]

Richard Kennaway, Jan Willem Klop, Ronan Sleep, and Fer-Jan de Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM Transactions on Programming Languages and Systems, 16(3):493--523, 1994.

....(dags) rather than trees for efficiency reasons. Common subterms are structurally shared in a dag. In this way, multiple occurrences of a subterm may be simultaneously reduced to a common term. This reduction relation is called shared rewriting; it is a particular case of term graph rewriting (see [3, 16]) where graphs are acyclic. In a shared reduction step, subterms that correspond to the same variable in the left hand side of the rule are not copied but shared in the resulting dag, even if the right hand side of the rule has multiple occurrences of this variable. Formally, in order to define ....

....corresponds to a term rewriting sequence, but the converse does not hold in general. However, for orthogonal systems every term reduction sequence can be extended to a sequence which does correspond to a shared rewriting sequence. This is a consequence of a general theorem by Kennaway et al. [16] showing the adequacy of graph rewriting for simulating term rewriting in the case of orthogonal systems. Property 2.7 If R is an orthogonal term rewriting system then for any rewriting sequence t 0 t 1 : t n there exists a shared rewriting sequence t 0 0 s t 0 1 s : ....

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J. R. Kennaway, J. W. Klop, M. R. Sleep, and F. J. de Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM TOPLAS, 16(3):493--523, 1994.

....to termination and modularity, the graph model behaves even more friendly. Variations of the here presented rewriting model have been considered by various authors. See, for example, Sta80, Pad82, BvEG 87, GKM87, AK96] Extensions deal with cyclic term graphs and infinite computations [FW90, KKSdV94] and with term graph rewriting for logic programming and equation solving [CMR 91, CR93, CW94, HP96] An area related to term graph rewriting is graph reduction for the lambda calculus. From the numerous literature of this area we mention only [Wad71, Lam90, AL95] 3.2 Specification of an ....

Richard Kennaway, Jan Willem Klop, Ronan Sleep, and Fer-Jan de Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM Transactions on Programming Languages and Systems, 16(3):493--523, 1994.

....the family reductions in the former become reductions in the latter. Therefore, via that encoding, the results obtained here for affine SDRSs are applied to all SDRSs. Restriction to family reductions is inevitable when one studies adequate simulation of a duplicating system with an affine one [KKSV94]. In order to make the introduced concepts more meaningful computationally, we relativize them w.r.t. the semantics one may be interested in. For example, in the calculus, one might be interested in computing normal forms, head normal forms, weak head normal forms, etc. In [GK96] we have ....

Kennaway J. R., Klop J. W., Sleep M. R, de Vries F.-J. On the adequacy of Graph Rewriting for simulating Term Rewriting. ACM Transactions on Programming Languages and Systems, 16(3):493-523, 1994.

....of the proof. It consists of two parts. First we show that a sequence of shared rewrite steps (see [9] for details on shared rewriting) s t using rules in R can be simulated by a sequence of interactions from Theta 0 (s) using the translation of R. Then we use a result of Kennaway et al. [6] that shows that in the case of orthogonal systems, shared rewriting can simulate standard rewriting. This completes the proof. Non linearity and Overlappings. We address first the problem of encoding matchsequential non overlapping constructor systems with non linear left hand sides. Let us ....

....the new interaction system is sufficiently general to code a number of other features, such as references, communication and sharing. We are currently developing implementations of term rewriting systems based on parallel nets. Most implementations of term rewriting use graph rewriting (see e.g. [6]) a related formalism which also allows sharing of computation. Graph rewriting is more flexible than interaction nets in that left hand sides are not restricted. However, interaction nets have the advantage of being easier to implement (possibly distributed) since the rewrite rules are local and ....

J. R. Kennaway, J. W. Klop, M. R. Sleep, and F. J. de Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM TOPLAS, 16(3):493--523, 1994.

....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) x yields, when applied to certain cyclic graphs, different results in the mentioned approaches (see [KKSdV94] and [CD97] We are mainly interested, in this ....

....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) x yields, when applied to certain cyclic graphs, different results in the mentioned approaches (see [KKSdV94] and [CD97] We are mainly interested, in this paper, in con uence properties of term graph rewriting. We will address not only rewriting by applications of term rewrite rules, but also extensions with the operations of collapsing and copying, and with both operations together. These operations ....

[Article contains additional citation context not shown here]

Richard Kennaway, Jan Willem Klop, Ronan Sleep, and Fer-Jan de Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM Transactions on Programming Languages and Systems, 16(3):493-523, 1994.

....excludes certain rewrite sequences. In this paper, we consider acyclic term graph rewriting according to the approach of [Plu93b, Plu98] The definition of rewrite steps in this setting is as far as acyclic term graphs are concerned equivalent to the corresponding definitions 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) x yields, when applied to certain cyclic graphs, different results in the mentioned approaches (see [KKSdV94] and [CD97] We are mainly interested, in this ....

....to the corresponding definitions 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) x yields, when applied to certain cyclic graphs, different results in the mentioned approaches (see [KKSdV94] and [CD97] We are mainly interested, in this paper, in confluence properties of term graph rewriting. We will address not only rewriting by applications of term rewrite rules, but also extensions with the operations of collapsing and copying, and with both operations together. These operations ....

[Article contains additional citation context not shown here]

Richard Kennaway, Jan Willem Klop, Ronan Sleep, and Fer-Jan de Vries. On the adequacy of graph rewriting for simulating term rewriting. ACM Transactions on Programming Languages and Systems, 16(3):493--523, 1994.

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