Kennaway J. R., Klop J. W Sleep R., de Vries F.J., "Transfinite Reductions in Orthogonal Term Rewriting Systems", Info. and Comp. 119(1), 1995, 18-38.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a using the leftmost outermost rewriting strategy yields the strongly converging sequence: a f(a,a) f(f(a,a) a) Delta Delta Delta which does not converge to a normal form. In order to ensure correctness of nf S w.r.t. nf, we need to use an infinitary normalizing strategy S(see [16, 20, 25] for a discussion about the definition of such strategies. 3.2 Semantics of context sensitive computations The rewriting semantics cs nf for a TRS R computes the set of normal forms of each term t: cs nf (t) fs 2 NF We also consider the infinitary version: cs nf (t) fs 2 ....

....is no need to consider transfinite sequences at all [24] Theorem 4. Regarding the computation of infinite normal forms (i.e. nf) we need to restrict the attention to left bounded orthogonal TRSs, where a TRS is left bounded (LB) if the depth of the left hand sides of its rules is bounded [16, 24]. 14 Example 6 Consider the confluent TRS R [24] f(a) f(f(a) f(a) a where only symbols a and f belong to the underlying signature. Term f(a) has a finite (constructor) normal form a, and an infinite (non constructor) normal form f . Unicity of infinite normal forms obtained from ....

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R. Kennaway, J.W. Klop, M.R. Sleep, and F.-J. de Vries. Transfinite reductions in Orthogonal Term Rewriting Systems. Information and Computation 119(1):1838, 1995.

....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) representation. More type discipline would be useful, for instance as in PROGRES [24] Also, context free graph languages might be too restricted for specifying the shapes of graphs that occur in certain ....

R. Kennaway, J. W. Klop, R. Sleep, and F.-J. de Vries. Transfinite reductions in orthogonal term rewriting systems. Information and Computation, 119(1):18- 38, 1995. 14

.... value of from(0) According to this situation, some research has been done concerning infinitary rewriting, i.e. rewriting that also considers infinite reduction sequences, probably involving infinite terms, and even term rewriting systems built from infinite terms [Cor93, CG99, DKP91, KKSV95, Luc01c, Mid97] By an infinite sequence S of elements taken from a set A we mean a mapping S : N A. We denote the n th element of the sequence as Sn rather than as S(n) The definition of a notion of convergence to a limit of infinitary sequences on a set A can be done by introducing a ....

....a limit which is either a finite or an infinite term. In infinitary normalization, we consider infinite sequences of length (the first limit ordinal) whose limit is a (possibly infinite) normal form. Kennaway et al. have developped the notion of strongly converging (infinite) rewrite sequence [KKSV95] In sequences of this kind, the depth of the contracted redexes tends to infinite. Definition 4 [Mid97] An infinite rewrite sequence t 1 t 2 Delta Delta Delta is strongly converging if for all d 0, there is an index i 1 such that the depth of every redex contracted in t i t i 1 ....

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R. Kennaway, J.W. Klop, M.R. Sleep, and F.-J. de Vries. Transfinite reductions in Orthogonal Term Rewriting Systems. Information and Computation 119(1):18-38, 1995.

....3 Topology Given a set X, a topology T C 2 x is a way of adding structure to this set. Roughly speaking, a topology defines which points U C X are in the neighborhood of a point x U. In literature from the field of computer science, structure on sets is usually added by giving a metric. In [5, 11, 10], this metric is defined on the state space, while [2, 4] use a metric to define structure on the labels. Furthermore, this was, to our knowledge, never used with respect to bisimulation equivalence. Note that giving a metric on a set is only one way of inducing a topology. Alternatively, for ....

R. Kennaway, J.W. Klop, M.R. Sleep, and F.-J. de Vries. Transfinite reductions in orthogonal term rewriting systems. In R.V. Book, editor, Proceedings of the Fourth International Conference on Rewriting Techniques and Applications, volume 488 of Lecture Notes in Computer Science, pages 1-12, Como, Italy, April 1991. Springer-Verlag.

....properties of graphs with cycles, and rewriting rules that recognize or create cycles. In this respect our calculus goes farther than either [20] or [8] where only acyclic graphs are considered. Without cyclic graphs some important implementation ideas are ruled out. More recently, Klop et al. [15] have extended the Barendregt s graph rewriting system to deal with cycles. However, their approach is significantly different from ours in that they model cyclic graph rewriting as transfinite reduction of infinitary graph terms. In the following, we formally introduce a Graph Rewriting System ....

....for application; it is central to expressing the sharing of subexpressions. Our GRS includes cyclic terms and permits both non left linear rules and left cyclic rules. We prove that in the absence of interfering rules a GRS is confluent. This is a more general result than the confluence theorem in [15]. We think that our approach also leads to a simpler proof of confluence than in [15] We also develop a term model for a GRS without interfering rules along the lines of L evy s term model for the calculus. We introduce the notion of information content associated to a term, and show that the ....

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J. Kennaway, J. Klop, M. Sleep, and F. de Vries. Transfinite Reductions in Orthogonal Term Rewriting Systems. In Proc. RTA '91, Springer-Verlag LNCS, 1991.

....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. g 00 1 : g 00 2 ) We have seen in Section 1 that confluence (modulo bisimilarity) of AGRSs is not a straightforward extension of that of orthogonal term rewriting systems. In [11], it is proved that orthogonal graph rewriting systems (and thus AGRSs) are confluent modulo the equivalence of the socalled hypercollapsing graphs. A graph g is hypercollapsing if g g. We are not interested in confluence modulo the equivalence of hypercollapsing graphs in the present paper. ....

J. R. Kennaway, J. K. Klop, M. R. Sleep, and F. J. De Vries. Transfinite reduction in orthogonal term rewriting systems. Information and Computation, 119(1):18-- 38, 1995.

....slucas dsic.upv.es Lazy languages admit giving infinite values as the meaning of expressions. Infinite values are limits of converging infinite sequences of partially defined values which are more and more defined. This can be formalized by using strongly convergent infinitary rewrite sequences [KKSV95]. These are Cauchy convergent rewrite sequences in which redexes are (ultimately) contracted deeper and deeper. We only consider sequences of length at most issued from finite terms. The existence of a strongly convergent rewrite sequence leading from a term to another is undecidable. However, ....

R. Kennaway, J.W. Klop, M.R. Sleep, and F.-J. de Vries. Transfinite reductions in Orthogonal Term Rewriting Systems. Information and Computation 119(1):18-38, 1995.

....(Comp) even beyond non terminating traces. 5. TRANSFINITE SEMANTICS In order to find a compositional semantics which is useful for slicing we consider traces able to look beyond the infinite computations, namely we use semantics represented by transfinite state traces of programs (e.g. see [23]) For transfinite traces we mean traces whose length can be any # # O. In this way the finite computations are finite traces, while the infinite computations can have lengths which overcome the first infinite ordinal #. We can generalize the definition of finite trace obtaining the set of ....

....on transfinite traces which allows us to specify limit states for traces whose length is a limit ordinal. This is achieved by considering the standard topology on traces as induced by O : The limit ordinal are the open elements on O while the successor ones are the closed elements of the topology [23]. This induces a topology on # # and in the following we assume that this topology is metrizable and complete [31] i.e. any Cauchy sequence of transfinite traces has a limit in # # . The transfinite semantics is more concrete than the maximal trace semantics defined in the Cousot semantic ....

Kennaway, J. R., Klop, J. W., Sleep, M. R., and Vries, F. J. Transfinite reductions in orthogonal term rewriting systems. Information and computation 119, 1 (1995), 18--38.

....by term graph rewriting. In [22, 24, 23] it is shown how to simulate logic programming by term graph rewriting. While this survey is restricted to acyclic term graphs and finitary term rewriting, one may also consider cyclic graphs and infinitary term rewriting. The interested reader may consult [36, 37, 63, 64, 18, 20]. Further issues that have been considered are term graph rewriting over conditional term rewriting systems [82] the relation between term graph rewriting and event structures [62, 17] and the term generating power of context free jungle grammars [35] The area of graph reduction for the ....

Richard Kennaway, Jan Willem Klop, Ronan Sleep, and Fer-Jan de Vries. Transfinite reductions in orthogonal term rewriting systems. Information and Computation, 119:18--38, 1995.

.... reduction sequences of length of at most seem more adequate for real applications (but transfinite rewriting is suitable for modeling rewriting with finite cyclic graphs) There are two main frameworks for transfinite rewriting: DKP91] considers standard Cauchy convergent rewriting sequences; [KKSV95] only admits strongly convergent sequences which are Cauchy convergent sequences in which redexes are contracted at deeper and deeper positions. Cauchy convergent sequences are more powerful than strongly convergent ones w.r.t. their computational strength, i.e. the ability to compute canonical ....

....in which redexes are contracted at deeper and deeper positions. Cauchy convergent sequences are more powerful than strongly convergent ones w.r.t. their computational strength, i.e. the ability to compute canonical forms of terms (normal forms, values, etc. Example 1. Consider the TRS (see [KKSV95]) f(x,g) f(c(x) g) g a f(x,a) c(x) h c(h) and the derivation of length 2: f(a,g) f(c(a) g) Delta Delta Delta f(c ,g) f(c ,a) c No strongly convergent reduction rewrites f(a,g) into the infinite term c . This work has been partially supported by CICYT ....

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R. Kennaway, J.W. Klop, M.R. Sleep, and F.-J. de Vries. Transfinite reductions in Orthogonal Term Rewriting Systems. Information and Computation 119(1):18-38, 1995.

....results. Rootnormalization and normalization of reduction strategies are compared in Section 6. In Section 7 we show that under natural assumptions root normalization implies infinitary normalization. We compare our results with the infinitary normalization results in Kennaway et al. [9]. We conclude in Section 8 with a few remarks on (un)decidability. 2 Preliminaries In this preliminary section we recall the basic notions of term rewriting. Further details can be found in [5, 11] A signature is a set F of function symbols. Associated with every f 2 F is a natural number ....

....(and essentiality) when it comes to infinitary normalization. Before we can show this formally, we need to get a grip on the concept of infinitary normalization. We are interested in infinite rewrite sequences of length whose limit (with respect to the standard notion of Cauchy convergence [2, 9]) is an infinite normal form. Unlike Kennaway et al. 9] we don t consider transfinite rewrite sequences whose length exceeds since such sequences don t make sense from a practical computational point of view. Consider the infinite orthogonal TRS (based on a similar example in [9] f(g i ....

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J.R. Kennaway, J.W. Klop, M.R. Sleep, and F.-J. de Vries, Transfinite Reductions in Orthogonal Term Rewriting Systems, Information and Computation 119(1), pp. 18--38, 1995.

.... Applications of the infinite calculus to the study of the easy terms of the classical calculus are given in [10] Klop and his collaborators on the other hand were interested in generalizing their earlier work on transfinite reduction sequences in the context of term rewriting systems [30]. In [31] several versions of the infinite calculus are defined but it is shown that only three of them have good properties. These three calculi can be distinguished by the behavior of the element . In the version corresponding to the Bohm trees we get M = x: and can be interpreted ....

Kennaway J. R., Klop J. W Sleep R., de Vries F.-J., "Transfinite Reductions in Orthogonal Term Rewriting Systems", Info. and Comp. 119(1), 1995, 18-38.

....s = t[oe(r) p , for some rule l r 2 R, p 2 Pos(t) and substitution oe. This can eventually be detailed by writing t p s. The one step rewrite relation for R is . A term t is root stable if it cannot be rewritten to a redex. The set of infinite terms is denoted by T 1 ( Sigma; X ) see [KKSV95] for a formal definition of infinite term) An infinite rewrite sequence is an infinite sequence t 1 ; t 2 ; of terms such that t n t n 1 for all n 2 IN. Following [Mid97] in this paper we are only interested in infinite rewrite sequences of length (unlike [KKSV95] We write t s ....

....1 ( Sigma; X ) see [KKSV95] for a formal definition of infinite term) An infinite rewrite sequence is an infinite sequence t 1 ; t 2 ; of terms such that t n t n 1 for all n 2 IN. Following [Mid97] in this paper we are only interested in infinite rewrite sequences of length (unlike [KKSV95]) We write t s if either t s or s is the (possibly infinite) limit of an infinite rewrite sequence starting from t (according to the standard notion of Cauchy convergence, see [KKSV95] for details) If t s, then s is a reduct of t. NF R denotes the set of finite or infinite ....

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R. Kennaway, J.W. Klop, M.R. Sleep, and F.-J. de Vries. Transfinite reductions in Orthogonal Term Rewriting Systems. Information and Computation 119(1):18-38, 1995.

.... considered as the (infinite) value of from(0) According to this situation, some research has been done concerning infinitary rewriting, i.e. rewriting that also considers infinite reduction sequences, probably involving infinite terms, and even term rewriting systems built from infinite terms [DKP91,KKSV95,Mid97]. The redundancy of an argument of a function f in a TRS R depends on the semantic properties of R that we are interested in observing. In [AEL00] we consider different (reduction) semantics including the standard normalization semantics nf (typical of pure rewriting [Jou94] and the evaluation ....

R. Kennaway, J.W. Klop, M.R. Sleep, and F.-J. de Vries. Transfinite reductions in Orthogonal Term Rewriting Systems. Information and Computation 119(1):18-38, 1995.

....prove the completeness of narrowing [EJ97] The confluence of a rewrite relation allows to evaluate expressions in a deterministic and efficient way by using rewriting strategies. Such strategies have been well investigated in the setting of finite and infinite orthogonal TRSs (e.g. O D77, HL91, KKSV95] In [Ant92] a strategy that computes outermost needed redexes based on definitional trees has been designed in the framework of orthogonal constructor based TRSs. In this article, we show that Antoy s strategy can be extended to orthogonal constructor based GRSs with the same nice properties. ....

J. R. Kennaway, J. K. Klop, M. R. Sleep, and F. J. De Vries. Transfinite reduction in orthogonal term rewriting systems. Information and Computation, pages 18--38, 1995.

....for orthogonal rewrite systems. 30 The theory of term graph rewriting has been extended in the last years to the case of graphs with cycles, which can represent infinite, rational terms, and it has been shown that rewriting of these term graphs corresponds to infinite term rewriting sequences [KKSdV90] In [CD96] Corradini and Drewes take a different point of view, showing that if the rewriting of term graphs is performed using the double pushout approach, then a single graph rewriting can correspond to an infinite parallel rewriting, i.e. to the application of a rewrite rule to infinitely ....

J.R. Kennaway, J.W. Klop, M.R. Sleep, and F.-J. de Vries. Transfinite reductions in orthogonal Term Rewriting System. Technical Report Report CS-R9041, Centre for Mathematics and Computer Science, 1990.

....for orthogonal rewrite systems. The theory of term graph rewriting has been extended in the last years to the case of graphs with cycles, which can represent infinite, rational terms, and it has been shown that rewriting of these term graphs corresponds to infinite term rewriting sequences [KKSdV90] In [CD96] Corradini and Drewes take a different point of view, showing that if the rewriting of term graphs is performed using the double pushout approach, then a single graph rewriting can correspond to an infinite parallel rewriting, i.e. to the application of a rewrite rule to infinitely ....

J.R. Kennaway, J.W. Klop, M.R. Sleep, and F.-J. de Vries. Transfinite reductions in orthogonal Term Rewriting System. Technical Report Report CS-R9041, Centre for Mathematics and Computer Science, 1990.

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Kennaway J. R., Klop J. W Sleep R., de Vries F.J., "Transfinite Reductions in Orthogonal Term Rewriting Systems", Info. and Comp. 119(1), 1995, 18-38.

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J.R. Kennaway, J.W. Klop, M.R. Sleep, and F.J. de Vries. Transfinite reductions in orthogonal term rewriting systems. Information and Computation, 119(1):18--38, 1995.

....term. A finite sequence of graph reductions may correspond to a term reduction sequence of length greater than to. A precise account of the relationship between graph rewriting, including cyclic graphs, and term rewriting must therefore consider infinitary term rewriting. In a related paper [Ken93b] the authors have set out the foundations of infinitary term rewriting for orthogonal term rewrite systems. In this paper we define finitary and infmitary term and term graph rewriting, and a notion of one rewrite system implementing another. We show that for orthogonal systems of rewrite rules, ....

.... We require the left hand side of a role to be finite on both philosophical grounds (the question of whether a term is a redex by a given rule should be decidable in a finite time) and technical grounds (infinite left hand sides cause some of the properties of infinitary rewriting to fail see [Ken93b]) Infinite right hand sides do not cause problems. We will in fact require infinite righthand sides in order to model graph rewrite rules with cyclic right hand sides. Unbounded lefthand sides are also unproblematic that is, while we require each left hand side to be finite, we do not require ....

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KENNAWAY, J.R., KLOP, J.W., SLEEP, M.R., and DE VRIES, F.J., Transfinite reductions in orthogonal term rewriting systems, Technical Report, CWI, Amsterdam, and submitted to Information and Computation, (1993).

....the form #x 1 . #x n .yM 1 . M k . Several other classes of terms have also been proposed as formalizing the notion of undefinedness. 1 The Journal of Functional and Logic Programming 1999 1 Kennaway et al. Meaningless Terms in Rewriting 1 In our study of transfinite term rewriting ( KKSdV95] that is, orthogonal term rewriting in which terms may be infinitely large and rewrite sequences may have any ordinal length, we have encountered a class of terms having similar properties the so called hypercollapsing terms. In addition, we have found that the Church Rosser property of ....

....We consider left linear term rewrite systems and lambda calculus, in both finitary and transfinite forms. We assume the reader to be familiar with the basic theory of term rewriting [DJ90, Klo92] and lambda calculus [Bar84, HS86] The basic theory of transfinite rewriting has already been set out [KKSdV95, KKSdV97] We will show the usefulness of our axioms in several ways. They arise naturally from the notion of rewriting as computation of the meaning of terms. The axioms imply two standard lemmas: the Genericity Lemma and the Consistency Lemma. Genericity states that a meaningless ....

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J. R. Kennaway, J. W. Klop, M. R. Sleep, and F. J. de Vries. Transfinite reductions in orthogonal term rewriting systems. Information and Computation, 119:18--38, 1995.

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J.R. Kennaway, J.W. Klop, M.R. Sleep, and F.-J. de Vries "Transfinite reductions in orthogonal term rewrite systems" (in preparation, 199--).

....the concepts that arise in the finitary theory, such as notions of undefinedness of terms. In this connection, Berarducci and Intrigila ( Ber, BI94] have independently developed an infinitary lambda calculus and applied it to the study of consistency problems in the finitary lambda calculus. In [KKSdV95] we developed the basic theory of transfinite reduction for orthogonal term rewrite systems. In this paper we perform the same task for the lambda calculus. In contrast 2. Basic definitions 2 to the situation for term rewriting, in lambda calculus there turn out to be several different possible ....

....spoken informally of convergent reduction sequences but not yet defined them. The obvious definition is that a reduction sequence of length converges if the sequence of terms converges with respect to the metric. However, this proves to be an unsatisfactory definition, for the same reasons as in [KKSdV95]. There are two problems. Firstly, a certain property which is important for attaching computational meaning to reduction sequences longer than fails. Definition 2.4 A reduction system admitting transfinite sequences satisfies the Compression Property if for every reduction sequence from a term ....

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J.R. Kennaway, J.W. Klop, M.R. Sleep, and F.J. de Vries. Transfinite reductions in orthogonal term rewriting systems. Information and Computation, 1995. To appear. Available by ftp from ftp::/ftp.sys.uea.ac.uk/pub/kennaway/transfinite. fdvi,psg.Z.

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R. Kennaway, J. W. Klop, R. Sleep, and F.-J. d. Vries. Transfinite reductions in orthogonal term rewriting systems. Inform. and Comput., 119(1):18--38, 1995.

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R. Kennaway, J.W. Klop, R. Sleep, and F.J. de Vries. Transfinite Reductions in Orthogonal Term Rewriting. Information and Computation, 119(1):18 -- 38, 1995.

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