Vincent van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, Amsterdam, March 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....constructor headed term) In the latter case, s is rooted by a defined symbol and s must be evaluated to a constructor headed term. Otherwise, no reduction of t to true is possible. 2 The next desirable result is to show that needed reductions are normalizing. This is suggested from related works [27, 16], but is beyond the scope of this paper. 6.2 Lifting Rewriting to Narrowing We first take a closer look at the variables involved for a LNT computation. Lemma 6.14 If I(t) LNT S ) 0 LNT S 0 , then Dom( 0 ) FV( t) Proof The substitution is composed of (partial) ....

....6.14. Furthermore, the only other bindings computed are for the intermediate variables on the right, where no branching is needed. These bindings do not occur in the computed solution. 2 It is also conjectured that our notion of needed reductions is optimal (this is subject to current research [3, 27, 28]) Note, however, that sharing is needed for optimality, as shown for the first order case in [2] 8 Avoiding Function Synthesis Although the synthesis of functional objects by full higher order unification in LNT is very powerful, it can also be expensive and operationally complex. There is an ....

Vincent van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, 1994. Amsterdam.

....a constructor headed term. Theorem 9. If t reduces to true, then t has a needed redex at position p and t must be reduced at p eventually. Otherwise, t is not reducible to true. The next desirable result is to show that needed reductions are normalizing. This is suggested from related works [15, 11], but is beyond the scope of this paper. For a goal system S, we call the variables that do not occur in T (S) dummies. In particular, all variables on the right and all variables in selectors in patterns of some tree in S are dummies. Lemma 10. If S ) LNT fg, then S ) fg LNT fg. ....

....unification. 6 i.e. FV( FV(S) FV(T (S) Theorem14 (Optimality) If I(t) LNT fg and I(t) 0 LNT fg are two different derivations, then and 0 are incomparable. It is also conjectured that our notion of needed reductions is optimal (this is subject to current research [16, 15, 3]) Note, however, that sharing is needed for optimality, as shown for the first order case in [2] 8 Avoiding Function Synthesis Although the synthesis of functional objects by full higher order unification in LNT is very powerful, it can also be expensive and operationally complex. There is an ....

Vincent van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, 1994. Amsterdam.

....of Q. t denotes the empty reduction starting from t. We use U; V; W to denote sets of redexes of a term. DRSs are similar to Stark s Determinate Concurrent Transition Systems [Sta89] to Abstract Reduction Systems of Gonthier et al. GLM92] and to van Oostrom s Descendant Rewriting Systems [Oos94]. The main difference from DCTSs is that Stark considers a non duplicating residual relation (and we do not distinguish a start state) The difference from ARSs of [GLM92] Technical Report SYS C96 06 UEA Norwich, UK Discrete Normalization and Standardization in Stable DRSs is that we do not have ....

Van Oostrom V. Confluence for Abstract and Higher-Order Rewriting. Ph.D. Thesis, Free University of Amsterdam, 1994.

....term. The Strong Church Rosser (confluence) property is established for OERSs in [24, 27] the Finite Developments Theorem [5, 29] is proved first, from which strong confluence follows by a standard argument. Strong confluence for other higher order rewriting formats are obtained, among others, in [29, 48, 44, 37]. 5 or strongly equivalent, or permutation equivalent 7 Theorem 4 (Finite Developments) All complete developments of a set of redexes in a term t, in an OERS, end at the same term s, and the residuals in s of any redex in t along any complete development are the same. Theorem 5 (Strong ....

V. van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Free University, Amsterdam, 1994.

.... evaluation models, with the standard Church notion of residual [L ev78, HuL e91, Klo80, Kat90, Lam90, Kha92, KKSV93, Nip93, Oos94, Raa96, Gue96] Besides the CTSs of Stark [Sta89] and the ARSs of Gonthier et al. GLM92] closely related, but more syntactically oriented, models are studied in [Oos94, Mel96, Raa96]. Definition 2.2 (Deterministic Residual Structure) A Deterministic Residual Structure (DRS for short) is a pair R = A; where A is an ARS and = is a residual relation on redexes relating redexes in the source and target term of every reduction t u s 2 A, such that for v 2 t, the set v=u ....

Van Oostrom V. Confluence for Abstract and Higher-Order Rewriting. Ph.D. Thesis, Free University, Amsterdam, 1994.

....as in the calculus [Bar84] and its extensions. Restricted rewriting systems with substitutions were first studied in Pkhakadze [Pkh77] and Aczel [Acz78] Several interesting formalisms have been introduced later [Kha90, Nip93, Wol93, OR94] We refer to Klop et al. KOR93] and van Oostrom [Oos94] for a survey. Here we use a system of higher order rewriting, Expression Reduction Systems (ERSs) defined in Khasidashvili [Kha90, Kha92] ERSs are called CRSs in [Kha92] the present formulation is simpler. Definition 2.1 Let Sigma be an alphabet, comprising variables, denoted by x; y; z; ....

Van Oostrom V. Confluence for Abstract and Higher-Order Rewriting. Ph. D. Thesis, Free University of Amsterdam, 1994.

....which consists of lambdacalculus with constants and rewrite rules for these constants [10] Higher order rewrite systems were introduced with the aim to study the meta theory of systems like Isabelle and Prolog. The presentation of higher order rewriting systems in this paper is mainly based on [25, 32], which builds on [16, 22] However, I would like to stress that the actual format of higher order rewriting is not of the utmost importance here, since the paper is informal in nature. Moreover, the essence of concepts and proofs does not depend on the details of the chosen format. For various ....

....possible to give a complete overview in the present paper. Readers interested in the theory of equational reasoning and narrowing for higher order rewriting are referred to [23, 30] and the literature mentioned there. Further, results concerning confluence and termination can be found in detail in [16, 25, 19, 29, 32]. Acknowledgements. I am grateful to Jan Willem Klop and Vincent van Oostrom for discussions that in the course of the last years always have been, and still are, a source of inspiration. I wish to thank Jean Pierre Jouannaud, Aart Middeldorp, Jaco van de Pol, and Roel de Vrijer for helpful ....

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V. van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, Amsterdam, The Netherlands, March 1994.

....same limit. ## We cannot provide an abstract analogue of the approximate result of Theorem 6, since our axioms do not provide any way of defining the # relation. 5 Higher order rewriting 5. 1 Finitary higher order rewriting There have been many definitions of finitary higher order rewriting ([11, 13, 14, 15, 16]) 16] contains a comparison of several of these, the conclusion of which is that the di#erences are insignificant. For the present paper we will use Khasidashvili s notation, slightly simplified in inessential ways. The terms of expression reduction systems are constructed from function symbols ....

V. van Oostrom. Confluence for abstract and higher-order rewriting. PhD thesis, Vrije Universiteit, Amsterdam, 1994.

....Systems (DCTSs) Sta89] and the Abstract Reduction Systems of Gonthier et al. GLM92] Unlike DCTSs, the residual relation in DRSs may be duplicating, and unlike the ARSs of [GLM92] we do not have a nesting relation on redexes. Several refined concepts of abstract rewriting are studied in [Oos94, Mel96, Raa96]. DRS are intended to model all orthogonal term and graph rewrite systems, both first and higherorder, and other conflict free transition systems, with a reasonable level of abstraction, while the ARSs of [GLM92] do not cover e.g. orthogonal term graph rewriting systems [KKSV94] as observed by ....

Van Oostrom V. Confluence for Abstract and Higher-Order Rewriting. Ph.D. Thesis, Free University, Amsterdam, 1994.

.... from [Kha94c] CERSs extend Expression Reduction Systems [Kha92] a formalism of higher order (rather, second order) rewriting close to Combinatory Reduction Systems [Klo80] We refer to [Raa96] for an extensive survey of the relationship among various formats of higher order rewriting (such as [Klo80, Kha92, Wol93, Nip93, OR94, Oos94]) and refer to [Klo92] for a survey of results on conditional TRSs. Terms in CERSs are built from the alphabet like in the first order case, except some symbols may have binding power. For example, a fi redex in the calculus appears as Ap(x t; s) where Ap is a function symbol of arity 2, and ....

Van Oostrom V. Confluence for Abstract and Higher-Order Rewriting. Ph.D. Thesis, Free University, Amsterdam, 1994.

....extended higher order rewriting systems. The present paper is rather concise in nature; for a detailed account the interested reader is referred to [Raa96] 2 Preliminaries In this section we recall the definition of higher order rewriting systems [Nip91, MN94] following the presentation in [Oos94, Raa96]. We further give the definitions of almost orthogonality and full extendedness. The reader is supposed to be familiar with simply typed calculus with fi reduction (denoted by fi ) and restricted j expansion (denoted by j ) see for instance [Bar92, Aka93] Simple types, written as A; B; C; ....

....types and the binary type constructor . We suppose that for every type A there are infinitely many variables of type A, written as x A ; y A ; z A ; Higher Order Rewriting Systems. The meta language of higher order rewriting systems, which we call the substitution calculus as in [Oos94, OR94, Raa96], is simply typed calculus. A higher order rewriting system is specified by a pair (A; R) consisting of a rewrite alphabet and a set of rewrite rules over A. A rewrite alphabet is a set A consisting of simply typed function symbols. A preterm of type A over A is a simply typed term of type A ....

Vincent van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, Amsterdam, March 1994. Available at http://www.cs.vu.nl/~oostrom.

....2 Proposition 9.15 Let s be a term and E a sequence of equations. Then, hC[s] j Ei = dgc hC[s E ] j Ei : Proof: By structural induction on C[2] and Proposition 9.14. 2 Intermezzo 9. 16 In the proof of confluence of OE 1 we will use the decreasing diagram method proposed by van Oostrom [vO94]. The method consists of associating a label to each reduction step and giving a well founded order on these labels. If all weakly confluent diagrams turn out to be of a specific kind, namely decreasing, then confluence is guaranteed. Definition 9.17 Let j:j be a measure from strings of labels to ....

....b m a n is decreasing if ffa; bgg jab 1 : b m j and ffa; bgg jba 1 : a n j. Theorem 9.18 If a labelled reduction system is weakly confluent and all weakly confluent diagrams are decreasing with respect to a well founded order on labels then the system is confluent. Proof: See [vO94]. 2 Theorem 9.19 OE 1 is confluent. Proof: We call the external and acyclic substitution reductions s reductions, and the remaining reductions, except fi reduction, o reductions (written as Gamma o ) Since the black hole rule is strongly normalizing, and does not change the depth of a box, ....

V. van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, 1994.

....confluent. 8 Related Work As already indicated in the introduction, HRS are closely related to CRS [11, 13] Confluence of CRS has been investigated for orthogonal systems. This has lead to a notion of 25 higher order rewriting system (HORS) which is parameterized by a substitution calculus [25, 23]. HORS generalize both CRS and HRS. In the case of HRS the substitution calculus is the simply typed calculus. It has been shown that all weakly orthogonal HORS are confluent [25, 23] Although no notion of critical pair has been defined for HORS, weak orthogonality for HRS can be translated as ....

....lead to a notion of 25 higher order rewriting system (HORS) which is parameterized by a substitution calculus [25, 23] HORS generalize both CRS and HRS. In the case of HRS the substitution calculus is the simply typed calculus. It has been shown that all weakly orthogonal HORS are confluent [25, 23]. Although no notion of critical pair has been defined for HORS, weak orthogonality for HRS can be translated as follows: Definition 8.1 An HRS R is called weakly orthogonal iff it is left linear and all of its critical pairs are of the form hu; ui. By the above result it follows that all weakly ....

Vincent van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, Amsterdam, 1994.

....study of residuals. See Figure 2, displaying several assumptions about nesting of redexes (OE) and residuals. Actually, several of these occur already in Newman [New42] In recent years such studies have been taken up again by, among others, Plotkin, Gonthier, L evy, Melli es and van Oostrom [Plo78, GLM92, Oos94, Mel97, Mel98]. The use of labels to trace subterms through a reduction was, in the form of underlining , an important ingredient in the early work of Barendregt. In [Bar71] he developed the technique of underlining into a sophisticated tool for the study of various systems of calculus and Combinatory ....

Vincent van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, Amsterdam, March 1994.

....[59] Ghilezan [18] and Statman [72] The theorem has also been proved in several ways for various notions of higher order rewrite systems. Klop [38] proves it for orthogonal combinatory reduction systems by means of his technique to reduce weak normalization to strong normalization. Van Oostrom [49, 51] proves finiteness of 6 See the end of [13] or the beginning of Chapter V of [12] 36 developments for orthogonal higher order rewriting systems and for pattern rewriting systems. Each of these two results implies finite developments for orthogonal combinatory reduction systems. Melli es [45] ....

V. van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit Amsterdam, 1994.

....are terminating and all complete developments of the same set of redexes are Hindley equivalent. This stronger version follows easily from the weaker version (i.e. termination of all developments) and the strong commutativity of co initial steps. PERPETUALITY AND UNIFORM NORMALIZATION 13 HRSs [54], one can in orthogonal CCERSs use FD and strong commutativity to define for any co initial reductions P and Q the residual of P under Q, written P=Q. We write P Theta L Q if P=Q = Theta L is the L evy embedding relation) P and Q are called L evy equivalent or permutation equivalent (written ....

Van Oostrom, V. (1994), Confluence for abstract and higher-order rewriting, Ph.D. Thesis, Vrije Universiteit, Amsterdam.

....combinatory reduction systems (CRSs) Khasidashvili s expression reduction systems (ERSs) and Nipkow s higher order pattern rewriting systems (PRSs) 1 Introduction This paper is concerned with a method to prove confluence of rewriting systems. It s an extension of some confluence results in [CR36, Hue80, Toy88, Klo80, Kha92, Raa93, Tak, MN94, Oos94, ORb] and we refer the reader to these papers and to the handbook chapters [DJ, Klo] for motivation and for standard definitions as well. Here we will mainly be concerned with proving our result: Left linear development closed PRSs are confluent. Let s explain the terminology used. A rewrite system ....

....our result: Left linear development closed PRSs are confluent. Let s explain the terminology used. A rewrite system for which the rewrite rules do not depend on one another is called orthogonal. Formalising this notion can be quite involved depending on the rewrite formalism it is applied to ([Hue80, Klo80, HL, GLM, MN94, Oos94]) but the intuition to be captured is always the same: an application of a rule replaces some substructure by another one, and in orthogonal systems we moreover have that if two distinct substructures can be replaced then these substructures are independent. Some (non)examples are: 1. The rules F ....

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Vincent van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, Amsterdam, March 1994. Available at http://www.cs.vu.nl/oostrom.

....is one whose right hand side is a variable. For example, the rules Head(Cons(x, y) # x and I(x) # x are collapsing, but F (A) # A is not. For lambda calculus, the definition implies that beta reduction is collapsing. The definition also makes sense for higher order rewriting (see e.g. Oos94] but that is beyond the scope of this paper. An example of a root active term is the term A, given the rule A # A. Less trivially, consider the rules Last(Cons(x, y) # Last(y) and Ones # Cons(1, Ones) For these rules, the term Last(Ones) is root active. Neither of these terms is ....

V. van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, Amsterdam, March 1994.

.... order term rewriting system (see e.g. Klo] one can associate a PRS via so called currying, i.e. by associating to every m ary function symbol f of the TRS, a function symbol f: o Delta Delta Delta o z m o and translating terms and rules in the obvious homomorphic way (see e.g. Oos94, Sec. 3.3.1] In fact, for rules we take the closure of the translated rule, i.e. if X is the (non repetative) list of free variables from left to right in the result l r of the translation, then the resulting PRS rule will be X:l X:r. Hence the arity of the PRS rule is just the ....

....Finite Family Developments in Section 4 distinct residuals can be contracted by distinct rules. Finiteness of developments (and strengthened versions of it) have been studied extensively in the literature for various classses of rewriting systems (see e.g. CR36, Sch65, Hin78, Klo80, Kha92, Raa, Oos94, Melon] In Subsection 3.1 a simple proof of FD is presented for the class of PRSs. In Subsection 3.2 upperbound information is added to the termination proof of Subsection 3.1, yielding (exact) upperbounds on the lengths of marked rewrite sequences. As a consequence this yields upperbounds on ....

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Vincent van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, Amsterdam, March 1994.

....rewriting rule 2 Theta x x x to the term 2 Theta 3 gives rise to the duplication of the term 3 in the result 3 3. How this duplication is actually performed (for example, using sharing) depends on the designer s implementation of substitution. This decomposition has been shown useful in [OR94, Oos94] in the case of first order term rewriting systems (TRSs, DJ90, Klo92] and higher order term rewriting systems (Klop s combinatory reduction systems (CRSs) Klo80] Nipkow s higher order rewrite systems (HRSs) Nip93] We will indicate how, using this decomposition, results can be proved ....

....case that all the non determinism of the rewriting process is due to parallelism. That is, any two distinct steps that can be performed take place in disjoint parallel parts of the structure. In the case of term rewriting such orthogonal systems are known to be confluent ( Klo80, Nip93] In [OR94, Oos94] it was shown that confluence of orthogonal term rewriting systems can be viewed as depending on the underlying calculus for substitutions. In particular, it was shown that the Finite Developments theorem ( Klo80, Bar84] can be reduced to normalisation of the substitution calculus. The Finite ....

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Vincent van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, Amsterdam, March 1994.

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Vincent van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, Amsterdam, March 1994.

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V. van Oostrom. Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit, 1994. Amsterdam.

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V. van Oostrom, Confluence for abstract and higher-order rewriting, Ph.D. Thesis, Free University of Amsterdam, 1994.

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V. van Oostrom, "Confluence for Abstract and Higher-Order Rewriting ", Th`ese de l'Universit'e Libre d'Amsterdam, Pays-Bas (1994).

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Vincent van Oostrom (1994a). Confluence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije Universiteit te Amsterdam.

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