P. Aczel. A general Church-Rosser theorem. Technical report, University of Manchester, july 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... introduced Combinatory Reduction Systems (CRSs) in [Klo80] to provide a uniform framework for reductions with substitutions (also referred to as higherorder rewriting) as in the A calculus [Bar84] Restricted rewriting systems with substitutions were first studied in Pkhakadze [Pkh77] and Aczel [Acz78]. Several interesting formalisms have been introduced later [Nip93, Wo193, OR94] We refer to Klop et al. KOR93] and van Oostrom [Oos94] for a survey. Here we use a system of higher order rewriting, Ezprcssion Reduction Systems (ERSs) defined in Khasidashvili [Kha90, Kha92] ERSs are called CRSs ....

Aczel P. A general Church-Rosser theorem. Preprint, University of Manch- ester, 1978.

.... out in [115] An illuminating investigation of parallel computations in the lambda calculus using rewriting logic has been carried out by Laneve and Montanari [87, 88] who have considered the even more general case of orthogonal, left normal combinatory reduction systems as formalized by Aczel [1], that contain the lambda calculus as a special case. They show that such systems exactly correspond to rewrite theories R whose equational part E consists of explicit substitution equations. They then prove that the traditional model of parallel rewriting in such systems generalizing parallel ....

P. Aczel. A general Church-Rosser theorem. Manuscript, University of Manchester, 1978.

....the extended General Schema can also be applied to HRSs. Second, we show how Nipkow s higher order critical pair analysis technique for proving local confluence can be applied to IDTSs. Appendices A, B and C (proofs) are available from the web page. 1 Introduction In 1980, after a work by Aczel [1], Klop introduced the Combinatory Rewrite Systems (CRSs) 15, 16] to generalize both first order term rewriting and rewrite systems with bound variables like Church s calculus. In 1991, after Miller s decidability result of the pattern unification problem [20] Nipkow introduced Higher order ....

P. Aczel. A general Church-Rosser theorem. Technical report, University of Manchester, United Kingdom, 1978.

....of sinfor#jI descr#ICx . of the evaluation ofter#4 [33, 34] and ther#I y developed the concept of a structured lazy evaluation system, which is essentially the same as that of an evaluation system used in thepr#C I t pap er# The syntax of evaluation systems has itsor#C in the wor# of Aczel [2]. Combinatory reduction systems [29] and exp ession reduction systems [28]ar# also developments d Paraphrased. 37 fr#r Aczel s wor# Wear# also indebted to Howefor the definition of bisimulationlikepr #.jFC equivalences andfor the pr# of that theyar# congr#4 Cx. The completenessr#mpletenes ....

P. Aczel, "A general Church-Rosser theorem," Technical Report, University of anchester, 1978.

....is appropriate for that purpose. One can also choose not to work with a notion of descendants at all, and to prove the diamond property directly for an inductively defined notion of complete development, by induction over that definition. This method (and variations) one can find described in [Acz78, Raa93, Tak, Nip, MN94, Raa96] and in many ad hoc confluence proofs in literature as well. The advantages of keeping the descendants around is that it yields such nice byproducts as the theory of permutation equivalence. The advantage of not using descendants is that it yields short confluence proofs. A paper displaying both ....

Peter Aczel. A general Church-Rosser theorem. Technical report, University of Manchester, July 1978.

....Combinatory Reduction Systems (CRSs) in [29] to provide a uniform framework for reductions with substitutions (also referred to as higher order rewriting) as in the # calculus [5] and its extensions. Restricted rewriting systems with substitutions were first studied in Pkhakadze [42] and Aczel [1]. Several interesting formalisms have been introduced later [24, 51, 36, 48] We refer to van Raamsdonk [49] for a survey. Expression Reduction Systems Here we use Expression Reduction Systems (ERSs) defined in [24] under the name of CRSs) The present formulation follows [27] and is simpler. ....

P. Aczel. A general Church-Rosser theorem. Technical report, University of Manchester, 1978.

....Reduction Systems (CRSs) in [Klo80] to provide a uniform framework for reductions with substitutions (also referred to as higher order rewriting) 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 ....

Aczel P. A general Church-Rosser theorem. Preprint, University of Manchester, 1978.

....of combinatory reduction systems. Another impulse for the study of rewriting with bound variables originates in the work by Nipkow, who defines in [22] the class of higher order rewrite systems. Combinatory reduction systems form a generalisation of the class of contraction schemes defined in [1], and can more generally be seen as standing in a tradition where extensions of lambda calculus are studied. Examples of such extensions are lambda calculus extended with infinitely many ffi rules that test for equality of closed normal forms, defined by Church (see [3] lambda calculus with ....

P. Aczel. A general Church-Rosser theorem. University of Manchester, July 1978.

....Expression Reduction Systems Klop introduced Combinatory Reduction Systems (CRSs) 38] to provide a uniform framework for reductions with substitutions like that in the calculus [4] and its extensions. Restricted rewriting systems with substitutions were first studied by Pkhakadze [52] and Aczel [1]. Several interesting formalisms were introduced later [24,47,68,51] See [57] for a survey. We will refer to such systems using a collective name higher order rewriting . Here we use Expression Reduction Systems (ERSs) 24,26] 1 ERSs are based on the syntax of Pkhakadze [52] and were ....

....arguments of u, if any, are in normal form (resp. are strongly normalizing) A reduction is NF erasing ( SN erasing) if it contracts only NF erasing (SNerasing) redexes. Definition 16 Let t be a term in an OTRS ( Sigma ; R ) and let v be a redex in t. 1) We say that t satisfies property [1], written [1] t) if any redex in t is main. 12 (2) We say that v satisfies property [r Gamma 2] written [r Gamma 2] v) if there are no symbols in quasi main arguments of v. We say that t satisfies property [2] written [2] t) if any redex in t satisfies [r Gamma 2] We show in the next ....

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P. Aczel, A general Church-Rosser theorem. Preprint, University of Manchester, 1978.

....Systems) Khasidashvili [8] Expression Reduction Systems) and Nipkow [16] Higher order Rewrite Systems) They are extensions of Term Rewriting Systems [5, 12] by means of variable binding and substitution mechanisms. Restricted notions of CRSs were first introduced in Pkhakadze [18] and Aczel [1]. A comparison of some formalisms of rewriting systems with bound variables and substitution mechanism (referred to also as higher order rewrite systems) can be found in van Oostrom and van Raamsdonk [17] A survey paper is Klop et al. 13] Here we describe a system for higher order rewriting as ....

Aczel P. A general Church-Rosser theorem. Preprint, University of Manchester, 1978.

....notions of equivalence coincide, and conclude that the equational description in TR (X) is considerably simpler than that provided by the residual calculus. Laneve and Montanari then go on to consider the general case of orthogonal, left normal combinatory reduction systems as formalized by Aczel [1], that contain the lambda calculus as a special case. They show that such systems exactly correspond to rewrite theories R whose equational part E consists of explicit substitution equations. They then prove that the traditional model of parallel rewriting in such systems generalizing parallel ....

Peter Aczel. A general Church-Rosser theorem. Manuscript, University of Manchester, 1978.

....closely [12] because they are similar to functional programs with pattern matching. The key property of orthogonal systems is their confluence, regardless of whether they terminate or not. We show that this holds for OHRS as well. 20 6. 1 The Classical Proof In this section we generalize Aczel s [1] confluence proof from his consistent sets of contraction schemes to arbitrary OHRS. Note that the former are a proper subset of the latter. The proof proceeds roughly like the one for the untyped calculus due to Tait and Martin Lof [3] Although we want to prove the confluence of OHRS, the ....

Peter Aczel. A general Church-Rosser theorem. Technical report, University of Manchester, 1978.

.... Phi Phi ffifl fflfi H H ffifl fflfi Gamma j Figure 4: Graphical representations of expressions 4 Preliminaries (residuals and standardization) The aim of this section is to fix the notation, rather than to provide an accurate account of the properties of IS s. We refer to [Aczel 1978, Klop 1980] for all the details. We shall denote redexes through the access path to the corresponding subexpression, defined as follows: ffl (the empty string) is the access path of the root; ffl if u is the access path of f( x 1 : t 1 ; Delta Delta Delta ; x n : t n ) then u Delta ....

P. Aczel (1978) A general Church-Rosser theorem. Draft, Manchester.

....has to analyze only the last rules in the derivation of the premise, using Lemma 3.4 to handle term substitutions. Theorem 4.3 (CR) F on untyped terms is Church Rosser. Proof Standard. The combinator reductions rules for F are left linear and non overlapping, and one can apply the result in (Aczel, 1978) (see also (Klop, 1980) Corollary 4.4 (CR) F on typable terms is Church Rosser. Proof Immediate from SR and CR on untyped terms. Theorem 4.5 (SN ) If ; t : then t is strongly normalizing. Proof We prove SN for a system more powerful than FML, functorial F (brie y FF) which can type ....

Aczel, P. (1978). A general Church-Rosser theorem. Tech. rept. Univ. of Manchester.

....one has to analyse only the last rules in the derivation of the premise, using Lemma 4 to handle term substitutions. ut Theorem 7 CR. F on untyped terms is Church Rosser. Proof. Standard. The combinatory reductions rules for F are left linear and non overlapping, and one can apply the result in [Acz78] (see also [Klo80] ut Corollary 8 CR. F on typable terms is Church Rosser. Proof. Immediate from SR and CR on untyped terms. ut Theorem 9 SN. If ; t: then t is strongly normalising. Proof. We prove SN for a system more powerful than FML, functorial F (brie y FF) which can type every ....

P. Aczel. A general Church-Rosser theorem. Technical report, Univ. of Manchester, 1978.

....are also finite. It is interesting to note that, where developments correspond naturally with the notion of parallel reduction employed in the confluence proof of Tait Martin Lof, superdevelopments correspond to the parallel reduction employed in a slight variant of that proof, by Aczel [Acz78]. See Appendix A. 4.7 Standardization and a duality The next main theorem in fi calculus to be discussed is the Standardization Theorem. Again its formulation and proof crucially depend on the notion of descendant. Standardizing a reduction sequence can be compared to sorting a sequence of ....

....that needed reduction is hypernormalizing and in pointing out some errors. 12 Appendices Appendix A: Parallel reduction a la Aczel We compare the notions of parallel reduction as it usually employed in proofs of CR due to Tait and Martin Lof, and the amended notion that was proposed by Aczel [Acz78]. For an extensive discussion see van Raamsdonk [Raa96] We use the notation Gamma ffi for parallel reduction. In the style of Tait and Martin Lof, it is defined by the inductive clauses in Table 10. It characterizes complete developments, in the sense that M Gamma ffi N if and only if there is ....

[Article contains additional citation context not shown here]

P. Aczel. A general Church-Rosser theorem. Technical report, University of Manchester, july 1978.

.... by Klop [37] under the name of Combinatory Reduction Systems (CRSs) Since then, several other interesting formalisms have been introduced [27, 71, 52, 45, 57] Restricted rewriting systems PERPETUALITY AND UNIFORM NORMALIZATION 5 with substitutions were first studied by Pkhakadze [58] and Aczel [2]. See van Raamsdonk [61] for a detailed comparison of various forms of higher order rewriting. It is often of interest to have the possibility of putting restrictions on the generation of either the terms or the rewrite relation or both. For example, many typed lambda calculi (such as the simply ....

Aczel, P. (1978), "A general Church-Rosser theorem," Preprint, University of Manchester.

....This paper was presented at the Third International Workshop on Extensions of Logic Programming, February 1992. 1 Introduction Higher order rewrite systems extend first order rewrite systems and provide a mechanism for reasoning about equality in languages that include notions of bound variables [1, 9, 12, 5]. First order conditional rewrite systems extend firstorder rewrite systems, providing more expressive power by allowing conditions to be placed on rewrite rules [2, 8] Such conditions must be satisfied before a particular rewrite can be applied. In this paper, we extend these two notions to ....

Peter Aczel. A general church-rosser theorem. Technical report, University of Manchester, 1978.

....Research Group AN IMPLEMENTATION OF HIGHER ORDER REWRITING (EXTENDED ABSTRACT) D.A. Wolfram PRG TR 8 93 Delta Oxford University Computing Laboratory Wolfson Building, Parks Road, Oxford OX1 3QD 1 Higher Order Rewriting Combinatory Reduction Systems of Aczel [1], and Klop [8] generalize first order rewriting by allowing bound variables to occur in rewrite rules. When restricted to the simply typed calculus [2, 4] the definitions of rewriting by Nipkow [10] and Wolfram [15, 16] are of increasing generality. Combinatory Reduction Systems have rewrite ....

P. Aczel, A general Church-Rosser theorem, Technical Report, University of Manchester, 1978.

....systems have been able to exploit this knowledge to implement effective strategies for reasoning about equality between first order terms. More recently, the study of rewrite systems has included the more expressive higher order rewrite systems. One direction involves extending early work by Aczel [1] and Klop [16] which uses terms as a meta language for expressing rewrite systems for object languages that include notions of bound variables. Such terms can be used to elegantly express the higher order abstract This paper appears in Proceedings of the 1991 International Workshop on ....

Peter Aczel. A general church-rosser theorem. Technical report, University of Manchester, 1978.

.... introduced long ago by Klop [Klo80] under the name of Combinatory Reduction Systems (CRSs) Since then, several other interesting formalisms have been introduced [Kha92, Wol93, Nip93, Lor93, OR94] Restricted rewriting systems with substitutions were first studied by Pkhakadze [Pkh77] and Aczel [Acz78]. See van Raamsdonk [Raa96] for a detailed comparison of various forms of higher order rewriting. It is often of interest to have the possibility of putting restrictions on the generation of either the terms or the rewrite relation or both. For example, many typed lambda calculi (such as the ....

Aczel P., A general Church-Rosser theorem. Preprint, University of Manchester, 1978.

....they constitute a generalisation of the rewriting done by functional programming evaluation steps . The CRS formalism was invented by Klop (1980) for a systematic treatment of combinations of term rewrite systems (TRS) with the calculus, inspired by the definable extensions of the calculus of Aczel (1978). However, CRS can also be understood as a form of higher order rewriting (Nipkow 1991, van Oostrom and van Raamsdonk 1995) This section gives a brief introduction to CRS. The presentation follows the (highly recommended) survey of Klop, van Oostrom and van Raamsdonk (1993) closely, with only a ....

Aczel, P. (1978). A general Church-Rosser theorem. Technical report. Univ. of Manchester.

....has to analyse only the last rules in the derivation of the premise, using Lemma 4 to handle term substitutions. ut Theorem7 CR. Ffi on untyped terms is Church Rosser. Proof. Standard. The combinatory reductions rules for Ffi are left linear and non overlapping, and one can apply the result in [Acz78] (see also [Klo80] ut Corollary 8 CR. Ffi on typable terms is Church Rosser. Proof. Immediate from SR and CR on untyped terms. ut Theorem9 SN. If Delta; Gamma t: then t is strongly normalising. Proof. This is proved using semantic techniques (reducibility candidates) The details are ....

P. Aczel. A general church-rosser theorem. Technical report, Univ. of Manchester, 1978.

....by NWO SION project 612 316 606. 1. Introduction This paper is concerned with a comparison of two formats of higher order rewriting: Combinatory Reduction Systems (CRSs) as introduced by Klop [Klo80] and Higher order Rewrite Systems (HRSs) as introduced by Nipkow [Nipa] Inspired by Aczel [Acz78], Klop defined CRSs in [Klo80] as first order term rewriting systems possibly with bound variables, so as to include both first order rewrite systems such as Curry s Combinatory Logic and rewrite systems with bound variables such as Church s calculus. The point was that a large amount of ....

....of systems until 1980 see also [Klo80, pp. 132,133] In (as far as we know) historical order we have: ffl TRS = Term Rewriting System. We don t know who introduced this name, but they were known at the end of the seventies. cf. also Rosen [Ros73] ffl CS = Contraction Scheme. Introduced by Aczel [Acz78]. ffl (a) reductions were introduced by Hindley [Hin78] ffl CRS = Combinatory Reduction System. Introduced by Klop [Klo80] ffl HOTRS = Higher Order Term Rewriting System. Introduced by Wolfram in his PhD thesis, see [Wol93] ffl ERS = Expression Reduction System. Introduced by Khasidashvili ....

Peter Aczel. A general Church-Rosser theorem. Technical report, University of Manchester, July 1978.

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P. Aczel. A general Church-Rosser theorem. Technical report, University of Manchester, july 1978.

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