Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Re- duction Systems. Report 1825, INRIA Rocquencourt, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 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 is, ML is complete with r#th ect to# C wasfir#I sketched in [23] ....

Z. Khasidashvili, "The Church-Rosser theorem in orthogonal combinatory reduction systems," INRIA Research Report 1825, INRIA, 1992.

....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 ....

....It took two years before looking at it again and realising that a straightforward adaptation of the measure function did the trick. The proof is modular in the following sense. The basis of the result is formed by a more or less abstract theory of independence of redexes as found at many places ([CR36, Klo80, HL, Kha92, Hue93, Oos94, Mel95]) and briefly recapitulated here. For orthogonal systems this immediately yields confluence. The development closed condition requires on top of that also a term structure of the objects of the rewriting system. The known relaxation of orthogonality to weak orthogonality (having only trivial ....

Zurab Khasidashvili. The Church-Rosser theorem in orthogonal combinatory reduction systems. Rapports de Recherche 1825, INRIA-Rocquencourt, December 1992. 8 Question asked by Aart Middeldorp.

....in 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 ....

....is terminable by the induction hypothesis, hence terminating. 2 7 Not so many proofs of FD are known for classes of higher order rewriting systems. In the seminal thesis [Klo80] a first such proof 2 was presented for the class of combinatory reduction systems. Later proofs can be found in [Kha92, Oos94, Melon] The proof in [Kha92] is for the class of expression reduction systems, roughly equivalent to the class of combinatory reduction systems (cf. Raa96, Sec. 4.6] The proof in [Oos94] is more abstract in the sense that it s based on a number of conditions which were verfied there to ....

[Article contains additional citation context not shown here]

Zurab Khasidashvili. The Church-Rosser theorem in orthogonal combinatory reduction systems. Rapports de Recherche 1825, INRIA-Rocquencourt, December 1992.

....than HRSs. This can be considered both as an advantage or a disadvantage, depending on one s point of view or needs. In the figure we have referred to the more explicit (i.e. slower ) way of CRSs to evaluate substitutions as lazy simulation . The format of higher order rewriting developed by [Kha90, Kha92] is equivalent to that of CRSs but the set up is closer to the one of calculus and of first order logic. Extensions of calculus by means of conditions are studied in [Tak89, Tak93] These conditional calculi comprise many CRSs; in personal communication we have learned that a slight ....

Z. Khasidashvili. Church-Rosser Theorem in Orthogonal Combinatory Reduction Systems. INRIA Rocquencourt report no. 1825, 1992.

....an OTRS possesses: residuals of disjoint components of a term in an OERS remain disjoint, and this allows for simpler proofs. The rest of the paper is organized as follows. In the next section, we review Expression Reduction Systems (ERS) a formalism for higher order rewriting that we use here [Kha90, Kha92]; define the descendant relation for components, and show that it is invariant under Lvy equivalence. Section 3 establishes equivalence of Maranget s neededhess and our essentiality for OERSs. In section 4, we introduce the relative notion of neededhess. In section 5, we sketch some prop erties ....

....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 in [Kha92] the present formulation is simpler. Definition 2.1 Let be an alphabet, comprising variables, denoted by z, y, z; function symbols, also called simple operators; and operator signs or quantifier signs. Each function symbol has an arity k C N, and each operator ....

[Article contains additional citation context not shown here]

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Re- duction Systems. Report 1825, INRIA Rocquencourt, 1992.

....redexes in a term t will lead to an S normal form of t whenever there is one. Roughly, S is stable if it is closed under reduction (this condition can be relaxed slightly) and any step u entering S is S needed. This work is done in the context of orthogonal Expression Reduction Systems (OERS) [Kha92], a form of higher order rewriting which subsumes Term Rewriting and the calculus. In [GlKh96] the authors further generalized the theory of relative normalization by abstracting from the structure of terms. They study relative normalization in Deterministic Residual Structures (DRSs) and ....

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992.

....address the question of normalization relative to a desired set of final terms, considering the properties that a set S of terms must possess in order for the neededness theory of Huet and L evy still to make sense. This work is done in the context of orthogonal Expression Reduction Systems (OERS) [Kha92], a form of higher order rewriting which subsumes Term Rewriting and the calculus. Natural conditions are imposed on S, called stability, that are necessary and sufficient for the following Relative Normalization (RN for short) theorem to hold: each S normalizable term not in S (not in S normal ....

.... u = Q=u) u=Q W 2 s N Q=P = Q=u) N s 0 3 W N= Q=u) One can verify that the calculus [L ev78] orthogonal TRSs [HuL e91] orthogonal Higher Order Rewriting Systems (HORSs) OR94, Oos94] and hence other orthogonal higher order rewrite systems, such as CRSs [Klo80] ERSs [Kha92], and HRSs [Nip93] and Orthogonal Term Graph Rewriting Systems [KKSV93] form DRSs; the latter are equivalent to Maranget s orthogonal DAG (Directed Acyclic Graphs) rewriting systems, defined via labelled orthogonal TRSs. All these are stable, and can be shown to be so just using an appropriate ....

[Article contains additional citation context not shown here]

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992. UEA Norwich, UK Technical Report SYS-C96-05 John Glauert and Zurab Khasidashvili 21

....think that contracting only the S needed redexes of S needed families could yield S optimal reductions that are S minimal at the same time. We show however that this is not the case either in the # calculus or in OTRSs. Overview In the next section, we review Expression Reduction Systems [23, 24, 27]. In Section 3, we introduce the relative notion of neededness. In Section 4, we sketch some properties of our labelling system for OERSs needed to define a family relation among redexes. We prove correctness of the S needed strategy for computing terms of S, for all stable S, in Section 5, ....

....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. Definition 1 Let # be an alphabet comprising variables x, y, z, ....

[Article contains additional citation context not shown here]

Z. Khasidashvili. The Church-Rosser theorem in orthogonal combinatory reduction systems. Technical Report 1825, INRIA Rocquencourt, 1992.

No context found.

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992.

....reductions that are S minimal at the same time. We show however that this is not the case either in the calculus or in OTRSs. The rest of the paper is organized as follows. In the next section, we review Expression Reduction Systems, a formalism for higher order rewriting that we use here [Kha90, Kha92]. Section 3 establishes equivalence of Maranget s neededness and our essentiality for OERSs. In section 4, we introduce the relative notion of neededness. In section 5, we sketch some properties of the labelling system of Kennaway Sleep [KeSl89] for OERSs needed to define a family relation among ....

....[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; function symbols, also called simple operators; and operator signs or quantifier signs. Each function symbol has an arity k 2 N , and ....

[Article contains additional citation context not shown here]

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992. UEA Norwich, UK Technical Report SYS-C94-06 J. Glauert and Z. Khasidashvili 29

....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 introduced by the present author independently from other formats of higher order rewriting. The present formulation of ERSs is 1 ERSs are called CRSs in [26] 4 simpler than in [24,26] Definition 1 Let Sigma be an alphabet ....

.... higher order rewriting . Here we use Expression Reduction Systems (ERSs) 24,26] 1 ERSs are based on the syntax of Pkhakadze [52] and were introduced by the present author independently from other formats of higher order rewriting. The present formulation of ERSs is 1 ERSs are called CRSs in [26]. 4 simpler than in [24,26] Definition 1 Let Sigma be an alphabet comprising variables x; y; z; function symbols, also called simple operators; and operator signs or quantifier signs. Each function symbol has an arity k 2 N , and each operator sign oe has an arity (m; n) with m;n 6= 0 ....

[Article contains additional citation context not shown here]

Z. Khasidashvili, The Church-Rosser theorem in orthogonal combinatory reduction systems, Report 1825, INRIA Rocquencourt, 1992.

....of fi K redexes in every context. Klop [Klo80] generalized Church s Theorem to all non erasing orthogonal Combinatory Reduction Systems (CRSs) by showing that the latter are UN, and Khasidashvili [Kha94c] generalized the Conservation Theorem to all orthogonal Expression Reduction Systems (ERSs) [Kha92], by proving that all non erasing redexes are perpetual in orthogonal ERSs. For orthogonal Term Rewriting Systems (OTRSs) a very powerful perpetuality criterion was obtained by Klop [Klo92] in terms of critical redexes. These are redexes that are not perpetual, i.e. reduce 1 terms to strongly ....

....results of this paper, and we will demonstrate their usefulness in applications. We obtain our results in the framework of Orthogonal Conditional Expression reduction Systems (OCERSs) KO95] CERS is a format for higher order rewriting, or to be precise, second order rewriting, which extends ERSs [Kha92] by allowing restrictions both on arguments of redexes and on the contexts in which the redexes can be contracted. Various interesting typed calculi, including the simply typed calculus and the system F [Bar92] can directly be encoded as OCERSs (see also [KOR93] calculi with specific ....

[Article contains additional citation context not shown here]

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992. 5

....various kinds of redex creation are reflected in syntactic properties of rewriting systems such as normalization, perpetuality, decidability of weak and strong normalization (that is, termination of all reductions) expressive power, etc. Some results in this direction are obtained in [10] In [9], we introduced a formalism for higher order rewriting (i.e. term rewriting systems with bound variables and substitution mechanism) which is close to Combinatory Reduction Systems (CRSs) of Klop [11] Our syntax is more close to the syntax of calculus and First Order Logic. For example, the ....

.... 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 defined in Khasidashvili [9]; it is based on the syntaxs of [18] Definition 2.1 (1) Let Sigma be an alphabet, comprising variables v 0 ; v 1 ; function symbols, also called simple operators; and operator signs or quantifier signs. Each function symbol has an arity k 2 N , and each operator sign oe has an arity (m; ....

[Article contains additional citation context not shown here]

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992.

.... To unify our results with the ones already in the literature for different orthogonal rewrite systems, we first introduce a framework of Context sensitive Conditional Expression Reduction Systems (CCERSs) This framework provides a format for higher order rewriting which extends ERSs [27] by allowing restrictions on term formation, on arguments of redexes, and on the contexts in which the redexes can be contracted. Various interesting typed calculi (such as the simply typed calculus [6] its extension with pairing [68] and system F [6] and calculi with specific reduction ....

....case there are several conceptually similar, but notationally often quite different, proposals. The first general higher order format was introduced long ago 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 ....

Khasidashvili, Z. (1992), "The Church-Rosser theorem in orthogonal combinatory reduction systems," Report 1825, INRIA Rocquencourt.

.... To unify our results with the ones already in the literature for different orthogonal rewrite systems, we first introduce a framework of Context sensitive Conditional Expression Reduction Systems (CCERSs) This framework provides a format for higher order rewriting which extends ERSs [Kha92] by allowing restrictions on term formation, on arguments of redexes, and on the contexts in which the redexes can be contracted. Various interesting typed calculi (such as the simply typed calculus [Bar92] its extension with pairing [TS96] and system F [Bar92] and calculi with specific ....

....there are several conceptually similar, but notationally often quite different, proposals. The first general higher order format was 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 ....

Khasidashvili Z., The Church-Rosser theorem in orthogonal combinatory reduction systems. Report 1825, INRIA Rocquencourt, 1992.

....applications. To unify our results with the ones already in the literature for different orthogonal rewrite systems, we first introduce a framework of Context sensitive Conditional Expression reduction Systems (CCERSs) This framework provides a format for higher order rewriting which extends ERSs [Kha92, Kha97] by allowing restrictions on term formation, on arguments of redexes, and on the contexts in which the redexes can be contracted. Various interesting typed calculi, including the simply typed calculus and the system F, can be directly encoded as CCERSs (see also [KOR93] as can calculi with ....

....there are several conceptually similar, but notationally often quite different, proposals. The first general higher order format was 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 ....

[Article contains additional citation context not shown here]

Khasidashvili Z., The Church-Rosser theorem in orthogonal combinatory reduction systems. Report 1825, INRIA Rocquencourt, 1992.

....and Combinatory Logic, for example. This paper is a part of a general study of how various kinds of redex creation are reflected in syntactic properties of rewriting systems such as normalization, perpetuality, expressive power, etc. Some results in this direction are obtained in [7, 8, 10] In [6], we introduced a formalism for higher order rewriting (i.e. term rewriting systems (TRSs) with bound variables and substitution mechanism) which is close to the Combinatory Reduction Systems (CRSs) of Klop [11] The syntax of our Expression Reduction Systems (ERSs, called CRSs in [6] is ....

.... 8, 10] In [6] we introduced a formalism for higher order rewriting (i.e. term rewriting systems (TRSs) with bound variables and substitution mechanism) which is close to the Combinatory Reduction Systems (CRSs) of Klop [11] The syntax of our Expression Reduction Systems (ERSs, called CRSs in [6]) is closer to the syntax of calculus and First Order Logic. For example, the fi rule is written as fi : Ap(aA; B) B=a)A; where a is to be instantiated by a variable and A and B are to be instantiated by terms; an instance (t=x)s of (B=a)A denotes the result of substitution of the term t for ....

[Article contains additional citation context not shown here]

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. INRIA Report 1825, 1992.

....S normal form of t whenever there is one. It is shown also that if a stable S is regular, i.e. if S unneeded redexes cannot duplicate S needed ones, then the S needed strategy is hypernormalizing as well. This work was performed in the context of Orthogonal Expression Reduction Systems (OERSs) [Kha92], a form of higher order rewriting which subsumes TRSs and the calculus and is similar to Klop s Combinatory Reduction systems (CRSs) Klo80] Most of these results were later generalized to an abstract framework of Deterministic Residual Structures [GlKh96] Normalization theory has developed in ....

.... Expression Reduction Systems Klop introduced Combinatory 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] Several interesting formalisms have been introduced later [Kha92, Nip93, OR94]. We refer to van Raamsdonk [Raa96] for a survey. Here we use a system of higher order rewriting, Expression Reduction Systems (ERSs) defined in [Kha92] under the name of CRSs) the present formulation follows [GlKh94] and is simpler. Definition 2.1 Let Sigma be an alphabet, comprising ....

[Article contains additional citation context not shown here]

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992.

No context found.

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992.

....binding) there exist several conceptually similar, but notationaly often quite different proposals. Long ago, the first general higher order format was introduced by Klop [Klo80] under the name of Combinatory Reduction Systems. Since then, several other interesting formalisms have been introduced [Kha92,Nip93,Wol93,OR94,Tak93]. This paper is based on the notion of Expression Reduction System introduced by the first author [Kha92] but our results also apply to the other higher order formats. Often it is of interest to have the possibility to put restrictions on the generation of either the terms or the rewrite relation ....

....higher order format was introduced by Klop [Klo80] under the name of Combinatory Reduction Systems. Since then, several other interesting formalisms have been introduced [Kha92,Nip93,Wol93,OR94,Tak93] This paper is based on the notion of Expression Reduction System introduced by the first author [Kha92], but our results also apply to the other higher order formats. Often it is of interest to have the possibility to put restrictions on the generation of either the terms or the rewrite relation (or both) For example, many typed lambda calculi can be viewed as untyped lambda calculus with ....

[Article contains additional citation context not shown here]

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992. Khasidashvili, Van Oostrom

....(that is, TRSs with bound variables and substitution In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, Logic at St. Petersburg , LFCS 94, Springer LNCS, vol. 813, Narode A. Matiyasevich Yu. V. eds. St. Petersburg, 1994. p. 191 203. mechanism [6]) a strongly normalizable subterm may also be a potentially infinite redex after contraction of an outer redex a term can be substituted in it that makes the subterm no longer strongly normalizable. Thus innermost reductions and complete reductions are no longer perpetual in OERSs. Therefore ....

....for reductions with substitutions (also referred to as higher order rewriting) as in the calculus [1] Several other formalisms have been introduced later. We refer to Klop et al. 12] for a survey. Here we use a system of higher order rewriting, Expression Reduction Systems (ERSs) defined in [6] (under the name of CRSs) Definition 2.1 (1) Let Sigma be an alphabet, comprising variables v 0 ; v 1 ; function symbols, also called simple operators; and operator signs or quantifier signs. Each function symbol has an arity k 2 N , and each operator sign oe has an arity (m; n) with m; ....

[Article contains additional citation context not shown here]

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992.

....binding) there exist several conceptually similar, but notationaly often quite different proposals. Long ago, the first general higher order format was introduced by Klop [Klo80] under the name of Combinatory Reduction Systems. Since then, several other interesting formalisms have been introduced [Kha92, Nip93, Wol93, OR94, Tak93, Lor93]. This paper is based on the notion of Expression Reduction System introduced by the first author [Kha92] but our results also apply to the other higher order formats. Often it is of interest to have the possibility to put restrictions on the generation of either the terms or the rewrite ....

....format was introduced by Klop [Klo80] under the name of Combinatory Reduction Systems. Since then, several other interesting formalisms have been introduced [Kha92, Nip93, Wol93, OR94, Tak93, Lor93] This paper is based on the notion of Expression Reduction System introduced by the first author [Kha92], but our results also apply to the other higher order formats. Often it is of interest to have the possibility to put restrictions on the generation of either the terms or the rewrite relation (or both) For example, many typed lambda calculi can be viewed as untyped lambda calculus with ....

[Article contains additional citation context not shown here]

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992.

.... [Kha93] and Orthogonal Expression Reduction Systems (OERS) Kha94] a form of higher order rewriting similar to Klop s CRSs This work was supported by the Engineering and Physical Sciences Research Council of Great Britain under grant GR H (which subsumes Term Rewriting and the calculus) [Kha92]. Sekar and Ramakrishnan [SeRa93] study a normalizing strategy which in each multi step contracts a necessary set of redexes. A different approach to normalization is developed in Kennaway [Ken89] and in Antoy and Middeldorp [AnMi94] Antoy et al. AEH94] designed a needed narrowing strategy. ....

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992.

....terms must possess in order for the neededness theory of Huet and L evy still to make sense. This work is This work was supported by the Engineering and Physical Sciences Research Council of Great Britain under grant GR H done in the context of orthogonal Expression Reduction Systems (OERS) [Kha92], a form of higher order rewriting which subsumes Term Rewriting and the calculus. Natural conditions are imposed on S, called stability, that are necessary and sufficient for the following Relative Normalization (RN) theorem to hold: each S normalizable term not in S (not in S normal form) has ....

....the underlying DRS is not stable. Lemma 4.2 Let S be stable, t 62 S, t u t 0 , UNS (u; t) and let u 0 2 t 0 be a redex created by u, in a DFS F . Then UN S (u 0 ; t 0 ) Proof By Lemma 3.2 and Lemma 4.1. Now we can generalize the RN theorem, proved in [GlKh94] for orthogonal ERSs [Kha92], to all DFSs. We now allow for arbitrary stable sets S. Below FAM (P ) denotes the set of families (whose member redexes are) contracted in P . Theorem 4.1 (Relative Normalization) Let S be a stable set of terms in a Deterministic Family Structure F , and let t 62 S be S normalizable. Then (1) ....

Khasidashvili Z. The Church-Rosser theorem in Orthogonal Combinatory Reduction Systems. Report 1825, INRIA Rocquencourt, 1992.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC