F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197-240, 1993. 24  Home/Search   Document Details and Download   Summary   Related Articles   Check

....fundamental asymmetry in this situation. 3.1 The Composition Lemma Let us consider the following rule Mhx = Nihy = Li Mhy = Lihx = Nhy = Lii which we call composition. It abstracts the composition one nds in systems like [1, 11] namely the rule called Map) or in the extension xkc of x [6, 7, 14, 15] (namely the rule kc of [6] see also [21] page 75) We would like to see that the converse of the composition rule preserves (strong, head, or leftmost) normalization. Unfortunately this rule does not commute in a nice way with reduction, essentially due to the duplication of substitutions in ....

F. Kamereddine and R.P. Nederpelt. On stepwise explicit substitutions. International Journal of Foundations of Computer Science, 4(3):197240, 1993.

....operator: the conventions governing the correspondence between types and the string resources they represent are too inexpressive. 1 For a detailed treatment of structural resource management options yielding local or global versions of fi reduction, we refer to Kamareddine and Nederpelt [7]. quantifier rules. This deficiency can be overcome in part in special cases. For example, consider the type constructor A B for quantifiers proposed in Moortgat [11] This is governed by the rules QR and QL below: Gamma ) u : A Gamma ) v:vu : A B QR Delta; v : A; Delta 0 ) u : B ....

Kamareddine, F. (1993), `On stepwise explicit substitution'. International Journal of Foundations of Computer Science, 4(3), 197--240.

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F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197-240, 1993. 24

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Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

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Kamareddine, F., and Nederpelt, R.P., (1993) On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

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F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

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F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

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F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

.... s e as well as and the suspension calculus are non comparable while s e is more adequate than the suspension calculus. Keywords: Calculi of explicit substitutions, lambda calculi, eta reduction. 1 Introduction Recent years have witnessed an explosion of work on expliciting substitutions [1,7,9,14,15,17,19] and on establishing its usefulness to computation: e.g. to Partially supported by the Brazilian CNPq research council grant number 47488101 6. First author partially suported by the FEMAT Brazilian foundation for research in mathematics, second author supported by the CAPES Brazilian ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197-240, 1993.

.... s e as well as and the suspension calculus are non comparable while s e is more adequate than the suspension calculus. Keywords: Calculi of explicit substitutions, lambda calculi, eta reduction. 1 Introduction Recent years have witnessed an explosion of work on expliciting substitutions [1,7,9,14,15,17,19] and on establishing its usefulness to computation: e.g. to Partially supported by the Brazilian CNPq research council grant number 47488101 6. First author partially suported by the FEMAT Brazilian foundation for research in mathematics, second author supported by the CAPES Brazilian ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197-240, 1993.

.... substitution and new rules to handle these operators have been proposed (e.g. 10, 2, 17, 30, 4, 5, 21, 22, 28, 13] Amongst these calculi we mention C (cf. 14] the calculi of categorical combinators (cf. 10] SP (cf. 2, 11, 30] referred to as the family; BLT (cf. [20]) cf. 4] and (cf. 28] which are descendants of the family; s (cf. 21] and s e (cf. 22] Most of these calculi are described in de Bruijn notation and can roughly be classi ed under two styles: the [2, 17] and the s styles [21, 22] The new symbols added by explicit ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197-240, 1993.

....partially substituted terms, we must render the latter from being a metalevel notion to an object level notion. It turns out that our new notation will enable such rendering efficiently and will enable the representation of various forms of substitution: local, global, implicit and explicit. KN 93] introduces substitution which is object level but which can evaluate terms fully obtaining the result of the metalevel substitution. More precisely, we introduce the process of stepwise substitution, which is meant to refine reduction procedures. Since substitution is the fundamental operation ....

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

No context found.

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-140, 1993.

....term reshuffling. The work carried out in this paper will have many applications. We mentioned the semantics of lazy evaluation and the new reduction strategies which may lead to further optimal results. These points are under investigation. The new notation moreover deserves attention. KN 93] and [NK 94] have shown many of its advantages for formulating and generalising type theory and for rendering substitution explicit in the calculus. Further advantages are also studied in [KN 9z] ....

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....as (I(B)ffi)I(A) here is ffi a special symbol used for application) and I(O x:A :B) is written as (I(A)O x )I(B) where O = or Pi. Both (tffi) and (tO x ) t being a term in item notation, are called items. For reasons explaining the usefulness of such a notation, the reader is referred to [KN 93] and [KN 96a] For this paper however, the reader is to notice that redexes and definitions can be easily generalised and introduced with item notation. A traditional redex is a term that starts with a ffi item next to a item. A definition is itself a certain form of a ffi item next to a item. ....

....the Cube with definitions only, or with both definitions and generalised reduction. When using generalised reduction without definitions, one must remain in the and as the other systems lose their SR. 2 The item notation For a detailed description of item notation, the reader is referred to [KN 93] KN 94] KN 95] and [KN 96a] We will introduce in this section the minimum machinery needed to represent the Cube in item notation and for introducing generalised reduction and definitions. The systems of the Cube are based on a set of pseudo expressions T defined by: T = V j C j (T ffi)T j ....

Kamareddine, F., and Nederpelt, R.P. (1993), On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240.

....partially substituted terms, we must render the latter from being a metalevel notion to an object level notion. It turns out that our new notation will enable such rendering efficiently and will enable the representation of various forms of substitution: local, global, implicit and explicit. [19] introduces substitution which is object level but which can evaluate terms fully obtaining the result of the metalevel substitution. More precisely, we introduce the process of stepwise substitution, which is meant to refine reduction procedures. Since substitution is the fundamental operation ....

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....the local and global way Fairouz Kamareddine School of Mathematical and Computational Sciences Heriot Watt Univ. Riccarton Edinburgh EH14 4AS, Scotland Alejandro R os Department of Computer Science University of Buenos Aires Buenos Aires, Argentina Abstract Kamareddine and Nederpelt [9], resp. Kamareddine and R os [11] gave two calculi of explicit of substitutions highly in uenced by de Bruijn s notation of the calculus. These calculi added to the explosive pool of work on explicit substitution in the past 15 years. As far as we know, calculi of explicit substitutions: a) are ....

....substitution in the past 15 years. As far as we know, calculi of explicit substitutions: a) are unable to handle local substitutions, and b) have answered (positively or negatively) the question of the termination of the underlying calculus of substitutions. The exception to a) is the calculus of [9] where substitution is handled both locally and globally. However, the calculus of [9] does not satisfy properties like con uence and termination. The exception to b) is the s e calculus [11] for which termination of the s e calculus, the underlying calculus of substitutions, remains unsolved. ....

[Article contains additional citation context not shown here]

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197-240, 1993.

....using implicit rather than explicit substitution. Implementations of the calculus provide their own explicit substitution procedures as in HOL [GM 93] Nuprl [Con 86] and Authomath [NGdV 94] Furthermore, research on theories of explicit substitution has been striving lately ( HL 89] ACCL 91] KN 93] Mel 95] BBLR 95] and [KR 95] In this paper, we extend the calculus of [KN 93] which is influenced by Authomath) giving B, a calculus which uses de Bruijn indices and where reduction and substitution are step wise and explicit. The species of variable names is cultivated and ordered so ....

.... their own explicit substitution procedures as in HOL [GM 93] Nuprl [Con 86] and Authomath [NGdV 94] Furthermore, research on theories of explicit substitution has been striving lately ( HL 89] ACCL 91] KN 93] Mel 95] BBLR 95] and [KR 95] In this paper, we extend the calculus of [KN 93] which is influenced by Authomath) giving B, a calculus which uses de Bruijn indices and where reduction and substitution are step wise and explicit. The species of variable names is cultivated and ordered so that a fine inter marriage between de Bruijn s indices and variable names takes place. ....

[Article contains additional citation context not shown here]

Kamareddine, F., and Nederpelt, R.P., (1993) On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....indeed it is easy to study this status in item notation. Finally, we show that for a substitution calculus a la de Bruijn with open terms, it is simpler to describe normal forms using item notation. There are further advantages of item notation that are studied elsewhere. For example, in [9], we show that explicit substitution is easily built in item notation and that global and local strategies of substitution can be accommodated. In [10] we show that with item notation, one can give a unified approach to type theory. An implementation of this item notation with most of the ....

....over Definition 1.3. But just imagine that in the calculus you had not only and ffi as internal operators but also oe for substitution, for typing and so on. In fact, internalising substitution (i.e. making it explicit) has been a topic of research in the last decade (see [1] 8] 7] [9]) Now, internalising extra operators means that in classical notation, in Definition 1.3, two extra rules are added for each new operator. In item notation on the other hand, Definition 2.3 does not depend on the number of operators. Simply, the set of operators to which belongs will increase. ....

[Article contains additional citation context not shown here]

F. Kamareddine and R. Nederpelt, On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....rather than implicit substitution. Implementations of the calculus provide their own explicit substitution procedures as in Nuprl and Automath . Furthermore, research on theories of explicit substitution has been striving lately 5;12;13;22;4;18 . In this paper, we extend the calculus of [13] (which is influenced by Automath) giving B, a calculus which uses de Bruijn indices and where reduction and substitution are step wise and explicit. The species of variable names is cultivated and ordered so that a fine inter marriage between de Bruijn s indices and variable names takes place. We ....

....of terms remains simple to describe and enables one to define reduction and substitution in a step wise fashion where at every step it is clear which item moves inside (or over) which one. This step wise fashion gives explicit substitution and enables local and global reduction as shown in [13]. We provide a method which takes any term of the calculus with named variables and implicit substitution, into B such that all ff equivalent terms in are mapped into a unique element of B. The other direction however, of mapping elements of B into elements of is more difficult. This is ....

[Article contains additional citation context not shown here]

F. Kamareddine, and R. Nederpelt, On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), (1993) 197--240.

....Several calculi including new operators to denote substitution and new rules to handle these operators have been proposed. Amongst these calculi we mention COE (cf. 6] the calculi of categorical combinators (cf. 4] oe, oe , oe SP (cf. 1, 5, 14] referred to as the oe family; oeBLT (cf. [7]) AE (cf. 3] and i (cf. 13] which are descendants of the oe family; s (cf. 8] s e (cf. 11] and t (cf. 10] This article will focus on oe, oe , AE, s, t and u which is an efficient version of s presented here for the first time. All these calculi are rewriting systems on a set of ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

....calculi including new operators to denote substitution have been proposed. Amongst these calculi we mention COE [dB78] the calculi of categorical combinators [Cur86] oe [ACCL91] oe [CHL92] oe SP [R io93] referred to as the oe family; AE [BBLRD95] a descendant of the oe family; oeBLT [KN93], exp [Blo95] s [KR95a] s e [KR96a] and i [MH95] All these calculi (except exp) are described in a de Bruijn setting where natural numbers play the role of variables and the set of terms on which substitution will be made explicit is defined by: IN j ( j ( But, why so many varieties ....

....of AE on open terms. Hence, i preserves strong normalisation and is itself confluent on open terms. Unfortunately, i is not able to simulate one step fi reduction as shown in [MH95] Instead, it simulates only a big step fi reduction. Another line of expliciting substitutions has been made in [KN93, KR95a, KR96a, KR96, KRW97]. In [KN93] the calculus was rewritten using a notation influenced strongly by de Bruijn s notation for Automath [NGdV94] In that notation [KN95] every term is simply a sequence of items followed by a variable. This item notation, allowed also the introduction of so called substitution items ....

[Article contains additional citation context not shown here]

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

....to handle these operators have been proposed. Amongst these calculi we mention COE (cf. dB78b] the calculi of categorical combinators (cf. Cur86] oe, oe , oe SP (cf. ACCL91] CHL92] R io93] referred to as the oe family; AE (cf. BBLRD95] a descendant of the oe family and oeBLT (cf. [KN93]) The basic features of these systems of substitution depart quite extensively from the classical calculus while in this paper we propose a system which remains as close as possible to it. Furthermore, for the above systems either strong normalisation (SN) has not been studied (as for COE and ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

....is hardly visible. Reshuffling A and C however, makes this claim visible. That is, TS(A) j TS(C) j TS(B) and so all three terms A, B and C are reductionally equivalent. The classical notation cannot extend the notion of redexes or enable reshuffling in an easy way. Item notation however (see [KN 93] KN 94] and [KN 96b] can. In item notation, complex terms of the cube are of the form (A )B where 2 fffig[fO x ; x is a variable, a or a 2 and O = or Pig. We call (A ) an item and (Affi)B means apply B to A (note the order) A redex starts with a ffi item next to a item ( KN 96b] ....

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....classical notation. Moreover, we showed in the same paper that accounting for bound and free variables in a term is only a matter of a very simple calculation and demonstrated that term construction can be done via trees which are at the same time proofs of the well typedness of the term. 5. In [KN 93] we embedded stepwise substitution in the new calculus showing how the new notation facilitates the introduction of substitution as an object level notation in the calculus resulting in a system which can accommodate most substitution strategies. 6. in [KN 9z] we show that reduction can be ....

....where not all the occurrences of the name of the function are replaced by the body of the function. In many mathematical proofs, we need to keep the name instead of the body of the function. This will be facilitated by our notation and using our explicit substitution and reduction rules of [KN 93] Example 3.5 In this example we use two s which we denote Pi and respectively. Now the following introduces as a term of type , as a term of type and defines ) as the product ( a ) b ) a Pi x )b. This states that, given c and d of type , the term (dffi) cffi) fi reduces to the ....

[Article contains additional citation context not shown here]

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993. 61

....calculi, including new operators to denote substitution, have been proposed. Among these calculi we mention COE [dB78] the calculi of categorical combinators [Cur86] oe [ACCL91] oe [CHL96] and oe SP [R io93] referred to as the oe family; AE [BBLRD96] a descendant of the oe family; oeBLT [KN93]; exp [Blo95] s [KR95a] s e [KR97] and i [Hur96a] All of these calculi (except exp) are described in a de Bruijn setting, where natural numbers play the role of variables. In [KR95a] we extended the calculus with explicit substitutions by turning de Bruijn s meta operators into object ....

....only present in the meta language of the calculus. By doing so, we believe that our calculi are closer to the calculus from an intuitive rather than a categorical point of view. A calculus accommodating explicit substitution via explicit rewrite rules in the calculus was first presented in [KN93]. In that article, the intention was to introduce the philosophy in general, and the calculus obtained did not possess either confluence or preservation of strong normalization. In [KR95a] the part of the calculus that was confluent and preserved strong normalization was singled out. In this ....

F. Kamareddine and R. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4:197--240, 1993.

....as (I(B)ffi)I(A) here is ffi a special symbol used for application) and I(O x:A :B) is written as (I(A)O x )I(B) where O = or Pi. Both (tffi) and (tO x ) t being a term in item notation, are called items. For reasons explaining the usefulness of such a notation, the reader is referred to [KN 93] and [KN 9z] For this paper however, the reader is to notice that redexes and definitions can be easily generalised and 4 introduced with item notation. A traditional redex is a term that starts with a ffi item next to a item. A definition is itself a certain form of a ffi item next to a ....

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....new operators to denote substitution and new rules to handle these operators have been proposed. Amongst these calculi we mention COE (cf. dB78b] the calculi of categorical combinators (cf. Cur86] oe, oe , oe SP (cf. ACCL91] CHL92] R io93] referred to as the oe family; oeBLT (cf. [KN93]) AE (cf. BBLRD95] and i (cf. MH95] which are descendants of the oe family; s (cf. KR95a] and s e (cf. KR96] All the calculi above mentioned are described in de Bruijn notation (cf. dB72] and [dB78a] This formalism consists in replacing the usual variable names with natural numbers ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

....this presentation, called calculus a la de Bruijn, the argument appears before the function and terms are structured in a di erent manner to the classical calculus. The calculus a la de Bruijn can also be written using de Bruijn indices instead of variable names, and we refer the reader to [23] for further details. In Section 3.1 we present the pure type systems framework in the classical notation of the calculus using variable names. In Section 3.2 we present the pure type systems in classical notation using de Bruijn indices and establish their isomorphism to the version with ....

.... Terms of the calculus a la de Bruijn are also constructed using application (as in (B)A) or abstraction (as in [v]A if variable names are used, or [ A if de Bruijn indices are used) The calculus a la de Bruijn is only given using variable names, for the version using de Bruijn indices see [23]. 2.1 Rewriting notions All the systems of this paper have a common feature. First, the syntax (the set of terms, types, substitutions, etc. is given and then a set of rules that work on the syntax is presented. Those rules are rewrite rules and are of the form A R B or (A; B) 2 R if we ....

F. Kamareddine and R.P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science 4(3), 197-240, 1993.

....and used in practical languages and theorem provers such as prolog and Isabelle [35, 37] In most of these approaches, the notion of substitution plays an important role. The importance of the notion of substitution led to an explosion of work on making substitutions explicit in recent years [1, 7, 24, 26, 19, 9, 21]. Moreover, a number of works have been devoted to establishing the usefulness of explicit substitution to automated deduction and theorem proving [32, 34] to proof theory [43] to programming languages [29, 6, 8] and to HOU [16] The latter paper [16] shows that in the HOU framework, if ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197-240, 1993.

....and used in practical languages and theorem provers such as prolog and Isabelle ( NW90,Pau90] In most of these approaches, the notion of substitution plays an important role. The importance of the notion of substitution led to an explosion of work on expliciting substitutions in recent years [ACCL91,BBLRD96,KN93,KR95,FKP96,Blo97,Gui99b]. Also, a number of work has been devoted to establish the usefulness of explicit substitution to automated deduction and theorem proving [Mag95,Mu n97b] to proof theory [VW99] to programming languages [KRW98,Ben97,BLR96] and to HOU [DHK95] The latter paper shows that in the HOU framework, if ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197-240, 1993.

....have been proposed. Amongst these calculi we mention C [dB78] the calculi of categorical combinators [Cur86] ACCL91] CHL96] SP [R o93] referred to as the family; BBLRD96] the calculi of [FKP99] and [Mu n97c] which are descendants of the family; BLT [KN93], calculus [LRD95] x [BR96] s [KR95] t [KR98] s e [KR97] and l [Gui99b, Gui99a] All these calculi (except x) are described in a de Bruijn setting where natural numbers play the role of variables and the set of terms on which substitution will be made explicit is de ned by: ....

....is itself con uent on open terms. Unfortunately, is not able to simulate one step reduction as shown in [Mu n97c] Instead, it simulates only a big step reduction. This is our reason for not discussing it further in this paper. Another line of expliciting substitutions has been made in [KN93, KR95, KR97, KRW98]. In [KN93] the calculus was rewritten using a notation in uenced strongly by de Bruijn s notation for Automath [NGdV94] In that notation [KN93] every term is simply a sequence of items followed by a variable. This item notation, allowed also the introduction of so called substitution ....

[Article contains additional citation context not shown here]

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197-240, 1993.

....Several calculi including new operators to denote substitution and new rules to handle these operators have been proposed. Amongst these calculi we mention C OE (cf. 6] the calculi of categorical combinators (cf. 4] oe, oe , oe SP (cf. 1, 5, 14] referred to as the oe family; oeBLT (cf. [7]) AE (cf. 3] and i (cf. 13] which are descendants of the oe family; s (cf. 8] s e (cf. 11] and t (cf. 10] This article will focus on oe, oe , AE, s, t and u which is an efficient version of s presented here for the first time. All these calculi are rewriting systems on a set of ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

....explicitly; various calculi including new operators to denote substitution have been proposed. Amongst these calculi we mention C OE (cf. dB78] the calculi of categorical combinators (cf. Cur86] oe, oe , oe SP (cf. ACCL91] CHL92] R io93] referred to as the oe family; oeBLT (cf. [KN93]) AE (cf. BBLRD95] a descendant of the oe family; s (cf. KR95a] exp (cf. Blo95] and i (cf. MH95] These calculi (except exp) are described in a de Bruijn setting where natural numbers play the role of the classical variables. Classical terms are coded as closed terms in these This ....

....Theorem 6 (Local confluence) The s e and s e calculi are locally confluent on s op . We give now further motivation for the rules of s e . Motivation behind the rules of Figure 2 was given in [KR95a] and motivation for explicit substitution rules that belong to the same family can be found in [KN93]. Hence, we concentrate on the rules of Figure 3. We gave already some motivation for the oe oe transition rule where we said that such a rule helps to re establish confluence. The other rules were also introduced as a necessity to close critical pairs. Notice now the following symetries: there ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

....partially substituted terms, we must render the latter from being a metalevel notion to an object level notion. It turns out that our new notation will enable such rendering efficiently and will enable the representation of various forms of substitution: local, global, implicit and explicit. KN 93] introduces substitution which is object level but which can evaluate terms fully obtaining the result of the metalevel substitution. More precisely, we introduce the process of stepwise substitution, which is meant to refine reduction procedures. Since substitution is the fundamental operation ....

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....using implicit rather than explicit substitution. Implementations of the calculus provide their own explicit substitution procedures as in HOL [GM 93] Nuprl [Con 86] and Authomath [NGdV 94] Furthermore, research on theories of explicit substitution has been striving lately ( HL 89] ACCL 91] KN 93] Mel 95] BBLR 95] and [KR 95] In this paper, we extend the calculus of [KN 93] which is influenced by Authomath) giving B, a calculus which uses de Bruijn indices and where reduction and substitution are step wise and explicit. The species of variable names is cultivated and ordered so ....

.... their own explicit substitution procedures as in HOL [GM 93] Nuprl [Con 86] and Authomath [NGdV 94] Furthermore, research on theories of explicit substitution has been striving lately ( HL 89] ACCL 91] KN 93] Mel 95] BBLR 95] and [KR 95] In this paper, we extend the calculus of [KN 93] which is influenced by Authomath) giving B, a calculus which uses de Bruijn indices and where reduction and substitution are step wise and explicit. The species of variable names is cultivated and ordered so that a fine inter marriage between de Bruijn s indices and variable names takes place. ....

[Article contains additional citation context not shown here]

Kamareddine, F., and Nederpelt, R.P., (1993) On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....explicitly; various calculi including new operators to denote substitution have been proposed. Amongst these calculi we mention C OE [12] the calculi of categorical combinators [9] oe [1] oe [10] oe SP [43] referred to as the oe family; AE [5] a descendant of the oe family; oeBLT [20], exp [6] s [23] s e [26] and i [36] All these calculi (except exp) are described in a de Bruijn setting where natural numbers play the role of variables. In [23] we extended the calculus with explicit substitutions by turning de Bruijn s meta operators into object operators, thus offering a ....

....only present in the meta language of the calculus. By doing so, we believe that our calculi are closer to the calculus from an intuitive point of view, rather than a categorical one. A calculus accommodating explicit substitution via explicit rewrite rules in the calculus was first presented in [20]. In that article, the intention was to introduce the philosophy in general and the calculus obtained did not possess neither confluence nor preservation of strong normalisation. In [23] the part of the calculus that was confluent and preserved strong normalisation was singled out. In this paper, ....

F. Kamareddine and R. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

....including new operators to denote substitution have been proposed. Amongst these calculi we mention C OE (cf. 10] the calculi of categorical combinators (cf. 8] oe, oe , oe SP (cf. 1] 9] 33] referred to as the oe family; AE (cf. 4] a descendant of the oe family; oeBLT (cf. [18]) exp (cf. 5] s (cf. 20] s e (cf. 22] and i (cf. 28] All these calculi (except exp) are described in a de Bruijn setting where natural numbers play the role of the classical variables. In [20] we extended the calculus with explicit substitutions by turning de Bruijn s metaoperators ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

....based on but where de Bruijn s indices and explicit substitution are used. For this, we start by introducing de Bruijn s indices. Such indices have the practical advantages that they avoid all the need to deal with variable renaming in terms (see [de Bruijn 72] Abadi et al. 91] CH 88] and [KN 93] The calculus based on and on de Bruijn s indices will be called Omega Xi for Xi being the set of variables which are de Bruijn s indices together with a special variable. In the first instance, Omega is taken to be f; ffig. In order to accommodate substitution explicitly and in order to ....

....that this is the first precise formulation of terms, variables and reductions. Furthermore, we believe that this formulation not only enables explicit and local substitution as we show in this paper, but also enables a generalisation over all branches of calculus and type theory (see [KN 93] NK 94] and [KN 9x] To sum up, we provide , a calculus which uses item notation, variable names and explicit substitution. We extend to Omega Xi where item notation is used with de Bruijn indices instead of variable names and explicit rather than implicit substitution. We provide the ....

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Kamareddine, F., and Nederpelt, R.P., (1993) On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....calculi including new operators to denote substitution have been proposed. Amongst these calculi we mention C OE [dB78] the calculi of categorical combinators [Cur86] oe [ACCL91] oe [CHL92] oe SP [R io93] referred to as the oe family; AE [BBLRD95] a descendant of the oe family; oeBLT [KN93], exp [Blo95] s [KR95a] s e [KR96a] and i [MH95] All these calculi (except exp) are described in a de Bruijn setting where natural numbers play the role of variables and the set of terms on which substitution will be made explicit is defined by: IN j ( j ( But, why so many varieties ....

....of AE on open terms. Hence, i preserves strong normalisation and is itself confluent on open terms. Unfortunately, i is not able to simulate one step fi reduction as shown in [MH95] Instead, it simulates only a big step fi reduction. Another line of expliciting substitutions has been made in [KN93, KR95a, KR96a, KR96, KRW97]. In [KN93] the calculus was rewritten using a notation influenced strongly by de Bruijn s notation for Automath [NGdV94] In that notation [KN95] every term is simply a sequence of items followed by a variable. This item notation, allowed also the introduction of so called substitution items ....

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F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

....is hardly visible. Reshuffling A and C however, makes this claim visible. That is, TS(A) j TS(C) j TS(B) and so all three terms A, B and C are reductionally equivalent. The classical notation cannot extend the notion of redexes or enable reshuffling in an easy way. Item notation however (see [KN 93] KN 94] and [KN 96b] can. In item notation, complex terms of the cube are of the form (A )B where 2 fffig[fO x ; x is a variable, a or a 2 and O = or Pig. We call (A ) an item and (Affi)B means apply B to A (note the order) A redex starts with a ffi item next to a item ( KN 96b] ....

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....term reshuffling. The work carried out in this paper will have many applications. We mentioned the semantics of lazy evaluation and the new reduction strategies which may lead to further optimal results. These points are under investigation. The new notation moreover deserves attention. KN 93] and [NK 94] have shown many of its advantages for formulating and generalising type theory and for rendering substitution explicit in the calculus. Further advantages are also studied in [KN 9z] ....

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....new operators to denote substitution and new rules to handle these operators have been proposed. Amongst these calculi we mention C OE (cf. dB78b] the calculi of categorical combinators (cf. Cur86] oe, oe , oe SP (cf. ACCL91] CHL92] R io93] referred to as the oe family; oeBLT (cf. [KN93]) AE (cf. BBLRD95] and i (cf. MH95] which are descendants of the oe family; s (cf. KR95a] and s e (cf. KR96] All the calculi above mentioned are described in de Bruijn notation (cf. dB72] and [dB78a] This formalism consists in replacing the usual variable names with natural numbers ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

....as (I(B)ffi)I(A) here is ffi a special symbol used for application) and I(O x:A :B) is written as (I(A)O x )I(B) where O = or Pi. Both (tffi) and (tO x ) t being a term in item notation, are called items. For reasons explaining the usefulness of such a notation, the reader is referred to [KN 93] and [KN 9z] For this paper however, the reader is to notice that redexes and definitions can be easily generalised and introduced with item notation. A traditional redex is a term that starts with a ffi item next to a item. A definition is itself a certain form of a ffi item next to a ....

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....the main Pi items are removed. For example, if S j (xffi) y z ) z Pi r ) then S j (y z ) z r ) and S Pi j (xffi) y Pi z ) With these notations, Gamma; Sx) j S Pi ( GammaS ; x) This item notation has been used to study, extend and clarify many notions of the calculus (see [KN 93] and [KN 9y] Remark 5.9 Note that typability of subterms fails for . That is, can be defined for some A without being defined for all its subterms. For example, x: x)y) j ( Pi x: y, but ( y) is not defined. Note also that unicity of types fails for . That is, we can have ....

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....calculi including new operators to denote substitution have been proposed. Amongst these calculi we mention C OE (cf. 10] the calculi of categorical combinators (cf. 8] oe, oe , oe SP (cf. 1, 9, 34] referred to as the oe family; AE (cf. 4] a descendant of the oe family; oeBLT (cf. [15]) exp (cf. 5] s (cf. 17] s e (cf. 19] and i (cf. 29] All these calculi (except exp) are described in a de Bruijn setting where natural numbers play the role of the classical variables. In [17] we extended the calculus with explicit substitutions by turning de Bruijn s meta operators ....

F. Kamareddine and R. P. Nederpelt. On stepwise explicit substitution. International Journal of Foundations of Computer Science, 4(3):197--240, 1993.

....at present. The notation presented in this paper has further advantages than generalising reduction and term reshuffling. These advantages are studied in our articles mentioned in the bibliography. Of these advantages however, we mention the ability to describe substitution explicitly as in [KN 93] and of generalising type systems as in [KN 94] There is moreover the advantage of being able to make normal order reduction more efficient than in the classical calculus. The reason for this being that when searching for the leftmost outermost redex in a term, we need to make less recursive ....

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....partially substituted terms, we must render the latter from being a metalevel notion to an object level notion. It turns out that our new notation will enable such rendering efficiently and will enable the representation of various forms of substitution: local, global, implicit and explicit. [19] introduces substitution which is object level but which can evaluate terms fully obtaining the result of the metalevel substitution. More precisely, we introduce the process of stepwise substitution, which is meant to refine reduction procedures. Since substitution is the fundamental operation ....

Kamareddine, F., and Nederpelt, R.P., On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240, 1993.

....as (I(B)ffi)I(A) here is ffi a special symbol used for application) and I(O x:A :B) is written as (I(A)O x )I(B) where O = or Pi. Both (tffi) and (tO x ) t being a term in item notation, are called items. For reasons explaining the usefulness of such a notation, the reader is referred to [KN 93] and [KN 96a] For this paper however, the reader is to notice that redexes and definitions can be easily generalised and introduced with item notation. A traditional redex is a term that starts with a ffi item next to a item. A definition is itself a certain form of a ffi item next to a item. ....

....the Cube with definitions only, or with both definitions and generalised reduction. When using generalised reduction without definitions, one must remain in the and as the other systems lose their SR. 2 The item notation For a detailed description of item notation, the reader is referred to [KN 93] KN 94] KN 95] and [KN 96a] We will introduce in this section the minimum machinery needed to represent the Cube in item notation and for introducing generalised reduction and definitions. The systems of the Cube are based on a set of pseudo expressions T defined by: T = V j C j (T ffi)T ....

Kamareddine, F., and Nederpelt, R.P. (1993), On stepwise explicit substitution, International Journal of Foundations of Computer Science 4 (3), 197-240.

....to denote substitution have been proposed. Amongst these calculi we mention C OE (cf. de Bruijn, 1978) the calculi of categorical combinators (cf. Curien, 1986) oe, oe , oe SP (cf. Abadi et al. 1991) Curien et al. 1992) R ios, 1993) referred to as the oe family; oeBLT (cf. (Kamareddine Nederpelt, 1993)) AE (cf. Benaissa et al. 1995) a descendant of the oe family; s (cf. Kamareddine R ios, 1995a) exp (cf. Bloo, 1995) and i (cf. Mu noz Hurtado, 1996) These calculi (except exp) are described in a de Bruijn setting where natural numbers play the role of the classical variables. ....

....terms 11 The s e and s e calculi are locally confluent on s op . We give now further motivation for the rules of s e . Motivation behind the rules of Figure 2 was given in (Kamareddine R ios, 1995a) and motivation for explicit substitution rules that belong to the same family can be found in (Kamareddine Nederpelt, 1993). Hence, we concentrate on the rules of Figure 3. We gave already some motivation for the oe oe transition rule where we said that such a rule helps to re establish confluence. The other rules were also introduced as a necessity to close critical pairs. Remark now the following symetries: there ....

Kamareddine, F., & Nederpelt, R. P. (1993). On stepwise explicit substitution. International journal of foundations of computer science, 4(3), 197--240.

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