Benaissa, Z., Briaud, D., Lescanne, P., and Rouyer-Degli, J. (1996). ##, a calculus of explicit substitutions which preserves strong normalisation. Journal of Functional Programming, 6(5):699--722.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 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 ....

Z.-el-A. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. , a Calculus of Explicit Substitutions which Preserves Strong Normalization. Journal of Functional Programming, 6(5):699-722, 1996.

.... 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 ....

Z.-el-A. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. , a Calculus of Explicit Substitutions which Preserves Strong Normalization. Journal of Functional Programming, 6(5):699-722, 1996.

....been an interest in formalising substitution explicitly in order to provide a theoretical framework for the implementation of programming languages and theorem provers. Several calculi including new operators to denote 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 ....

.... 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 substitution calculi ....

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Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. , a calculus of explicit substitutions which preserves strong normalisation. Journal of Functional Programming, 6(5):699-722, September 1996.

.... calculi of explicit substitutions which are confluent on open terms: the oe calculus (cf. HL89] and [CHL91] but the non preservation of strong normalisation for oe has recently been proved (cf. Mel95] There are also calculi which satisfy the preservation property: the AE calculus (cf. [BBLRD95]) but this calculus is not confluent on open terms. Moreover, in order to get a confluent extension, the introduction of a composition operator for substitutions seems unavoidable, but precisely this operator is the cause of the non preservation of strong normalisation as shown in [Mel95] 1 The ....

.... calculus: Theorem 5 (Soundness) Let a; b 2 , if a s b then a fi b. 2 The typed s calculus The proof of strong normalisation of well typed terms that we give in this section follows an original idea of Melli es (personal communication) which is based on the technique developped in [BBLRD95] to prove the preservation of strong normalisation in the AE calculus. We shall recall first the syntax and typing rules for the simply typed calculus in de Bruijn notation. The types are generated from a set of basic types K with the binary type operator . Environments are lists of types. ....

Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. AE, a calculus of explicit substitutions which preserves strong normalisation. Personal communication, 1995.

....the set of fi SN terms, but also that we can actually use fi e with explicit substitution. In fact, explicit substitution, is an important topic of research and PSN is an important property for any calculus extended with explicit substitution. In fact, lately, much research has been carried out ( BLR 95] and [KR 95] in order to find systems of explicit substitution which are both confluent and have the PSN property (if M is fi SN then M is s SN where s is the lambda calculus extended with 3 explicit substitution) This is the reason for our interest in PSN of fi e (which is confluent by the ....

....e SN. Preservation of Strong Normalisation (PSN) is a property that has to be established for any extension of a reduction relation that is strongly normalising. For example, a lot of research has been carried out lately to establish PSN for fi reduction extended with explicit substitution (see [BLR 95] KR 95] and [MN 95] The results of this paper establish that fi e is indeed a safe extension of fi. Finally, it is worth noting that we used item notation in this paper in order to reach the results desired. There is a reason for this. In the usual notation, generalised redexes are not ....

Benaissa, Briaud, Lescanne, Rouyer-Degli, AE, a calculus of explicit substitutions which preserves strong normalisation, personel communication, 1995.

....of s become available as [ i . Finally, 13] introduced a slash operator of sort term substitution which transforms a term a into a substitution a= This operator may be considered as consing with id (in the jargon) and was rst introduced and exploited in the calculus (cf. [2]) Here is the formalisation of this syntax and the rewriting rules of : De nition 3.18 The set of terms of the calculus, is de ned as [ where are mutually de ned as follows (j 1 and i 0) Terms : IN j j [ j : The ....

Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. , a calculus of explicit substitutions which preserves strong normalisation. Functional Programming, 6(5), 1996.

....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 that a fine ....

....to order the list of bound variables which we call l imposing ff conversion. In Appendix B, we provide a semantics where all ff equivalent terms are identifiable. In [KR 95] s, the subsystem of B where oe generation does not preserve the ffi couple, has been studied. s along with the system of [BBLR 95] are the first calculi of explicit substitution which enjoy confluence on closed terms and preserve strong normalisation. In [KR 9x] it was shown that in the simply typed version of s, well typed terms are strongly normalising. In [KR 9y] it was shown that s extended with open terms is ....

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Benaissa Z., Briaud D., Lescanne P. and Rouyer-Degli J., (1995), AE, a calculus of explicit substitutions which preserves strong normalisation, Personal communication.

....to order the list of bound variables which we call l imposing ff conversion. In Appendix B, we provide a semantics where all ff equivalent terms are identifiable. In [18] s, the subsystem of B where oe generation does not preserve the ffi couple, has been studied. s along with the system of [4] are the first calculi of explicit substitution which enjoy confluence on closed terms and preserve strong normalisation. In [19] it was shown that in the simply typed version of s, welltyped terms are strongly normalising. In [20] it was shown that s extended with open terms is confluent. At ....

....s extended with open terms is confluent. At the moment, we are extending the work of [18,19,20] to study the properties of s where oe generation preserves the ffi couple, hence resulting in the system B of this paper. Finally, Daniel Briaud noted our attention that adding intersection types to [4] is problematic as there will be terms that are strongly normalising but not typable. This is not the case when intersection types are added to s. This could be seen as an advantage to our framework of remaining close to the calculus rather than using combinators as in [1,4] Acknowledgements I ....

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Z. Benaissa, D. Briaud, P. Lescanne and J. Rouyer-Degli, AE, a calculus of explicit substitutions which preserves strong normalisation, Functional programming 6(5), (1997).

....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 terms that ....

....concerned. The oe calculus is a variation of the oe calculus that is confluent on open terms (terms with variables of sort term and substitution) As all the calculi in the oe family were shown in [12] not to possess the Preservation of Strong Normalisation property (PSN) the AE calculus (cf. [3]) removes the composition of substitutions to guarantee PSN. For every , we use a; b; c; to range over the set of terms , and s; t; to range over the set of substitutions . We use to denote the set of rules of the calculus (which contains a rule (Beta) and take the ....

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Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. AE, a calculus of explicit substitutions which preserves strong normalisation. Journal of Functional Programming, 1995.

....in formalising substitution explicitly; various 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: ....

....of s are available in our new syntax as Gamma[ i . Finally, we introduce a slash operator of sort term substitution wich transform a term a into a substitution a= This operator may be considered as consing with id (in the oe jargon) and has been exploited in the AE calculus (cf. [BBLRD95]) Here is the formalisation of this syntax and the rewriting rules of : Definition 10 The set of terms of the calculus, noted , is defined as [ where are mutually defined as follows: Terms : IN j [ j : where j 1 and i 0. The set, ....

Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. AE, a calculus of explicit substitutions which preserves strong normalisation. Personal communication, 1995.

....[ACCL91] and therefore the relation between both calculi will be made explicit. The confluence of the s calculus is obtained by the interpretation method ( Har89] CHL92] The proof of the preservation of normalisation follows the lines of an analogous result for the AE calculus (cf. [BBLRD95]) The relation between s and AE is also studied. 1 Introduction Most literature on the calculus considers substitution as an implicit operation. It means that the computations to perform substitution are usually described with operators which do not belong to the language of the calculus. ....

....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; 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 ....

[Article contains additional citation context not shown here]

Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. AE, a calculus of explicit substitutions which preserves strong normalisation. Personal communication, 1995.

....in formalizing substitution explicitly; various 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 ....

....oe [CHL96] and i [Hur96a] but they also have important disadvantages. Melli es proved that oe (as well as the rest of the oe family and the categorical combinators) does not preserve strong normalization [Mel95] There are also calculi that preserve strong normalization, e.g. the AE calculus [BBLRD96], but this calculus is not confluent on open terms. Recently, the i calculus (cf. Hur96a] has been proposed as a calculus that preserves strong normalization and is itself confluent on open terms. The i calculus works with two new applications that allow the passage of substitutions within ....

[Article contains additional citation context not shown here]

Z.-E.-A. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. AE, a calculus of explicit substitutions which preserves strong normalisation. Journal of Functional Programming, 6, 1996.

....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 which account for the ....

....9.3, we have: u(a) exp u(a 1 ) exp u(a 2 ) exp : exp u(a 2n 1 ) exp u(a 2n 2 ) exp : and this contradicts the fact that u(a) 2 exp SN. Therefore, a 2 t SN. 2 20 7 Comparison with oe and AE For the syntax and rules of the oe and AE calculi see [ACCL91] and [BBLRD95], respectively. The t calculus can be interpreted into the oe calculus using a similar translation as the one presented in [KR95a] to interpret the s calculus into oe. However, in the case of the t calculus the interpretation works better: now t derivations are preserved (only s derivations and ....

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Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. AE, a calculus of explicit substitutions which preserves strong normalisation. Personal communication, 1995.

....important in its implementation. A good starting point for an adequate intensional representation of lambda terms is the de Bruijn notation for lambda terms [3] This notation eliminates names for bound variables, thus simplifying identity checking modulo renaming. Explicit substitution notations [1,2,4,8,13] that build on the de Bruijn scheme provide the basis for meeting several of the other mentioned requirements. There are di erences in the speci c characteristics of such notations and choices must also be made in the speci c manner in which these are to be deployed in the context of ....

....Of these, combinability of substitutions seems to be the most important for metalanguage implementations. Unfortunately, most explicit substitution calculi seem not to include this facility. Particular calculi sacri ce other properties as well. The calculus preserves strong normalizability [2] but it does not admit meta variables. The s e calculus permits meta variables and is con uent even with this addition [8] but does not preserve strong normalizability [6] The wso calculus alone both admits meta variables and preserves strong normalizability [4] The two calculi that do ....

Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. , a calculus of explicit substitutions which preserves strong normalization. Journal of Functional Programming, 6(5):699-722, 1996.

....index 1. Further, the indices of the free variables in the terms that appear in the environment must themselves be incremented by 1. Explicit substitution notations that have been developed in recent years for the lambda calculus o er a complete treatment of this kind of encoding of substitutions [1, 7, 16, 21, 39]. We outline here a version of such a notation that we have developed for use speci cally in the implementation of our higher order language [31] This notation builds on the traditional de Bruijn notation by adding a new category of terms called a suspension. A suspension represents a skeletal ....

Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. , a calculus of explicit substitutions which preserves strong normalization. Journal of Functional Programming, 6(5):699-722, 1996.

.... only in the names chosen for bound variables is, for instance, to transform them into a nameless form using a scheme due to de Bruijn [4] Similarly, several new notations for the lambda calculus have been described in recent years that have the purpose of making substitutions explicit (e.g. see [1, 3, 10, 18]) However, the actual manner in which all these devices should be deployed in a practical context is far from clear. In particular, there are tradeo s involved with di erent choices and determining the precise way in which to make them requires experimentation with an actual system: the ....

....annotated version of t can be rewritten using our rules to a term s that is itself a consistently annotated version of s. The ( s ) rule is redundant to our collection if our sole purpose is to simulate contraction; indeed, omitting this rule yields a calculus that is similar to those in [3] and [10] However, the ( s ) rule is the only one in our system that permits substitutions arising from di erent contractions to be combined into one environment and, thereby, to be carried out in the same walk over the structure of the term being substituted into. The rule (r11) is also ....

Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. , a calculus of explicit substitutions which preserves strong normalization. Journal of Functional Programming, 6(5):699-722, 1996. 14

....Hence we consider in this paper a calculus based on DILL rather than on the subset. 2. It is also possible to construct linear calculi with explicit substitution based on other linear calculi like the one by Bierman et al. 5] and other calculi with explicit substitutions like x [6] or [4]. This construction follows a general schema: We take a linear calculus and extend it rstly by the term substitutions of the calculus with explicit substitutions (e.g. x or ) and secondly by the let substitutions discussed above. The correctness theorem, which states that the reductions of ....

Z.-E.A. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. , a calculus of explicit substitutions which preserves strong normalisation. Journal of Functional Programming, 6(5), 1996.

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Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. ##, a calculus of explicit substitutions which preserves strong normalisation. Journal of Functional Programming, 6(5):699--722, 1996.

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Benaissa, Z., Briaud, D., Lescanne, P., and Rouyer-Degli, J. (1996). ##, a calculus of explicit substitutions which preserves strong normalisation. Journal of Functional Programming, 6(5):699--722.

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Z.-el-A. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. , a Calculus of Explicit Substitutions which Preserves Strong Normalization. J. of Func. Programming, 6(5):699-722, 1996.

No context found.

Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. AE, a calculus of explicit substitutions which preserves strong normalisation. Personal communication, 1995.

No context found.

Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. ##, a calculus of explicit substitutions which preserves strong normalisation. Personal communication, 1995.

No context found.

Z.E.A. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. , a calculus of explicit substitutions which preserves strong normalisation. Journal of Functional Programming, 6(5):699-722, 1996.

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Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. AE, a calculus of explicit substitutions which preserves strong normalisation. Personal communication, 1995.

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Z. Benaissa, D. Briaud, P. Lescanne, and J. Rouyer-Degli. v, a calculus of explicit substitutions which preserves strong normalization. Journal of Functional Programming, 6(5):699-- 722, 1996.

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