N.G. de Bruijn. A namefree lambda calculus with facilities for internal de nition of expressions and segments. Technical Report TH-Report 78-WSK03, Department of Mathematics, Technical University of Eindhoven, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 of these calculi are described in de Bruijn ....

N. de Bruijn. A namefree lambda calculus with facilities for internal de nition of expressions and segments. Technical report, Department of Mathematics , University of Eindhoven, Netherlands, 1978.

....with this as its root. 8 Describing normal forms in a substitution calculus Lambda calculi with explicit substitutions attempt to close the gap between the classical calculus and concrete implementations. Recently, there has been various attempts at providing calculi of explicit substitution ([6], 7] 9] 13] 14] Most of the above mentioned work (except [9] uses classical notation. 13] provided s, a calculus of substitution a la de Bruijn, which remains as close as possible to the classical calculus. Here is a descrition of s (we assume familiarity with de Bruijn indices) ....

N.G. de Bruijn, A namefree lambda calculus with facilities for internal definition of expressions and segments, Technical Research Report, 78-WSK-03, Eindhoven University of Technology, Department of Mathematics, 1978.

....substitution explicitly in order to provide a theoretical framework for the implementation of functional programming languages. 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 ....

N. G. de Bruijn. A namefree lambda calculus with facilities for internal definition of expressions and segments. Technical Report TH-Report 78-WSK-03, Department of Mathematics, Eindhoven University of Technology, 1978.

....proofs, which are believed to need explicit substitutions [MH95, Mag95] The last fifteen years have seen an increasing interest 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 ....

N. G. de Bruijn. A namefree lambda calculus with facilities for internal definition of expressions and segments. Technical Report TH-Report 78WSK -03, Department of Mathematics, Eindhoven University of Technology, 1978.

....incomplete proofs, which are believed to need explicit substitutions [Hur96b, Mag95] The last 15 years have seen an increasing interest 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 ....

N. G. de Bruijn. A name-free lambda calculus with facilities for internal definition of expressions and segments. Technical Report 78-WSK-03, Department of Mathematics, Eindhoven University of Technology, 1978.

....not applicable Type inference not applicable Weak Normalization Strong Normalization Technical Issues Uses MV yes Explicit Subst. Functional encoding Based on Untyped calculus de Bruijn Ind. no 3.2 Automath 3.2. 1 The calculus of segments The calculus was introduced by de Bruijn [9] and later further studied in [2] It provides facilities to introduce abbreviations for parts of terms containing a hole which can be lled with another term. Let us consider a small example 1 : d y l x l y x l x l y This diagram is a tree representation of the term x y x y:xy. In the ....

....for a segment. The binding between the segment and the segment variable is accomplished by an arti cially created redex ( x y: Second, there is a problem with the bound variables: in the original expression the variables x and y are bound and in the abbreviated expression they 1 In [9] and [2] the system is presented as a namefree calculus, but here we will illustrate it using named variables. 25 seem to be outside the scope of the binders. This situation is resolved by adding annotations to the segment variables: d l s w l x l y s(x1,y1) s(x2,y2) d y2 x2 Things now ....

N.G. de Bruijn. A namefree lambda calculus with facilities for internal definition of expressions and segments. Technical Report 78-WSK-03, Eindhoven University of Technology, 1978.

....avoiding names in the calculus is an e ective way of clarifying the meaning of terms and, for the uni cation process, of eliminating dummy and redundant renaming. N. de Bruijn developed a notation for the calculus where 2. PRELIMINARIES 527 names of bound variables were replaced by indices [13, 15, 14]. These indices relate bound variables to their corresponding abstractors. It is clear that the correspondence between an occurrence of a bound variable and its associated abstractor operator is uniquely determined by its depth, that is the number of abstractors between them. Hence, terms can ....

N. G. de Bruijn. A Namefree Lambda Calculus with Facilities for Internal Denition of Expressions and Segments. Technical Report TH-Report 78-WSK-03, Department of Mathematics, Eindhoven University of Technology, 1978.

....con uence GC, meta con uence MC, one step simulation 1SIM, and preservation of strong normalisation PSN. People are interested in these properties, because they insure that the whole behaviour of the calculus is preserved. The rst calculus of explicit substitution is the C of de Bruijn [dB78], of which [BBLRD96] has a presentation in today notations. But it is somewhat traditional to set the beginning of explicit substitutions with [ACCL91] where the authors de ne a one step simulating calculi they call . They are mainly interested on the meta con uence issue, but they are not aware ....

N. G. de Bruijn. A namefree lambda calculus with facilities for internal denition of expressions and segments. TH-Report 78-WSK03, Technological University Eindhoven, Netherlands, Department of Mathematics, 1978.

....in calculus is an e ective way to make clear meaning of terms and, for the uni cation process, to eliminate dummy, repetitively and redundant renaming. At the beginning of the seventies de Bruijn developed a notation for calculus where names of bounded variables where replaced with indices [dB72,dB78b,dB78a]. These indices relate bounded variables to their corresponding abstractors. De nition 215. Let t 2 (V) and i j positions in O(t) The number of abstractors between i and j is de ned as jfk j i k j and tj k has as lead symbol an abstractor operatorgj De nition 216. The depth of an ....

N. G. de Bruijn. A Namefree Lambda Calculus with Facilities for Internal Denition of Expressions and Segments. Technical Report TH-Report 78-WSK-03, Department of Mathematics, Eindhoven University of Technology, 1978.

....of explicit substitution. De Bruijn s approach which is also ours aims to describe faithfully the mechanism of substitution with the character of iextreme simplicityj advocated by Curry and Feys for combinatory logic. Historically, the rst calculus in this family was introduced by de Bruijn [13], see also [20] Another calculus belonging to this family, which is extensively studied in this paper was proposed by one of us in [22] Those calculi attempt to describe (perhaps naively) the principles of the implementation of calculus. They do not aim to eOEciency. Their main feature is that ....

....pJ . In D 00 all the rewrites from a J are below p J , especially the rewrite from t 0 f(c 0 )b N g pJ to t 0 f(c 0 )b N 1 g pJ . That contradicts the minimality of the derivation D 0 . 2 Corollary 7 Typed pure terms in Term AE are strongly AE normalisable. 6 The system COE In [13], N. G. de Bruijn presents the rst calculus of explicit substitutions which he calls COE. As his notations are somewhat diOEcult to read and dioeerent of these we are used to, we propose to describe his rules in notations similar to those used in the previous section. Starting from rule (B) de ....

N. G. de Bruijn. A namefree lambda calculus with facilities for internal denition of expressions and segments. TH-Report 78-WSK-03, Technological University Eindhoven, Netherlands, Department of Mathematics, 1978.

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N. G. de Bruijn. A namefree lambda calculus with facilities for internal definition of expressions and segments. Technical Report TH-Report 78-WSK-03, Department of Mathematics, Eindhoven University of Technology, 1978.

....which do not belong to the language of the calculus. There has however been an interest in formalising substitution explicitly; various calculi including 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 ....

N. G. de Bruijn. A namefree lambda calculus with facilities for internal definition of expressions and segments. Technical Report TH-Report 78-WSK-03, Department of Mathematics, Eindhoven University of Technology, 1978.

....which do not belong to the language of the calculus. There has however been an interest in formalising substitution explicitly; 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. [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 ....

N. G. de Bruijn. A namefree lambda calculus with facilities for internal definition of expressions and segments. Technical Report TH-Report 78-WSK-03, Department of Mathematics, Eindhoven University of Technology, 1978.

....for (G; D) then FV ( D) FV ( G) See [4] for details. 3 Simplified Expression Reduction Systems with Indices We introduce de Bruijn indices based higher order rewrite formalism SERSDB . 3. 1 De Bruijn metaterms and terms A classical way to avoid ff conversion is to use de Bruijn index notation [7, 8], where names of variables are replaced by natural numbers. Indeed, a given occurrence p of a variable in a term, say x, is replaced by the number of binder operators whose occurrences are between the binder of this x and p. For example, x: y: f(x; z: g(z; x) y) is written ( f(2; g(1; 3) ....

N. de Bruijn. A namefree lambda calculus with facilities for internal definition of expressions and segments. Technical Report 78-WSK-03, Eindhoven University of Technology, 1978.

No context found.

N.G. de Bruijn. A namefree lambda calculus with facilities for internal de nition of expressions and segments. Technical Report TH-Report 78-WSK03, Department of Mathematics, Technical University of Eindhoven, 1978.

No context found.

de Bruijn, N.G. (1978). A namefree lambda calculus with facilities for internal definition of expressions and segments. Technical Report TH-Report 78-WSK-03, Department of Mathematics, Technical University of Eindhoven.

No context found.

N. G. de Bruijn. A namefree lambda calculus with facilities for internal definition of expressions and segments. TH-Report 78-WSK-03, Department of Mathematics, Technological University Eindhoven, Netherlands, 1978.

No context found.

N. de Bruijn. A namefree lambda calculus with facilities for internal de nition of expressions and segments. T.H.-Report 78-WSK-03, Technological University Eindhoven, Eindhoven - The Netherlands, August 1978.

No context found.

N. G. de Bruijn. A namefree lambda calculus with facilities for internal definition of expressions and segments. Technical Report TH-Report 78-WSK-03, Department of Mathematics, Eindhoven University of Technology, 1978.

No context found.

N. G. de Bruijn. A namefree lambda calculus with facilities for internal definition of expressions and segments. Technical Report TH-Report 78-WSK-03, Department of Mathematics, Eindhoven University of Technology, 1978.

No context found.

N.G. de Bruijn, A namefree lambda calculus with facilities for internal definition of expressions and segments. Technical Report 78-WSK-03, Eindhoven University of Technology, the Netherlands, 1978.

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