Srensen, M.H., \Strong Normalization from Weak Normalization in Typed Lambda Calculi ", Report from CS Department, University of Copenhagen, 1996, available at URL: http://www.diku.dk/research-groups/topps/personal/rambo.html  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... of typed programming languages, construction of semantics denitions for languages with jumps [56, 61] exceptions, and concurrency primitives [26] embedding of classical logics in intuitionistic logics [31, 48] techniques to infer strong normalization from weak normalization in typed calculi [68, 74], and the construction of looping combinators in inconsistent pure type systems [15] Related Direct Style (DS) translations [17, 19, 58] have also been used in both theoretical [57] and implementation oriented applications [60] The range of these applications has been conned thus far by the ....

....than strong normalization, it is natural to develop techniques to infer the latter from the former. Indeed, several such techniques have been presented,most of which infer strong normalization of one notion of reduction from weak normalization of a more complicated notion of reduction, see [68] for references. However S#rensen [68] and Xi [74] recently developed techniques which infer strong normalization of fi reduction in a typed calculus from weak normalization of the same notion of reduction, i.e. fi reduction. These techniques provide some hope for a positive answer to a ....

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M.H. S#rensen. Strong normalization from weak normalization in typed -calculi. Information and Computation, 133(1):3571, February 1997.

....problem is reduced to the Partially supported by MURST grants. 1 normalization problem with respect to a new calculus ( 42, 35, 33] or to a new notion of reduction ( 25, 32] More recently, continuations have been used by Xi and S rensen to reduce strong normalization to normalization [63, 54] for the same notion of reduction: in some sense, at least in the cases where the method of [63, 54] applies (e.g. in the simply typed calculus) this gives relevance to normalization with respect to strong normalization and motivates the choice made in the present paper to give normalization ....

.... respect to a new calculus ( 42, 35, 33] or to a new notion of reduction ( 25, 32] More recently, continuations have been used by Xi and S rensen to reduce strong normalization to normalization [63, 54] for the same notion of reduction: in some sense, at least in the cases where the method of [63, 54] applies (e.g. in the simply typed calculus) this gives relevance to normalization with respect to strong normalization and motivates the choice made in the present paper to give normalization proofs for typed calculi rather than strong normalization ones. Semantic proof techniques. Tait [58] ....

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M.H. Srensen. Strong normalizations from weak normalization in typed lambdacalculi. Information and Computation 133(1):35-71, 1997.

....and Wells [14] use to reduce the problem of strong normalisation to the problem of weak normalisation (WN) for related reductions. Kfoury and Wells use and to reduce typability in the rank 2 restriction of system F to the problem of acyclic semi uni cation [13] Klop, S rensen, and Xi [16, 26, 24] use related reductions to reduce SN to WN. Finally, Ariola, Felleisen, Maraist, Odersky and Wadler use (called let C ) in [1] as a part of an analysis of how to represent sharing in a call by need language implementation in a formal calculus. All the research mentioned above is a living ....

....eI normalisable (i.e. e normalisable reducing only redexes that don t erase their arguments, so called I redexes, or strict redexes) then it is strongly normalisable. This is interesting in view of the ongoing interest of showing that strong normalisation can be reduced to weak normalisation [16, 24, 26]. 3. Postponement of K reduction. Generalised reduction allows the postponement of K reduction (which discards their arguments) after I reductions (which use their arguments in at least one place) Hence, generalised reduction allows unnecessary K redexes to be bypassed. From the implementation ....

M.H. Srensen. Strong normalization from weak normalization in typed -calculi. Journal of Information and Comuptation 133(1), 35-71, 1997.

....and Wells [30] use fl to reduce the problem of fi strong normalisation to the problem of weak normalisation (WN) for related reductions. Kfoury and Wells use and fl to reduce typability in the rank 2 restriction of system F to the problem of acyclic semi unification [28] Klop, S rensen, and Xi [31, 47, 45] use related reductions to reduce SN to WN. Finally, 2] uses (called let C ) as a part of an analysis of how to represent sharing in a call by need language implementation in a formal calculus. 1.2 The calculus with explicit substitution Most literature on the calculus treats substitution ....

M. H. Sørensen. Strong normalization from weak normalization in typed -calculi. Journal of Information and Comuptation. To appear.

....fl to reduce the problem of fi strong normalization to the problem of weak normalization (WN) for related reductions. Kfoury and Wells used and fl to reduce typability in the rank 2 restriction of system F to the problem of acyclic semi unification [KW94] Klop, S rensen, and Xi [Klo80, Xi96, S r97] used related reductions to reduce SN to WN. Finally, AFM 95] used (called let C ) as a part of an analysis of how to represent sharing in a call by need language implementation in a formal calculus. 1.2 The Calculus with Explicit Substitution Most literature on the calculus treats ....

.... et al. Calculi of Generalized fi Reduction x1.3 has been extended to s e , which is confluent on open terms (cf. KR97] and simulates one step fi reduction, but the preservation of strong normalization for the extension s e remained an open problem until it was shown at the end of November 1997 by Bruno Guillome that the property does not hold. 1.3 Combining Generalized Reduction and Explicit Substitution We have already explained the separate usefulness of generalized reduction and explicit substitutions. The main benefits of these concepts are similar: both emphasize flexibility in ....

M. H. Sørensen. Strong normalization from weak normalization in typed -calculi. Information and Computation, 133(1):35--71, 25 February 1997.

....notions of reduction different from fi reduction, deriving strong fi normalisation from weak normalisation of these newly introduced notions of reduction. For example, Klop s technique amounts to introducing a pairing constant [ Delta; Delta] a reduction ; and a reduction ; as follows [29]. M 1 ; M 2 ]N ; M 1 N;M 2 ] M 1 ; M 2 ] M 1 These techniques are successful when applied to calculi S for which there exist syntactic proofs of S j= WN fi . If one tries to prove 2 j= SN fi , it is doubtful that one can gain much (if there is any) by arguing that 2 with some newly ....

....= x [ x:M ) k 1 :k 1 (x:k 2 :h[M ] k 2 ) xi) M 1 (M 2 ) k 1 : M 0 ] k 2 :k 2 ( M 1 ] k 1 ) Note that h; i is a fresh variable and a term of form hM; N i stands for h; i(M ) N ) Delta] is a minor variant of Plotkin s call by name continuation passing style translation. S renson[29] then proved the equivalence between WN fi and SN fi in various typed calculi including the simply typed calculus, the simply typed calculus with positive recursive types, and . The author[32] showed the equivalence between WN fi and SN fi in and 2 with Church typing, and mentioned that ....

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M.H. Sørensen (1996), Strong Normalization from Weak Normalization by A-Translation in Typed lambda-Calculi, Manuscript announced on the types mailing list, February.

....where a rewrite system is said to be UN if each of its terms is so. Interest in the criteria for UN arises, for example, in the proofs of strong normalization of typed calculi, since these criteria are related to the work on reducing strong normalization proofs to proving weak normalization [Ned73, Klo80, Kar85, dVr87b, dGr93, Kha94c, KW95, KW95a, S r97, Xi97, MNS99]. Furthermore, the question: Which classes of terms are UN is posed by Bohm and Intrigila [BI94] in connection with finding UN solutions to fixed point equations, and with the representability of partial recursive functions by UN terms only, in the calculus. 1 A useful UN subclass of terms ....

Sørensen M.H., Strong Normalization from weak normalization in typed -calculi. Information and Computation, 133(1):35--71, 1997.

....(fi SN) In the absence of K redexes the two notions coincide. There is a long trail of results on how to reduce fi SN to fi WN without excluding K redexes since the late 1960 s, by Nederpelt, by Klop, and by many others in the 1980 s and 1990 s (see the references in [4] and [7] for example) We tackle this question once more, not to prove a result (Theorem 6.5) which is likely to be found in some form or other in the extensive literature, but to adapt it to our later needs (Section 7) Every term M which is not in fi nf contains a leftmost fi redex occurrence R j ( x:P ....

Sørensen, M.H., "Strong Normalization from Weak Normalization in Typed Lambda Calculi ", Report from CS Department, University of Copenhagen, 1996, available at URL: http://www.diku.dk/research-groups/topps/personal/rambo.html

....is strongly normalizable) The rewrite system is UN if every term is UN. Interest in the criteria for UN arises, for example, in the proofs of strong normalization of typed calculi, since these criteria are related to the work on reducing strong normalization proofs to proving weak normalization [Ned73, Klo80, Kar85, dVr87, dGr93, Kha94c, KW95, KW95a, S r97, Xi97]. Furthermore, the question: Which classes of terms are UN is posed by Bohm and Intrigila [BI94] in connection with finding UN solutions to fixed point equations, and with the representability of partial recursive functions by UN terms only, in the calculus. 1 Let us call a term t an 1 term ....

Sørensen M.H., Strong Normalization from weak normalization in typed -calculi. Information and Computation 133(1):35-71, 1997.

.... too numerous to be listed exhaustively here, include compilation, transformation, and analysis of typed languages [1, 5, 9, 15, 16, 17, 19] embedding of classical logics in intuitionistic logics [11, 14] techniques to infer strong normalization from weak normalization in pure type systems [20, 22], and the construction of looping combinators in inconsistent logical pure type systems [6] The range of these applications have been confined thus far by the fact that CPS translations are known only for non dependent type systems. Indeed, the most general class of systems with known CPS ....

....typed calculus, it also excludes some other well known systems (e.g. the calculus of constructions, and the LF calculus [12] which are dependent. More specifically, the need for more general CPS translations has appeared in several lines of recent work in which the authors are involved [3, 20]. Moreover, further applications of such general translations are emerging and may include conservativity results for classical logics and strong normalisation of pure type systems with definitions [18] In this paper we introduce CPS translations of the cube [2] including the calculus of ....

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M.H. Sørensen. Strong normalization from weak normalization in typed -calculi. Submitted, 1996.

....(fi SN) In the absence of K redexes the two notions coincide. There is a long trail of results on how to reduce fi SN to fi WN without excluding K redexes since the late 1960 s, by Nederpelt, by Klop, and by many others in the 1980 s and 1990 s (see the references in [6] and [11] for example) We tackle this question once more, not to prove a result (Theorem 3.6) which is likely to be a minor variation of an earlier one in the extensive literature, but to adapt it to our later needs (Theorem 3.15) Every standard term M which is not in fi nf contains a leftmost fi redex ....

Sørensen, M.H., "Strong Normalization from Weak Normalization in Typed Lambda Calculi ", Report from CS Department, University of Copenhagen, 1996, available at URL: http://www.diku.dk/research-groups/topps/personal/rambo.html

.... of typed programming languages, construction of semantics definitions for languages with jumps [56, 61] exceptions, and concurrency primitives [26] embedding of classical logics in intuitionistic logics [31, 48] techniques to infer strong normalization from weak normalization in typed calculi [68, 74], and the construction of looping combinators in inconsistent pure type systems [15] Related Direct Style (DS) translations [17, 19, 58] have also been used in both theoretical [57] and implementation oriented applications [60] The range of these applications has been confined thus far by the ....

....than strong normalization, it is natural to develop techniques to infer the latter from the former. Indeed, several such techniques have been presented,most of which infer strong normalization of one notion of reduction from weak normalization of a more complicated notion of reduction, see [68] for references. However S rensen [68] and Xi [74] recently developed techniques which infer strong normalization of fi reduction in a typed calculus from weak normalization of the same notion of reduction, i.e. fi reduction. These techniques provide some hope for a positive answer to a ....

[Article contains additional citation context not shown here]

M.H. Sørensen. Strong normalization from weak normalization in typed -calculi. Information and Computation, 133(1):35--71, February 1997.

.... to be listed exhaustively here, include compilation [1, 21] transformation [10, 44] and analysis [49, 50, 53] of typed programming languages, embedding of classical logics in intuitionistic logics [27, 42] techniques to infer strong normalization from weak normalization in typed calculi [56, 61], and the construction of looping combinators in inconsistent pure type systems [11] Related Direct Style (DS) translations [13, 15, 49] have also been used in both theoretical and implementation oriented applications. The range of these applications has been confined thus far by the fact that ....

....Indeed, Nederpelt [43] Klop [38] Khasidashvili [37] Karr [35] de Groote [17] and Kfoury and Wells [36] have invented such techniques. However, these techniques all infer strong normalization of one notion of reduction from weak normalization of a more complicated notion of reduction. S rensen [56] and Xi [61] recently developed techniques which infer strong normalization of fi reduction in a typed calculus from weak normalization of the same notion of reduction, i.e. fi reduction. These techniques provide some hope for a positive answer to a conjecture, presented by Barendregt at Typed ....

[Article contains additional citation context not shown here]

M.H. Sørensen. Strong normalization from weak normalization in typed -calculi. Information and Computation, 133(1):35--71, February 1997.

.... too numerous to be listed exhaustively here, include compilation, transformation, and analysis of typed languages [1, 13, 36, 64, 76, 77, 82] embedding of classical logics in intuitionistic logics [44, 59] techniques to infer strong normalization from weak normalization in typed calculi [85, 91], and the construction of looping combinators in inconsistent logical pure type systems [16] The range of these applications have been confined thus far by the fact that CPS translations are known only for non dependent type systems. Indeed, the most general class of systems with known CPS ....

.... and higher order typed calculus, it also excludes some other well known dependent systems, e.g. the calculus of constructions, and the LF calculus [45] More specifically, the need for more general CPS translations has appeared in several lines of recent work in which the authors are involved [10, 85]. Moreover, further applications of such general translations are emerging and may include conservativity results for classical logics and strong normalization of pure type systems with definitions [81] Address: CWI, PO Box 94079, 1090 GB Amsterdam, The Netherlands, gilles cwi.nl y Address: ....

[Article contains additional citation context not shown here]

M.H. Sørensen. Strong normalization from weak normalization in typed -calculi. Submitted, 1996. 4-30

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Srensen, M.H., \Strong Normalization from Weak Normalization in Typed Lambda Calculi ", Report from CS Department, University of Copenhagen, 1996, available at URL: http://www.diku.dk/research-groups/topps/personal/rambo.html

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Srensen, M.H., \Strong Normalization from Weak Normalization in Typed Lambda Calculi", Information and Computation, Vol. 133, no. 1, pp 35-71, February 1997.

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M.H. Srensen, Strong normalization from weak normalization in typed -calculi, Information and Computation Vol. 133 No. 1 (1997) 35-71.

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