Tait, W. W., A realizability interpretation of the theory of species, in: Logic Colloquium, Lecture Notes in Mathematics 453 (1975), pp. 240--251. 18  Home/Search   Document Not in Database   Summary   Related Articles   Check

....allowing an in nite sequence of reduction steps. In other words, strong normalization is inherited from that of F which in turn is a well known fact [Gir72] In that original work, only weak normalization has been proven but an extension to strong normalization is standard technology since [Tai75]. 15 Let = set n : j j, and de ne for jf j = n : F : F G G) Ran G (Y ) It : m s f r: rmsf in : t m s f : s m(It ms)f t : F : monF 9G : G F G) Lan G (Y ) Coit : m s f t: hm; packhs; packhf 1 ; packhf n ; ti : ....

William W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium Boston 1971/72, volume 453 of Lecture Notes in Mathematics, pages 240-251. Springer Verlag, 1975.

....traditional strong normalization proofs, e.g. by the method of logical predicates, such that they do not refer to reduction. Using suitable extensions of the vector notation, SN can also be de ned for system F and extensions such as by monotone inductive types. The saturated sets variant [Tai75] of Girard s normalization proof may be adapted, with the notion of saturatedness naturally arising from the de nition of SN. This is also true for systems with permutative conversions as shown, e.g. in [Mat00] 4 Strong normalization proof In this section we lift the proof of Lemma 2.1 to ....

William W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium Boston 1971/72, volume 453 of Lecture Notes in Mathematics, pages 240{ 251. Springer Verlag, 1975.

....the strong normalization property of which is sometimes even more useful. The standard method of proving strong normalization of typed calculi was invented by Tait [104] for simply typed calculus, generalized to secondorder typed calculus by Girard [44] and subsequently simplified by Tait [105]. Our presentation follows [8] we consider in this section terms a la Curry. 4.4.1. Definition. i) SN fi = fM 2 j M is strongly normalizing g. ii) For A; B , define A B = fF 2 j 8a 2 A : F a 2 Bg. 4.4. Strong normalization 69 (iii) For every simple type oe, define [ oe] by: ....

W.W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium, volume 453 of Lecture Notes in Mathematics, pages 240--251. Springer-Verlag, 1975.

....types and saturated sets, which is referred to as the reducibility method. The reducibility method is a generally accepted way for proving the strong normalisation property of various type systems such as the simply typed lambda calculus in Tait [23] and the polymorphic lambda calculus in Tait [24] and Girard [13] All the above mentioned papers characterising evaluation properties of terms and of terms in x by means of intersection types apply variants of this method. 1 The Calculus and the Type Assignment Following [10] we consider the set of terms x which uses names rather than De ....

W. W. Tait. A realizability interpretation of the theory of species. In Logic Colloquium, volume 453 of Lecture Notes in Mathematics, pages 240--251. Springer, 1975. 11

....the term eventually end in normal forms; that is, if the term has no innite reductions. The classical proof of strong normalization for fi reduction in simply typed calculus is due to Tait [69] It was generalized to second order typed calculus by Girard [30] and subsequently simplied by Tait [70]. The technique is very AEexible and has been generalized to a variety of calculi. 4 For some notions of reduction in some typed calculi there is a technique to prove weak normalization that is simpler than the Tait Girard technique to prove strong normalization. For instance, Turing [27] ....

W.W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium, volume 453 of Lecture Notes in Mathematics, pages 240251. Springer-Verlag, 1975.

....suitable sets of lambda terms which satisfy certain realizability properties. The reducibility method, based on realizability interpretations, was introduced in Tait [13] for proving the strong normalization property for the simply typed lambda calculus and further developed in Girard [6] and Tait [14] for proving the strong normalization property for polymorphic (second order) lambda calculus. There is an overview of these proofs in Barendregt [2] In Mitchell [10] and [11] this method is referred to as the logical relations and it is discussed that apart from the strong normalization this ....

.... M : M : M : M : 3 Reducibility Method for The reducibility method is a generally accepted way for proving the strong normalization property of various type systems such as the simply typed lambda calculus in Tait [13] the polymorphic lambda calculus in Tait [14] and Girard [6] and the pure intersection type assignment system in Krivine [9] This method was applied for the proof of the Church Rosser property (con uence) of the simply typed lambda calculus in Statman [12] Koltesos [7] and Mitchell [10] and [11] The general idea of the reducibility ....

Tait, W.W.: A realizability interpretation of the theory of species. In: Logic Colloquium (Boston). Lecture Notes in Mathematics, Vol. 453. Springer-Verlag, Berlin (1975) 240-251. 15

....is used in order to prove the diamond property for one reduction relation whose transitive closure is reduction. 1 Introduction The reducibility method was introduced in [12] for proving the strong normalization property for the simply typed lambda calculus and further developed in [7] and [13] for proving the strong normalization property for the second order lambda calculus. There is an overiew of these proofs in [2] In [11] and [5] the reducibility method is applied in order to characterize all strongly normalizing lambda terms. This method is extended in various ways. On the one ....

Tait, W.W.: A realizability interpretation of the theory of species. In: Logic Colloquium (Boston). Lecture Notes in Mathematics, Vol. 453. Springer-Verlag, Berlin (1975) 240-251

....in F 2 are SN in spite the fact that F 2 can type a limited form of self application. The original proof of SN for terms typable in F 2 is due to Girard and was given simultaneously with his introduction of the system itself. Since then the proof has been simplified by a variety of authors, e.g. [Tai75, Mit86]. An extensive overview of the SN proofs for F 2 can be found in [Gal90] The inference rules of F 2 (and other type systems) allow one to derive sequents of the form Sigma . M : where Sigma is a set of variable declarations, M is a term, and is a type expression. Intuitively, a type denotes ....

William W. Tait. A realizability interpretation of the theory of species. In Rohit Parikh, editor, Logic Colloqium, '73, volume 453, pages 22--37. Springer-Verlag, 1975.

....implicit. ffl Proofs by reducibility use an interpretation of the types, but such interpretations are very syntactical. ffl Proofs by reducibility seem to involve the construction of certain kinds of models. ffl Proofs by reducibility use various inductive invariants (due to Girard [6, 7] Tait [24, 25], Krivine, 17] but it is hard to see what they have in common. These observations suggest the following two questions which are the primary concerns of this paper: 1. What is the connection between realizability and reducibility 2. Is is possible to give more semantic versions of proofs ....

W.W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium, volume 453 of Lecture Notes in Math., pages 240--251. Springer Verlag, 1975.

....Typically, it is used to prove strong normalization or normalization, but it can be used to prove other properties as well. The method was pioneered by Tait [22] for the simply typed calculus, and brilliantly extended to various higher order typed calculi by Girard [9, 10] see also Tait [23]) Various expositions and analyses of such proofs are given in Mitchell [15] Krivine [14] Huet [11] and Gallier [5, 6, 7, 8] among others. Another crucial concept is that of a partial equivalence relation, or PER. PER s were introduced by Hyland [12] and Mulry [17] PERs are a major tool in ....

W.W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium, volume 453 of Lecture Notes in Math., pages 240--251. Springer Verlag, 1975.

....we have that every term that type checks in D is strongly normalizing. The main technique involved is a kind of realizability argument known as reducibility . The crux of the reducibility method is to interpret every type oe as a set [ oe] of terms having certain closure properties (see Tait [18, 19], Girard [8, 9] Krivine [13] and Gallier [5, 6] One of the crucial properties is that for a nice type oe, the terms in [ oe] satisfy the predicate P (but this does not have to be the case for ugly types ) If the sets [ oe] are defined right, then the following realizability property ....

....a candidate. Inspired by Koletsos [12] we use the notion of a P candidate defined in definition 3.3. This notion has the advantage of not requiring the terms to be strongly normalizing (as in Girard [8, 9] or to involve rather strange looking terms such as M [N=x]N 1 : N k (as in Tait [19], Mitchell [15] or Krivine [13] By isolating the dual notions of I terms and simple terms, we can give a definition that remains invariant no matter what the definition of the sets [ oe] is. Also, the definition of a P candidate only requires that the predicate P be satisfied, but nothing to ....

W.W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium, volume 453 of Lecture Notes in Math., pages 240--251. Springer Verlag, 1975.

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Tait, W. W., A realizability interpretation of the theory of species, in: Logic Colloquium, Lecture Notes in Mathematics 453 (1975), pp. 240--251. 18

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Tait, W. W. (1975). A realizability interpretation of the theory of species. Pages 240-- 251 of: Parikh, R. (ed), Logic colloquium. Lecture Notes in Mathematics, vol. 453. Springer-Verlag.

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W. W. Tait. A realizability interpretation of the theory of species. In Logic Colloquium, R. Parikh Ed. LNM 453, Springer-Verlag, 1975.

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William W. Tait. A realizability interpretation of the theory of species. In Logic Colloquium, volume 453 of Lecture Notes in Mathematics, pages 240--251. Springer, 1975.

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W. W. Tait, A realizability interpretation of the theory of species, in: R. Parikh (Ed.), Logic Colloquium, Vol. 453 of Lecture Notes in Mathematics, SpringerVerlag, 1975, pp. 240251.

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W. W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium, volume 453 of Lecture Notes in Mathematics, pages 30 240--251, Berlin, 1975. Springer-Verlag.

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W. W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium, volume 453 of Lecture Notes in Mathematics, pages 240--251. Springer-Verlag, 1975. Cited on page 2.

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William W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium Boston 1971/72, volume 453 of Lecture Notes in Mathematics, pages 240-251. Springer Verlag, 1975.

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William W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium, volume 453 of Lecture Notes in Mathematics, pages 240--251. Springer, 1975.

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William W. Tait. A realizability interpretation of the theory of species. In Logic Colloquium, volume 453 of Lecture Notes in Mathematics, pages 240--251. Springer, 1975.

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W.Tait, A realizability interpretation of the theory of species, Logic Colloquium, Lecture Notes in Mathematics 453 (1975) 240-251.

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W. W. Tait. A realizability interpretation of the theory of species. In Logic Colloquium, volume 453 of Lecture Notes in Mathematics, pages 240--251. Springer, 1975.

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W.W. Tait. A realizability interpretation of the theory of species. In R. Parikh, editor, Logic Colloquium, volume 453 of Lecture Notes in Math., pages 240-251. Springer Verlag, 1975.

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Tait, W.W., A realizability interpretation of the theory of species, In Logic Colloquium, ed. R. Parikh, LNCS No. 453, Springer Verlag, 1975, 240-251.

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