J.L. Krivine, Lambda-calcul, Types et Mod`eles, Masson, Paris, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we summarize some definitions and results that we will need in the subsequent part of the paper. With regard to the lambda calculus we follow the notation and terminology of Barendregt (see [3] For the general theory of lambda calculus the reader may consult Barendregt [3] and Krivine [30]. For the general theory of universal algebras the reader may consult Burris and Sankappanavar [13] Gratzer [22] and McKenzie, McNulty and Taylor [32] The main references for topological algebras are Taylor [52, 53] Gumm [23] Bentz [8] and Coleman [14, 15] 2.1. Lambda theories denotes the ....

....the class of pomodels whose specialization order is a complete partial ordering and the representable functions are all the continuous ones w.r.t. the Scott topology. The graph model semantics (see [46] 19] 37] 38] 10, Section 5. 5] is a subclass of the K semantics isolated by Krivine (see [30], 10, Section 5.6.2] within the continuous semantics. The filter model semantics was defined by Coppo, Dezani, Honsell and Longo in [16] see also [4] within the continuous semantics. The stable semantics introduced by Berry [11] is the class of po models whose specialization order is a ....

J.L. Krivine, Lambda-Calcul, types et modeles, Masson, Paris (1990)

....fragment are exactly the ones provably total in Peano Arithmetic. This is inspired by the reduction of 1 1 comprehension to inductive de nitions presented in [Buch2] and this complements a result of [Leiv] The argument uses a nitary model of a fragment of the system AF2 considered in [Kriv,Leiv]. 1 The polymorphic calculus We let D be the set of all untyped, maybe open, terms, with conversion as equality. We let c n be the lambda term x f f n x: We consider the following types T : j T T j ( T where in the quanti cation, T has to be built using only and : We use T ....

J.L. Krivine. Lambda-calcul. Types et modeles. Masson, Paris, 1990.

....Plotkin on the sequentiality of calculus. They are used for modelling processes. We are grateful to G. Winskel for useful pointers to the basic literature on event structures. 15 The present de nition of morphisms arises from the de nition of continuous models of untyped calculus given in [18] in a more restricted context (such models are called K models in section 5) 15 happen to be a morphism relatively to distinct triples, and then will give rise to di erent pairs (q j ; ap j ) 16 . We keep it however since it is very convenient, and will manage ambiguous occurrences when ....

....2 ) The proof only depends on a symmetry property of the web of E 2 (or simply E) which can also be directly found in a lot of other models (but not all) or forced voluntarily during the construction of a model. The way for modelling 23 This is similar to a proof given for Scott s D1 in [18], which dates back to Scott. 23 other constructs can call for more complex webs, but the basic principle is the same. 5.1.3 The (ext K) 2 models. A K model is a re exive prime web of the form W : j) K models were isolated in [18] and are prime algebraic complete lattices. The ....

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J.L. Krivine, Lambda-calcul, types et modeles, Masson, 1990.

....fragment are exactly the one provably total in Peano Arithmetic. This is inspired by the reduction of Pi 1 1 comprehension to inductive definitions presented in [Buch] and this complements a result of [Leiv] The argument uses a finitary model of a fragment of the system AF 2 considered in [Kriv, Leiv]. 1 Polymorphic calculus We let D be the set of all untyped, maybe open, terms, with fi conversion as equality. We let c n be the lambda term xf f n x: We consider the following types T : ff j T T j ( Piff)T where in the quantification, T has to be built using only ff and : We use ....

J.L. Krivine. Lambda-calcul. Types et mod`eles. Masson, Paris, 1990.

....to investigate it in a future paper. Unless otherwise stated we shall use the terminology of Barendregt [1] for lambda calculus and that of Salibra and Goldblatt [30] for lambda abstraction algebras. For the general theory of lambda calculus the reader may consult Barendregt [1] and Krivine [16]. For lambda abstraction algebras and variable binding calculi the reader may consult Goldblatt [7, 8] Pigozzi and Salibra [25, 27, 26, 22] Salibra and Goldblatt [30] and Salibra [28, 29] For the general theory of universal algebras the reader may consult Burris and Sankappanavar [2] Gratzer ....

J.L. Krivine, Lambda-Calcul, types et modeles. Masson, Paris (1990)

....we obtain a generalization of the Genericity Lemma ( 3; 14.3.24] of finitary lambda calculus to the infinitary lambda calculus. We also give an algebraic proof of consistency of the infinitary lambda calculus. In the last section of the paper we generalize some algebraic constructions by Krivine [26]. We introduce the idempotent expansions of LAA s and show that every LAA is a retract of each of its idempotent expansions. We also show that the least idempotent expansion of an LAA is an LAA. Outline of paper. In the first section of this paper we review the basic definitions of the lambda ....

....ALGEBRAS 25 Lemma 37. Let C be a algebra and let t; s be combinatory polynomials over C. Then t jC s if and only if x(t) jC x(s) for every x 2 I . The proof of the above Lemma can be found in [29; Lemma 7. 12] A remarkable algebraic and simple proof was discovered by Krivine [26]. It is outlined at the beginning of the last Section in this paper. Only identities between closed combinatory terms are sufficient for axiomatizing algebras over combinatory algebras; hence, if C is a algebra and LA j= t = u, then we have t jC u. The following well known result shows that ....

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J.L. Krivine, Lambda-Calcul, types et modeles, Masson, Paris, 1990.

....to use one of these two local transformations: x :N)PQ ( x :NQ)P ( x : y :N)P y : x :N)P 1 Note that putting the argument before the function was rst introduced by de Bruijn in his Automath project [20] and has been used by many researchers since. For example, Krivine in [17] also puts the argument before the function. 2 These rules transform terms to make more redexes visible to the ordinary notion of reduction. For example, both the and rules make sure that y and Q in the term A of Example 1 can form a redex before the redex based on x and P is ....

J.L. Krivine. Lambda-calcul, types et modeles. Masson, 1990. 19

....the sequentiality of calculus. They are used for modelling processes. The second author is grateful to G. Winskel for useful pointers to the basic literature on event structures. 16 The present denition of morphisms is issued from the denition of continuous models of untyped calculus given in [18] in a more restricted context (such models are called K models in section 5) 14 3.5 Retraction pairs for application and abstraction. Suppose there is a morphism j from W V W 0 to W ; then S(W ) S(W 0 ) is a retract of S(W ) under the retraction pair (q j ; ap j ) dened by q j : S(j) ....

....or forced voluntarily during the construction of a model. The way for modelling other constructs can call for more complex webs, but the basic principle is the same. 5.1.3 The (ext K) 2 models. A K model is a reAEexive prime web of the form W : Omega ; j) K models were isolated in [18], and are prime algebraic complete lattices. The family includes in particular Scott s and Park s D1 models. If K is an extensional K model, then K 2 is an extensional model of F: This is a particular case of the remark which follows the denition of square models. Here types = T erms = Gamma ....

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J.L. Krivine, Lambda-calcul, types et mod#les, Masson, 1990. 30

....typing, and point out some available methods for removing type dependencies on terms. Finally, we mention some related work and draw some conclusions on our technique. 2 Preliminaries We give a brief explanation on the notions and terminology used in this paper. Most details can be found in [1] [19] and [2] Definition1. The set of terms is defined inductively as follows. variable) There are infinitely many variables x; y; z; in ; variables are subterms of themselves. abstraction) If M 2 then (x:M ) 2 ; N is a subterm of (x:M ) if N is (x:M ) or a subterm of M . ....

....k Delta k since one has to deal with a lot of administrative fi redexes introduced by CPS transformation [22] 6 Conclusion and Future Work We have demonstrated some applications of our technique. The reader is encouraged to verify that this technique also works on Gamma with Curry typing [19], Gamma with Church typing [20] and the calculi with positive recursive types [31] On the other hand, translation [ Delta] which exploits continuations, has troubles handling Curry typing [29] and is less robust than translation k Delta k. Besides, k Delta k in the author s opinion ....

J.L. Krivine (1990), Lambda-calcul, Types et Mod`eles, Masson, Paris.

....des termes typables par S est strictement contenu dans l ensemble des termes fortement normalisables: il suffit de considerer le terme de l exemple precedent. Proposition 1. 1 Soient t; t 0 deux termes tels que t fi t 0 alors si Gamma S t : A on a que Gamma S t 0 : A Preuve: Voir [Kri90] 2. 2 Sch emas de Types On va donner la d efinition du syst eme des types avec intersection. Soit A un ensemble d enombrable de variables propositionelles que on appellera atomes. On consid ere le language construit a partir de A avec les symboles ; Omega . On pose que les suivantes ....

....r eseaux par d interm ediarie de D Omega sont multiples: cela permet d ej a de faire le lien avec les travaux de l ecole de Turin, et de les utiliser. C est aussi une bonne transition vers la s emantique d enotationelle, puisque le syst eme D Omega y apparait naturellement (voir par exemple [Kri90], ou les espaces coh erents construits par Eric Duquesne dans sa th ese) Cela a ete aussi en partie a l origine de la nouvelle syntaxe des r eseaux avec un seul lien pour les contractions, par rapport a l ancienne syntaxe avec quatre liens: dereliction, affaiblissement, contraction, ....

J.L. Krivine. Lambda-calcul, types et mod`eles. Masson, 1990.

....: v] s) Gamma Gamma ] t : s) s [ u] t : s) t : u] s) Fig. 4. Translation to the simply typed calculus The fi rule (x Delta t)t 0 t[t 0 =x] defines a rewrite relation on terms that we again write Gamma . Recall that the simply typed calculus terminates [6]. We write Gamma , resp. Gamma , its (resp. reflexive) transitive closure. These notions are extended to lists: t 1 : t n : t n 1 Gamma t 0 1 : t 0 n : t 0 n 1 if and only if t i Gamma t 0 i for every i, 1 i n 1; and t 1 : t n : t ....

.... 0 Xi x Delta u : 0 Xi u : 0 Xi v : Xi uv : 0 Xi u : ff not free in Xi Xi ff Delta u : 8ff Delta Xi u : 8ff Delta Xi u 0 : 0 =ff] The typed 8 calculus then normalizes strongly (by standard reducibility arguments, see e.g. [4, 6]) Our translation extends trivially to oe 8 by: ff Delta u] s) ff Delta ( u] s) u ] s) u] s) We then need to check: Lemma 11. If u : Gamma is derivable in oe 8 , where Gamma = 1 ; n ; Delta and Xi t 1 : 1 , Xi t n : n , Xi t n 1 ....

J.-L. Krivine. Lambda-calcul, types et mod`eles. Masson, 1992.

....is [8] see also [9, 11] In particular, the papers [32, 86, 85, 10] define principal type schemes for various intersection type disciplines. An exposition of the proof theoretic properties of systems CDV and CDV 6 used as tools for the syntactical study of type free calculus can be found in [61] (there called systems D Omega and D respectively) which makes extensive use of the notion of saturated sets (adaptation of Tait computability predicate [89] The paper [17] gave rise to a number of studies relating intersection type theories to domain theory and models. In particular, every ....

J.L. Krivine, Lambda-calcul, Types et Mod`eles, Masson, Paris, 1990.

....to use a simplified format for conjunctive types, due to S. van Bakel [2] see also [21] where the type of any given term is unique modulo the choice of types for occurrences of variables and has a well defined arity: this is in contrast with the usual sort of conjunctive types (see e.g. [15, 8]) Define the strict intersection types by: B j : 1 ; n ] n 0) where [ 1 ; n ] is the multiset of types 1 , n . Intuitively, 1 ; n ] denotes the intersection of 1 , n . If n = 0, denotes the set of all terms. We ....

....weakly and strongly normalizable terms by conjunctive typings (this paper) all this in an infinite but regular first order equational formulation. Further work should investigate whether easier, direct proofs of the results presented here are possible, using variants of the reducibility method [15, 8]. 1 F. Lang notes that some of the results of this paper are wrong, and we have followed his remarks in the table (see http: www.ens lyon.fr flang papiers.html) ....

J.-L. Krivine. Lambda-calcul, types et mod`eles. Masson, 1992.

....reflexive, transitive closure and the reflexive, transitive, symmetric closure of the onestep reduction. Finally, when a c term M is typable with type A according to Griffin s system, we will write C M : A. 3 Parigot s Calculus Parigot s calculus is a classical extension of Krivine s AF 2 [11]. It is therefore a second order system. In this paper, we will focus on first order calculus. This restriction affects neither the syntax of the language, nor the reduction rules. Simply, it allows fewer terms to be typable. Terms are built from two distinct alphabets of variables: the set ....

J.-L. Krivine. Lambda-calcul, types et mod`eles. Masson, 1990.

....ff ; Delta ff: M : Gamma Gamma A; Delta When a judgement of the form M : A is derivable according to these typing rules, we will write M : A. If one forgets about the naming rules (and therefore, about abstractions and named terms) the above type system amounts to Krivine s AF 2 [7]. When dealing with pure calculus or with AF 2 , we will allow only for the notion of reduction fi. Then, we will respectively write , and = for the relations of one step reduction, reduction, and conversion induced by fi. When dealing with the intuitionistic sequents of AF 2 , we ....

....by simplifying, in the definition of the CPS translation, Clauses (i) and (v) as follows: i) x = x; v) ffi ] M = M ffi . With this definition, however, Lemma 3.3 does not hold any more and therefore some sort of j reduction would be needed. 3 The well typed terms of AF 2 are normalizable [7]. Therefore, we get the following proposition as a corollary. Corollary 5.3 (Normalization) Any well typed term is normalizable. ut 6 Conclusions We have presented a CPS translation of the calculus into the calculus that satisfies translation and simulation properties similar to the ones ....

J.-L. Krivine. Lambda-calcul, types et mod`eles. Masson, 1990.

....which is the second order # calculus (also called system F ) of Girard [2] rediscovered by Reynolds [16] in a computer science frame. We shall extend it in two ways: Types will be formulas of second order predicate calculus, and not only, as in system F, second order propositional calculus [5, 6]. In a certain sense, this is a harmless extension, since the # terms which are typable are the same. This kind of extension has already been considered by D. Leivant [11] A much more serious extension is the following: the underlying logic will be classical logic, and not only, as in system ....

....is su#cient to prove # #x (Int [x] # Int [fx] For example, a proof of # #x (Int [x] # Int [sx] gives a program for the successor (such as #n#f#x f. nfx) a proof of # #x (Int [x] # Int [px] from the equations p0 = 0; psx = x; gives a # term for the predecessor in Church integers [5, 6]. 3 Storage operators and the strategy of reduction for # terms The strategy of head reduction (call by name) has the following advantages: Its good mathematical properties, given by the standardization theorem: if a # term is solvable, then we obtain a head normal form by head reduction. ....

J.L. Krivine. Lambda-calcul, types et modeles. Masson, Paris (

.... cedex 05 e mail krivine logique.jussieu.fr 15 Juillet 1996 Introduction Il est bien connu que la correspondance de Curry Howard permet d associer un programme, sous la forme d un # terme, a toute preuve intuitionniste, formalisee dans le calcul des predicats du second ordre (voir, par exemple [3]) Cette correspondance a ete etendue, assez recemment, a la logique classique moyennant une extension convenable du # calcul (voir [1,4,5,6] Chaque theoreme formalise en logique du second ordre correspond donc a une specification de programme. Il se pose alors le probleme, en general tout a ....

....formule F , on pose 0 F # (F # O) et on designe par F # la traduction de Godel de F , obtenue en remplacant, dans F , chaque formule atomique A par 0 A. Il est bien connu que, si F est demontrable en logique classique du second ordre, alors F # l est en logique intuitionniste (voir [3]) On a donc # TC # , c est a dire : #M[Mod ## (M) #y( 0 Jy # 0 My) # 0 Ma] Ded( 0 J) # 0 Ja. Designons par Mod # 1 (M) Mod # 2 (M) Mod # 3 (M) Mod # 4 (M) les quatre formules (i) ii) iii) iv) de Mod # (M ) On a donc : Mod ## 1 (M) # #xy[ 0 M(x ##y) 0 Mx # 0 ....

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J.L. Krivine. Lambda-calcul, types et modeles. Masson, Paris (

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J.L. Krivine, Lambda-calcul, Types et Mod`eles, Masson, Paris, 1990.

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J.L. Krivine, Lambda-calcul, types et modeles, Masson 1990.

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Krivine, J.-L., Lambda-calcul, Types et Modeles, Masson, Paris, 1990.

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J.L. Krivine, Lambda-calcul, types et modeles, Masson 1990. 14

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J.L. Krivine, Lambda-calcul, types et modeles, Masson 1990.

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Krivine, J.-L., Lambda-calcul, Types et Modeles, Masson, Paris, 1990.

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Krivine J.L., Lambda-calcul, Types et Mod`eles. Masson, Paris, 1990.

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J.L. Krivine, Lambda-calcul, types et mod`eles, Masson, Paris 1990.

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J.L. Krivine, Lambda-calcul, Types et mod#les, Masson, Paris, 1990.

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Krivine, J.L.(1990) Lambda-calcul, Types et Mod`eles. Masson, Paris.

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