H.B. Curry, "Functionality in Combinatory Logic", Proceedings of Natural Academy of Sciences U.S.A. 20, 1934, 584--590.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the universal type , as presented by Coppo and Dezani in [CD80] give a characterization of strongly normalizable terms, in the sense that a term is typed in an intersection type system without if and only if it is strongly normalizable. These systems type more terms than the Curry type system ([Cur34]) or the type system of pure ML ( DM82] New attention was given to intersection type systems due to a result of Kfoury and Wells ( KW99] which proved that these systems are decidable for restrictions of nite rank, which correspond to a large class of typable terms. Although Kfoury and Wells s ....

....works the linear calculus is de ned as the computational interpretation of linear logic and the problem of transforming non linear terms into linear ones is not a central issue. 3 Types Here we describe the type systems used in the rest of the paper. 3. 1 Curry Types The Curry type system ([Cur34]) sometimes called the Simple type system) is the basis of every type system for the calculus. Here we describe this system along the lines presented in [Hin97] From now on terms of the calculus are considered module equivalence. We also assume that in a term M no variable is bound more ....

H.B. Curry. Functionality in combinatory logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584-590, 1934. 11

....to the possibility of adding boxes just replacing variables by variables) and so at a first look the problem of typability for the # calculus seems at most semi decidable. In [4] a procedure has been designed for assigning to a # term, that can be typed in the Curry s type assignment system [5 7], a type of EAL: the procedure takes as input a pair of a term and a Curry s type that can be assigned to it, and produces as output either a negative result or a typing, assigning to the term itself an EAL type related to the Curry type in input, in the sense that it expresses the same ....

Curry, H.B.: Functionality in combinatory logic. In: Proc. Nat. Acad. Science USA. Volume 20. (1934) 584--590

....normalisation properties can be characterised. The Intersection Type Discipline (ITD) as presented in [11] a more enhanced system was presented in [10] for an overview of the various existing systems, see [2] was introduced mainly to overcome the limitations of Curry s type assignment system [13, 14] and has been used to characterise normalisation using types. It is an extension of Curry s system, in that term variables (and terms) are allowed to have more than one type: in a certain context M , a term variable x can play different, even non unifyable, roles. This slight generalisation of ....

H.B. Curry. Functionality in Combinatory Logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584-- 590, 1934.

....are given to establish this relation. Finally, we give a simple characterisation of the proof trees that represent principal normal inhabitants of a type . 1 Introduction Typed versions of systems of calculus have been studied since their appearance in the beginning of this century in [6] and [5] due to their importance to areas of mathematical logic and more recently to computer science. One of the elds with direct application is the area of functional programming (e.g. programming languages ML, Miranda or Haskell) The concept of functional programming relies on the ....

H. B. Curry. Functionality in combinatory logic. Proc. Nat. Acad. Science USA, 20, 584-590, 1934.

....we will write f : t . In this de nition we will try to follow the line of the Principia as much as possible. For remarks, discussion and examples, see Section 3.7 below. The symbol in f : t is the same symbol that Frege used to assert a proposition. It enters Type Theory in 1934 [18], via Curry s combinatory logic. Curry de nes a functionality combinator F in such a way that FXY f holds, exactly if f is a function from X to Y . To denote the assertion of FXY f , Curry uses Frege s symbol . De nition 40 (Rami ed Theory of Types: RTT) The judgements f : is inductively ....

H.B. Curry. Functionality in combinatory logic. Proceedings of the National Academy of Science of the USA, 20:584-590, 1934.

....pu o scrivere un programma P , ed un tipo (se esiste) per P viene inferito nella fase di compilazione. L assegnazione di un tipo per P pu o essere visto come una interpretazione astratta del programma che pu o essere usata come criterio di correttezza. I tas sono stati introdotti da Curry [Cur34] per la Logica Combinatoria ed in seguito modificati da Curry, Feys, Hindley e Seldin [CF58, CHS72] per il calcolo. Nella sua forma pi u generale, un tas prova giudizi della forma 1.2. Sistemi Tipati per il calcolo 7 dove M e un termine del calcolo puro, OE e un tipo e Gamma e un ....

....assegnamo alle variabili libere di M i tipi specificati in Gamma. In letteratura esistono diversi tas: l insieme dei termini a cui possiamo assegnare un tipo e l insieme dei costruttori di tipi (e quindi di tipi) varia da un sistema all altro. Ad esempio, il Curry Type Assignment System (F1) Cur34] ha come unico costruttore di tipo la freccia o costruttore di funzioni. In questo sistema possiamo assegnare alla funzione identit a x:x tutti i tipi ottenibili per sostituzione dal tipo pi u generale ff ff, dove ff e una variabile di tipo. Diremo che ff ff e lo schema di tipo principale ....

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H.B. Curry. Functionality in Combinatory Logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584--590, 1934.

....The theory of Pure Type Systems is a generalization of the theory of the cube. Because the cube is easier to understand than the theory of PTSs we start with introducing the cube. The cube The cube is a generalization of a set of eight type systems including the well known systems [5][6], 2 [7] 18] 7] and C [19] We give a short description of these four systems below. The system is the basis of all type systems for functional programming. In this system it is possible to define terms that depend on other terms, for instance M j n : n with M : We call M a term ....

H. Curry. Functionality in combinatory logic. In Proc. Nat. Acad. Sci, U.S.A., volume 20, pages 584--590, 1934.

.... and expressive type system, using intersection types, has been developed in [5] for Curryfied Term Rewriting Systems (CuTRS, first order TRS extended with application) This type system is inspired by the Intersection Type Discipline defined in [8] see also [7, 1] an extension of Curry s system [10, 11] in that, essentially, terms are allowed to have more than one type (using the type constructor ) By introducing also the type constant a type system for LC is obtained that is closed under fi equality, and interpreting terms by their assignable types gives a filter lambda model [7, 3] ....

.... It is well known that in the study of normalization of reduction systems, the notion of types plays an important role, and that many of the now existing type assignment systems for Functional Programming Languages (FPL) are based on (extensions of) the Curry type assignment system for LC [10, 11]. The Intersection Type Discipline (ITD) as presented in [8] see also [7, 1] is an extension of Curry s system, in that, essentially, terms are allowed to have more than one type (using the type constructor ) By introducing also the type constant a system is obtained that is closed under ....

H.B. Curry. Functionality in Combinatory Logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584-- 590, 1934.

....Grant CRG 970285 Extended Rewriting and Types . been developed in [5] for Curryfied Term Rewriting Systems (CuTRS, first order TRS extended with application) This type system is inspired by the Intersection Type Discipline defined in [8] see also [7, 1] an extension of Curry s system [10, 11] in that, essentially, terms are allowed to have more than one type (using the type constructor ) By introducing also the type constant a type system for LC is obtained that is closed under fi equality, and interpreting terms by their assignable types gives a filter lambda model [7, 3] In ....

H.B. Curry. Functionality in Combinatory Logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584--590, 1934.

....also other combinatory complete CS are discussed) It is well known that in the study of normalization of reduction systems, the notion of types plays an important role. Many of the now existing type assignment systems for functional languages are based on (extensions of) the Curry system for LC [10, 11]. The intersection type discipline (see [8, 7, 1] is an extension of Curry s system, in that, essentially, terms and term variables are allowed to have more than one type (using the type constructor ) By introducing also the type constant a system is obtained that is closed under ....

H.B. Curry. Functionality in Combinatory Logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584--590, 1934.

....at all, and it is the task of the type algorithm to infer types for objects and to check consistency. Many of the now existing type assignment systems for functional programming languages that use the untyped approach are based on (extensions of) the Curry type assignment system [11, 12] for the pure, untyped lambda calculus [5] For example, the functional programming language ML [19] is in fact an extended lambda calculus and its type system is based on Curry s system. It is well known that in Curry s system, the problem of typeability Given a term M , are there B and oe such ....

H.B. Curry. Functionality in combinatory logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584--590, 1934.

....for inventing intuitionism, and Heyting s for inventing a formalisation [12] On the other hand DN and EM were thought to be denitely alien to computation, and to programming theory. The simplest intuitionistic logic where the Curry Howard correspondence makes sense is Curry s simple type system [4], namely propositional logic over implication only. The most fruitful ones are the intuitionistic restrictions of Second order propositional logic (Girard s System F ( 7] 8] 9] rediscovered by Reynolds [30] in a computer science context) and Second order predicate logic (Leivant [22] ....

H.B. Curry, Functionality in Combinatory Logic, in Proc. Nat. Acad. Sci., USA 20, 584-590, 1934.

....at all, and it is the task of the type algorithm to infer types for objects and to check consistency. Many of the now existing type assignment systems for functional programming languages that use the untyped approach are based on (extensions of) the Curry type assignment system [11, 12] for the pure, untyped lambda calculus [5] For example, the functional programming language ML [19] is in fact an extended lambda calculus and its type system is based on Curry s system. It is well known that in Curry s system, the problem of typeability Given a term M , are there B and oe such ....

H.B. Curry. Functionality in combinatory logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584--590, 1934.

....type assignment system can be obtained by applying E to every object occurring in the rules of R. The dependency free plane of TAS contains some type assignment systems already known in the literature, that are convertible to certain typed systems: the Curry type assignment system (F1) [9] that corresponds to , the polymorphic type assignment system (F2) 17] that corresponds to 2, and the higher order type assignment system (F ) 14] that corresponds to . The fact that in [13] also systems that contain dependencies were considered, was a first attempt to study dependent types ....

H.B. Curry. Functionality in Combinatory Logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584-- 590, 1934.

....type assignment to programs and objects is in fact a way of performing abstract interpretation that provides necessary information for both compilers and programmers. Many of the now existing type assignment systems for FPL are based on (extensions of) the Curry type assignment system for LC [11, 12]. The intersection type discipline as presented in [9] see also [7] and [1] is an extension of Curry s system, in that, essentially, terms and term variables are allowed to have more than one type. Intersection types are constructed by adding, next to the type constructor , the type ....

H.B. Curry. Functionality in Combinatory Logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584-- 590, 1934.

....conditions for types to be descriptions of terms. We then prove the soundness with respect to those conditions of the Curry and ML type systems. 1 Introduction The theoretical foundations of type systems was established by Alonzo Church, Chu40] with the typed calculus and by Haskell Curry in [Cur34]. Those two works led to two different views of types: the former corresponds to prescriptive types, where terms have a semantic in typed domains, and the latter to descriptive types, where terms and types have different semantics, and type systems define a relation of two different semantic ....

H.B. Curry. Functionality in combinatory logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584--590, 1934.

.... the function M applied to the argument N ; a term of the form x:M is 1 Church had been considerably helped by his students in the early development of the lambda calculus, notably by Kleene, see Kleene [1981] and Rosser [1982] Other important influences came from Curry [1930] and Curry [1934]. called an abstraction, with the intended interpretation the function that assigns to x the value M . In this interpretation the notion of function is to be taken intensional, i.e. as an algorithm. Scott [1972] succeeded to give lambda calculus also an extensional interpretation by ....

.... x : A; Gamma M : A B) Gamma N : A ) Gamma (MN) B; Gamma; x:A M : B ) Gamma (x:M) A B) 1.4. Example. i) x: A A B) y:A xyy : B: ii) xy:xyy : A A B) A B) This version of the simply typed lambda calculus has implicit types at each abstraction x and is studied by Curry [1934]. In Church [1940] a variant with explicit types at abstractions is introduced. In this theory the rule for introducing abstractions is Gamma; x:A M : B ) Gamma (x:A:M) A B) An essential difference between the two approaches is that in the explicit case the unique type of a term always ....

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Curry, H.B. [1934] Functionality in combinatory logic, Proceedings of the National Academy of Science of the USA 20, 584--590.

....basic properties of domain free pure type systems, establish their formal relationship with pure type systems and type assignment systems, and give a number of applications of these correspondences. 1 Introduction Typed versions of the calculus were introduced independently by Church (1940) and Curry (1934). More precisely, Curry (1934) introduced types into the theory of combinators, and Curry and Feys (1958) modified the system in a natural way to calculus. In Church s system abstractions have domains, i.e. are of the form x : D : t, whereas in Curry s system abstractions have no domain, i.e. ....

....pure type systems, establish their formal relationship with pure type systems and type assignment systems, and give a number of applications of these correspondences. 1 Introduction Typed versions of the calculus were introduced independently by Church (1940) and Curry (1934) More precisely, Curry (1934) introduced types into the theory of combinators, and Curry and Feys (1958) modified the system in a natural way to calculus. In Church s system abstractions have domains, i.e. are of the form x : D : t, whereas in Curry s system abstractions have no domain, i.e. are of the form x : t. Thus, in ....

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Curry, H. (1934). Functionality in combinatory logic. Proceedings of the National Academy of Science USA, 20, 584--590.

....that are denoted by , possibly indexed if necessary. If in a notion of type assignment for M there are basis B and type oe such that B M :oe, then M is typed with oe, and oe is assigned to M . 1 A historical perspective Type assignment for the Lambda Calculus was first studied by H.B. Curry in [13]. See also [14] Curry s system the first and most primitive one expresses abstraction and application and its major advantage is that the problem of type assignment is decidable. The types used in this system are those obtained from type variables and the type constructor arrow . ....

H.B. Curry. Functionality in combinatory logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584--590, 1934.

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Haskell Brooks Curry. Functionality in combinatory logic. Proceedings of the National Academy of Sciences of the United States of America, 20:584--590, 1934.

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H.B. Curry, "Functionality in Combinatory Logic", Proceedings of Natural Academy of Sciences U.S.A. 20, 1934, 584--590.

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H.B. Curry. Functionality in Combinatory Logic. In Proc. Nat. Acad. Sci. U.S.A., volume 20, pages 584--590, 1934.

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Haskell Brooks Curry. Functionality in combinatory logic. American Journal of Mathematics, 58:345--363, 1936.

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H. B. Curry. Functionality in combinatory logic. In Proc. of National Academy of Sciences, 1934.

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H. B. Curry, Functionality in combinatory logic, Proceedings of the National Academy of Science of the USA, vol. 20 (1934), pp. 584--590.

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