W. Howard. The formulas-as-types notion of construction. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda-Calculus, and Formalism, pages 479--490. Academic Press, NY, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Benthem Jutting [2] translated and veri ed Landau s Grundlagen der Analysis [23] in Automath and Zucker [29] formalised classical real analysis in Automath. As for goal 2, de Bruijn used types and a propositions as types (pat) principle that was somewhat di erent from Curry and Howard s [11,17]. De Bruijn spent a lot of e ort on goal 3 and studied the language of mathematics in depth [7] Automath features that helped him in goal 3 include: The use of books. Just like a mathematical text, Automath is written line by line. Each line may refer to de nitions or results given in ....

W.A. Howard. The formulas-as-types notion of construction. In [25], pages 479-490, 1980.

.... slash operators can be shown to induce a fragment #RG of the ordinary typed lambda calculus such that for every sequent # s G in RG provable by some proof # there will be a term M s G such that M is a construction modulo direction of # by a generalization of the Curry Howard isomorphism [7] to a homomorphism. The initial idea for this is found in van Benthem [18] 5 Definition 3.1 The mapping # from syntactic types in the type logic to semantic types in the lambda calculus is defined as follows. The semantic types that are mapped to in this way are said to be equivalent modulo ....

Howard, W. A., The formulas-as-types notion of construction, in: J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, New York, 1980 pp. 479-- 490.

....of proofs. An element of such a set of proofs is represented as a term of the corresponding type. This means that propositions are interpreted as types , and proofs of a proposition a as terms of type a. PAT was, independently from Heyting and Kolmogorov, discovered by Curry and Feys [13] Howard [25] follows the argument of Curry and Feys [13] and combines it with Tait s discovery of the correspondence between cut elimination and reduction of terms [32] Howard s discovery dates from 1969, but was not published until 1980. Independently of Curry and Feys and Howard, we nd a variant of ....

Howard, W.A., The formulas-as-types notion of construction, In To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, editors Seldin, J.P. and Hindley, J.R., Pages 479-490, 1980. Academic Press, New York.

....Language Syntax to recent work by Shao et al. 21] which was developed independently. In Section 7 we compare the two systems and discuss the methodological di#erences in their use. This paper assumes familiarity with linear logic [7, 24] and with the propositions as types correspondence [11]. Additional familiarity with LF and logical frameworks in general will be helpful, but is not required. 2 Intuitionistic LTT The LTT type theory consists of two parts: a proof sublanguage, and a computational programming language built around it. LTT is structured with a syntactic division ....

W. Howard. The formulas-as-types notion of construction. In J. P. Seldin and J. R. Hindley, editors, To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, pages 479--490. Academic Press, 1980.

....more generally an operational semantics for a programming language, describes formally the computation steps of the programs. It turns out that cut elimination in logic, and normalization in formal semantics of programming languages match closely. It has been remarked by Howard, who pointed out in [18] that there exists a one to one correspondence between proofs in natural deduction style and (typed) terms: this is the celebrated Curry Howard isomorphism. More recently, Herbelin [17] noticed that caclulus could be seen as matching Gentzen s sequent calculus closely and directly, which was a ....

W. Howard, The formulas-as-types notion of construction, in Curry Festschrift, Hindley and Seldin eds, 479-490 , Academic Press (

....BoyerMoore prover was used to create an extensive formal theory [19] and the Nuprl libraries provide a constructive theory. Section 9 Logic and the Peano Axioms Our approach to logic comes from Brouwer as formalized in Heyting [61] One of the most influential accounts historically is Howard [66] and also deBruijn [48, 49] for Automath. The Automath papers are collected in [85] The connection between propositions and types has found its analogue in set theory as well. The set theory of Anthony P. Morse from 1986 equated sets and propositions. He asserted a set by the claim that it was ....

W. Howard. The formulas-as-types notion of construction. In To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, pages 479--490. Academic Press, NY, 1980. REFERENCES 44

....variables ranging over types and allow various forms of abstraction or quantification over them. Historically, these type systems are based on fundamental insights in proof theory, particularly the formulas as types notion that evolved through the work of Curry and Feys [CF58] Howard [How80], de Bruijn [deB80] and Scott [Sco70] This notion provided the basis for Martin Lo f s formalizations of constructive logic as Intuitionistic Type Theory (ITT) M L71, M L74, M L82] and was utilized by Girard [Gir71] who introduced a form of second order typed lambda calculus as a tool in his ....

W. Howard, The formulas-as-types notion of construction, in To H. B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, Academic Press,

....justified by the following equivalences from classical logic, all subject to the condition that x does not appear free in P : #x .P # Q) P # (#x .Q) 1) #x .Q # P) #x .Q) # P (2) #x .P # Q) P # (#x .Q) 3) #x .Q # P) #x . Q) # P (4) Using the Curry Howard isomorphism [7] as a bridge between logic and type theory, we can use Equation 1 to justify the equivalence: #a.#x .T a # #) #a.T a # (#x .# ) This result tells us that we can convert an arbitrary term of one type to a corresponding term of the other type. In this case, the equivalence allows us to ....

W. Howard. The formulas-as-types notion of construction. In To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, pages 479--490. Academic Press, 1980.

....interested in the static semantics of LTT and treat its operational semantics only informally. We then present the framework for modular development of operational semantics in Section 6. This paper assumes familiarity with linear logic [6, 18] and with the propositions as types correspondence [9]. Additional familiarity with LF and logical frameworks in general will be helpful, but is not required. 2 Intuitionistic LTT The LTT type theory consists of two parts: a proof sub language, and a computational programming language built around it. LTT is structured with a syntactic division ....

W. Howard. The formulas-as-types notion of construction. In J. P. Seldin and J. R. Hindley, editors, To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, pages 479-490. Academic Press, 1980.

....a computation. The rules governing the function type constructor exhibit an intriguing similarity to the introduction and elimination rules for implication in Gentzen s system of natural deduction. This similarity is not accidental: according to the propositions as types principle [CF58, CHS72, How80] there is an isomorphism between propositions and types with the property that the natural deduction proofs of a proposition correspond to the elements of its associated type. This principle extends to the full range of logical connectives and quantifiers, including those of second and ....

William A. Howard. The formulas-as-types notion of construction. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus and Formalism, pages 479--490. Academic Press, 1980.

....property of natural deduction proofs was discovered by Prawitz [Pra65] some 30 years later. The Curry Howard isomorphism firmly bonds the mathematics of logic to the science of programming languages. Curry [CF58] noticed the relationship between natural deduction and combinatory logic, and Howard [How80] showed that there is an equivalent one toone correspondence between natural deduction and the simply typed lambda calculus. 52 Chapter 3 : Cut Elimination Propositions in the logic can be viewed as types, and proofs as terms in the lambda calculus. In addition, simplifying a proof corresponds to ....

W. Howard. The formulas-as-types notion of construction. In J.P. Seldin and J.R. Hindley, editors, To H.B. Curry: Essays on Combinatory Logic, LambdaCalculus and Formalism, pages 479--490. Academic Press, 1980. Bibliography 171

....of these systems possible. The weakest systems of the Cube is Church s simply typed calculus [7] and the strongest system is the Calculus of Constructions C [8] Girard s wellknown System F [10] gures on the Cube between and C. Moreover, via the Propositions as Types principle (see [13]) many logical systems can be described in the systems of the Cube, see [9] In the Cube, we have in addition to the usual abstraction, a type forming operator . Brie y, if A is a type, and B is a type possibly containing the variable x, then x:A:B is the type of functions that, given a ....

W.A. Howard. The formulas-as-types notion of construction. In [21], pages 479{ 490, 1980.

....11, 14, 2, 4, 19] to name but a few. One such system, the Calculus of Constructions (CC) was introduced by Coquand and Huet as a comprehensive basis for the formalization of constructive mathematics. 11, 14] CC may be viewed as the calculus associated, via the propositions as types principle [24], with natural deduction proofs in an extension of Church s higher order logic [6] The system has been proved both proof theoretically [11] and model theoretically [29, 17, 27] consistent, and the type checking problem has been proved decidable [14, 11] Although CC is an exceedingly rich ....

....representation of mathematical structures such as algebras, automata, and ordered sets. It is by now widely recognized [36, 10] that the appropriate type theoretic representation of mathematical structures is as elements of strong sum types 1 introduced by Martin Lof [35, 36, 37] and Howard [24]. Strong sums have also been used to model modularity constructs in programming languages [33, 41, 2, 4] Unfortunately, strong sums are, in a sense, incompatible with impredicativity [12, 23, 41] As a result, it is necessary to extend the calculus with a level of types, and to postulate the ....

William A. Howard. The formulas-as-types notion of construction. In Seldin and Hindley [45], pages 479--490.

....union over a family #, universal Q , product over a family Indeed, Martin Lof s original theory was intended as a constructive set theory; the logical interpretation is recovered if we consider a proposition to be represented by the set of all its proofs. This idea was written up by Howard [26]. It came from the suggestive similarity between the formal descriptions of, as one case, function application and 29 implication elimination and, as another case, abstraction within the # calculus and implication introduction. Rather than going into more details here, we direct the reader to ....

....It came from the suggestive similarity between the formal descriptions of, as one case, function application and 29 implication elimination and, as another case, abstraction within the # calculus and implication introduction. Rather than going into more details here, we direct the reader to [26], 37] and [43] 8 Mathematical considerations The axiom of choice (in an informal form of our syntax) #x : A) #y : B(x) C(x, y) # (#f : #x:A)B(x) #x : A)C(x, apply(f,x) is derivable in this system. The proof, following the one in [31] goes informally as follows. Assume z : #x:A) #y : ....

W.A. Howard, "The formula--as--types notion of construction", in To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism (J.P. Seldin and J.R. Hindley, eds), Academic Press, 1980.

....be extracted. Such systems involve constructive higher order logic and typically synthesize programs from constructive proofs of formula of the form (8x) P (x) oe (9y)R(x; y) where P is the input assertion and R is the input output relation. This approach is based on the Curry Howard isomorphism [20] and the propositions as types principle; the latter identifies logical propositions with types whose inhabitants are proofs of the proposition. This approach leads to the synthesis of total functions, although there are some ways to extend this to partial functions. A number of implementations of ....

W. Howard. The formulas-as-types notion of construction. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda-Calculus, and Formalism, pages 479--490. Academic Press, NY, 1980.

....languages and theorem provers. De Bruijn s new type systems were in uential in the discovery of new powerful type systems [6] De Bruijn re invented the Curry Howard isomorphism (also referred to as the Curry Howard de Bruijn isomorphism) Independently of Curry and Feys [7] and Howard [9] , we nd a variant of the propositions as types principle in the rst Automath system of de Bruijn (Aut 68 [12, 4] Though de Bruijn was probably in uenced by Heyting (see [5] in [12] p. 211) his ideas arose independently from Curry, Feys and Howard. This can be clearly seen in Section 2.4 of ....

W.A. Howard. The formulas-as-types notion of construction. In [13], pages 479-490, 1980.

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W. Howard. The formulas-as-types notion of construction. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda-Calculus, and Formalism, pages 479--490. Academic Press, NY, 1980.

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HOWARD, W. To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism. Academic, New York, 1980, ch. The formulas-as-types notion of construction, pp. 479--490.

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Howard, W. A., The formulas-as-types notion of construction, in: J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Academic Press, New York, 1980 pp. 479{ 490.

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W. Howard. The formulas-as-types notion of construction. In To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, pages 479--490. Academic Press, 1980.

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W. Howard. The formulas-as-types notion of construction. In To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism, pages 479--490. Academic Press, 1980. (p 98)

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W.A. Howard. The formulas-as-types notion of construction. In J.P. Seldin and J.R. Hindley, editors, To H.B. Curry: Essays on Combinatory Logic, - Calculus and Formalism, 479-490, 1980. Academic Press.

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W.A. Howard. The formulas-as-types notion of construction. In [25], pages 479{ 490, 1980.

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W.A. Howard. The formulas-as-types notion of construction. In [40], pages 479-490, 1980.

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HOWARD, W. The formulas-as-types notion of construction. In To H. B. Curry: Essays on Combznatory Logic, Lambda-Calculus and Formalism Academic Press, 1980, 479 490.

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