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Curry's Anticipation of the Types Used in Programming Languages
, 2003
"... This paper shows that H. B. Curry anticipated both the kind of data types used in computer programming languages and also the dependent function type. ..."
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This paper shows that H. B. Curry anticipated both the kind of data types used in computer programming languages and also the dependent function type.
Variants of the Basic Calculus of Constructions
, 2004
"... In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version i ..."
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In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version in early papers on the subject by Seldin. None of these results is very deep, but it seems useful to collect them in one place.
On the relation between Churchstyle typing and Currystyle typing ∗
, 2008
"... There are two versions of type assignment in λcalculus: Churchstyle, in which the type of each variable is fixed, and Currystyle (also called “domain free”), in which it is not. As an example, in Churchstyle typing, λx: A. x is the identity function on type A, and it has type A → A but not B → B f ..."
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There are two versions of type assignment in λcalculus: Churchstyle, in which the type of each variable is fixed, and Currystyle (also called “domain free”), in which it is not. As an example, in Churchstyle typing, λx: A. x is the identity function on type A, and it has type A → A but not B → B for a type B different from A. In Currystyle typing, λx. x is a general identity function with type C → C for every type C. In this paper, I will show how to interpret in a Currystyle system every Pure Type System (PTS) in the Churchstyle without losing any typing information. I will also prove a kind of comservative extension result for this interpretation, a result which implies that for most consistent PTSs of the Churchstyle, the corresponding Currystyle system is consistent. (This generalizes some unpublished work with
On the Role of Implication in Formal Logic
"... Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with no ..."
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Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a "classical" version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higherorder BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic. 1991 AMS (MOS) Classification: 03B40, 03F05, 03B20 Key words: Implication, negation, combinatory logic, lambda calculus, comprehension principle, normalization, cutelimination...
On the Role of Implication in Formal Logic
, 1998
"... Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with no ..."
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Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or #calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a "classical" version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higherorder BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic. 1991 AMS (MOS) Classification: 03B40, 03F05, 03B20 Key words: Implication, negation, combinatory logic, lambda calculus, comprehension principle, normalization, cutelimination...