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Inductive, Projective, and Retractive Types
, 1993
"... We give an analysis of classes of recursive types by presenting two extensions of the simplytyped lambda calculus. The first language only allows recursive types with builtin principles of wellfounded induction, while the second allows more general recursive types which permit nonterminating com ..."
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We give an analysis of classes of recursive types by presenting two extensions of the simplytyped lambda calculus. The first language only allows recursive types with builtin principles of wellfounded induction, while the second allows more general recursive types which permit nonterminating computations. We discuss the expressive power of the languages, examine the properties of reductionbased operational semantics for them, and give examples of their use in expressing iteration over large ordinals and in simulating both callbyname and callbyvalue versions of the untyped lambda calculus. The motivations for this work come from category theoretic models. 1 Introduction An examination of the common uses of recursion in defining types reveals that there are two distinct classes of operations being performed. The first class of recursive type contains what are generally known as the "inductive" types, as well as their duals, the "coinductive" or "projective" types. The distingui...
From Heyting's arithmetic to verified programs
, 1998
"... We discuss higher type constructions inherent to intuitionistic proofs. As an example we consider Gentzen's proof of transnite induction up to the ordinal 0 . From the constructive content of this proof we derive higher type algorithms for some ordinal recursive hierarchies of number theoretic funct ..."
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We discuss higher type constructions inherent to intuitionistic proofs. As an example we consider Gentzen's proof of transnite induction up to the ordinal 0 . From the constructive content of this proof we derive higher type algorithms for some ordinal recursive hierarchies of number theoretic functions.
3584 CS Utrecht
, 1992
"... By constructing a counter, model we show that a,.appealing certain equation E has no solution in Girard's [1972] second order lambdacalculus (the socalled polymorphic, lambda " calculus). The equation E ', = E(4i) ' (with 45 a type 3 variable) is a simple functional equation in thelkiguage of Gode ..."
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By constructing a counter, model we show that a,.appealing certain equation E has no solution in Girard's [1972] second order lambdacalculus (the socalled polymorphic, lambda " calculus). The equation E ', = E(4i) ' (with 45 a type 3 variable) is a simple functional equation in thelkiguage of Godel's [1958] system of higher order primitive recursive fanctiotals and` has an easy solution in Spector"s°[1962] system of bar recursive functionals This shows that the class of bar recursive functionals differs from the class _ of functionals definable ' in the polymorphic lambda calculus."The fact that the two calculi have different classes of definable functionals (at least of type 3), contrasts the metamathematical results from 'Spector [1962] and Girard [1972] which state that the twocalculi have the same class of definable functions, 'namely, the 'provably.total recursive:.functions. of `..analysis