Introduction to the constructive point of view in the foundations of mathematics, in particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation and formalization of logic is described. For constructive (intuitionistic) arithmetic, Kleene’s realizability interpretation is given; this provides an example of the possibility of a constructive mathematical practice which diverges from classical mathematics. The crucial notion in intuitionistic analysis, choice sequence, is briefly described and some principles which are valid for choice sequences are discussed. The second half of the article deals with some aspects of proof theory, i.e., the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions are outlined: his introduction of the so-called Gentzen systems which use sequents instead of formulas and his result on first-order arithmetic showing that (suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in first-order arithmetic.

this paper we study some extensions of the Kleene-Kreisel continuous functionals [7, 8] and show that most of the constructions and results, in particular the crucial density theorem, carry over from nite to dependent and transnite types. Following an approach of Ershov we dene the continuous functionals as the total elements in a hierarchy of Ershov-Scott-domains of partial continuous functionals. In this setting the density theorem says that the total functionals are topologically dense in the partial ones, i.e. every nite (compact) functional has a total extension. We will extend this theorem from function spaces to dependent products and sums and universes. The key to the proof is the introduction of a suitable notion of density and associated with it a notion of co-density for dependent domains with totality. We show that the universe obtained by closing a given family of basic domains with totality under some quantiers has a dense and co-dense totality provided the totalities on the basic domains are dense and co-dense and the quantiers preserve density and co-density. In particular we can show that the quantiers and have this preservation property and hence, for example, the closure of the integers and the booleans (which are dense and co-dense) under and has a dense and co-dense totality. We also discuss extensions of the density theorem to iterated universes, i.e. universes closed under universe operators. From our results we derive a dependent continuous choice principle and a simple order-theoretic characterization of extensional equality for total objects. Finally we survey two further applications of density: Waagb's extension of the Kreisel-Lacombe-Shoeneld-Theorem showing the coincidence of the hereditarily eectively continuous hierarchy...

We describe a constructive theory of computable functionals, based on the partial continuous functionals as their intendend domain. Such a task had long ago been started by Dana Scott [28], under the well-known abbreviation

It is a fact of experience from the study of higher type computability that a wide range of approaches to defining a class of (hereditarily) total functionals over N leads in practice to a relatively small handful of distinct type structures. Among these are the type structure C of Kleene-Kreisel continuous functionals, its effective substructure C eff, and the type structure HEO of the hereditarily effective operations. However, the proofs of the relevant equivalences are often non-trivial, and it is not immediately clear why these particular type structures should arise so ubiquitously. In this paper we present some new results which go some way towards explaining this phenomenon. Our results show that a large class of extensional collapse constructions always give rise to C, C eff or HEO (as appropriate). We obtain versions of our results for both the “standard” and “modified” extensional collapse constructions. The proofs make essential use of a technique due to Normann. Many new results, as well as some previously known ones, can be obtained as instances of our theorems, but more importantly, the proofs apply uniformly to a whole family of constructions, and provide strong evidence that the above three type structures are highly canonical mathematical objects.

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Abstract. This paper investigates what is essentially a call-by-value version of PCF under a complexity-theoretically motivated type system. The programming formalism, ATR, has its first-order programs characterize the polynomial-time computable functions, and its second-order programs characterize the type-2 basic feasible functionals of Mehlhorn and of Cook and Urquhart. (The ATR-types are confined to levels 0, 1, and 2.) The type system comes in two parts, one that primarily restricts the sizes of values of expressions and a second that primarily restricts the time required to evaluate expressions. The size-restricted part is motivated by Bellantoni and Cook’s and Leivant’s implicit characterizations of polynomial-time. The time-restricting part is an affine version of Barber and Plotkin’s DILL. Two semantics are constructed for ATR. The first is a pruning of the naïve denotational semantics for ATR. This pruning removes certain functions that cause otherwise feasible forms of recursion to go wrong. The second semantics is a model for ATR’s time complexity relative to a certain abstract machine. This model provides a setting for complexity recurrences arising from ATR recursions, the solutions of which yield second-order polynomial time bounds. The time-complexity semantics is also shown to be sound relative to the costs of interpretation on the abstract machine. 1.

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