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59
Constructivism and Proof Theory
, 2003
"... 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. F ..."
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Cited by 162 (4 self)
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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 socalled Gentzen systems which use
sequents instead of formulas and his result on firstorder arithmetic showing that
(suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in
firstorder arithmetic.
Program extraction from normalization proofs
 Typed Lambda Calculi and Applications, number 664 in Lecture Notes in Computer Science
, 1993
"... This paper describes formalizations of Tait’s normalization proof for the simply typed λcalculus in the proof assistants Minlog, Coq and Isabelle/HOL. From the formal proofs programs are machineextracted that implement variants of the wellknown normalizationbyevaluation algorithm. The case stud ..."
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Cited by 60 (5 self)
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This paper describes formalizations of Tait’s normalization proof for the simply typed λcalculus in the proof assistants Minlog, Coq and Isabelle/HOL. From the formal proofs programs are machineextracted that implement variants of the wellknown normalizationbyevaluation algorithm. The case study is used to test and compare the program extraction machineries of the three proof assistants in a nontrivial setting. 1
Local stability of ergodic averages
 Transactions of the American Mathematical Society
"... We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesguemeasure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages An ..."
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Cited by 27 (4 self)
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We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesguemeasure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L2([0,1]). In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, we show that, for any nonexpansive linear operator T on a separable Hilbert space, and any element f, it is possible to compute a bound on the rate of convergence of (Anf) from T, f, and the norm ‖f ∗ ‖ of the limit. In particular, if T is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space X and f is any computable element of L2(X), then there is a computable bound on the rate of convergence of the sequence (Anf). The mean ergodic theorem is equivalent to the assertion that for every function K(n) and every ε> 0, there is an n with the property that the ergodic averages Amf are stable to within ε on the interval [n, K(n)]. Even in situations where the sequence (Anf) does not have a computable limit, one can give explicit bounds on such n in terms of K and ‖f‖/ε. This tells us how far one has to search to find an n so that the ergodic averages are “locally stable ” on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general prooftheoretic methods falling under the heading of “proof mining.” 1
Modified Realizability Toposes and Strong Normalization Proofs (Extended Abstract)
 Typed Lambda Calculi and Applications, LNCS 664
, 1993
"... ) 1 J. M. E. Hyland 2 C.H. L. Ong 3 University of Cambridge, England Abstract This paper is motivated by the discovery that an appropriate quotient SN 3 of the strongly normalising untyped 3terms (where 3 is just a formal constant) forms a partial applicative structure with the inherent appl ..."
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) 1 J. M. E. Hyland 2 C.H. L. Ong 3 University of Cambridge, England Abstract This paper is motivated by the discovery that an appropriate quotient SN 3 of the strongly normalising untyped 3terms (where 3 is just a formal constant) forms a partial applicative structure with the inherent application operation. The quotient structure satisfies all but one of the axioms of a partial combinatory algebra (pca). We call such partial applicative structures conditionally partial combinatory algebras (cpca). Remarkably, an arbitrary rightabsorptive cpca gives rise to a tripos provided the underlying intuitionistic predicate logic is given an interpretation in the style of Kreisel's modified realizability, as opposed to the standard Kleenestyle realizability. Starting from an arbitrary rightabsorptive cpca U , the tripostotopos construction due to Hyland et al. can then be carried out to build a modified realizability topos TOPm (U ) of nonstandard sets equipped with an equali...
A proof of strong normalisation using domain theory
 In LICS’06
, 2006
"... U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is. The goal of this paper is to show tha ..."
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U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is. The goal of this paper is to show that, using ideas from the theory of intersection types [2, 6, 7, 21] and MartinLöf’s domain interpretation of type theory [18], we can in turn simplify U. Berger’s argument in the construction of such a domain model. We think that our domain model can be used to give modular proofs of strong normalization for various type theory. As an example, we show in some details how it can be used to prove strong normalization for MartinLöf dependent type theory extended with bar recursion, and with some form of proofirrelevance. 1
Continuous Functionals of Dependent and Transfinite Types
, 1995
"... this paper we study some extensions of the KleeneKreisel 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 functi ..."
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Cited by 10 (2 self)
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this paper we study some extensions of the KleeneKreisel 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 ErshovScottdomains 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 codensity 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 codense totality provided the totalities on the basic domains are dense and codense and the quantiers preserve density and codensity. 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 codense) under and has a dense and codense 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 ordertheoretic characterization of extensional equality for total objects. Finally we survey two further applications of density: Waagb's extension of the KreiselLacombeShoeneldTheorem showing the coincidence of the hereditarily eectively continuous hierarchy...
Strongly Uniform Bounds from SemiConstructive Proofs
, 2004
"... In [12], the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie nonconstructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and nor ..."
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Cited by 10 (6 self)
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In [12], the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie nonconstructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and normed linear spaces and guarantee the independence of the bounds from parameters raging over metrically bounded (not necessarily compact!) spaces. The use of classical logic imposes some severe restrictions on the formulas and proofs for which the extraction can be carried out. In this paper we consider similar metatheorems for semiintuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain nonconstructive principles. Contrary to
Density Theorems for the DomainsWithTotality Semantics of Dependent Types
 Applied Categorical Structures
, 2000
"... . We study a semantics of dependent types and universe operators based on parametrized domains with totality. The main results are generalizations of the Kleene/Kreisel density theorem for the continuous functionals. This continues work of E. Palmgren and V. Stoltenberg{Hansen on the domain interpre ..."
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Cited by 10 (0 self)
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. We study a semantics of dependent types and universe operators based on parametrized domains with totality. The main results are generalizations of the Kleene/Kreisel density theorem for the continuous functionals. This continues work of E. Palmgren and V. Stoltenberg{Hansen on the domain interpretation of dependent types, and of D. Normann on universes of wellfounded types with density. Key words: Continuous functionals, Domains, Totality, Dependent types, Universes 1. Introduction In Mathematical Logic and Computer Science there is growing interest in constructive type theories as developed by Martin{Lof [8]. This paper is concerned with a semantics of such theories within the realm of Ershov{Scott domains [5] with totality [10]. Erik Palmgren and Viggo Stoltenberg{Hansen [15], [17] developed a semantics for a partial type theory (modelling partial functions and functionals) based on the notion of a parametrization, i.e. a domain depending on parameters. Since this semantics wa...
The Meaning of Types  From Intrinsic to Extrinsic Semantics
 Department of Computer Science, University of Aarhus
, 2000
"... A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic " if all phrases have meanings that are independent of their typings, while typing ..."
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A definition of a typed language is said to be "intrinsic" if it assigns meanings to typings rather than arbitrary phrases, so that illtyped phrases are meaningless. In contrast, a definition is said to be "extrinsic " if all phrases have meanings that are independent of their typings, while typings represent properties of these meanings. For a simply typed lambda calculus, extended with recursion, subtypes, and named products, we give an intrinsic denotational semantics and a denotational semantics of the underlying untyped language. We then establish a logical relations theorem between these two semantics, and show that the logical relations can be "bracketed" by retractions between the domains of the two semantics. From these results, we derive an extrinsic semantics that uses partial equivalence relations.