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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 165 (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, volume 664 of LNCS
, 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 64 (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 29 (5 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 Bar Recursion and Classical Dependent Choice
 In Logic Colloquium 2001
"... We introduce a variant of Spector's bar recursion in nite types to give a realizability interpretation of the classical axiom of dependent choice allowing for the extraction of witnesses from proofs of 1 formulas in classical analysis. We also give a bar recursive denition of the fan functional ..."
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Cited by 27 (17 self)
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We introduce a variant of Spector's bar recursion in nite types to give a realizability interpretation of the classical axiom of dependent choice allowing for the extraction of witnesses from proofs of 1 formulas in classical analysis. We also give a bar recursive denition of the fan functional and study the relationship of our variant of bar recursion with others. x1.
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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Cited by 14 (1 self)
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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...
Uniform Heyting arithmetic
 Annals Pure Applied Logic
, 2005
"... Abstract. We present an extension of Heyting Arithmetic in finite types called Uniform Heyting Arithmetic (HA u) that allows for the extraction of optimized programs from constructive and classical proofs. The system HA u has two sorts of firstorder quantifiers: ordinary quantifiers governed by the ..."
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Cited by 13 (0 self)
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Abstract. We present an extension of Heyting Arithmetic in finite types called Uniform Heyting Arithmetic (HA u) that allows for the extraction of optimized programs from constructive and classical proofs. The system HA u has two sorts of firstorder quantifiers: ordinary quantifiers governed by the usual rules, and uniform quantifiers subject to stronger variable conditions expressing roughly that the quantified object is not computationally used in the proof. We combine a Kripkestyle Friedman/Dragalin translation which is inspired by work of Coquand and Hofmann and a variant of the refined Atranslation due to Buchholz, Schwichtenberg and the author to extract programs from a rather large class of classical firstorder proofs while keeping explicit control over the levels of recursion and the decision procedures for predicates used in the extracted program. §1. Introduction. According to the BrouwerHeytingKolmogorov interpretation of constructive logic a proof is a construction providing evidence for the proven formula [20]. Viewing this interpretation from a dataoriented perspective one arrives at the socalled proofsasprograms paradigm associating a constructive proof with a program ‘realizing ’ the proven formula. This paradigm has been
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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Cited by 13 (1 self)
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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.
The Modified Realizability Topos
 Journal of Pure and Applied Algebra
, 1996
"... The modified realizability topos is the semantic (and higher order) counterpart of a variant of Kreisel's modified realizability (1957). These years, this realizability has been in the limelight again because of its possibilities for modelling type theory (Streicher, HylandOngRitter) and ..."
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Cited by 10 (1 self)
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The modified realizability topos is the semantic (and higher order) counterpart of a variant of Kreisel's modified realizability (1957). These years, this realizability has been in the limelight again because of its possibilities for modelling type theory (Streicher, HylandOngRitter) and strong normalization.
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 (7 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