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Functional interpretation and inductive definitions
 Journal of Symbolic Logic
"... Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finitetype functionals defined using transfinite recursion on wellfounded trees. 1. ..."
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Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finitetype functionals defined using transfinite recursion on wellfounded trees. 1.
Light Dialectica program extraction from a classical Fibonacci proof
 PROCEEDINGS OF DCM’06 AT ICALP’06, ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE (ENTCS, 2007), 10PP., ACCEPTED FOR PUBLICATION, DOWNLOADABLE @ HTTP://WWW.BRICS.DK/ EDANHER
, 2007
"... We demonstrate program extraction by the Light Dialectica Interpretation (LDI) on a minimal logic proof of the classical existence of Fibonacci numbers. This semiclassical proof is available in MinLog’s library of examples. The term of Gödel’s T extracted by the LDI is, after strong normalization, ..."
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We demonstrate program extraction by the Light Dialectica Interpretation (LDI) on a minimal logic proof of the classical existence of Fibonacci numbers. This semiclassical proof is available in MinLog’s library of examples. The term of Gödel’s T extracted by the LDI is, after strong normalization, exactly the usual recursive algorithm which defines the Fibonacci numbers (in pairs). This outcome of the Light Dialectica metaalgorithm is much better than the Tprogram extracted by means of the pure Gödel Dialectica Interpretation. It is also strictly less complex than the result obtained by means of the refined Atranslation technique of Berger, Buchholz and Schwichtenberg on an artificially distorted variant of the input proof, but otherwise it is identical with the term yielded by Berger’s Kripkestyle refined Atranslation. Although syntactically different, it also has the same computational complexity as the original program yielded by the refined Atranslation from the undistorted input classical Fibonacci proof.
THE BOUNDED FUNCTIONAL INTERPRETATION OF THE DOUBLE NEGATION SHIFT
"... Abstract. We prove that the (nonintuitionistic) law of the double negation shift has a bounded functional interpretation with bar recursive functionals of finite type. As an application, we show that full numerical comprehension is compatible with the uniformities introduced by the characteristic p ..."
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Abstract. We prove that the (nonintuitionistic) law of the double negation shift has a bounded functional interpretation with bar recursive functionals of finite type. As an application, we show that full numerical comprehension is compatible with the uniformities introduced by the characteristic principles of the bounded functional interpretation for the classical case. §1. Introduction and background. In 1962 [14], Clifford Spector gave a remarkable characterization of the provably recursive functionals of full secondorder arithmetic (a.k.a. analysis). The central result of his paper is an extension, from arithmetic to analysis, of the (then quite recent) dialectica interpretation of Gödel of 1958 [7]. Spector’s extension relies on a form of wellfounded recursion
Harrington’s conservation theorem redone
 In: Archive for Mathematical Logic
, 2008
"... Leo Harrington showed that the secondorder theory of arithmetic WKL0 is Π 1 1conservative over the theory RCA0. Harrington’s proof is modeltheoretic, making use of a forcing argument. A purely prooftheoretic proof, avoiding forcing, has been eluding the efforts of researchers. In this short paper ..."
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Leo Harrington showed that the secondorder theory of arithmetic WKL0 is Π 1 1conservative over the theory RCA0. Harrington’s proof is modeltheoretic, making use of a forcing argument. A purely prooftheoretic proof, avoiding forcing, has been eluding the efforts of researchers. In this short paper, we present a proof of Harrington’s result using a cutelimination argument. 1
A most artistic package of a jumble of ideas
"... In the course of ten short sections, we comment on Gödel’s seminal “Dialectica ” paper of fifty years ago and its aftermath. We start by suggesting that Gödel’s use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics and in tune with his defen ..."
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In the course of ten short sections, we comment on Gödel’s seminal “Dialectica ” paper of fifty years ago and its aftermath. We start by suggesting that Gödel’s use of functionals of finite type is yet another instance of the realistic attitude of Gödel towards mathematics and in tune with his defense of the postulation of ever increasing higher types in foundational studies. We also make some observations concerning Gödel’s recasting of intuitionistic arithmetic via the “Dialectica ” interpretation, discuss the extra principles that the interpretation validates, and comment on extensionality and higher order equality. The latter sections focus on the role of majorizability considerations within the “Dialectica ” and related interpretations for extracting computational information from ordinary proofs in mathematics. I Kurt Gödel’s realism, a stance “against the current ” of his time, is now wellknown
Bounded Modified Realizability
, 2005
"... We define a notion of realizability, based on a new assignment of formulas, which does not care for precise witnesses of existential statements, but only for bounds for them. The novel form of realizability supports a very general form of the FAN theorem, refutes Markov’s principle but meshes well w ..."
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We define a notion of realizability, based on a new assignment of formulas, which does not care for precise witnesses of existential statements, but only for bounds for them. The novel form of realizability supports a very general form of the FAN theorem, refutes Markov’s principle but meshes well with some classical principles, including the lesser limited principle of omniscience and weak König’s lemma. We discuss some applications, as well as some previous results in the literature. 1
PROOF INTERPRETATIONS AND MAJORIZABILITY
"... Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpret ..."
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Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finitetype arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpretation of Ulrich Kohlenbach, has been at the center of a vigorous program in applied proof theory dubbed proof mining. We discuss some of the traditional and majorizability interpretations, including the recent bounded interpretations, and focus on the main theoretical techniques behind proof mining. Contents