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Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
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This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
Synthesis of moduli of uniform continuity by the Monotone Dialectica Interpretation
 in the proofsystem MINLOG. Electronic Notes in Theoretical Computer Science
, 2007
"... We extract on the computer a number of moduli of uniform continuity for the first few elements of a sequence of closed terms t of Gödel’s T of type (N→N)→(N→N). The generic solution may then be quickly inferred by the human. The automated synthesis of such moduli proceeds from a proof of the heredi ..."
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We extract on the computer a number of moduli of uniform continuity for the first few elements of a sequence of closed terms t of Gödel’s T of type (N→N)→(N→N). The generic solution may then be quickly inferred by the human. The automated synthesis of such moduli proceeds from a proof of the hereditarily extensional equality (≈) of t to itself, hence a proof in a weakly extensional variant of BergerBuchholzSchwichtenberg’s system Z of t ≈(N→N)→(N→N) t. We use an implementation on the machine, in Schwichtenberg’s MinLog proofsystem, of a nonliteral adaptation to Natural Deduction of Kohlenbach’s monotone functional interpretation. This new version of the Monotone Dialectica produces terms in NbEnormal form by means of a recurrent partial NbEnormalization. Such partial evaluation is strictly necessary.
Dialectica program extraction from a classical Fibonacci proof, in
 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.
Can proofs be animated by games?
 . URZYCZYN ED., TLCA 2005, LNCS 3461
, 2005
"... Proof animation is a way of executing proofs to find errors in the formalization of proofs. It is intended to be "testing in proof engineering". Although the realizability interpretation as well as the functional interpretation based on limitcomputations were introduced as means for pro ..."
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Proof animation is a way of executing proofs to find errors in the formalization of proofs. It is intended to be "testing in proof engineering". Although the realizability interpretation as well as the functional interpretation based on limitcomputations were introduced as means for proof animation, they were unrealistic as an architectural basis for actual proof animation tools. We have found game theoretical semantics corresponding to these interpretations, which is likely to be the right architectural basis for proof animation.
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
Light Functional Interpretation
 Lecture Notes in Computer Science, 3634:477 – 492, July 2005. Computer Science Logic: 19th International Workshop, CSL
, 2005
"... an optimization of Gödel’s technique towards the extraction of (more) efficient programs from (classical) proofs ..."
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an optimization of Gödel’s technique towards the extraction of (more) efficient programs from (classical) proofs
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