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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
A quadratic rate of asymptotic regularity for CAT(0)spaces
, 2005
"... In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the KrasnoselskiMann iterations of nonexpansive mappings in CAT(0)spaces, whereas previous results guarantee only exponential bounds. The method we use is to extend to the more general setting of uniformly convex hy ..."
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Cited by 8 (0 self)
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In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the KrasnoselskiMann iterations of nonexpansive mappings in CAT(0)spaces, whereas previous results guarantee only exponential bounds. The method we use is to extend to the more general setting of uniformly convex hyperbolic spaces a quantitative version of a strengthening of Groetsch’s theorem obtained by Kohlenbach using methods from mathematical logic (socalled “proof mining”).
A variant of the doublenegation translation
, 2006
"... An efficient variant of the doublenegation translation explains the relationship between Shoenfield’s and Gödel’s versions of the Dialectica interpretation. Fix a classical firstorder language, based on the connectives ∨, ∧, ¬, and ∀. We will define a translation to intuitionistic (even minimal) l ..."
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An efficient variant of the doublenegation translation explains the relationship between Shoenfield’s and Gödel’s versions of the Dialectica interpretation. Fix a classical firstorder language, based on the connectives ∨, ∧, ¬, and ∀. We will define a translation to intuitionistic (even minimal) logic, based on the usual connectives. The translation maps each formula ϕ to the formula ϕ ∗ = ¬ϕ∗, so ϕ ∗ is supposed to represent an intuitionistic version of the negation of ϕ. The map from ϕ to ϕ ∗ is defined recursively, as follows: ϕ ∗ = ¬ϕ, when ϕ is atomic (¬ϕ) ∗ = ¬ϕ∗ (ϕ ∨ ψ) ∗ = ϕ ∗ ∧ ψ∗ (ϕ ∧ ψ) ∗ = ϕ ∗ ∨ ψ∗ (∀x ϕ) ∗ = ∃x ϕ∗ Note that we can eliminate either ∨ or ∧ and retain a complete set of connectives.
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
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"... and/or the philosophy of mathematics? At the age of 13 or so some initial interest in philosophy and Aristotelian logic was prompted by my classes in Ancient Greek language which was a main emphasis of study at my high school. My real interest in the foundations of mathematics, however, started at t ..."
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and/or the philosophy of mathematics? At the age of 13 or so some initial interest in philosophy and Aristotelian logic was prompted by my classes in Ancient Greek language which was a main emphasis of study at my high school. My real interest in the foundations of mathematics, however, started at the age of 17 during my last year at high school. Our mathematics teacher had the idea to have each of us to write an extended essay on some period in the history of mathematics. He designed a list of 20 topics starting from ancient mathematics to the beginning 20th century. The very day the topics could be chosen I was ill and could not attend school. When I finally was back in school I had to learn that only topic no. 20 on ‘Cantor, Dedekind, Hilbert ’ was left, apparently because everybody had figured out that a topic touching on comparatively recent mathematics would be more difficult to deal with than, say, Babylonian mathematics. After I had overcome some initial shock I went to the university library in Frankfurt to get hold of the collected works of G. Cantor, D. Hilbert as well as R. Dedekind’s ‘Was sind und was sollen die Zahlen ’ and some popular treatments of the ‘foundational crisis’ at the early 20’s century. Immediately, I got excited about the topic. After having finished the essay I was determined to study philosophy and mathematics with the aim to become a logician.