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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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Cited by 16 (2 self)
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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
Proof mining in L_1approximation
, 2001
"... In this paper we present another case study in the general project of proof mining which means the logical analysis of prima facie noneffective proofs with the aim of extracting new computationally relevant data. We use techniques based on monotone functional interpretation (developed in [17]) to a ..."
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In this paper we present another case study in the general project of proof mining which means the logical analysis of prima facie noneffective proofs with the aim of extracting new computationally relevant data. We use techniques based on monotone functional interpretation (developed in [17]) to analyze Cheney's simplification [6] of Jackson's original proof [10] from 1921 of the uniqueness of the best L 1 approximation of continuous functions f # C[0, 1] by polynomials p # Pn of degree # n. Cheney's proof is noneffective in the sense that it is based on classical logic and on the noncomputational principle WKL (binary Konig lemma). The result of our analysis provides the first e#ective (in all parameters f, n and #) uniform modulus of uniqueness (a concept which generalizes `strong uniqueness' studied extensively in approximation theory). Moreover, the extracted modulus has the optimal #dependency as follows from Kroo [21]. The paper also describes how the uniform modulus of uniqueness can be used to compute the best L 1 approximations of a fixed f # C[0, 1] with arbitrary precision. We use this result to give a complexity upper bound on the computation of the best L 1 approximation in [24].
Strong Uniqueness
, 2010
"... This is a survey paper on the subject of strong uniqueness in approximation theory. ..."
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This is a survey paper on the subject of strong uniqueness in approximation theory.
On the Computational Complexity of Best L1Approximation
, 2001
"... We give an upper bound on the complexity of the sequence (p n ) n#N of best L1approximations of a polynomialtime computable function on the interval [0, 1] from the space of polynomials of bounded degree Pn . The analysis makes essential use of the modulus of uniqueness for L1approximation presen ..."
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Cited by 2 (1 self)
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We give an upper bound on the complexity of the sequence (p n ) n#N of best L1approximations of a polynomialtime computable function on the interval [0, 1] from the space of polynomials of bounded degree Pn . The analysis makes essential use of the modulus of uniqueness for L1approximation presented in [KO01].
Mathematical Logic Quarterly c © WILEYVCH Verlag Berlin GmbH 2001 On the computational complexity of best L1approximation
"... Abstract. It is well known that for a given continuous function f: [0, 1] ! R and a number n there exists a unique polynomial pn 2 Pn (polynomials of degree n) which best L1approximates f. We establish the rst upper bound on the complexity of the sequence (pn)n2N, assuming f is polynomialtime com ..."
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Abstract. It is well known that for a given continuous function f: [0, 1] ! R and a number n there exists a unique polynomial pn 2 Pn (polynomials of degree n) which best L1approximates f. We establish the rst upper bound on the complexity of the sequence (pn)n2N, assuming f is polynomialtime computable. Our complexity analysis makes essential use of the modulus of uniqueness for L1approximation presented in [13].
Applied Foundations: Proof Mining in Mathematics
"... A central theme in the foundations of mathematics, dating back to D. Hilbert, can be paraphrased by the following question ‘How is it that abstract methods (‘ideal elements’) can be used to prove ‘real ’ statements e.g. about the natural numbers and is this use necessary in principle?’ Hilbert’s aim ..."
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A central theme in the foundations of mathematics, dating back to D. Hilbert, can be paraphrased by the following question ‘How is it that abstract methods (‘ideal elements’) can be used to prove ‘real ’ statements e.g. about the natural numbers and is this use necessary in principle?’ Hilbert’s aim was to show that the use of such ideal elements can be shown to be consistent by finitistic means (‘Hilbert’s program’). Hilbert’s program turned out to be impossible in the original form by the seminal results of K. Gödel. However, more recent developments show it can be carried out in a partial form in that one can design formal systems A which are sufficient to formalize substantial parts of mathematics and yet can be reduced prooftheoretically to primitive recursive arithmetic PRA, a formal system usually associated with ‘finitism’. These systems