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Explicit Provability And Constructive Semantics
 Bulletin of Symbolic Logic
, 2001
"... In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is noth ..."
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Cited by 139 (25 self)
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In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a BrouwerHeytingKolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #calculus.
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 35 (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.
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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Cited by 10 (4 self)
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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].
Proof Mining in Analysis: Computability and Complexity
, 2001
"... Proof Mining consists in extracting from prima facie nonconstructive proofs constructive information. For that purpose many techniques mostly based on negative translation, realizability, functional interpretation, Atranslation and combinations thereof have been developed. After introducing variou ..."
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Proof Mining consists in extracting from prima facie nonconstructive proofs constructive information. For that purpose many techniques mostly based on negative translation, realizability, functional interpretation, Atranslation and combinations thereof have been developed. After introducing various different approaches to Proof Mining we present an application of `monotone functional interpretation' to a modification of a classical proof from 1921 of uniqueness of L 1 approximation of continuous functions by polynomials of degree # n. From that proof we extracted the first effective uniform rate of strong uniqueness (also called `uniform modulus of uniqueness') for L 1 approximation, which existence had been studied for 80 years but always in a nonconstructive manner. For that reason, previous results only stated the dependencies of the modulus but explicit constants had never been obtained. We also show how this modulus of uniqueness can be used to compute the actual best L 1 approximation of the given continuous functions on a compact interval. Finally, we analyze the complexity of the resulting algorithm using the tools of Computable Analysis (cf. [Ko91] and [Wei00]). The first part of this work (the extraction of the uniform modulus of uniqueness) has been concluded, the last two parts are under development. This report also includes indications of future areas of work. 1