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58
On the Arithmetical Content of Restricted Forms of Comprehension, Choice and General Uniform Boundedness
 PURE AND APPLIED LOGIC
, 1997
"... In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems T n in all finite types which are suited to formalize substantial parts of analysis but ..."
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Cited by 9 (4 self)
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In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems T n in all finite types which are suited to formalize substantial parts of analysis but nevertheless have provably recursive function(al)s of low growth. We reduce the use of instances of these principles in T n proofs of a large class of formulas to the use of instances of certain arithmetical principles thereby determining faithfully the arithmetical content of the former. This is achieved using the method of elimination of Skolem functions for monotone formulas which was introduced by the author in a previous paper. As
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 9 (1 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
Computational interpretations of analysis via products of selection functions
 CIE 2010, INVITED TALK ON SPECIAL SESSION “PROOF THEORY AND COMPUTATION
, 2010
"... We show that the computational interpretation of full comprehension via two wellknown functional interpretations (dialectica and modified realizability) corresponds to two closely related infinite products of selection functions. ..."
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Cited by 9 (8 self)
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We show that the computational interpretation of full comprehension via two wellknown functional interpretations (dialectica and modified realizability) corresponds to two closely related infinite products of selection functions.
Continuous functions on final coalgebras
, 2007
"... In a previous paper we have given a representation of continuous functions on streams, both discretevalued functions, and functions between streams. the topology on streams is the ‘Baire ’ topology induced by taking as a basic neighbourhood the set of streams that share a given finite prefix. We ga ..."
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Cited by 9 (1 self)
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In a previous paper we have given a representation of continuous functions on streams, both discretevalued functions, and functions between streams. the topology on streams is the ‘Baire ’ topology induced by taking as a basic neighbourhood the set of streams that share a given finite prefix. We gave also a combinator on the representations of stream processing functions that reflects composition. Streams are the simplest example of a nontrivial final coalgebras, playing in the coalgebraic realm the same role as do the natural numbers in the algebraic realm. Here we extend our previous results to cover the case of final coalgebras for a broad class of functors generalising (×A). The functors we deal with are those that arise from countable signatures of finiteplace untyped operators. These have many applications. The topology we put on the final coalgebra for such a functor is that induced by taking for basic neighbourhoods the set of infinite objects which share a common prefix, according to the usual definition of the final coalgebra as the limit of a certain inverse chain starting at �. 1
Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces, arXiv:0707.1626v2 [math.FA
 J. of the European Math. Soc
, 2007
"... This paper provides a fixed point theorem for asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces as well as new effective results on the KrasnoselskiMann iterations of such mappings. The latter were found using methods from logic and the paper continues a case study in the g ..."
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Cited by 8 (7 self)
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This paper provides a fixed point theorem for asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces as well as new effective results on the KrasnoselskiMann iterations of such mappings. The latter were found using methods from logic and the paper continues a case study in the general program of extracting effective data from primafacie ineffective proofs in the fixed point theory of such mappings. 1
The Peirce Translation and the Double Negation Shift
"... We develop applications of selection functions to proof theory and computational extraction of witnesses from proofs in classical analysis. The main novelty is a translation of classical minimal logic into minimal logic, which we refer to as the Peirce translation, and which we apply to interpret b ..."
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Cited by 5 (5 self)
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We develop applications of selection functions to proof theory and computational extraction of witnesses from proofs in classical analysis. The main novelty is a translation of classical minimal logic into minimal logic, which we refer to as the Peirce translation, and which we apply to interpret both a strengthening of the doublenegation shift and the axioms of countable and dependent choice, via infinite products of selection functions.
A Note on Spector's QuantifierFree Rule of Extensionality
 Arch. Math. Logic
, 1999
"... In this note we show that the socalled weakly extensional arithmetic in all nite types, which is based on a quanti erfree rule of extensionality due to C. Spector and which is of signi cance in the context of Godel's functional interpretation, does not satisfy the deduction theorem for add ..."
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Cited by 5 (3 self)
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In this note we show that the socalled weakly extensional arithmetic in all nite types, which is based on a quanti erfree rule of extensionality due to C. Spector and which is of signi cance in the context of Godel's functional interpretation, does not satisfy the deduction theorem for additional axioms. This holds already for 1  axioms. Previously, only the failure of the stronger deduction theorem for deductions from (possibly open) assumptions (with parameters kept xed) was known.
Continuous semantics for strong normalization
 In CiE’05, volume 3526 of LNCS
, 2005
"... Abstract. We prove a general strong normalization theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domaintheoretic semantics. The theorem applies to extensions of G"odel's system T, but also to various forms of bar recursion for ..."
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Cited by 4 (2 self)
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Abstract. We prove a general strong normalization theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domaintheoretic semantics. The theorem applies to extensions of G"odel's system T, but also to various forms of bar recursion for which strong normalization was hitherto unknown.
On the Computational Content of the BolzanoWeierstraß Principle
, 2009
"... We will apply the methods developed in the field of ‘proof mining’ to the BolzanoWeierstraß theorem BW and calibrate the computational contribution of using this theorem in proofs of combinatorial statements. We provide an explicit solution of the Gödel functional interpretation (combined with nega ..."
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Cited by 4 (4 self)
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We will apply the methods developed in the field of ‘proof mining’ to the BolzanoWeierstraß theorem BW and calibrate the computational contribution of using this theorem in proofs of combinatorial statements. We provide an explicit solution of the Gödel functional interpretation (combined with negative translation) as well as the monotone functional interpretation of BW for the product space ∏i∈N[−k i, k i] (with the standard product metric). This results in optimal program and bound extraction theorems for proofs based on fixed instances of BW, i.e. for BW applied to fixed sequences in ∏i∈N[−k i, k i].