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Analyzing Proofs in Analysis
 LOGIC: FROM FOUNDATIONS TO APPLICATIONS. EUROPEAN LOGIC COLLOQUIUM (KEELE
, 1993
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Mathematically Strong Subsystems of Analysis With Low Rate of Growth of Provably Recursive Functionals
, 1995
"... This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of ..."
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Cited by 34 (21 self)
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This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. We establish the following extraction rule for an extension of GnA # by quantifierfree choice ACqf and analytical axioms # having the form #x # #y ## sx#z # F0 (including also a `non standard' axiom F  which does not hold in the full settheoretic model but in the strongly majorizable functionals): From a proof GnA # +ACqf + # # #u 1 , k 0 #v ## tuk#w 0 A0(u, k, v, w) one can extract a uniform bound # such that #u 1 , k 0 #v ## tuk#w # #ukA0 (u, k, v, w) holds in the full settheoretic type structure. In case n = 2 (resp. n = 3) #uk is a polynomial (resp. an elementary recursive function) in k, u M := #x. max(u0, . . . , ux). In the present paper we show that for n # 2, GnA # +ACqf+F  proves a generalization of the binary Knig's lemma yielding new conservation results since the conclusion of the above rule can be verified in G max(3,n) A # in this case. In a subsequent paper we will show that many important ine#ective analytical principles and theorems can be proved already in G2A # +ACqf+# for suitable #. 1
New Effective Moduli of Uniqueness and Uniform aPriori Estimates for Constants of Strong Unicity by Logical Analysis of Known Proofs in Best Approximation Theory
, 1993
"... Let U and V be complete separable metric spaces, Vu compact in V and G : U IR a continuous function. For a large class of (usually nonconstructive) proofs of uniqueness theorems Vu G(u, v1 ) = inf G(u, v) = G(u, v2) v1 = v2 one can extract an e#ective modulus of uniqueness # by ..."
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Cited by 16 (12 self)
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Let U and V be complete separable metric spaces, Vu compact in V and G : U IR a continuous function. For a large class of (usually nonconstructive) proofs of uniqueness theorems Vu G(u, v1 ) = inf G(u, v) = G(u, v2) v1 = v2 one can extract an e#ective modulus of uniqueness # by logical analysis, i.e.
Elimination of Skolem functions for monotone formulas in analysis
 ARCHIVE FOR MATHEMATICAL LOGIC
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On the Uniform Weak König's Lemma
, 1999
"... The socalled weak König's lemma WKL asserts the existence of an in nite path b in any in nite binary tree (given by a representing function f ). Based on this principle one can formulate subsystems of higherorder arithmetic which allow to carry out very substantial parts of classical mathematics b ..."
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Cited by 10 (5 self)
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The socalled weak König's lemma WKL asserts the existence of an in nite path b in any in nite binary tree (given by a representing function f ). Based on this principle one can formulate subsystems of higherorder arithmetic which allow to carry out very substantial parts of classical mathematics but are 2  conservative over primitive recursive arithmetic PRA (and even weaker fragments of arithmetic). In [10] we established such conservation results relative to nite type extensions PRA of PRA (together with a quanti erfree axiom of choice schema). In this setting one can consider also a uniform version UWKL of WKL which asserts the existence of a functional which selects uniformly in a given in nite binary tree f an in nite path f of that tree. This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in [10] actually can be used to eliminate even this uniform weak König's lemma provided that PRA only has a quanti erfree rule of extensionality QFER instead of the full axioms (E) of extensionality for all nite types. In this paper we show that in the presence of (E), UWKL is much stronger than WKL: whereas WKL remains to be 2 conservative over PRA, PRA + (E)+UWKL contains (and is conservative over) full Peano arithmetic PA.
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].
Intuitionistic Choice and Restricted Classical Logic
, 2000
"... Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic in all finite types together with various forms of the axiom of choice and a numerical omniscience schema (NOS) which implies classical logic for arithmetical formulas. Feferman subsequently observed that the proof theore ..."
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Cited by 7 (1 self)
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Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic in all finite types together with various forms of the axiom of choice and a numerical omniscience schema (NOS) which implies classical logic for arithmetical formulas. Feferman subsequently observed that the proof theoretic strength of such systems can be determined by functional interpretation based on a nonconstructive operator and his wellknown results on the strength of this operator from the 70's. In this note we consider a weaker form LNOS (lesser numerical omniscience schema) of NOS which su#ces to derive the strong form of binary Konig's lemma studied by Coquand/Palmgren and gives rise to a new and mathematically strong semiclassical system which, nevertheless, can proof theoretically be reduced to primitive recursive arithmetic PRA. The proof of this fact relies on functional interpretation and a majorization technique developed in a previous paper. # Basic Research in Computer Science, Centre...
Things that can and things that can't be done in PRA
, 1998
"... It is wellknown by now that large parts of (nonconstructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the BolzanoW ..."
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Cited by 3 (1 self)
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It is wellknown by now that large parts of (nonconstructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the BolzanoWeierstra principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmetical comprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). In this paper
The Computational Strength of Extensions of Weak König's Lemma
, 1998
"... The weak Konig's lemma WKL is of crucial signi cance in the study of on the other hand have a low prooftheoretic and computational strength. In addition to the restriction to binary trees (or equivalently bounded trees), WKL is also `weak' in that the tree predicate is quanti erfree. Whereas in g ..."
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Cited by 1 (0 self)
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The weak Konig's lemma WKL is of crucial signi cance in the study of on the other hand have a low prooftheoretic and computational strength. In addition to the restriction to binary trees (or equivalently bounded trees), WKL is also `weak' in that the tree predicate is quanti erfree. Whereas in general the computational and prooftheoretic strength increases when logically more complex trees are allowed, we show that this is not the case for trees which are given by formulas in a class 1 where we allow an arbitrary function quanti er pre x over bounded functions in front of a 1 formula. This results in a schema 1WKL. Another way of looking at WKL is via its equivalence to the principle 8x9y 18z A 0 (x; y; z) ! 9f x:18x; z A 0 (x; fx; z); where A 0 is a quanti erfree formula (x; y; z are natural number variables). We generalize this to 1 formulas as well and allow function quanti ers `9g s' instead of `9y 1', where g s is de ned pointwise. The resulting schema is Basic Research in Computer Science, Centre of the Danish National Research Foundation. called 1 bAC In the absence of functional parameters (so in particular in a second order context), the corresponding versions of 1WKL and 1 bAC turn out to be equivalent to WKL. This changes completely in the presence of functional variables of type 2 where we get proper hierarchies of principles n WKL and . Variables of type 2 however are necessary for a direct representation of analytical objects and { sometimes { for a faithful representation of such objects at all as we will show in a subsequent paper. By a reduction of 1WKL and 1 bAC to a nonstandard axiom F (introduced in a previous paper) and a new elimination result for F relative to various fragment of arithmetic in...