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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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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.
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
Extracting Computable Bounds (and Algorithms) from Classical Existence Proofs: Girard Domains Enable Us to Go Beyond Local Compactness
"... In classical mathematics, the existence of a solution is often proven indirectly, nonconstructively, without an efficient method for constructing the corresponding object. In many cases, we can extract an algorithm from a classical proof: e.g., when an object is (nonconstructively) proven to be un ..."
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In classical mathematics, the existence of a solution is often proven indirectly, nonconstructively, without an efficient method for constructing the corresponding object. In many cases, we can extract an algorithm from a classical proof: e.g., when an object is (nonconstructively) proven to be unique in a locally compact space (or when there are two such objects with a known lower bound on the distance between them). In many other practical situations, a (seemingly) natural formalization of the corresponding practical problem leads to a noncompact set. In this paper, we show that often, in such situations, we can extract efficient algorithms from classical proofs – if we explicitly take into account (implicit) knowledge about the situation. Specifically, we show that if we consistently apply Heisenberg’s operationalism idea and define objects in terms of directly measurable quantities, then we get a Girarddomain type representation in which a natural topology is, in effect, compact – and thus, uniqueness implies computability. 1
Cantor’s Paradise Regained: Constructive Mathematics from Brouwer to Kolmogorov to
"... Abstract. Constructive mathematics, mathematics in which the existence of an object means that that we can actually construct this object, started as a heavily restricted version of mathematics, a version in which many commonly used mathematical techniques (like the Law of Excluded Middle) were forb ..."
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Abstract. Constructive mathematics, mathematics in which the existence of an object means that that we can actually construct this object, started as a heavily restricted version of mathematics, a version in which many commonly used mathematical techniques (like the Law of Excluded Middle) were forbidden to maintain constructivity. Eventually, it turned out that not only constructive mathematics is not a weakened version of the classical one – as it was originally perceived – but that, vice versa, classical mathematics can be viewed as a particular (thus, weaker) case of the constructive one. Crucial results in this direction were obtained by M. Gelfond in the 1970s. In this paper, we mention the history of these results, and show how these results affected constructive mathematics, how they led to new algorithms, and how they affected the current activity in logic programmingrelated research.
BAR RECURSION AND PRODUCTS OF SELECTION FUNCTIONS
"... Abstract. We show how two iterated products of selection functions can both be used in conjunction with system T to interpret, via the dialectica interpretation and modified realizability, full classical analysis. We also show that one iterated product is equivalent over system T to Spector’s bar re ..."
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Abstract. We show how two iterated products of selection functions can both be used in conjunction with system T to interpret, via the dialectica interpretation and modified realizability, full classical analysis. We also show that one iterated product is equivalent over system T to Spector’s bar recursion, whereas the other is Tequivalent to modified bar recursion. Modified bar recursion itself is shown to arise directly from the iteration of a different binary product of ‘skewed ’ selection functions. Iterations of the dependent binary products are also considered but in all cases are shown to be Tequivalent to the iteration of the simple products. §1. Introduction. Gödel’s [13] socalled dialectica interpretation reduces the consistency of Peano arithmetic to the consistency of a quantifierfree calculus of functionals T. In order to extend Gödel’s interpretation to full classical analysis PA ω + CA, Spector [18] made use of the fact that PA ω + CA can be embedded, via the negative translation, into HA ω + ACN + DNS. Here PA ω
Adding Constraints – A (Seemingly Counterintuitive but) Useful Heuristic in Solving Difficult Problems
"... Abstract. Intuitively, the more constraints we impose on a problem, the more difficult it is to solve it. However, in practice, difficulttosolve problems sometimes get solved when we impose additional constraints and thus, make the problems seemingly more complex. In this methodological paper, we ..."
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Abstract. Intuitively, the more constraints we impose on a problem, the more difficult it is to solve it. However, in practice, difficulttosolve problems sometimes get solved when we impose additional constraints and thus, make the problems seemingly more complex. In this methodological paper, we explain this seemingly counterintuitive phenomenon, and we show that, dues to this explanation, additional constraints can serve as a useful heuristic in solving difficult problems.