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28
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 39 (24 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
Recursively Enumerable Reals and Chaitin Ω Numbers
"... A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from b ..."
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Cited by 39 (3 self)
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A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal selfdelimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number islike. In this paper we show that the converse implication is true as well: any Ωlike real in the unit interval is the halting probability of a universal selfdelimiting Turing machine.
Randomness Space
 AUTOMATA, LANGUAGES AND PROGRAMMING, PROCEEDINGS OF THE 25TH INTERNATIONAL COLLOQUIUM, ICALP’98
, 1998
"... MartinLöf defined infinite random sequences over a finite alphabet via randomness tests which describe sets having measure zero in a constructive sense. In this paper this concept is generalized to separable topological spaces with a measure, following a suggestion of Zvonkin and Levin. After st ..."
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Cited by 24 (4 self)
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MartinLöf defined infinite random sequences over a finite alphabet via randomness tests which describe sets having measure zero in a constructive sense. In this paper this concept is generalized to separable topological spaces with a measure, following a suggestion of Zvonkin and Levin. After studying basic results and constructions for such randomness spaces a general invariance result is proved which gives conditions under which a function between randomness spaces preserves randomness. This corrects and extends a result by Schnorr. Calude and Jürgensen proved that the randomness notion for real numbers obtained by considering their bary representations is independent from the base b. We use our invariance result to show that this notion is identical with the notion which one obtains by viewing the real number space directly as a randomness space. Furthermore, arithmetic properties of random real numbers are derived, for example that every computable analytic function pres...
The Effective Riemann Mapping Theorem
, 1997
"... The main results of the paper are two effective versions of the Riemann mapping theorem. The first, uniform version is based on the constructive proof of the Riemann mapping theorem by Bishop and Bridges and formulated in the computability framework developed by Kreitz and Weihrauch. It states which ..."
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Cited by 12 (1 self)
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The main results of the paper are two effective versions of the Riemann mapping theorem. The first, uniform version is based on the constructive proof of the Riemann mapping theorem by Bishop and Bridges and formulated in the computability framework developed by Kreitz and Weihrauch. It states which topological information precisely one needs about a nonempty, proper, open, connected, and simply connected subset of the complex plane in order to compute a description of a holomorphic bijection from this set onto the unit disk, and vice versa, which topological information about the set can be obtained from a description of a holomorphic bijection. The second version, which is derived from the #rst by considering the sets and the functions with computable descriptions, characterizes the subsets of the complex plane for which there exists a computable holomorphic bijection onto the unit disk. This solves a problem posed by PourEl and Richards. We also show that this class of sets is strictl...
The emperor’s new recursiveness: The epigraph of the exponential function in two models of computability
 In Masami Ito and Teruo Imaoka, editors, Words, Languages & Combinatorics III
, 2003
"... In his book “The Emperor’s New Mind ” Roger Penrose implicitly defines some criteria which should be met by a reasonable notion of recursiveness for subsets of Euclidean space. We discuss two such notions with regard to Penrose’s criteria: one originated from computable analysis, and the one introdu ..."
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Cited by 12 (0 self)
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In his book “The Emperor’s New Mind ” Roger Penrose implicitly defines some criteria which should be met by a reasonable notion of recursiveness for subsets of Euclidean space. We discuss two such notions with regard to Penrose’s criteria: one originated from computable analysis, and the one introduced by Blum, Shub and Smale. 1
Effective Metric Spaces and Representations of the Reals
 THEORETICAL COMPUTER SCIENCE, 2002 (CCA'99 SPECIAL ISSUE
, 2000
"... Based on standard notions of classical recursion theory, a natural model of approximate computability for partial functions between effective metric spaces is presented. It generalizes the KoFriedman approach to computability of real functions by means of oracle Turing machines and avoids the ex ..."
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Cited by 9 (1 self)
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Based on standard notions of classical recursion theory, a natural model of approximate computability for partial functions between effective metric spaces is presented. It generalizes the KoFriedman approach to computability of real functions by means of oracle Turing machines and avoids the explicit use of representations. The topological arithmetical hierarchy is introduced and shown to be strict if the underlying space contains an effectively discrete sequence. The domains of computable functions are just the \Pi 2 sets of that hierarchy if the space admits a finitary stratification. Finally, this framework is used to investigate and characterize the standard representations of the real numbers. They are just those functions from the name space onto the reals which have both computable extensions and inversions that are computable as relations.