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  1.  78
    Effective Choice and Boundedness Principles in Computable Analysis.Vasco Brattka & Guido Gherardi - 2011 - Bulletin of Symbolic Logic 17 (1):73-117.
    In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice (...)
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  2.  14
    Weihrauch Degrees, Omniscience Principles and Weak Computability.Vasco Brattka & Guido Gherardi - 2011 - Journal of Symbolic Logic 76 (1):143 - 176.
    In this paper we study a reducibility that has been introduced by Klaus Weihrauch or, more precisely, a natural extension for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice. It turns out that parallelization is a closure operator for this semi-lattice and that the parallelized Weihrauch degrees even form a lattice into which the Medvedev lattice and the Turing degrees can be embedded. The (...)
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  3.  21
    Effective Borel Measurability and Reducibility of Functions.Vasco Brattka - 2005 - Mathematical Logic Quarterly 51 (1):19-44.
    The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. (...)
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  4.  11
    The Bolzano–Weierstrass Theorem is the Jump of Weak Kőnig’s Lemma.Vasco Brattka, Guido Gherardi & Alberto Marcone - 2012 - Annals of Pure and Applied Logic 163 (6):623-655.
  5.  15
    Closed Choice and a Uniform Low Basis Theorem.Vasco Brattka, Matthew de Brecht & Arno Pauly - 2012 - Annals of Pure and Applied Logic 163 (8):986-1008.
  6.  7
    Completion of Choice.Vasco Brattka & Guido Gherardi - 2021 - Annals of Pure and Applied Logic 172 (3):102914.
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  7.  4
    On the Uniform Computational Content of the Baire Category Theorem.Vasco Brattka, Matthew Hendtlass & Alexander P. Kreuzer - 2018 - Notre Dame Journal of Formal Logic 59 (4):605-636.
    We study the uniform computational content of different versions of the Baire category theorem in the Weihrauch lattice. The Baire category theorem can be seen as a pigeonhole principle that states that a complete metric space cannot be decomposed into countably many nowhere dense pieces. The Baire category theorem is an illuminating example of a theorem that can be used to demonstrate that one classical theorem can have several different computational interpretations. For one, we distinguish two different logical versions of (...)
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  8.  10
    Connected Choice and the Brouwer Fixed Point Theorem.Vasco Brattka, Stéphane Le Roux, Joseph S. Miller & Arno Pauly - 2019 - Journal of Mathematical Logic 19 (1):1950004.
    We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than (...)
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  9.  16
    Borel Complexity and Computability of the Hahn–Banach Theorem.Vasco Brattka - 2008 - Archive for Mathematical Logic 46 (7-8):547-564.
    The classical Hahn–Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari Downey, Ishihara and others and it is known that the theorem does not admit a general computable version. We prove a new computable version of this theorem without unrolling the classical proof of the (...)
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  10.  28
    Computability and Analysis: The Legacy of Alan Turing.Jeremy Avigad & Vasco Brattka - unknown
    We discuss the legacy of Alan Turing and his impact on computability and analysis.
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  11.  8
    A Computable Version of Banach’s Inverse Mapping Theorem.Vasco Brattka - 2009 - Annals of Pure and Applied Logic 157 (2-3):85-96.
    Given a program of a linear bounded and bijective operator T, does there exist a program for the inverse operator T−1? And if this is the case, does there exist a general algorithm to transfer a program of T into a program of T−1? This is the inversion problem for computable linear operators on Banach spaces in its non-uniform and uniform formulation, respectively. We study this problem from the point of view of computable analysis which is the Turing machine based (...)
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  12.  7
    Isaac Newton Institute, Cambridge, UK July 2–6, 2012.George Barmpalias, Vasco Brattka, Adam Day, Rod Downey, John Hitchcock, Michal Koucký, Andy Lewis, Jack Lutz, André Nies & Alexander Shen - 2013 - Bulletin of Symbolic Logic 19 (1).
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  13.  9
    Foreword.Ulrich Berger, Vasco Brattka, Andrei S. Morozov & Dieter Spreen - 2012 - Annals of Pure and Applied Logic 163 (8):973-974.
  14.  11
    Addendum To: “The Bolzano–Weierstrass Theorem is the Jump of Weak Kőnig's Lemma” [Ann. Pure Appl. Logic 163 623–655]. [REVIEW]Vasco Brattka, Andrea Cettolo, Guido Gherardi, Alberto Marcone & Matthias Schröder - 2017 - Annals of Pure and Applied Logic 168 (8):1605-1608.
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  15.  16
    Computability of Compact Operators on Computable Banach Spaces with Bases.Vasco Brattka & Ruth Dillhage - 2007 - Mathematical Logic Quarterly 53 (4‐5):345-364.
    We develop some parts of the theory of compact operators from the point of view of computable analysis. While computable compact operators on Hilbert spaces are easy to understand, it turns out that these operators on Banach spaces are harder to handle. Classically, the theory of compact operators on Banach spaces is developed with the help of the non-constructive tool of sequential compactness. We demonstrate that a substantial amount of this theory can be developed computably on Banach spaces with computable (...)
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  16.  8
    Computability of Finite-Dimensional Linear Subspaces and Best Approximation.Vasco Brattka & Ruth Dillhage - 2010 - Annals of Pure and Applied Logic 162 (3):182-193.
    We discuss computability properties of the set of elements of best approximation of some point xX by elements of GX in computable Banach spaces X. It turns out that for a general closed set G, given by its distance function, we can only obtain negative information about as a closed set. In the case that G is finite-dimensional, one can compute negative information on as a compact set. This implies that one can compute the point in whenever it is uniquely (...)
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  17.  12
    Editorial: Math. Log. Quart. 5/2008.Vasco Brattka, Hajime Ishihara, Matthias Schröder & Ning Zhong - 2008 - Mathematical Logic Quarterly 54 (5):453-453.
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  18.  12
    Order‐Free Recursion on the Real Numbers.Vasco Brattka - 1997 - Mathematical Logic Quarterly 43 (2):216-234.
  19.  10
    Preface: MLQ ‐ Math. Log. Quart. 4–5/2004.Vasco Brattka, Peter Hertling, Ker-I. Ko & Ning Zhong - 2004 - Mathematical Logic Quarterly 50 (4‐5):327-328.
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  20.  6
    Preface: MLQ ‐ Math. Log. Quart. Supplement 1/2002.Vasco Brattka, Peter Hertling, Mariko Yasugi & Ning Zhong - 2002 - Mathematical Logic Quarterly 48 (S1):III-III.
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  21.  11
    Preface: MLQ - Math. Log. Quart. 4–5/2004.Vasco Brattka, Peter Hertling, Ker-I. Ko & Ning Zhong - 2004 - Mathematical Logic Quarterly 50 (45):327-328.
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  22.  2
    Weihrauch Goes Brouwerian.Vasco Brattka & Guido Gherardi - 2020 - Journal of Symbolic Logic 85 (4):1614-1653.
    We prove that the Weihrauch lattice can be transformed into a Brouwer algebra by the consecutive application of two closure operators in the appropriate order: first completion and then parallelization. The closure operator of completion is a new closure operator that we introduce. It transforms any problem into a total problem on the completion of the respective types, where we allow any value outside of the original domain of the problem. This closure operator is of interest by itself, as it (...)
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  23.  5
    Contents.Dieter Spreen, Hannes Diener & Vasco Brattka - 2014 - In Dieter Spreen, Hannes Diener & Vasco Brattka (eds.), Logic, Computation, Hierarchies. De Gruyter.
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  24.  5
    Index.Dieter Spreen, Hannes Diener & Vasco Brattka - 2014 - In Dieter Spreen, Hannes Diener & Vasco Brattka (eds.), Logic, Computation, Hierarchies. De Gruyter. pp. 411-414.
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  25.  7
    Logic, Computation, Hierarchies.Dieter Spreen, Hannes Diener & Vasco Brattka (eds.) - 2014 - De Gruyter.
  26.  4
    Preface.Dieter Spreen, Hannes Diener & Vasco Brattka - 2014 - In Dieter Spreen, Hannes Diener & Vasco Brattka (eds.), Logic, Computation, Hierarchies. De Gruyter.
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  27.  10
    Approaches to Effective Semi‐Continuity of Real Functions.Xizhong Zheng, Vasco Brattka & Klaus Weihrauch - 1999 - Mathematical Logic Quarterly 45 (4):481-496.
    For semi-continuous real functions we study different computability concepts defined via computability of epigraphs and hypographs. We call a real function f lower semi-computable of type one, if its open hypograph hypo is recursively enumerably open in dom × ℝ; we call f lower semi-computable of type two, if its closed epigraph Epi is recursively enumerably closed in dom × ℝ; we call f lower semi-computable of type three, if Epi is recursively closed in dom × ℝ. We show that (...)
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