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Unique Existence and Computability in Constructive Reverse Mathematics

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Computation and Logic in the Real World (CiE 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4497))

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Abstract

We introduce, and show the equivalences among, relativized versions of Brouwer’s fan theorem for detachable bars (FAN), weak König lemma with a uniqueness hypothesis (WKL!), and the longest path lemma with a uniqueness hypothesis (LPL!) in the spirit of constructive reverse mathematics. We prove that a computable version of minimum principle: if f is a real valued computable uniformly continuous function with at most one minimum on {0,1}N, then there exists a computable α in {0,1}N such that \(f(\alpha) = \inf f(\{0,1\}^\mathbf{N})\), is equivalent to some computably relativized version of FAN, WKL! and LPL!.

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Author information

Authors and Affiliations

  1. School of Information Science, Japan Advanced Institute of Science and Technology, Nomi, Ishikawa 923-1292, Japan

    Hajime Ishihara

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  1. Hajime Ishihara
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Editors and Affiliations

  1. Dept. of Pure Mathematics, University of Leeds, LS2 9JT, Leeds, UK

    S. Barry Cooper

  2. Institue for Logic, Language and Computation (ILLC), Universiteit van Amsterdam, Plantage Muidergracht 24, 1018, TV Amsterdam, The Netherlands

    Benedikt Löwe

  3. Dipartimento di Scienze Matematiche ed Informatiche “R. Magari”, University of Siena, 53100, Siena, Italy

    Andrea Sorbi

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Ishihara, H. (2007). Unique Existence and Computability in Constructive Reverse Mathematics. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_38

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  • DOI: https://doi.org/10.1007/978-3-540-73001-9_38

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73000-2

  • Online ISBN: 978-3-540-73001-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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