Unique Existence and Computability in Constructive Reverse Mathematics

  • Authors:

  • Hajime Ishihara

  • Affiliations:

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

  • Venue:
  • CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
  • Year:
  • 2007

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Abstract

We introduce, and show the equivalences among, relativizedversions of Brouwer's fan theorem for detachable bars (FAN), weakKönig lemma with a uniqueness hypothesis (WKL!), and thelongest path lemma with a uniqueness hypothesis (LPL!) in thespirit of constructive reverse mathematics. We prove that acomputable version of minimum principle: if fis a realvalued computable uniformly continuous function with at most oneminimum on {0,1}N, then there exists a computableα in {0,1}Nsuch that $f(\alpha) = \inff(\{0,1\}^\mathbf{N})$, is equivalent to some computablyrelativized version of FAN, WKL! and LPL!.