The notions of a low and high sets were introduced, I think by Soare, in the context of the dense structure of degrees of those sets which are neither r.e. neither recursive. My question is:

Why are all examples of "natural" decision problems (the spectral gap problem for example) reducible to the halting language and none are low or high?

  • $\begingroup$ Can you find a natural decision problem reducible to the halting language and neither Turing complete nor recursive? $\endgroup$
    – 喻 良
    May 15, 2022 at 2:43
  • $\begingroup$ @喻良 I don't think I could $\endgroup$
    – H.C Manu
    May 16, 2022 at 8:04


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