On the Turing degrees of minimal index sets

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We study generalizations of shortest programs as they pertain to Schaefer’s MIN∗ problem. We identify sets of m-minimal and T-minimal indices and characterize their truth-table and Turing degrees. In particular, we show MINm ⊕ ∅ ≡T ∅ , MINT(n) ⊕ ∅(n+2) ≡T ∅(n+4) , and that there exists a Kolmogorov numbering ψ satisfying both MINm ψ ≡tt ∅ and MINT(n) ψ ≡T ∅(n+4) . This Kolmogorov numbering also achieves maximal truth-table degree for other sets of minimal indices. Finally, we show that the set of shortest descriptions, SD, is 2-c.e. but not co-2-c.e. Some open problems are left for the reader. 1 The MIN∗ problem The set of shortest programs is f-MIN := {e : (∀j < e) [ϕj = ϕe]}. In 1972, Meyer demonstrated that f-MIN admits a neat Turing characterization, namely f-MIN ≡T ∅ [10]. In Spring 1990 (according to the best recollection of the author), John Case issued a homework assignment with the following deﬁnition [1]: f-MIN∗ := {e : (∀j &...

Jason Teutsch

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