On the Turing degrees of minimal index sets

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 MIN ⊕ ∅′′ ≡T ∅′′′, MIN (n) ⊕ ∅(n+2) ≡T ∅(n+4), and that there exists a Kolmogorov numbering ψ satisfying both MINψ ≡tt ∅′′′ and MIN (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.