Index Sets and Universal Numberings

Abstract

This paper studies the Turing degrees of various properties defined for universal numberings, that is, for numberings which list all partial-recursive functions. In particular properties relating to the domain of the corresponding functions are investigated like the set DEQ of all pairs of indices of functions with the same domain, the set DMIN of all minimal indices of sets and DMIN^* of all indices which are minimal with respect to equality of the domain modulo finitely many differences. A partial solution to a question of Schaefer is obtained by showing that for every universal numbering with the Kolmogorov property, the set DMIN^* is Turing equivalent to the double jump of the halting problem. Furthermore, it is shown that the join of DEQ and the halting problem is Turing equivalent to the jump of the halting problem and that there are numberings for which DEQ itself has 1-generic Turing degree.

This research was supported in part by NUS grant numbers R252-000-212-112 (all authors), R252-000-308-112 (S. Jain and F. Stephan) and R146-000-114-112 (F. Stephan). Missing proofs can be found in Technical Report TRA3/09 of the School of Computing at the National University of Singapore.