Index Sets and Universal Numberings

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.

[1]  Richard M. Friedberg,et al.  Three theorems on recursive enumeration. I. Decomposition. II. Maximal set. III. Enumeration without duplication , 1958, Journal of Symbolic Logic.

[2]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[3]  Marcus Schaefer.  A guided tour of minimal indices and shortest descriptions , 1998, Arch. Math. Log..

[4]  H. Rice.  Classes of recursively enumerable sets and their decision problems , 1953 .

[5]  James C. Owings,et al.  A cardinality version of Beigel's nonspeedup theorem , 1989, Journal of Symbolic Logic.

[6]  Jason Teutsch.  Noncomputable Spectral Sets , 2007, ArXiv.

[7]  P. Odifreddi.  Classical recursion theory , 1989 .

[8]  George Barmpalias,et al.  Post's Programme for the Ershov Hierarchy , 2007, J. Log. Comput..

[9]  Valentina S. Harizanov,et al.  Frequency Computations and the Cardinality Theorem , 1992, J. Symb. Log..

[10]  Lance Fortnow,et al.  Enumerations of the Kolmogorov function , 2006, Journal of Symbolic Logic.

[11]  Stuart A. Kurtz,et al.  Extremes in the Degrees of Inferability , 1994, Ann. Pure Appl. Log..

[12]  Martin Kummer.  A Proof of Beigel's Cardinality Conjecture , 1992, J. Symb. Log..

[13]  John Case,et al.  Control structures in hypothesis spaces: the influence on learning , 2002, Theor. Comput. Sci..

[14]  Jason Teutsch.  On the Turing degrees of minimal index sets , 2007, Ann. Pure Appl. Log..

[15]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[16]  Wolfgang Merkle,et al.  Kolmogorov Complexity and the Recursion Theorem , 2006, STACS.

[17]  Martin Kummer,et al.  Cuppability of Simple and Hypersimple Sets , 2007, Notre Dame J. Formal Log..

[18]  William I. Gasarch,et al.  Book Review: An introduction to Kolmogorov Complexity and its Applications Second Edition, 1997 by Ming Li and Paul Vitanyi (Springer (Graduate Text Series)) , 1997, SIGACT News.

[19]  Masahiro Kumabe,et al.  Degrees of generic sets , 1996 .

[20]  Albert R. Meyer.  Program Size in Restricted Programming Languages , 1972, Inf. Control..