A c.e. real that cannot be sw-computed by any \(\Omega\) number. (English) Zbl 1113.03035

Barmpalias, George; Lewis, Andrew E. M.

Notre Dame J. Formal Logic 47, No. 2, 197-209 (2006).

Summary: The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky, and Weinberger on applications of computability to differential geometry. We study the sw-degrees of c.e. reals and construct a c.e. real which has no random c.e. real (i.e., \(\Omega\) number) sw-above it.

MSC:

03D30	Other degrees and reducibilities in computability and recursion theory
03D15	Complexity of computation (including implicit computational complexity)
03D80	Applications of computability and recursion theory

Keywords:

strong weak truth table reducibility; measure of relative randomness; sw-degrees of c.e. reals