Trivial Reals

Solovay showed that there are noncomputable reals such that H( n) H(1n) + O(1), where H is prefix-free Kolmogorov complexity. Such H-trivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an H-trivial real. We also analyze various computability-theoretic properties of the H-trivial reals, showing for example that no H-trivial real can compute the halting problem. Therefore, our construction of an H-trivial computably enumerable set is an easy, injury-free construction of an incomplete computably enumerable set. Finally, we relate the H-trivials to other classes of "highly nonrandom" reals that have been previously studied. Some of the material in this paper was presented by Downey in his talk Algorithmic Randomness and Computability at the 8th Asian Logic Meeting in Chongqing, China. A preliminary version of this peared as an extended abstract in Brattka, Schr

• Type: Journal
• Year: 2002
• Where: ENTCS
• Authors: Rodney G. Downey, Denis R. Hirschfeldt, André Nies, Frank Stephan
• Keywords: ENTCS 2002, H-trivial Real, Noncomputable Reals, Such H-trivial Reals