Trivial Reals

Solovay showed that there are noncomputable reals α such that H(α ↾ n) 6 H(1) + 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 (which means that our construction of an H-trivial computably enumerable set is a particularly easy, injury-free construction of an incomplete c.e. set). Finally, we relate the H-trivials to other classes of “highly nonrandom” reals that have been previously studied. 1 Supported by the Marsden fund of New Zealand. 2 Supported by the Heisenberg program of the Deutsche Forschungsgemeinschaft (DFG), grant no. Ste 967/1–1. c ©2002 Published by Elsevier Science B. V. Downey, Hirschfeldt, Nies, and Stephan