@MISC{Downey_trivialreals, author = {Rod G. Downey and et al.}, title = {Trivial Reals }, year = {} }

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Abstract

Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 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.