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.

We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.

Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefix-free set of binary strings.

We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real if A is a computably enumerable pre x-free set of strings such that = .

Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability.