On the Construction of Effective Random Sets

Abstract

We give a direct and rather simple construction of Martin-Lof random and rec-random sets with certain additional properties. First, reviewing the result of Gacs and Ku?era, given any set X we construct a Martin-Lof random set R from which X can be decoded effectively. Second, by essentially the same construction we obtain a Martin-Lof random set R that is computably enumerable selfreducible. Alternatively, using the observation that a set is computably enumerable selfreducible if and only if its associated real is computably enumerable, the existence of such a set R follows from the known fact that every Chaitin ? real is Martin-Lof random and computably enumerable. Third, by a variant of the basic construction we obtain a rec-random set that is weak truthtable autoreducible.The mentioned results on self- and autoreducibility complement work of Ebert, Merkle, and Vollmer [7,8,9], from which it follows that no Martin-Lof random set is Turing-autoreducible and that no rec-random set is truth-table autoreducible.