Kolmogorov Complexity of Generalized Computably Enumerable Sets

The main purpose of this work is to characterize computably enumerable (c.e.) sets and generalized c.e. sets according to Kolmogorov complexity hierarchy. Given a set A, we consider A m, the initial segment of length m of the characteristic sequence of A. We show that the characteristic sequence of a 2 x-c.e. set can be random in the sense of Martin-LL of. We show that for x n-c.e. sets A, the uncoditional (conditional , uniform, resp.) Kolmogorov complexity of A m has optimal upper bound (n + 2) log m ((n + 1) log m, (n + 1) log m, resp.). We show that the corresponding optimal upper bounds for n-c.e. sets are the same as those for c.e. sets. We show that there is a strongly eeectively simple set A such that the unconditional (conditional, uniform, resp.) Kolmogorov complexity of A m is as high as what is possible for a computably enumerable set; but not all strongly eeectively simple sets have this property. We show that for any hypersimple set A, the uniform Kolmogorov complexity of A m cannot reach the upper bound of that for computably enumerable sets. In addition, we show that for any GG odel numbering ', the set of all nonrandom strings with respect to ' is Q-complete.