Journal of Computer and System Sciences
Turing degrees of reals of positive effective packing dimension
Information Processing Letters
Time-bounded incompressibility of compressible strings and sequences
Information Processing Letters
Analogues of Chaitin's Omega in the computably enumerable sets
Information Processing Letters
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The way in which way Kolmogorov complexity and instance complexity affect properties of recursively enumerable (r.e.) sets is studied. The well-known $2\log n$ upper bound on the Kolmogorov complexity of initial segments of r. e. sets is shown to be optimal, and the Turing degrees of r. e. sets which attain this bound are characterized. The main part of the paper is concerned with instance complexity, introduced by Ko, Orponen, Schöning, and Watanabe in 1986, as a measure of the complexity of individual instances of a decision problem. They conjectured that for every r. e. nonrecursive set, the instance complexity is infinitely often at least as high as the Kolmogorov complexity. The conjecture is refuted by constructing an r. e. nonrecursive set with instance complexity logarithmic in the Kolmogorov complexity. This bound is optimal up to an additive constant. In the other extreme, the conjecture is established for many classes of complete sets, such as weak-truth-table-complete (wtt-complete) and Q-complete sets. However, there is a Turing-complete set for which it fails.