The instance complexity conjecture

Abstract

This paper is concerned with instance complexity introduced by Ko, Orponen, Schoning, and Watanabe (1986) as a measure of the complexity of individual instances of a decision problem. They conjectured that for every nonrecursive r.e. set the instance complexity is infinitely often at least as high as the Kolmogorov complexity. We refute this conjecture by constructing a nonrecursive r.e. set with instance complexity logarithmic in the Kolmogorov complexity. This bound is optimal up to a constant. In the other extreme, we show that the conjecture can indeed be established for many classes of complete sets. In addition we consider Kolmogorov complexity of initial segments of r.e. sets and show that the well-known upper bound 2 log n is optimal.