# Hausdorff reductions to sparse sets and to sets of high information content

## Abstract

We investigate the complexity of sets that have a rich internal structure and at the same time are reducible to sets of either low or very high information content. In particular, we show that every length-decreasing or word-decreasing self-reducible set that reduces to some sparse set via a non-monotone variant of the Hausdorff reducibility is low for *Δ*_{2}^{p}.

Measuring the information content of a set by the space-bounded Kolmogorov complexity of its characteristic sequence, we further investigate the (non-uniform) complexity of sets *A* in EXPSPACE/poly that reduce to some set having very high information content. Specifically, we show that if the reducibility used has a certain property, called “reliability,” then *A* in fact is reducible to a sparse set (under the same reducibility). As a consequence of our results, the existence of hard sets (under “reliable” reducibilities) of very high information content is unlikely for various complexity classes as for example NP, PP, and PSPACE.

## Keywords

Kolmogorov Complexity High Information Content Reliable Reduction Conjunctive Reducibility Polynomial Time Computable Function## Preview

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