Splittings, robustness and structure of complete sets

Abstract

We investigate the structure of EXP and NEXP complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in te set. To address this question for a given set A, we construct a sparse set S, and ask whether A−S is still hard. It turns out, that for most of the reductions considered and for an arbitrary given sparseness condition, there is a single subexponential time computable set S that meets this condition, such that A−S is not hard for any A. On the other hand we show that for any polynomial time computable sparse set 5, the set A−S remains hard. In the second part of the paper we address the question whether the information that is evidently abundantly present can be used to produce two disjoint complete sets from a single complete set A, i.e. the question whether exponential-time complete sets are milotic. It turns out that the many-one complete sets are, yet there is strong evidence that other complete sets may not be. In particular we show the existence of a 3-tt complete set that can not be split into two many-one complete sets. Finally we show a complexity theoretic counterpart to Sacks' splitting theorem, i.e. we show that any many-one complete set for EXP can be split into two incomplete sets.