Limitations of Efficient Reducibility to the Kolmogorov Random Strings

We show the following results for polynomial-time reducibility to RC, the set of Kolmogorov random strings. 1. If P 6= NP, then SAT does not dtt-reduce to RC. 2. If PH does not collapse, then SAT does not n--reduce to RC for any α < 1. 3. If PH does not collapse, then SAT does not n-T-reduce to RC for any α < 2 . 4. There is a problem in E that does not dtt-reduce to RC. 5. There is a problem in E that does not n--reduce to RC, for any α < 1. 6. There is a problem in E that does not n-T-reduce to RC, for any α < 2 . These results hold for both the plain and prefix-free variants of Kolmogorov complexity and are also independent of the choice of the universal machine.