Lower bounds for reducibility to the Kolmogorov random strings

Abstract

We show the following results for polynomial-time reducibility to R C, the set of Kolmogorov random strings. 1 If P ≠ NP, then SAT does not dtt-reduce to RC. 2 If PH does not collapse, then SAT does not nα-tt-reduce to RC for any α < 1. 3 If PH does not collapse, then SAT does not nα-T-reduce to R C for any α < 1/2. 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 α-tt-reduce to RC , for any α< 1. 6. There is a problem in E that does not nα-T-reduce to RC, for any α < 1/2. 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. © 2010 Springer-Verlag Berlin Heidelberg.