Power from Random Strings

We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and non-uniform reductions. These sets are provably not complete under the usual many-one reductions. Let RC, RKt, RKS, RKT be the sets of strings x having complexity at least |x|/2, according to the usual Kolmogorov complexity measure C, Levin’s time-bounded Kolmogorov complexity Kt [Lev84], a space-bounded Kolmogorov measure KS, and a new time-bounded Kolmogorov complexity measure KT, respectively. Our main results are: 1. RKS and RKt are complete for PSPACE and EXP, respectively, under P/poly-truth-table reductions. Similar results hold for other classes with PSPACE-robust Turing complete sets. 2. EXP = NPKt . 3. PSPACE = ZPPKS ⊆ PC . 4. The Discrete Log, Factoring, and several lattice problems are solvable in BPPKT . Our hardness result for PSPACE gives rise to fairly natural problems that are complete for PSPACE under ≤T reductions, but not under ≤log m reductions. Our techniques also allow us to show that all computably enumerable sets are reducible to RC via P/poly-truth-table reductions. This provides the first “efficient” reduction of the halting problem to RC.