Abstract:
We consider sets of strings with high Kolmogorov complexity, mainly in resource-bounded settings but also in the traditional recursion-theoretic sense. We present efficient reductions, showing that these sets are hard and complete for various complexity classes. In particular, in addition to the usual Kolmogorov complexity measure K, we consider the time-bounded Kolmogorov complexity measure KT that was introduced by Allender, as well as a space-bounded measure KS, and Levin's time-bounded Kolmogorov complexity Kt. Let R_K, R_KT, R_KS, R_Kt be the sets of strings x having complexity at least |x|/2, according to each of these measures. Our main results are: o R_KS and R_Kt are complete for PSPACE and EXP, respectively, under P/poly-truth-table reductions. o EXP = NP^R_Kt. o PSPACE = ZPP^R_KS, which is contained in P^R_K. o The Discrete Log is in BPP^R_KT. Our hardness results for EXP and PSPACE rely on nonrelativizing proof techniques. Our techniques also allow us to show that all recursively-enumerable sets are reducible to RK via P/poly-truth-table reductions. Our hardness result for PSPACE gives rise to fairly natural problems that are complete for PSPACE under poly-time Turing reductions, but not under logspace-many-one reductions. In spite of the EXP- and PSPACE-completeness of R_Kt and R_KS, it remains unknown if either of these problems is in logspace.

Abstract:
We consider sets of strings with high Kolmogorov complexity, mainly in resource-bounded settings but also in the traditional recursion-theoretic sense. We present efficient reductions, showing that these sets are hard and complete for various complexity classes. In particular, in addition to the usual Kolmogorov complexity measure K, we consider the time-bounded Kolmogorov complexity measure KT that was introduced by Allender, as well as a space-bounded measure KS, and Levin's time-bounded Kolmogorov complexity Kt. Let R_K, R_KT, R_KS, R_Kt be the sets of strings x having complexity at least |x|/2, according to each of these measures. Our main results are: o R_KS and R_Kt are complete for PSPACE and EXP, respectively, under P/poly-truth-table reductions. o EXP = NP^R_Kt. o PSPACE = ZPP^R_KS, which is contained in P^R_K. o The Discrete Log is in BPP^R_KT. Our hardness results for EXP and PSPACE rely on nonrelativizing proof techniques. Our techniques also allow us to show that all recursively-enumerable sets are reducible to RK via P/poly-truth-table reductions. Our hardness result for PSPACE gives rise to fairly natural problems that are complete for PSPACE under poly-time Turing reductions, but not under logspace-many-one reductions. In spite of the EXP- and PSPACE-completeness of R_Kt and R_KS, it remains unknown if either of these problems is in logspace.