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Power from Random Strings

2006; Society for Industrial and Applied Mathematics; Volume: 35; Issue: 6 Linguagem: Inglês

10.1137/050628994

1095-7111

Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, Detlef Ronneburger,

Algorithms and Data Compression

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 nonuniform reductions. These sets are provably not complete under the usual many-one reductions. Let ${{R_{\rm C}}}, {{R_{\rm Kt}}}, {{R_{\rm KS}}}, {{R_{\rm KT}}}$ be the sets of strings x having complexity at least $|x|/2$, according to the usual Kolmogorov complexity measure ${\mbox{\rm C}}$, Levin's time-bounded Kolmogorov complexity ${\mbox{\rm Kt}}$ [L. Levin, Inform. and Control, 61 (1984), pp. 15-37], a space-bounded Kolmogorov measure ${\mbox{\rm KS}}$, and a new time-bounded Kolmogorov complexity measure ${\mbox{\rm KT}}$, respectively. Our main results are as follows: \begin{remunerate} \item ${{R_{\rm KS}}}$ and ${{R_{\rm Kt}}}$ are complete for ${{\rm{PSPACE}}}$ and {\mbox{\rm EXP}}, respectively, under ${\mbox{\rm P/poly}}$-truth-table reductions. Similar results hold for other classes with ${{\rm{PSPACE}}}$-robust Turing complete sets. \item ${\mbox{\rm EXP}} = {\mbox{\rm NP}}^{{{R_{\rm Kt}}}}.$ \item ${{\rm{PSPACE}}} = {\mbox{\rm ZPP}}^{{{R_{\rm KS}}}} \subseteq {\mbox{\rm P}}^{{{R_{\rm C}}}}$. \item The Discrete Log, Factoring, and several lattice problems are solvable in ${\mbox{\rm BPP}}^{{{R_{\rm KT}}}}$. \end{remunerate} Our hardness result for ${{\rm{PSPACE}}}$ gives rise to fairly natural problems that are complete for ${{\rm{PSPACE}}}$ under ${\mbox{$\leq^{\rm p}_{\rm T}$}}$ reductions, but not under ${\mbox{$\leq^{\rm log}_{\rm m}$}}$ reductions. Our techniques also allow us to show that all computably enumerable sets are reducible to ${{R_{\rm C}}}$ via ${\mbox{\rm P/poly}}$-truth-table reductions. This provides the first "efficient" reduction of the halting problem to ${{R_{\rm C}}}$.