2013 IEEE 54th Annual Symposium on Foundations of Computer Science (2002)
Nov. 16, 2002 to Nov. 19, 2002
ISSN: 0272-5428
ISBN: 0-7695-1822-2
pp: 669
Detlef Ronneburger , Rutgers University
Harry Buhrman , CWI and University of Amsterdam
Michal Koucký , Rutgers University
Dieter van Melkebeek , University of Wisconsin
Eric Allender , Rutgers University
ABSTRACT
<p>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.</p> <p>Let R<sub>K</sub>; R<sub>Kt</sub>; RK<sub>S</sub>;R<sub>KT</sub> be the sets of strings x having complexity at least {{\left| x \right|} \mathord{\left/ {\vphantom {{\left| x \right|} 2}} \right. \kern-\nulldelimiterspace} 2}, according to the usual Kolmogorov complexity measure K, Levin?s time-bounded Kolmogorov complexity Kt [27], a space-bounded Kolmogorov measure KS, and the time-bounded Kolmogorov complexity measure KT that was introduced in [4], respectively.</p> <p>Our main results are: <li>1. R<sub>KS</sub> and R<sub>Kt</sub> are complete for PSPACE and EXP, respectively, under P/poly-truth-table reductions.</li> <li>2. EXP = NP<sup>R<sub>Kt</sub></sup>.</li> <li>3. PSPACE = ZPP^{R_{ks} } \subseteq P^{R_k }.</li> <li>4. The Discrete Log, Factoring, and several lattice problems are solvable in BPP<sup>R<sub>KT</sub></sup>.</li></p>
INDEX TERMS
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CITATION
Detlef Ronneburger, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, Eric Allender, "Power from Random Strings", 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, vol. 00, no. , pp. 669, 2002, doi:10.1109/SFCS.2002.1181992