Limits on the computational power of random strings

How powerful is the set of random strings? What can one say about a set A that is efficiently reducible to R, the set of Kolmogorov-random strings? We present the first upper bound on the class of computable sets in P^R and NP^R. The two most widely-studied notions of Kolmogorov complexity are the ''plain'' complexity C(x) and ''prefix'' complexity K(x); this gives rise to two common ways to define the set of random strings ''R'': R"C and R"K. (Of course, each different choice of universal Turing machine U in the definition of C and K yields another variant R"C"""U or R"K"""U.) Previous work on the power of ''R'' (for any of these variants) has shown:*BPP@?{A:A=<"t"t^pR}. *PSPACE@?P^R. *NEXP@?NP^R. Since these inclusions hold irrespective of low-level details of how ''R'' is defined, and since BPP,PSPACE and NEXP are all in @D"1^0 (the class of decidable languages), we have, e.g.: NEXP@?@D"1^0@?@?"UNP^R^"^K^"^"^"^U. Our main contribution is to present the first upper bounds on the complexity of sets that are efficiently reducible to R"K"""U. We show:*BPP@?@D"1^0@?@?"U{A:A=<"t"t^pR"K"""U}@?PSPACE. *NEXP@?@D"1^0@?@?"UNP^R^"^K^"^"^"^U@?EXPSPACE. Hence, in particular, PSPACE is sandwiched between the class of sets polynomial-time Turing- and truth-table-reducible to R. As a side-product, we obtain new insight into the limits of techniques for derandomization from uniform hardness assumptions.

[1]  Ronald V. Book.  On Languages Reducible to Algorithmically Random Languages , 1994, SIAM J. Comput..

[2]  Andrej Muchnik,et al.  Kolmogorov entropy in the context of computability theory , 2002, Theor. Comput. Sci..

[3]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[4]  John M. Hitchcock.  Lower Bounds for Reducibility to the Kolmogorov Random Strings , 2010, CiE.

[5]  Lance Fortnow,et al.  BPP has subexponential time simulations unlessEXPTIME has publishable proofs , 2005, computational complexity.

[6]  Ronald V. Book,et al.  On the Robustness of ALMOST-R , 1996, RAIRO Theor. Informatics Appl..

[7]  Martin Kummer.  On the Complexity of Random Strings (Extended Abstract) , 1996, STACS.

[8]  Jack H. Lutz,et al.  An observation on probability versus randomness with applications to complexity classes , 2005, Mathematical systems theory.

[9]  Adam R. Day.  On the computational power of random strings , 2009, Ann. Pure Appl. Log..

[10]  Jin-Yi Cai,et al.  Circuit minimization problem , 2000, STOC '00.

[11]  Eric Allender,et al.  The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory , 2009, J. Comput. Syst. Sci..

[12]  Eric Allender,et al.  Limits on the computational power of random strings , 2011, Inf. Comput..

[13]  Luca Trevisan,et al.  Notions of Reducibility between Cryptographic Primitives , 2004, TCC.

[14]  William I. Gasarch,et al.  Book Review: An introduction to Kolmogorov Complexity and its Applications Second Edition, 1997 by Ming Li and Paul Vitanyi (Springer (Graduate Text Series)) , 1997, SIGACT News.

[15]  Eric Allender,et al.  What can be efficiently reduced to the Kolmogorov-random strings? , 2006, Ann. Pure Appl. Log..

[16]  Avi Wigderson,et al.  Randomness vs. time: de-randomization under a uniform assumption , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[17]  Salil P. Vadhan,et al.  Limitations of Hardness vs. Randomness under Uniform Reductions , 2008, APPROX-RANDOM.

[18]  Lance Fortnow,et al.  Derandomizing from Random Strings , 2009, 2010 IEEE 25th Annual Conference on Computational Complexity.

[19]  Eric Allender,et al.  Exposition of the Muchnik-Positselsky Construction of a Prefix Free Entropy Function that is not Complete under Truth-Table Reductions , 2010, Electron. Colloquium Comput. Complex..

[20]  Avi Wigderson,et al.  Randomness vs Time: Derandomization under a Uniform Assumption , 2001, J. Comput. Syst. Sci..

[21]  Luca Trevisan,et al.  Pseudorandomness and Average-Case Complexity Via Uniform Reductions , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[22]  Noam Nisan,et al.  Hardness vs Randomness , 1994, J. Comput. Syst. Sci..

[23]  Eric Allender,et al.  Power from Random Strings , 2006, SIAM J. Comput..