International Symposium on Mathematical Foundations of Computer Science

MFCS 2015: Mathematical Foundations of Computer Science 2015 pp 459-471 | Cite as

On Probabilistic Space-Bounded Machines with Multiple Access to Random Tape

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9235)

Abstract

We investigate probabilistic space-bounded Turing machines that are allowed to make multiple passes over the random tape. As our main contribution, we establish a connection between derandomization of such probabilistic space-bounded classes to the derandomization of probabilistic time-bounded classes. Our main result is the following.

  • For some integer \(k>0\), if all the languages accepted by bounded-error randomized log-space machines that use \(O(\log n \log ^{(k+3)} n)\) random bits and make \(O(\log ^{(k)} n)\) passes over the random tape is in deterministic polynomial-time, then \(\mathrm{ BPTIME}(n) \subseteq \mathrm{ DTIME}(2^{o(n)})\). Here \(\log ^{(k)} n\) denotes \(\log \) function applied k times iteratively.

This result can be interpreted as follows: If we restrict the number of random bits to \(O(\log n)\) for the above randomized machines, then the corresponding set of languages is trivially known to be in P. Further, it can be shown that (proof is given in the main body of the paper) if we instead restrict the number of passes to only O(1) for the above randomized machines, then the set of languages accepted is in P. Thus our result implies that any non-trivial extension of these simulations will lead to a non-trivial and unknown derandomization of \(\mathrm{ BPTIME}(n)\). Motivated by this result, we further investigate the power of multi-pass, probabilistic space-bounded machines and establish additional results.

References

  1. 1.
    Adleman, L.M.: Two theorems on random polynomial time. In: Proceedings of the 19th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 75–83 (1978)Google Scholar
  2. 2.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press, Cambridge (2009)CrossRefMATHGoogle Scholar
  3. 3.
    Borodin, A., Cook, S., Pippenger, N.: Parallel computation for well-endowed rings and space-bounded probabilistic machines. Inf. Control 58(1–3), 113–136 (1983)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chung, K.M., Reingold, O., Vadhan, S.: S-T connectivity on digraphs with a known stationary distribution. ACM Trans. Algorithms 7(3), 30 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    David, M., Nguyen, P., Papakonstantinou, P.A., Sidiropoulos, A.: Computationally Limited Randomness. In: Proceedings of the Innovations in Theoretical Computer Science (ITCS) (2011)Google Scholar
  6. 6.
    David, M., Papakonstantinou, P.A., Sidiropoulos, A.: How strong is Nisan’s pseudo-random generator? Inf. Process. Lett. 111(16), 804–808 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fortnow, L., Klivans, A.R.: Linear advice for randomized logarithmic space. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 469–476. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  8. 8.
    Gill, J.: Computational complexity of probabilistic Turing machines. SIAM J. Comput. 6(4), 675–695 (1977)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gutfreund, D., Viola, E.: Fooling parity tests with parity gates. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 381–392. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  10. 10.
    Hopcroft, J.H., Paul, W.J., Valiant, L.G.: On Time Versus Space. J. ACM 24(2), 332–337 (1977)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Impagliazzo, R., Nisan, N., Wigderson, A.: Pseudorandomness for network algorithms. In: Proceedings of the 26th ACM Symposium on Theory of Computing (STOC), pp. 356–364 (1994)Google Scholar
  12. 12.
    Karakostas, G., Lipton, R.J., Viglas, A.: On the complexity of intersecting finite state automata and NL versus NP. Theor. Comput. Sci. 302, 257–274 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Karpinski, M., Verbeek, R.: There is no polynomial deterministic space simulation of probabilistic space with a two-way random-tape generator. Inf. Control 67(1985), 158–162 (1985)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lipton, R.J., Viglas, A.: Non-uniform depth of polynomial time and space simulations. In: Lingas, A., Nilsson, B.J. (eds.) FCT 2003. LNCS, vol. 2751, pp. 311–320. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  15. 15.
    Nisan, N.: Pseudorandom generators for space-bounded computation. Combinatorica 12(4), 449–461 (1992)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Nisan, N.: On read once vs. multiple access to randomness in logspace. Theor. Comput. Sci. 107(1), 135–144 (1993)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Raz, R., Reingold, O.: On recycling the randomness of states in space bounded computation. In: Proceedings of the 31st ACM Symposium on Theory of Computing (STOC), pp. 159–168 (1999)Google Scholar
  18. 18.
    Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), 1–24 (2008)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Reingold, O., Trevisan, L., Vadhan, S.: Pseudorandom walks on regular digraphs and the \({\rm RL}\) vs. \({\rm L}\) problem. In: Proceedings of the 38th ACM Symposium on Theory of Computing (STOC), p. 457 (2006)Google Scholar
  20. 20.
    Saks, M.: Randomization and derandomization in space-bounded computation. In: Proceedings of the 11th IEEE Conference on Computational Complexity (1996)Google Scholar
  21. 21.
    Saks, M., Zhou, S.: \({\rm BPSPACE}(S) \subseteq {\rm DSPACE}(S^{3/2})\). J. Comput. Syst. Sci. 403, 376–403 (1999)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Santhanam, R., van Melkebeek, D.: Holographic proofs and derandomization. SIAM J. Comput. 35(1), 59–90 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Debasis Mandal
    • 1
  • A. Pavan
    • 1
  • N. V. Vinodchandran
    • 2
  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA
  2. 2.Department of Computer Science and EngineeringUniversity of Nebraska-LincolnLincolnUSA

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