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On randomized versus deterministic computation

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Automata, Languages and Programming (ICALP 1993)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 700))

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Abstract

In contrast to deterministic or nondeterministic computation, it is a fundamental open problem in randomized computation how to separate different randomized time classes (at this point we do not even know how to separate linear randomized time from O(n log n) randomized time) or how to compare them relative to corresponding deterministic time classes. In another words we are far from understanding the power of random coin tosses in the computation, and the possible ways of simulating them deterministically.

In this paper we study the relative power of linear and polynomial randomized time compared with exponential deterministic time. Surprisingly, we are able to construct an oracle A such that exponential time (with or without the oracle A) is simulated by linear time Las Vegas algorithms using the oracle A. For Las Vegas polynomial time (ZPP) this will mean the following equalities of the time classes:

$$ZPP^A = EXPTIME^A = EXPTIME\left( { = DTIME\left( {2^{poly} } \right)} \right).$$

Furthermore, for all the sets

$$M \subseteq \Sigma ^ * :M \leqslant _{UR} \bar A \Leftrightarrow M \in EXPTIME$$

(≤ ur being unfaithful polynomial random reduction, c.f. [Jo 90]).

Thus A is ≤ ur complete for EXPTIME, but interestingly not NP-hard under (deterministic) polynomial reduction unless EXPTIME=NEXPTIME. We are also able to prove, for the first time, that randomized reductions are exponentially more powerful than deterministic or nondeterministic ones (cf. [AM 77]). Moreover, a set B is constructed such that Monte Carlo polynomial time (BPP) under the oracle B is exponentially more powerful than deterministic time with nondeterministic oracles, more precisely: BPP B=δ2 EXPTIME B=δ2 EXPTIME(DTIME(2poly(NTIME(n)).

This strengthens considerably a result of Stockmeyer [St 85] about the polynomial time hierarchy that for some decidable oracle B, BPP B & δ2PB. Under our oracle BPP B is exponentially more powerful than δ 2PB, and B does not add any power to δ 2 EXPTIME.

Supported in part by the Leibniz Center for Research in Computer Science, by the DFG Grant KA 673/4-1 and by the ESPRIT BR Grant 7097.

Part of the research was done while visiting the International Computer Science Institute, Berkeley, California.

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References

  1. Adleman, L., Two Theorems on Random Polynomial Time, Proc. 19th IEEE FOCS, 1972, pp. 75–83.

    Google Scholar 

  2. Adleman, L., Manders, K., Reducibility, Randomness, and Intractibility, Proc. 9th ACM STOC, 1977, pp. 151–163.

    Google Scholar 

  3. Allender, E., Beigel, R., Hertrampf, U., Homer, S., Almost-Everywhere Complexity Hierarchies for Nondeterministic Time, Manuscript, 1992; A preliminary version has appeared in Proc. STACS '90, LNCS 415, Springer-Verlag, 1990, pp. 1–11.

    Google Scholar 

  4. Bennett, Ch. H., Gill, J., Relative to a Random Oracle A, P A ≠ N P A ≠ co-N P A with Probability 1, SIAM J. on Computing 10, 1981, pp. 96–113.

    Google Scholar 

  5. Fortnow, L., Personal Communication, 1992.

    Google Scholar 

  6. Fortnow, L., Sipser, M., Probabilistic Computation and Linear Time, Proc. 21st ACM STOC, 1989, pp. 148–166.

    Google Scholar 

  7. Freivalds, R., Fast Probabilistic Algorithms, Proc. MFCS'79, LNCS 75, 1979, Springer-Verlag, pp. 57–69.

    Google Scholar 

  8. Johnson, P.S., A Catalog of Complexity Classes, in Handbook of Theoretical Computer Science, Vol. A., Algorithms and Complexity, Elsevier-MIT Press, 1990, pp. 69–161.

    Google Scholar 

  9. Kannan, R., Circuit-Size Lower Bounds and Non-reducibility to Sparse Sets, Information and Control 55, 1982, pp. 40–46.

    Google Scholar 

  10. Karpinski, M., Verbeek, R., On the Monte Carlo Space Constructible Functions and Separation Results for Probabilistic Complexity Classes, Information and Computation 75, 1987, pp. 178–189.

    Google Scholar 

  11. Karpinski, M., Verbeek, R., Randomness, Provability, and the Separation of Monte Carlo Time and Space, LNCS 270, Springer-Verlag, 1988, pp. 189–207.

    Google Scholar 

  12. Rackoff, C, Relational Questions Involving Probabilistic Algorithms, J. ACM 29, 1982, pp. 261–268.

    Google Scholar 

  13. Sipser, M., A Complexity Theoretic Approach to Randomness, Proc. 15th ACM STOC, 1983, pp. 330–335.

    Google Scholar 

  14. Stockmeyer, L., On Approximation Algorithms for #P, SIAM J. Comput. 14, 1985, pp. 849–861.

    Google Scholar 

  15. Wilson, C., Relativized Circuit Complexity, 24th IEEE FOCS, 1983, pp. 329–334.

    Google Scholar 

  16. Yao, A. C., Coherent Functions and Program Checkers, Proc. 22nd ACM STOC, 1990, pp. 84–94.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dept. of Computer Science, University of Bonn, 5300, Bonn 1

    Marek Karpinski

  2. International Computer Science Institute, Berkeley, California

    Marek Karpinski

  3. Dept. of Computer Science, Fern Universität Hagen, 5800, Hagen 1

    Rutger Verbeek

Authors
  1. Marek Karpinski
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  2. Rutger Verbeek
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Andrzej Lingas Rolf Karlsson Svante Carlsson

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© 1993 Springer-Verlag Berlin Heidelberg

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Karpinski, M., Verbeek, R. (1993). On randomized versus deterministic computation. In: Lingas, A., Karlsson, R., Carlsson, S. (eds) Automata, Languages and Programming. ICALP 1993. Lecture Notes in Computer Science, vol 700. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56939-1_75

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  • DOI: https://doi.org/10.1007/3-540-56939-1_75

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56939-8

  • Online ISBN: 978-3-540-47826-3

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