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Mathematical Logic and Applications pp 189–202Cite as

Computational complexity of languages counting random oracles

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1388))

Abstract

The nature of the recursive real μ(UB(x)) in Structural Complexity Theory is discussed, in which μ is Lebesgue measure of random oracles and UB(x) denotes the set of every oracle Z such that the polynomial-time computable oracle-dependent language Bz accepts x. To formalize this argument we introduced the class Lℬ of any language Lm(B) expressing a lower bound of its measure of random oracles on an oracle-dependent language Bz in a relativized class ℬz, and we investigated the nature of this class. The class Cℬ of languages was further introduced to examine the circumstances of Lℬ, each language L of which indicates a lower bound of the cardinality of elements in L by exponentially-many searchings. Originating from the P-hierarchy, the related LP- and CP-hierarchies was obtainable and their structures were also studied. Our best result for the lower and upper bounds of the computational complexity of LP is that PP ≤ pm LP⊆PSPACE. More from an angle of relativization, a natural relativization of each Lℬ and Cℬ is treated, and the probability-one separations of the first levels of the LP- and CP-hierarchies are shown.

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References

  1. K. AMBOS-SPIES, Randomness, relativizations, and polynomial reducibility, Structure in Complexity Theory in Lecture Notes in Computer Science, 223(1986), 23–34.

    Article  MathSciNet  MATH  Google Scholar 

  2. K. AMBOS-SPIES, Minimal pairs for polynomial time reducibilities, Computer Theory and Logic in Lecture Notes in Computer Science, 270 (1987), 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  3. T. BAKER, J. GILL, and R. SOLOVAY, Relativizations of the P=?NP question, SIAM Journal on Computing, 4(1975), 431–442.

    Article  MathSciNet  MATH  Google Scholar 

  4. T.P. BAKER and A.L. SELMAN, A second step toward the polynomial hierarchy, Theoretical Computer Science, 8(1979), 177–187.

    Article  MathSciNet  MATH  Google Scholar 

  5. C.H. BENNETT and J. GILL, Relative to a random oracle A, PA≠NPA≠co-NPA with probability 1, SIAM Journal on Computing, 10(1981), 96–113.

    Article  MathSciNet  Google Scholar 

  6. R.V. BOOK, T.J. LONG, and A.L. SELMAN, Quantitative relativizations of complexity classes, SIAM Journal on Computing, 13(1984), 461–487.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. CAI, Probability one separation of the Boolean hierarchy, STACS 87 in Lecture Notes in Computer Science, 247(1987), 148–158.

    Article  MathSciNet  Google Scholar 

  8. M. FURST, J.B. SAXE, and M. SIPSER, Parity, circuites, and the polynomial-time hierarchy, Mathematical Systems Theory, 17(1984), 13–27.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. GILL, Computational complexity of probabilistic Turing machines, SIAM Journal on Computing, 6(1977), 675–695.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. HARTMANIS, Some observations about NP complete sets, Fundamentals of Computation Theory in Lecture Notes in Computer Science, 278(1987), 185–196.

    Article  MATH  Google Scholar 

  11. R.E. LADNER, On the structure of polynomial time reducibility, Journal of the Association for Computing Machinery, 22(1975), 153–171.

    Article  MathSciNet  MATH  Google Scholar 

  12. U.SCHÖNING, Complexity and Structure in Lecture Notes in Computer Science, 211(1986).

    Google Scholar 

  13. M.SIPSER, Borel sets and circuit complexity, Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 1983, 61–69.

    Google Scholar 

  14. L.J. STOCKMEYER, The polynomial-time hierarchy, Theoretical Computer Science, 3(1977), 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  15. L. STOCKMEYER, Classifying the computational complexity of problems, The Journal of Symbolic Logic, 52(1987), 1–43.

    Article  MathSciNet  MATH  Google Scholar 

  16. L.G. VALIANT, The complexity of computing the permanent, Theoretical Computer Science, 8(1979), 189–202.

    Article  MathSciNet  MATH  Google Scholar 

  17. A.C.YAO, Separating the polynomial-time hierarchy by oracles, Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science, 1985, 1–10.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Rikkyo University, Nishi-ikebukuro, 171, Tokyo, Japan

    Tomoyuki Yamakami

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  1. Tomoyuki Yamakami
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Juichi Shinoda Tosiyuki Tugué Theodore A. Slaman

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© 1989 Springer-Verlag

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Yamakami, T. (1989). Computational complexity of languages counting random oracles. In: Shinoda, J., Tugué, T., Slaman, T.A. (eds) Mathematical Logic and Applications. Lecture Notes in Mathematics, vol 1388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083671

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  • DOI: https://doi.org/10.1007/BFb0083671

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51527-2

  • Online ISBN: 978-3-540-48220-8

  • eBook Packages: Springer Book Archive

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