Notre Dame Journal of Formal Logic

Complexity of the $r$-query Tautologies in the Presence of a Generic Oracle

Toshio Suzuki


Extending techniques of Dowd and those of Poizat, we study computational complexity of $r\mathit{TAUT}[A]$ in the case when $A$ is a generic oracle, where $r$ is a positive integer, and $r\mathit{TAUT}[A]$ denotes the collection of all $r$-query tautologies with respect to an oracle $A$. We introduce the notion of ceiling-generic oracles, as a generalization of Dowd's notion of $t$-generic oracles to arbitrary finitely testable arithmetical predicates. We study how existence of ceiling-generic oracles affects behavior of a generic oracle, by which we show that $\{ X: co\mathit{NP}[X] $ is not a subset of $\mathit{NP}[r\mathit{TAUT}[X]]\}$ is comeager in the Cantor space. Moreover, using ceiling-generic oracles, we present an alternative proof of the fact (Dowd) that the class of all $t$-generic oracles has Lebesgue measure zero.

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Notre Dame J. Formal Logic, Volume 41, Number 2 (2000), 142-151.

First available in Project Euclid: 25 November 2002

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Zentralblatt MATH identifier

Primary: 68Q15: Complexity classes (hierarchies, relations among complexity classes, etc.) [See also 03D15, 68Q17, 68Q19]
Secondary: 03D15: Complexity of computation (including implicit computational complexity) [See also 68Q15, 68Q17]

computational complexity generic oracle random oracle t-generic oracle


Suzuki, Toshio. Complexity of the $r$-query Tautologies in the Presence of a Generic Oracle. Notre Dame J. Formal Logic 41 (2000), no. 2, 142--151. doi:10.1305/ndjfl/1038234608.

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  • Ambos-Spies, K., H. Fleischhack, and H. Hagen, "$p$-generic sets", pp. 58–68 in Automata, Languages and Programming (Antwerp, 1984), edited by J. Paredaens, Springer-Verlag, Berlin, 1984.
  • Baker, T., J. Gill, and R. Solovay, "Relativizations of the $\mathcal{P}=$?$\mathcal{N\!P}$ question", SIAM Journal on Computing, vol. 4 (1975), pp. 431–42.
  • Balcázar, J. L., R. V. Book, and U. Sch öning, "On bounded query machines", Theoretical Computer Science, vol. 40 (1985), pp. 237–43.
  • Balcázar, J. L., J. Díaz, and J. Gabarró, Structural Complexity. I, Springer-Verlag, Berlin, 2d edition, 1995.
  • Bennett, C. H., and J. Gill, "Relative to a random oracle ${A}$, P$^{A}\not=$ NP$^{A}\not=$ co-NP$^{A}$ with probability $1$", SIAM Journal on Computing, vol. 10 (1981), pp. 96–113.
  • Blum, M., and R. Impagliazzo, "Generic oracles and oracle classes", pp. 118–26 in 28th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, 1987.
  • Book, R. V., "Bounded query machines: on N"P and PSPACE, Theoretical Computer Science, vol. 15 (1981), pp. 27–39.
  • Cohen, P., "The independence of the continuum hypothesis", Proceedings of the National Academy of the Sciences of the United States of America, vol. 50 (1963), pp. 1143–48.
  • Cohen, P. J., "The independence of the continuum hypothesis. I"I, Proceedings of the National Academy of the Sciences of the United States of America, vol. 51 (1964), pp. 105–10.
  • Dowd, M., "Forcing and the ${P}$-hierarchy", Technical Report LCSR-TR-35, Rutgers University, New Brunswick, 1982.
  • Dowd, M., "Generic oracles, uniform machines, and codes", Information and Computation, vol. 96 (1992), pp. 65–76.
  • Feferman, S., "Some applications of the notions of forcing and generic sets", Fundamenta Mathematicae, vol. 56 (1964/1965), pp. 325–45.
  • Hinman, P. G., "Some applications of forcing to hierarchy problems in arithmetic", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 341–52.
  • Jockusch, C. G., Jr., "Degrees of generic sets", pp. 110–39 in Recursion Theory: Its Generalisation and Applications (Proceedings of the Logic Colloquium, Leeds, 1979), edited by F. R. Drake and S. S. Wainer, Cambridge University Press, Cambridge, 1980.
  • Kunen, K., Set Theory. An Introduction to Independence Proofs, North-Holland, Amsterdam, 1980.
  • Mehlhorn, K., "On the size of sets of computable functions", pp. 190–96 in 14th Annual IEEE Symposium on Switching and Automata Theory, IEEE Computer Society, Northridge, 1973.
  • Odifreddi, P., "Forcing and reducibilities", The Journal of Symbolic Logic, vol. 48 (1983), pp. 288–310.
  • Poizat, B., "$\mathcal{Q}=\mathcal{N\!Q}$?", The Journal of Symbolic Logic, vol. 51 (1986), pp. 22–32.
  • Rogers, H., Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967.
  • Suzuki, T., "Recognizing tautology by a deterministic algorithm whose while-loop's execution time is bounded by forcing", Kobe Journal of Mathematics, vol. 15 (1998), pp. 91–102.
  • Tanaka, H., and M. Kudoh, "On relativized probabilistic polynomial time algorithms", Journal of the Mathematical Society of Japan, vol. 49 (1997), pp. 15–30.