Notre Dame Journal of Formal Logic

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

Toshio Suzuki

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 25 November 2002

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1038234608

Digital Object Identifier
doi:10.1305/ndjfl/1038234608

Mathematical Reviews number (MathSciNet)
MR1932226

Zentralblatt MATH identifier
1015.03045

Subjects
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]

Keywords
computational complexity generic oracle random oracle t-generic oracle

Citation

Suzuki, Toshio. Complexity of the -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. https://projecteuclid.org/euclid.ndjfl/1038234608


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