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computational complexity

, Volume 14, Issue 4, pp 341–361 | Cite as

Languages to diagonalize against advice classes

  • Chris Pollett
Original Paper

Abstract.

Variants of Kannan’s Theorem are given where the circuits of the original theorem are replaced by arbitrary recursively presentable classes of languages that use advice strings and satisfy certain mild conditions. Let poly k denote those functions in O(n k ). These variants imply that \(\mathsf{DTIME}(n^{k})^{\mathsf{NE}}/\mathsf{poly} _{k}\) does not contain \(\mathsf{P}^{\mathsf{NE}},\;\mathsf{DTIME}\left(2^{n^{k'}}\right)/\mathsf{poly}_{k}\) does not contain \(\mathsf{EXP},\;\mathsf{SPACE}\left(n^{k'}\right)/\mathsf{poly}_{k}\) does not contain PSPACE, uniform TC 0/poly k does not contain CH, and uniform ACC/poly k does not contain ModPH. Consequences for selective sets are also obtained. In particular, it is shown that \(\mathsf{R}_{T}^{\mathsf{DTIME}(n^{k})}(\mathsf{NP}\hbox{-}\mathsf{sel})\) does not contain \(\mathsf{P}^{\mathsf{NE}},\;\mathsf{R}_{T}^{\mathsf{DTIME}(n^{k})}(\mathsf{P}\hbox{-}\mathsf{sel})\) does not contain EXP, and \(\mathsf{R}_{T}^{\mathsf{DTIME}(n^{k})}(\mathsf{L}\hbox{-}\mathsf{sel})\) does not contain PSPACE. Finally, a circuit size hierarchy theorem is established.

Keywords.

Advice classes EXP NEXP NE CH ModPH p-selective complexity classes nonuniform computation 

Subject classification.

68Q15 03D15 

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Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.Department of Computer ScienceSan Jose State UniversitySan JoseUSA

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