Languages to diagonalize against advice classes

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.