On pseudorandomness and resource-bounded measure

Abstract

In this paper we extend a key result of Nisan and Wigderson (J. Comput. System Sci. 49 (1994) 149–167) to the nondeterministic setting: for all $\text{α>0}$ we show that if there is a language in $\text{E}\text{=}\text{DTIME}\text{(2}{}^{\mathrm{<mtext>O</mtext><mtext>(n)</mtext>}}\text{)}$ that is hard to approximate by nondeterministic circuits of size $\text{2}{}^{\mathrm{\alpha n}}$, then there is a pseudorandom generator that can be used to derandomize $\text{BP}\text{·}\text{NP}$ (in symbols, $\text{BP}\text{·}\text{NP}\text{=}\text{NP}$). By applying this extension we are able to answer some open questions in Lutz (Theory Comput. Systems 30 (1997) 429–442) regarding the derandomization of the classes $\text{BP}\text{·}\text{Σ}{}^{\mathrm{<mtext>P</mtext>}}{}_{\mathrm{k}}$ and $\text{BP}\text{·}\text{Θ}{}^{\mathrm{<mtext>P</mtext>}}{}_{\mathrm{k}}$ under plausible measure theoretic assumptions. As a consequence, if $\text{Θ}{}^{\mathrm{<mtext>P</mtext>}}{}_2$ does not have p-measure $\text{0}$, then $\text{AM}\text{∩}\text{coAM}$ is low for $\text{Θ}{}^{\mathrm{<mtext>P</mtext>}}{}_2$. Thus, in this case, the graph isomorphism problem is low for $\text{Θ}{}^{\mathrm{<mtext>P</mtext>}}{}_2$. By using the Nisan–Wigderson design of a pseudorandom generator we unconditionally show the inclusion $\text{MA}\text{⊆}\text{ZPP}{}^{\mathrm{<mtext>NP</mtext>}}$ and that $\text{MA}\text{∩}\text{coMA}$ is low for $\text{ZPP}{}^{\mathrm{<mtext>NP</mtext>}}$.