### Relativizing complexity classes with sparse oraclesRelativizing complexity classes with sparse oracles

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 Author Long, Timothy J. ♦ Selman, Alan L. Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©1986 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Abstract Baker, Gill, and Solovay constructed sparse sets $\textit{A}$ and $\textit{B}$ such that $P(\textit{A})$ ≠ $NP(\textit{A})$ and $NP(\textit{B})$ ≠ $co-NP(\textit{B}).$ In contrast to their results, we prove that P = NP if and only if for every tally language $\textit{T},$ $P(\textit{T})$ = NP( $\textit{T}),$ and that NP = co-NP if and only if for every tally language $\textit{T},$ $NP(\textit{T})$ = $co-NP(\textit{T}).$ We show that the polynomial hierarchy collapses if and only if there is a sparse set $\textit{S}$ such that the polynomial hierarchy relative to $\textit{S}$ collapses. Similar results are obtained for several other complexity classes. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 1986-05-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 33 Issue Number 3 Page Count 10 Starting Page 618 Ending Page 627

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Source: ACM Digital Library