68Q05-LevinReduction.tex

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\pmtitle{Levin reduction}
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\begin{document}
If $R_1$ and $R_2$ are search problems and $\mathcal{C}$ is a complexity class then a $\mathcal{C}$ \emph{Levin reduction} of $R_1$ to $R_2$ consists of three functions $g_1, g_2, g_3\in\mathcal{C}$ which satisfy:

\begin{itemize}
\item $g_1$ is a $\mathcal{C}$ Karp reduction of $L(R_1)$ to $L(R_2)$

\item If $R_1(x,y)$ then $R_2(f(x),g(x,y))$

\item If $R_2(f(x),z)$ then $R_1(x,h(x,z))$
\end{itemize}

Note that a $\mathcal{C}$ Cook reduction can be constructed by calculating $f(x)$, using the oracle to find $z$, and then calculating $h(x,z)$.

$\mathcal{P}$ Levin reductions are just called Levin reductions.
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