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\begin{document}

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   \noindent
   \begin{center}
   \framebox{
      \vbox{\vspace{2mm}
    \hbox to 6.28in { {\bf CS 262: Computational Complexity
                        \hfill Due: February 5, 2010} }
       \vspace{4mm}
       \hbox to 6.28in { \hfill {\Large Homework 3} \hfill
               }
       \vspace{2mm}
       \hbox to 6.28in { {\it Instructor: Sandy Irani} \hfill }
      \vspace{2mm}}
   }
   \end{center}
   \vspace*{4mm}

\begin{enumerate}
\item
Suppose $L_1$ and $L_2$ are in $\NP \cap \coNP$. Then show that $L_1 \oplus L_2$ is in
$\NP \cap \coNP$, where
$L_1 \oplus L_2 = \{ x \mid x \mbox{ is in exactly one of } L_1 \mbox{~and~} L_2 \}$.

\item Prove that if {\em every} unary language in $\NP$ is also in \classP, then
\EXP = \NEXP. Recall that a language is unary if and only if it is a subset of $1^*$.

\item Say that class $C_1$ is {\em superior} to $C_2$ if there is a language $L_1$
in $C_1$ such that for every language $L_2 \in C_2$ and every $n$ sufficiently large, there 
is an input whose length is between $n$ and $n^2$ on which $L_1$ and $L_2$ differ. That is,
there is a string $x$ whose length is between $n$ and $n^2$ such that either $x \in L_1$ and
$x \not\in L_2$ or $x \not\in L_1$ and $x \in L_2$.
\begin{enumerate}
\item Is $\bf{DTIME}(n^4)$ superior to $\bf{DTIME}(n)$?
\item Why does our proof of the Non-deterministic Time Hierarchy not prove that
$\bf{NTIME}(n^{1.5})$ is superior to $\bf{NTIME}(n)$?

\end{enumerate}
\end{enumerate}

\end{document}
