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\begin{document}

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   \noindent
   \begin{center}
   \framebox{
      \vbox{\vspace{2mm}
    \hbox to 6.28in { {\em CS 262: Computational Complexity}
                        \hfill {\em Instructor: Sandy Irani} }
       \vspace{4mm}
       \hbox to 6.28in { \hfill {\Large Homework 1} \hfill
               }
       \vspace{2mm}
       \hbox to 6.28in { \hfill  {\bf Due: January  15, 2010} }
      \vspace{2mm}}
   }
   \end{center}
   \vspace*{4mm}

\begin{enumerate}
\item
The definition of $\sc{3SAT}$ allows for clauses to have fewer than
three literals. We will define a resticted version of {\sc 3SAT} called
{\sc EXACT-3SAT} in which the literals in each clause must be
distinct and every clause must have exactly three literals.
(Recall that a {\em literal} is a variable or the negation of a
variable). Prove that {\sc 3SAT} reduces to {\sc EXACT-3SAT}.

\item The {\em Kleene closure} of a set $L$ is the set
$L^* = \{ x_1x_2\cdots x_k: k \ge 0; x_1,x_2,\ldots,x_k \in L \}$.
Notice that the notation $\Sigma^*$ is compatible with this notation.
Show that {\sc PSPACE} is closed under Kleene closure.

\item A complexity class {\cal C} is said to be
{\em closed under reductions} if whenever $L$ reduces
to $L'$ and $L' \in {\cal C}$, then $L \in {\cal C}$.
Prove that {\sc P} and {\sc PSPACE} are closed under 
reductions. Is ${\sc TIME}(n^2)$ closed under
reductions? Justify your answer.

\item Suppose that a Turing Machine can insert and delete symbols
from its tape in addition to overwriting them.
\begin{enumerate}
\item Define carefully the transition function and computation
of such machines.
\item Show that such a machine can be simulated by an ordinary
Turing Machine with a quadratic loss of efficiency.
\end{enumerate}
\end{enumerate}

\end{document}
