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{
\fancyhead[L,LO]{
\textit{\small Math 387 Group problems, Wednesday Week 13}
}
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}

\begin{document}
\thispagestyle{firststyle}
\bigskip

\begin{prob} 
  Here is the Euclidean algorithm for finding the greatest common divisor of two
  integers:



  \textbf{Input:} $(u,v)\in\Z^{2}$.
  \medskip

  \textbf{def} \textsc{euclid}$(u,v)$: 

  \hspace{5mm} If $v=0$, then
  
  \hspace{10mm} return $u$

  \hspace{5mm} else

  \hspace{10mm} return \textsc{euclid}$(v,u \% v)$.
  \bigskip

  Here, $u \% v$ denotes the remainder of~$u$ upon division by~$v$.

  Find the lexicographically smallest input $(u,v)$ with $u>v\geq0$ such that the
  Euclidean algorithm takes $n$ division steps.  Start with the cases
  $n=0,1,2,3,4$.
\end{prob}
\bigskip

\textsc{Review} 
\bigskip

\begin{prob}
  Is $P$ closed under concatenation?  Prove by describing a
  decider, or give a counterexample.
\end{prob}

\begin{prob}
  Is $P$ closed under the Kleene star operator? Prove by describing a decider,
  or give a counterexample.
\end{prob}

\begin{prob}
  What is the statement of Savitch's theorem?
\end{prob}

\begin{prob}
  Is $\text{NP}\subseteq \text{PSPACE}$?
\end{prob}

\begin{prob}
  \begin{enumerate}
    \item What is the complexity class L?  NL?
    \item Prove that $\text{NL}\subsetneq \text{PSPACE}$.
  \end{enumerate}
\end{prob}

\begin{prob}
  Prove that $\text{NP}\neq \text{TIME}(n)$.
  Generalize this result.
\end{prob}

\begin{prob}
  What is the Church-Turing thesis?
\end{prob}

\begin{prob}
  Find a match in the following instance of the Post Correspondence Problem:
  \[
    \left\{
      \left[\frac{a}{baa}\right],
      \left[\frac{ab}{aa}\right],
      \left[\frac{bba}{bb}\right]
    \right\}.
  \]
\end{prob}
\end{document}
