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\fancyhead[L,LO]{
\textit{\small Math 387 Group problems, Wednesday Week 10}
}
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\begin{document}
\thispagestyle{firststyle}
\bigskip

\begin{prob} Review.
  \begin{enumerate}
    \item Suppose that~$f\colon\N\to\N$ where~$f(n)$ is at least~$O(\log(n))$. What
      does it mean to say \emph{space constructible}?
    \item Name a function~$f(n)$ that is in $O(n)$ but not $o(n)$.  Show the relevant calculation.
    \item State the space hierarchy theorem.
    \item Where is space constructibility used in the proof of the space
      hierarchy theorem?
    \item For any two rational numbers $0\leq a<b$, show that~$n^{a}\in
      o(n^{b})$ and hence, via the space hierarchy theorem,
      $\SPACE(n^{a})\subsetneq\SPACE(n^{b})$.  Show the relevant calculation.
      (Here, we are using the fact that~$n^{c}$ is space constructible for any
      rational number~$c$.)
    \item Savitch's theorem says $\NL\subseteq\SPACE(\log^{2}(n))$.  Use this to
      prove that~$\NL\subsetneq\PSPACE$. Show the relevant calculation.
    \item State the time hierarchy theorem.
    \item Prove that~$\text{TIME}(n^{2})\subsetneq \text{TIME}(n^{3})$. Show the relevant calculation.
  \end{enumerate}
\end{prob}

\begin{prob}
  \begin{enumerate}
    \item We know that PATH is NL-complete, i.e., everything in NL is log space
	  reducible to PATH.  Prove that $\overline{\text{PATH}}$ is coNL-complete.
    \item We have seen that $\overline{\text{PATH}}$ is in NL.  Explain why
      this implies NL $=$ coNL.
  \end{enumerate}
\end{prob}

\begin{prob}
  \begin{enumerate}
    \item Prove that $\text{TIME}(2^{n})=\text{TIME}(2^{n+1})$.
    \item Prove that $\text{TIME}(2^{n})\subsetneq\text{TIME}(2^{2n})$.
  \end{enumerate}
\end{prob}

\begin{prob}
  \begin{enumerate}
    \item Explain why $A_{\text{NFA}}$ is in NL.
    \item Prove that $A_{\text{NFA}}$ is NL-complete by giving a reduction from
      PATH.
  \end{enumerate}
\end{prob}
\end{document}

