
\documentstyle[12pt,twoside,amssymb]{article}
\input{defs-6045}

\newcommand{\defi}{\stackrel{\mbox{\tiny def}}{=}}

\newcommand{\poly}{{\Bbb P}}
\newcommand{\npoly}{{\Bbb NP}}
\newcommand{\Lsp}{{\Bbb L}}
\newcommand{\NL}{{\Bbb NL}}
\newcommand{\pspace}{{\Bbb PSPACE}}
\newcommand{\npspace}{{\Bbb NPSPACE}}

\newcommand{\RP}{{\Bbb RP}}
\newcommand{\nat}{{\Bbb N}}

\newcommand{\fa}{\forall}
\newcommand{\sig}{\Sigma}
\newcommand{\arr}{\rightarrow}
\newcommand{\pless}{\leq_{P}}
\newcommand{\mless}{\leq_m}
\newcommand{\Lless}{\leq_L}

\newcommand{\spa}{\mbox{SPACE}}
\newcommand{\nspa}{\mbox{NSPACE}}
\newcommand{\ttime}{\mbox{TIME}}
\newcommand{\ntime}{\mbox{NTIME}}
\newcommand{\llog}{\mbox{log}}
\newcommand{\sllog}{\mbox{{\tiny log}}}



\begin{document}

\handout{final-joke}{Final Exam Rough Draft}{May 8, 1992}


\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
This is the final exam we were thinking of giving you.
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%5



\begin{enumerate} 
\item Answer True or False:



\begin{enumerate}

\item $\ntime(n^{4}\lg n)$ is a proper superset of $\ttime(n^{4})$.

\item $\npoly \cap co-\npoly \subseteq P/\mbox{poly}$

\item If $\poly = \npoly$, then there is a cubic-time factoring algorithm.

\item If $\poly \not= \npoly$, then there is no cubic-time factoring algorithm
running on a four-counter machine.

\item If $\poly \not= \npoly$, then there is no cubic-time factoring algorithm
running on an Euclidean four-counter machine.

\item There is an algorithm deciding GSAT which runs in time {\em exactly\/}
$n^{4}+8n+\lceil{\sqrt{n \lg n \lg \lg n}\rceil}$ on a three-stack, two-counter
machine.

\item $\npoly \cap co-\npoly \subset P/\lg \lg n$, but this is not provable
by elementary methods.

\item (For Extra Credit) $\mbox{AC}^{0} \not= \poly$ 

\item (For Extra Credit) GSAT is decidable.
\end{enumerate}




\item {\bf Language Examples}

Give examples of the following languages, or proof-sketches that none exist.

\begin{enumerate}

\item Give a language \(L\) such that \(L\) can be decided in time $O(n^{\lg
n \lg \lg n })$ on a three-stack two-counter machine, but cannot be decided in
this time bound on a two-stack, three-counter machine.

\item Give a nonempty class of languages which is a proper subset of \(\poly\) which is
closed under the following operations:
\begin{itemize}
\item $\cup$
\item $\cap$
\item $\neg$
\item $*$
\item Intersection with a language in $\ntime(n^{4})$
\item Direct homomorphism
\item Inverse homomorphism
\item Polynomial-time filtering
\item Context-free Isomorphism
\end{itemize}

\item A nondeterministic fourty-two-counter machine which accepts GSAT in
time $O(n^{8})$.



\item A language \(L\) such that \(L\) is context-free but no homomorphic
image of \(L\) can be decided in $\lg n \lg \lg n$ on a CRCW 2n-processor
PRAM.

\end{enumerate}



\item {\bf Language Classification}

Classify the following languages as Regular, Context-Free,
Co-Context-Free, Linear-time Decidable, $\poly$, $\npoly$-complete,
$\ttime(2^{n})$-complete, doable in time $\lg n \lg \lg n$ on a CRCW
2n-processor PRAM,  doable in time {\em exactly\/} $n^{4}+8n+\lceil{\sqrt{n
\lg n \lg \lg n}\rceil}$ on a three-stack, two-counter machine, recursive, r.e.,
co-r.e., $\Sigma^{0}_{i}$, and hyperarithmetical.  List all
that apply.

\begin{enumerate}

\item The set of Turing machines \(M\) which run forever, and on precisely
the steps of the form $2^{n}$, \(M\)'s worktape contains a satisfiable
formula of propositional calculus.

\item The set of context-sensitive grammars which print sets of valid proofs of
second-order $\lambda$-calculus.

\item The set of Turing machines which compute (in time $2^{7^{n^{7}}}$ and
space $4^{n^2}$) reductions from GSAT to HAM, with the property that
\[\begin{array}{rlll} 
|f(x)| & = & 2|x|^{2} & \mbox{ if } |x| \mbox{ prime}, \\
       &   & 3|x|^{3} & \mbox{ if } |x| \mbox{ composite},\\
       &   & 18921 & \mbox{ if } |x| = 1,\\
       &   & 14231 & \mbox{ if } |x| = 0.
\end{array}
\]

\item The set of formulas of arithmetic which follow from precisely one of
$\poly=\npoly$ and $\poly\not=\npoly$.

\item The set of homology groups of 3-manifolds embeddable in 5-dimensional
projective tori.

\item Buchi automata \(A\) accepting precisely the set of (coded) oracles \(B\) which
distinguish $\poly$ and $\npoly$.


\end{enumerate}



\item 
%\rm

Define XTISP to be the class of languages accepted on \(k\)-counter,
\(l\)-tape Turing machines (where the tapes have 2 one-way heads each) in
time $c^{n^\epsilon}$ and space $\lg(n)^{\epsilon}$.  (Note: the same
$\epsilon$ appears in both bounds.)   

Define YTISP to be the class of languages accepted on on \(k\)-counter,
\(l\)-tape Turing machines (where the tapes have 2 one-way heads each) in
time precisely $c^{n^\epsilon}$ and space precisely $\lg(n)^{\epsilon'}$.
where $\epsilon' \leq \sqrt{\epsilon}$.



Find oracles \(A\), \(B\) such that $\mbox{XTISP}^{(A)} = \mbox{YTISP}^{(A)}$
and $\mbox{XTISP}^{(B)} \not= \mbox{YTISP}^{(B)}$.

%\end{enumerate}

%\begin{enumerate}
\item
%\rm

Assume the Poincar\'{e} conjecture and the extended Riemann hypothesis.
Prove that $\ttime(n^{2})$ and \(AC^{40}\) do not coincide.


\end{enumerate}


\end{document}


