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                {\Large \bf Math 441, Computability and Logic}\\[.2in]
        {\large \bf Final exam, Spring 2000}\\[.2in]
\end{centering}\vskip .25in \vskip .25in
A GENERAL REMARK: QUOTE ANY THEOREM THAT YOU ARE MAKING USE OF.\\ \vskip .25in \vskip .25in
\begin{centering}
        {\large \bf Part I\\[.2in]}
\end{centering}


ANSWER 4 QUESTIONS, GIVE A SHORT EXPLANATION TO YOUR ANSWER.

 \vskip .25in

1) Define the notion of a context-free language. Is it true that every regular language is a
context-free language ? \vskip .25in

2) Define the notion of a push-down automaton.\vskip .25in

3) Prove or refute: If $L$ is a context-free language then the number of equivalence classes of
$\approx_L$ is finite. \vskip .25in

4) Prove or refute: If $L$ and $K$ are regular languages then $LK\cap KL$ is a context-free
language.
%3) Explain why the language $L=\{w\in \{a,b\}^* \vert\ w={a^n}{b^{2n}}$ for some $n\geq 0 \}$ is
%context-free.

\vskip .25in

5) Prove or refute: there is a push-down automaton $M$ which accepts the language $L=\{0^n1^n
\vert$ $n$ is a natural number $\}$.


\vskip .25in \vskip .25in
\begin{centering}
        {\large \bf Part II\\[.2in]}
\end{centering}

ANSWER 4 QUESTIONS.\\

\vskip .25in
 1) Prove that the language $L=\{w\in \{a,b\}^* \vert$ the number of $a$-s in $w$ is not divisible
 by 3 $\}$ is regular.
\vskip .25in

 2) Construct a push-down automaton which accepts the following language: $L=\{ w\in \{a,b,c\}^*
 \vert\ w=xc^nx^R$ for some natural number $n$ and $x\in \{a,b\}^* \}$.
\vskip .25in
 3) Find a regular expression for the language $L=\{w\in \{a,b\}^* \vert$ $w$ has odd number of b-s $\}$.

\vskip .25in
 4) Prove that if $L$ is a regular language and $a_0, a_1,...,a_n,...$ is an infinite
 arithmetic progression of natural numbers then $\bigcup_i L^{a_i}$ is regular.
\vskip .25in
 5) Let $L=\{w\in \{a,b\}^* \vert$ the number of a-s appears in $w$ is bigger than
 twice the number of b-s appears in $w$ $\}$. Show that $L$ is not regular.
\vskip .25in

$\ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ $
GOOD LUCK !!


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