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                {\Large \bf Math 441, Computability and Logic}\\[.2in]
        {\large \bf Midterm Exam 1, Spring 2000}\\[.2in]
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REMARK: QUOTE ALL THEOREMS YOU ARE APPLYING.
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1.a) Define a regular expression over an alphabet $\sum$.

1.b) Suppose that $L\neq \sum^*$ is a regular language over an alphabet $\sum$. Is it true that
there is necessarily a regular language $K\ne \sum^*$ such that $L\subseteq K$ and $K\neq L$ ?
Prove your answer.

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2) Find a regular expression for the following languages (where, $\sum=\{a,b\}$):\\

(i) $L_1=\{w\in \sum^* \vert$ whenever $i<j\leq \vert w\vert$ and $w(i)=a$ and $w(j)=a$ and for all
$k$ with $i<k<j$, $w(k)=b$ the difference $j-i$ is odd $\}$

(ii) $L_2=\{w\in \sum^* \vert$ $w$ does not contain the string aaa as substring $\}$

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3) Find a deterministic finite automaton which accepts only words in the following language over
$\sum=\{a_1,a_2,a_3\}$:

$L=\{w\in \sum^* \vert$ for some $1\leq i\leq 3$, $a_i$ does not appear in $w\ \}$
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4) Let $L$ be a regular language over $\sum$. Prove that the language $K=\bigcup_{k}
(\sum^*\backslash L)^{3k+1}$ is accepted by a deterministic finite automaton.

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5) What is the language accepted by the following deterministic finite automaton (you may either
characterize this language or describe it by a regular expression) ?
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