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                {\Large \bf Math 441, Computability and Logic}\\[.2in]
        {\large \bf Midterm Exam 2, Spring 2000}\\[.2in]
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$ $\\$ $\\$ $\\
 CHOOSE 4 QUESTIONS.\\
 REMARK: QUOTE ALL THEOREMS YOU ARE APPLYING.

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1.a) Define the equivalence relation $\approx_L$ for a language $L\subseteq \sum^*$.\\

1.b) Describe the minimal deterministic finite automaton(i.e., with minimal number of states)
 which accepts a regular language $L$.

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2) Show that $L=\{w\in {\{a,b\}}^* \vert$ the number of a-s appears in $w$ is greater than the
number b-s appears in $w$ $\}$ is not regular.

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3) Show that $L=\{ w\in {\{0,1,...,9\}}^* \vert$ $w$ is a decimal notation for a number divisible
by 6 $\}$ is regular.

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4) Find the standard deterministic finite automaton which accepts the language $L=\{ w\in \{a,b\}^*
\vert$ aba is a substring of $w \}$.

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5) Prove or refute: If $L$ is regular then $L'=\{xy \vert\ x\in L$ and $y\not\in L \}$ is regular.


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