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\handout{quiz1review}{Review Problems for Quiz 1}{February 28, 1992}

{\bf Problem Set Statistics}.  Here are some statistics for the first
three problem sets (not including problem sets that were turned in late).

\begin{center}
\begin{tabular}{lccc}
		& PS 1 		& PS 2 		& PS 3 \\
		& (out of 75)	& (out of 100)	& (out of 100) \\
Average		& 54.82		& 63.00		& 65.56	\\
Standard Dev.	& 16.41		& 17.82		& 20.87 \\
\end{tabular}
\end{center}


\medskip

{\bf Quiz Information}.  Quiz 1 will be held this coming Thursday,
March 5, in room 4-370 (our normal lecture hall).  The quiz is 2 hours
long, and you may choose to take the exam either from 5-7 PM, or from
7-9 PM.  If you choose the earlier exam, you {\it must} stay until 7
PM, and if you choose the later exam, you {\it must} arrive by  7 PM.
The exam will be open book and open notes.

\vspace{.5in}

{\bf Quiz Review Problems.} Here are some questions from last year's
quiz to help you prepare.  In addition, we list some questions from
Martin about context-free languages, since we haven't had a problem
set about CFLs yet.

\medskip

{\bf True or False}.  For each of the following statements, indicate
whether it is true or false.  You do not need to give your reasoning.

\begin{enumerate}
\item There exists a regular language with a non-regular subset.

\item There exists a regular language $A$ such that $A^*$ is not
regular.

\item There exists a non-regular language $A$ such that $A^*$ is
regular.

\item An infinite union of regular languages is never regular.

\item If $M$ is an NFA accepting the language $L$, then exchanging the
final and nonfinal states in $M$ gives a new NFA accepting the
complement of $L$.

\item There exists a regular language whose complement is not
context-free.

\item There exists a regular language which is the intersection of a
context-free language and a regular language.

\item There exists an infinite context-free language which does not
contain an infinite regular language as a subset.

\end{enumerate}

\medskip

{\bf NFA $\rightarrow$ DFA  and NFA $\rightarrow$ Regular Expression}.
Below is a state transition diagram for an NFA $M$.

\vspace{1in}  


\begin{enumerate}
\item Using the subset construction, give an equivalent DFA, $M'$ that
accepts the same language.

\item Give a regular expression for $L(M)$.
\end{enumerate}

{\bf Regular or Not Regular}.  For each of the following languages,
prove if it is regular or not.

\begin{enumerate}

\item $L_1 = \{ a^n|n > 1854 \mbox{ and $n$ is divisible by 4.} \}$.

\item $L_2 = \{w | w \in \{0,1\}^*$  such that $w$ is at least 10
symbols long, and the 10th and 9th symbols from the right end of $w$
are 10 $\}$ (e.g., the string 0110101010101010 is in the language.)

\item The language $L$, where $K$ is a regular language and $LK$ (the
concatenation of $L$ and $K$) is regular.

\item $L_3 = \{ (a^nb)^n | n \geq 0 \}$.

\end{enumerate}

\medskip

{\bf Martin CFL Questions}.  Here are questions from Martin about CFLs
that might be similar to those on a quiz.

\begin{enumerate}



\item 9.9

\item 9.12

\item 10.2

\item 12.4 (c)

\item 12.10

\item 15.1

\end{enumerate}

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