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\begin{document}

\centerline{\bf HW 05 CMSC 452}
\centerline{\bf Morally Due TUES March 4 11:00AM} 
\centerline{\bf Dead-Cat Due THU March 6 at 11:00AM} 


\begin{enumerate}

\item
For this problem
\begin{itemize}
\item
The {\it size} of a DFA is the number of states.
\item
The {\it size} of an NFA is the number of states.
\item
The {\it size} of a regex is its length.
\end{itemize} 


This is just a FILL IN THE BLANK. You may use $O$-notation.
I give two examples.

EXAMPLE ONE

If the question was:

{\it There is an algorithm that will,
given two regex's $\alpha_1,\alpha_2$ of sizes $n_1,n_2$, returns 
a regex for $L(\alpha_1)L(\alpha_2)$
of size FILLIN.}

The answer would be $n_1+n_2+O(1)$. 

EXAMPLE TWO

{\it There is an algorithm that will,
given a DFA $M$ of size $n$, returns 
a regex for $L(M)$
of size FILLIN.}

the answer is $2^{O(n)}$. 
(This will be one of the questions below and I've just given you
the answer. Yeah for you!)

The ACTUAL QUESTIONS are on the next page

\newpage

\begin{enumerate}
\item
There is an algorithm that will,
given two DFA's $M_1,M_2$ of sizes $n_1,n_2$, returns a DFA for $L(M_1)\cap L(M_2)$
of size FILLIN.

\item
There is an algorithm that will,
given two DFA's $M_1,M_2$ of sizes $n_1,n_2$, returns an NFA for $L(M_1)\cdot L(M_2)$
of size FILLIN.

\item
There is an algorithm that will,
given a regex $\alpha$ of length $n$, returns an NFA for $L(\alpha)$
of size FILLIN.

\item 
There is an algorithm that will,
given a DFA $M$ of size $n$, returns a regex for $L(M)$
of size FILLIN.
\end{enumerate}

\newpage

\item 
PROVE the following statements by giving an algorithm
(your algorithm may use the algorithms in problem 1 as subroutines)
and fill in where it says FILLIN. No proof of the FILLIN is needed. 
\begin{enumerate}

\item
There is an algorithm that will,
given two DFA's $M_1,M_2$ of sizes $n_1,n_2$, returns a DFA for $L(M_1)\cdot L(M_2)$
of size FILLIN.
(NOTE: in Problem 1 we asked for going from two DFA's to an NFA. Here we are asking
to go from two DFA's to a DFA.)

\item
There is an algorithm that will,
given two  regex's $\alpha_1,\alpha_2$ of sizes $n_1,n_2$, returns a regex for 
$L(\alpha_1)\cap L(\alpha_2)$ 
of  size FILLIN.
\end{enumerate}


\newpage

\item
For each of the following state if its is REGULAR or NOT REGULAR.

If you say REGULAR then give either a DFA or REGEX for it.

If you say NOT REGULAR than prove it using the pumping lemma. 

\begin{enumerate}
\item
$\{ a^{\floor{\log_2(n)}} \colon n\ge 1 \}$

\item 
$\{ a^{2n} \colon n\ge 1\}$

\item 
$\{ a^{2^n} \colon n\ge 1 \}$
\end{enumerate}


\end{enumerate}
\end{document}