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\centerline{\bf HW 02 CMSC 452}
\centerline{\bf Morally Due TUES Feb 11 11:00AM} 
\centerline{\bf Dead-Cat Due THU Feb 13 at 11:00AM} 


\begin{enumerate} %B1

\item
(50 points) 
For this problem you might want to write a program that will,
given $n$, compute $10^0 \pmod n$, $10^1 \pmod n$, $10^2\pmod n$ etc and find a pattern.
If you write such  a program KEEP IT - you may need it later in the course.

Write a nice clean easy-to-grade table for the answer. I give an example later. 

$\overline{a}$ means $(a,a,\ldots)$. 

$\overline{a,b}$ means $(a,b,a,b,\ldots)$. 

For $n=14,15,16,17,18,19,20$ give the following

\begin{enumerate}
\item
The weights needed for a DFA classifier for mod $n$.  E.g.,

If $n=2$ the answer is $(2,\overline{0})$

If $n=3$ the answer is $(\overline{1})$ 

If $n=4$ the answer is $(1,2,\overline{0})$

If $n=11$ the answer is $(\overline{1,-1})$

\item
The size of a DFA classifier for determining what a number is congruent to mod $n$. 

(For this problem, UNLIKE CLASS, we will count the start state as having input 0
so $\equiv 0\pmod n$.  E.g., 

If $n=2$ the answer is 2.

If $n=3$ the answer is 3.

If $n=4$ I leave that to you to figure out. 

If $n=11$ the answer is 22.

\end{enumerate}

The format for the answer should be like this:

\[
\begin{array}{|l|l|l|l} 
\hline
n & \hbox{pattern} & \hbox{Numb of States} \cr 
\hline 
2   &  (2,\overline{0}) & 2 \cr 
3   &  (\overline{1}) & 3 \cr 
4   & (1,2,\overline{0}) & 18 \hbox{ NOT the real answer}\cr 
11  & (\overline{1,-1}) & 22 \cr
\hline
\end{array}
\]


\newpage

\item
(50 points) 
In this problem $\Sigma=\{a\}$.
\begin{enumerate}
\item
(10 points) 
Write a DFA for 

$$L_1=\{ a^n \colon n\equiv 0 \pmod 6 \}.$$

How many states does it have? 
\item
(10 points) 
Write a DFA for 

$$L_2=\{ a^n \colon n\equiv 0 \pmod 9 \}.$$

How many states does it have? 

\item
(10 points) 
IF you were to use the construction to get a DFA for $L_1\cap L_2$ 
how many states would it have?
(DO NOT do the construction.)

\item
(20 points) 
Give a DFA for $L_1\cap L_2$ that uses fewer states then the one from
the construction. 


\item 
(THOUGHT QUESTION- DO NOT SUBMIT)
IF you were to use the construction to get a DFA for $L_1\cup L_2$ 
how many states would it have?
(DO NOT do the construction.)
IS THERE a DFA for $L_1\cup L_2$ with fewer states than the one from construction?
\end{enumerate} 
\end{enumerate}

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