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\centerline{\bf HW 03 CMSC 452}
\centerline{\bf Morally Due TUES Feb 18 11:00AM} 
\centerline{\bf Dead-Cat Due THU Feb 20 at 11:00AM} 


\begin{enumerate} %B1

\item
(30 points) 
Let $L$ be regular via DFA $(Q,\Sigma,\delta,s,F)$. 

Write down an NFA $(Q',\Sigma,\delta',F')$ for $L^*$. 


\newpage

\item
(30 points) 
In this problem $\Sigma=\{a,b,c\}$.
Let $L$ be the set of all $w$ such that the following hold:

\begin{itemize}
\item 
$\#_a(w)\equiv 1 \pmod 3$, AND 
\item
$\#_b(w)\equiv 2 \pmod 4$, AND
\item 
$\#_c(w)\equiv 3 \pmod 5\}$. 
\end{itemize} 

Write a DFA for $L$ in table form. Give $Q,\delta,s,F$. (We already know $\Sigma$.) 

\newpage

\item
(40 points) 
Give an NFA for

$$L= \{ a^i \colon i\ne 100 \}.$$

that has substantially less than 100 states.

A few points about this:

\begin{itemize}
\item
You should use the technique on the SMALL NFA slides.
(The next two points make that point.) 

\item
You NEED to use the Chicken McNugget Theorem.
You may use it without proof. 

\item 
You NEED to use the following Theorem.

{\it 
Let $n\in\N$. Let $p_1,\ldots,p_m$ be primes such that $p_1\cdots p_L \ge m$. 
Let $i\in\N$. Let $X$ be the set of all $i$ such that 

$i\not\equiv n \pmod {p_1}$

$i\not\equiv n \pmod {p_2}$

$\vdots \qquad \vdots$

$i\not\equiv n \pmod {p_m}$

Then (1) $X$ contains $\{0,1,\ldots,n-1\}$, (2) $X$ does not contain $n$,
and
(3) it will contain some elements $>n$ but we don't care. }

\item
You can give either a NEATLY DRAWN diagram or a table.

\item
DO NOT optimize. Lets say you get it to be 50 states. DO NOT try to make it 49. 
\end{itemize} 

\end{enumerate}

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