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\centerline{\bf HW 04 CMSC 452}
\centerline{\bf Morally Due TUES Feb 25 11:00AM} 
\centerline{\bf Dead-Cat Due THU Feb 27 at 11:00AM} 


\begin{enumerate}

\item
(30 points) 
(HINT: DO NOT use the algorithm that, given a DFA, finds the regex for it.)

Let the alphabet be $\Sigma=\{a,b\}$.

Give a regex  for the following.

\begin{enumerate}
\item
$\{ w \colon \#_a(w)\equiv 0 \pmod 3 \}$

(example: $aaababaa$)

\item
$\{ w \colon \#_a(w)\equiv 1 \pmod 3 \}$

(example: $aaababaaa$)

\item
$\{ w \colon \#_a(w)\equiv 2 \pmod 3 \}$

(example: $aaabaabaaa$)

\end{enumerate} 

\newpage


\item
(30 points) 
Let $n\ge 11$. 

Assume that the sequence 

$10^0 \pmod n$, 

$10^1 \pmod n$, 

$\vdots$ $\qquad$ $\vdots$

has the pattern:

$(a_0,\ldots,a_{L-1},\overline{b_0,\ldots,b_{m-1}})$. 

(You can assume the $b_i$'s are nonzero.) 

Give an UPPER BOUND on the number of states in the DFA-classifier for mod $n$.
Explain your upper bound.

(You DO NOT NEED TO and probably SHOULD NOT be to clever. That is, you do not want
to use that the first tier has 10 states. You can assume it has $n$ states.) 


\newpage

\item
(40 points)
\begin{enumerate}
\item 
Find a value of $n$ such that the sequence 

$10^0 \pmod n$, 

$10^1 \pmod n$, 

$\vdots$ $\qquad$ $\vdots$

has the pattern:

$(a_0,\ldots,a_{L-1},\overline{b_0,\ldots,b_{m-1}})$. 

with $L\ge 3$, $m\ge 3$, and the $b_i$'s are nonzero. 

\item 
Give a table for the DFA classifier mod $n$. 
\end{enumerate}

\end{enumerate}

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