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\begin{document}

\centerline{\bf HW 06 CMSC 452}
\centerline{\bf Morally Due TUES March 11 11:00AM} 
\centerline{\bf Dead-Cat Due THU March 13 at 11:00AM} 


\begin{enumerate}

\item
(30 points) 
Give a CFG for the following languages

The alphabet is $\{a,b\}$. 

\begin{enumerate}
\item
$\{ w \colon \#_a(w) \equiv 0 \pmod 3 \}$.
\item
$\{ w \colon \#_a(w) \equiv 1 \pmod 3 \}$.
\end{enumerate}


\newpage

\item
(30 points) 
Give a context free grammar for the following language


$\overline{\{a^n b^n c^n \colon n\in\N\}}$.


\newpage
\item
(40 points) 
\begin{enumerate}
\item
(20 points) 
Give a function $f$ such that the following is true, and prove it:

If $\alpha$ is a regex of length $n$ then there exists
a context free grammar $G$ such that $L(G)=L(\alpha)$
 with $f(n)$ rules ($f(n)$ may use $O$-notation).

(The CFG DOES NOT have to be in Chomsky Normal Form.)

(Your proof should be an algorithm that takes $n$ and produces the grammar.
You may use any theorem in class as a subroutine for this algorithm.) 

\item 
(20 points) 
Give a function $g$ such that the following is true, and prove it:

The language

$$L_n=\{ w \colon \#_a(w) \equiv 0 \pmod n  \wedge \#_b(w) \equiv 0 \pmod n \}$$

has a context free grammar with $g(n)$ rules ($g(n)$ may use $O$-notation).

(The CFG DOES NOT have to be in Chomsky Normal Form.)

(Your proof should be an algorithm that takes $n$ and produces the grammar.
You may use any theorem in class as a subroutine for this algorithm.) 


\end{enumerate}

\end{enumerate}

\end{document}