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\section*{Homework 9}

\HEADER

\begin{enumerate}
\item Describe what is meant by \emph{eliminating tail
      recursion}, when such an optimization can be applied and
      why it is a benefit?

%  \item It is true (I confirmed it) that
%  
%  \begin{center} if $\varnothing$ does not occur in $r$
%  \;\;then\;\;$L(r) \not= \{\}$ 
%  \end{center}
%  
%  \noindent
%  holds, or equivalently
%  
%  \begin{center}
%  $L(r) = \{\}$ \;\;implies\;\; $\varnothing$ occurs in $r$.
%  \end{center}
%  
%  \noindent
%  You can prove either version by induction on $r$. The best way to
%  make more formal what is meant by `$\varnothing$ occurs in $r$', you can define
%  the following function:
%  
%  \begin{center}
%  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
%  $occurs(\varnothing)$      & $\dn$ & $true$\\
%  $occurs(\epsilon)$           & $\dn$ &  $f\!alse$\\
%  $occurs (c)$                    & $\dn$ &  $f\!alse$\\
%  $occurs (r_1 + r_2)$       & $\dn$ &  $occurs(r_1) \vee occurs(r_2)$\\ 
%  $occurs (r_1 \cdot r_2)$ & $\dn$ &  $occurs(r_1) \vee occurs(r_2)$\\
%  $occurs (r^*)$                & $\dn$ & $occurs(r)$ \\
%  \end{tabular}
%  \end{center}
%  
%  \noindent
%  Now you can prove 
%  
%  \begin{center}
%  $L(r) = \{\}$ \;\;implies\;\; $occurs(r)$.
%  \end{center}
%  
%  \noindent
%  The interesting cases are $r_1 + r_2$ and $r^*$.
%  The other direction is not true, that is if $occurs(r)$ then $L(r) = \{\}$. A counter example
%  is $\varnothing + a$: although $\varnothing$ occurs in this regular expression, the corresponding
%  language is not empty. The obvious extension to include the not-regular expression, $\sim r$,
%  also leads to an incorrect statement. Suppose we add the clause
%    
%  \begin{center}
%  \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}}
%  $occurs(\sim r)$      & $\dn$ & $occurs(r)$
%  \end{tabular}
%  \end{center}  
%  
%  \noindent
%  to the definition above, then it will not be true that
%  
%  \begin{center}
%  $L(r) = \{\}$ \;\;implies\;\; $occurs(r)$.
%  \end{center}
%  
%  \noindent
%  Assume the alphabet contains just $a$ and $b$, find a counter example to this
%  property.
  
\end{enumerate}

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