\begin{frame}{Removal of Useless Variables} \begin{exampleblock}{} \vspace{-1ex} \begin{align*} S &\to aSb \mid BC \mid \lambda & A &\to Sb & B &\to a & C &\to C \end{align*} \pause Which variables are non-productive? \begin{itemize} \pause \item $C$ is not productive \end{itemize} \pause\medskip We remove all rules containing non-productive variables: \begin{talign} S &\to aSb \mid \lambda & A &\to Sb & B &\to a \end{talign} \pause Which variables are reachable from $S$? \begin{itemize} \pause \item only $S$ is reachable \end{itemize} \pause\medskip We remove all rules containing non-reachable variables: \begin{talign} S &\to aSb \mid \lambda \end{talign} \pause Hence only $S$ is useful, the variables $A,B,C$ are not useful. \end{exampleblock} \end{frame}