\begin{frame}{Ambiguous Grammars}
  \begin{block}{}
    A context-free grammar $G$ is \emph{ambiguous} 
    if there exists a word $w \in L(G)$ for which there are multiple derivation trees. 
  \end{block}
  \pause\medskip
  
  \begin{exampleblock}{}
    Is the following grammar ambiguous?
    \begin{talign}
      S \to S+S \mid 0
    \end{talign}
    \pause
    Yes, there are two derivations trees for $0+0+0$:\pause
    \begin{center}
      \begin{tikzpicture}[default,>=latex,l/.style={yshift=-9mm},node distance=10mm,thin]
        \node (1) {$S$};
        \node (2a) [left of=1,l] {$S$};
        \node (2b) [at=(1),l] {$+$};
        \node (2c) [right of=1,l] {$S$};
        \begin{scope}[node distance=6mm]
        \node (3a) [left of=2a,l] {$S$};
        \node (3b) [at=(2a),l] {$+$};
        \node (3c) [right of=2a,l] {$S$};
        \end{scope}
        \node (3d) [at=(2c),l] {$0$};
        \node (4a) [at=(3a),l] {$0$};
        \node (4c) [at=(3c),l] {$0$};
        
        \draw (1) to (2a);
        \draw (1) to (2b);
        \draw (1) to (2c);
        \draw (2a) to (3a);
        \draw (2a) to (3b);
        \draw (2a) to (3c);
        \draw (2c) to (3d);
        \draw (3a) to (4a);
        \draw (3c) to (4c);
      \end{tikzpicture}
      \hspace{1cm}
      \begin{tikzpicture}[default,>=latex,l/.style={yshift=-9mm},node distance=10mm,thin]
        \node (1) {$S$};
        \node (2a) [left of=1,l] {$S$};
        \node (2b) [at=(1),l] {$+$};
        \node (2c) [right of=1,l] {$S$};
        \node (3a) [at=(2a),l] {$0$};
        \begin{scope}[node distance=6mm]
        \node (3b) [left of=2c,l] {$S$};
        \node (3c) [at=(2c),l] {$+$};
        \node (3d) [right of=2c,l] {$S$};
        \end{scope}
        \node (4a) [at=(3b),l] {$0$};
        \node (4c) [at=(3d),l] {$0$};
        
        \draw (1) to (2a);
        \draw (1) to (2b);
        \draw (1) to (2c);
        \draw (2a) to (3a);
        \draw (2c) to (3b);
        \draw (2c) to (3c);
        \draw (2c) to (3d);
        \draw (3b) to (4a);
        \draw (3d) to (4c);
      \end{tikzpicture}
    \end{center}
  \end{exampleblock}
\end{frame}