\begin{frame}{Acceptance with Empty Stack}
  All automata we have seen so far had the following property:
  \begin{talign}
    (q_0,w,z) \vdash^* (q',w',u') \quad\implies\quad (q' \in F \iff u' = \lambda)
  \end{talign}
  They reach an accepting state if and only if the stack is empty.
  \medskip
    
  \begin{exampleblock}{}
    \begin{center}
      \input{tikz/npda1.tex}
    \end{center}
  \end{exampleblock}
  \medskip\pause

  \begin{block}{Acceptance with Empty Stack}
    \emph{Empty stack language of} NPDA 
    $M=(Q,\Sigma,\Gamma,\delta,q_0,z,F)$ is
    \begin{talign}
      \alert{L_\lambda}(M) = \{\, w \in \Sigma^* \mid (q_0,w,z) \vdash^* (q',\lambda,\alert{\lambda}) \,\}.
    \end{talign}
  \end{block}
  (No need for final states in this definition.)
\end{frame}