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\textit{\small Math 387 Friday class summary Week 4}
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\begin{document}
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\bigskip

Some things covered in today's class discussion:
\bigskip

\begin{enumerate}[label=\arabic*.]
  \item Bonus problem discussion.
  \item General questions from the class.
  \item Review decidability of various languages for the three classes of
    machines we have considered.  Fill in the rows of the following
    table with $D$ for ``T-decidable'' by a TM, $R$ for ``T-recognizable but not T-decidable'', or
    $U$ for ``not T-recognizable''.
    \bigskip

    \begin{center}
      \begin{tabular}{c|c|c|c}
	&Acceptance&Emptiness&Equality\\\hline
	DFA &&&\\\hline
	PDA &&&\\\hline
	TM &&&\\
      \end{tabular}
    \end{center}
    \bigskip

    \noindent Recall, for a machine type~$W$,

    \begin{align*}
      \text{Acceptance:}&\quad A_{W}=\{\langle M,w\rangle: \text{$M$ is a machine of type~$W$ and $M$ accepts~$w$}\}\\[5pt]
      \text{Emptiness:}&\quad E_{W}=\{\langle M\rangle: \text{$M$ is a machine of type~$W$ and $L(M)=\emptyset$}\}\\[5pt]
      \text{Equality:}&\quad EQ_{W}=\{\langle M, N\rangle: \text{$M, N$ are machines of type~$W$ and $L(M)=L(N)$}\}.
    \end{align*}
    \medskip

  \item Review the theorem: A language is decidable if and only if it is
    recognizable and co-recognizable.
  \item What about the co-recognizability of the languages considered above?
  \end{enumerate}
\end{document}
