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

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

\begin{enumerate}[label=\arabic*.]
  \item Nathan demonstrated that there is an NFA with~$n+1$ states for which an
    equivalent DFA requires $2^n$ states.  (This was a bonus problem.)
  \item Quick discussion of the Church-Turing thesis.
  \item General questions from the class.
  \item Is every language Turing recognizable?
  \item Discuss the hierarchy of languages and machines we have seen so far in
    order of increasing strength.  
    \begin{enumerate}
      \item Give examples showing each successive class of languages in the
	hierarchy is larger than the previous.
      \item Discuss the closure properties for each model.  For instance,
	consider checking the boxes in the following table if the listed
	language is closed under the given property.  Try to fill in this table
	row-wise, starting with the top row.
	\medskip

	\begin{center}
	  \begin{tabular}{c|c|c|c|c}
	    &R& CF & TD& TR\\\hline
	    union&\checkmark&&&\\\hline
	    concatenation&&&&\\\hline
	    star&&&&\\\hline
	    intersection&&&&\\\hline
	    complementation&&&&\\\hline
	    reversal&&&&\\\hline
	    rotational closure&&&&
	  \end{tabular}
	\end{center}
	\bigskip

	R = regular, CF = context-free, TD = Turing-decidable, TM = Turing recognizable
    \end{enumerate}
\end{enumerate}
\end{document}
