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\fancyhead[L,LO]{
\textit{\small Math 387 Group problems, Wednesday Week 5}
}
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\begin{document}
\thispagestyle{firststyle}
\bigskip

\begin{prob}
  \begin{enumerate}
    \item Find a match in the following instance of the Post Correspondence Problem (PCP):
      \[
	\left\{
	  \left[\frac{a}{baa}\right],
	  \left[\frac{ab}{aa}\right],
	  \left[\frac{bba}{bb}\right]
	\right\}.
      \]
    \item We have seen that PCP is undecidable.  Prove that $\overline{\text{PCP}}$ is unrecognizable.
  \end{enumerate}
\end{prob}

\begin{prob} We start with some definitions.  A \emph{property} $P$ of
  recognizable languages is a set of Turing-recognizable languages.  In other
  words, for each~$A\in P$ there must exist some TM $M$ such that~$L(M)=A$.  We
  say~$P$ is \emph{non-trivial} if (i) it is non-empty, and (ii) there exists
  some recognizable language~$B$ such that~$B\not\in P$.  We say a TM $M$
  \emph{has property~$P$}, if $L(M)\in P$.
  \medskip

  \textbf{Rice's theorem.} Let $P$ be a non-trivial property of recognizable languages.
  Then the language
  \[
    P_{TM} = \{\langle M\rangle: \text{$M$ is a Turing machine with $L(M)\in P$}\}
  \]
  is undecidable.
  \begin{enumerate}
    \item Give an example of a set of languages that does not form a property of
      Turing recognizable languages.
    \item Indicate, with explanation, which of the following are properties of recognizable languages:
      \[
	\text{(i)}\ \  \{\emptyset\},\qquad \text{(ii)}\ \
	\{\Sigma^{*}\},\qquad \text{(iii)}\   \{\text{all
	decidable languages}\}.
      \]
    \item Give all examples of trivial properties of recognizable languages.
  \end{enumerate}
\end{prob}

\begin{prob}
  Decide whether Rice's theorem applies to the following languages.  If so,
  formally define the property (i.e., $P = \{ \text{blah}\}$) and show it is
  non-trivial by giving two examples of recognizable languages---one in~$P$ and
  one not in~$P$. Conclude the
  language is not decidable.  If Rice's theorem does not apply, explain why the property in question is
  not a property of recognizable languages or is trivial.
  \begin{enumerate}
    \item $T=\{\langle M\rangle: \text{$M$ is a TM and $L(M)=L(M)^{R}$}\}$.
    \item $R=\{\langle M\rangle: \text{$M$ is a TM and $L(M)$ is regular}\}$.
    \item $S = \{\langle M\rangle: \text{$M$ is a TM and $L(M)=\{\langle
      M\rangle\}$}\}$. (Be careful/precise with this one.)
    \item $H=\{\langle M\rangle: \text{$M$ is a TM and $M$ never writes a blank
      over a non-blank symbol}\}$.
  \end{enumerate}
\end{prob}

\begin{prob}\label{prob:05monprobT}
      Let $T=\{\langle M\rangle: \text{$M$ is a TM and $L(M)=L(M)^{R}$}\}$.
      Describe a reduction from $A_{TM}$ in order to prove that~$T$ is
      undecidable.  (Hint: mimic the proof of Theorem~5.3 in our text.)
\end{prob}

\begin{prob} (bonus)
  Find a match in the following instance of the Post Correspondence Problem:
  \[
    \left\{
      \left[\frac{aa}{abab}\right],
      \left[\frac{b}{a}\right]
      \left[\frac{aba}{b}\right]
      \left[\frac{aa}{a}\right]
    \right\}.
  \]
  \end{prob}
  \let\clearpage\relax
\end{document}
