<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[11pt]{amsart}

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{
\fancyhead[L,LO]{
\textit{\small Math 387 Group problems, Monday Week 5}
}
\renewcommand{\headrulewidth}{0pt}
}

\begin{document}
\thispagestyle{firststyle}
\bigskip

\begin{prob}
  \begin{enumerate}
    \item What is a \emph{computable function}?  (Give a precise definition that
      includes stating the domain and codomain.).
    \item Let~$A,B$ be languages.  What does it mean to say~$A$ is
      \emph{reducible} to~$B$?  What does it mean to say~$A$ is \emph{mapping
      reducible} to~$B$?
    \item Why is $A$ always reducible to $\overline{A}$?  
    \item Give an example, with explanation, of a language that is reducible to
      its complement but not mapping reducible to its complement. 
  \end{enumerate}
\end{prob}

\begin{prob}
  Show that a language $A$ is recognizable if and only if $A\leq_{m} A_{TM}$.
  (Divide your proof into two parts.  In the first, assume that~$A$ is
  recognizable and provide an explicit mapping reduction.  This will be a
  computable function~$f\colon\Sigma^{*}\to\Sigma^{*}$. So $f$ must account for
  every possible string, and its output must be a string.)
\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}
    We have seen that $A_{TM}$ is reducible to $E_{TM}$.  Review the proof.
    \begin{enumerate}
      \item Is $E_{TM}$ mapping reducible to $A_{TM}$?  
      \item (bonus) Is it reducible to~$A_{TM}$?
    \end{enumerate}
   \end{prob}

\begin{prob}
  Is the language~$T$ from Problem~\ref{prob:05monprobT} recognizable?
\end{prob}

\begin{prob} (bonus)
   Let $S = \{\langle M\rangle: \text{$M$ is a TM and $L(M)=\{\langle
   M\rangle\}$}\}$.  Show that neither $S$ nor $\overline{S}$ is recognizable.
\end{prob}
\end{document}
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