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\def\Tot{\mathop{\rm Tot}\nolimits}
\def\rng{\mathop{\rm rng}\nolimits}
\def\grf{\mathop{\varphi}\nolimits}     
\def\gro{\mathop{\omega}\nolimits}     
   


\begin{document}
\begin{center} {\bf Midterm Exam}\\
Math 610\\
(Recursion Theory)  \end{center}

\noindent You have one hour to complete this exam.
Answer the following questions as clearly and completely as possible. 
You may use your notes and Soare as references.


\begin{enumerate}
\item Let $A,$ $B \subseteq \gro.$ Prove that
$A \leq_1 B$ implies $A \leq_m B$ and that 
$A \leq_m B$ implies $A \leq_T B.$
Show that if $A \leq_m B$ and $B$ is recursive (respectively,
recursively enumerable) then so is $A.$
\item Let $\Tot = \{ e : \grf_e$ is total $\}$ be the index set
of the recursive functions. Prove using a diagonalization
argument that $\Tot$ is not recursive. Give a $\Pi_2$ definition of
$\Tot.$
\item Show that an infinite recursively enumerable subset 
$A \subseteq \gro$
is recursive if and only if it is the range of an increasing
recursive function.
\item Let $W_e = \rng \grf_e$ be an infinite recursively enumerable set.
Prove there exists a recursive subset $A_e \subseteq \gro$
such that $\grf_e$ restricted to  $A_e$ is 1:1 and onto $W_e.$
\item Let $W$ be an infinite recursively enumerable set. By the two previous
exercises, $W = \rng f$ where $f$ is a 1:1 recursive function. 
Let $B = \{ s : (\exists t > s) [f(t) < f(s)]\; \}$ be the {\em deficiency} 
set of $W.$ Show that if $W$ is not recursive, then $\bar{W}$ is infinite.
\item Denote by $\cal{E}$ the lattice of the recursively enumerable
sets under inclusion. Verify that the set \newline
$\cal{F}$ $:= \{ A \subseteq \gro : A$ is simple or cofinite $\}$ forms
a filter in $\cal{E}.$

\end{enumerate}


\end{document}