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\section*{Computability and Logic Homework 7}

\noindent
\textbf{Due: } Thursday, November 3, 2005

\subsection*{Practice Problems (Don't hand in)}

Exercises 13.1.4a on p.~773 and 13.2.2a on p.~788 of the text.

\subsection*{Problems to Hand In}

\paragraph{1.} Exercise 13.1.6 on p.~773.

\paragraph{2.} Construct a Turing machine to decide the language 
$\{ww\ |\ w \in \{a,b\}^*\}$.  If you wish, you may construct a
multitape machine.  Describe your machine informally but carefully.

\paragraph{3.} Exercise 13.2.2b on p.~788.

\paragraph{4.} Let $f : \mathbb{N}^m+1 \to \mathbb{N}$ be a recursive
function.  Show that the function $h$ defined by
\[
h(\bar{x},y) = 
\left\{\begin{array}{r@{\quad}l}
0 & \textit{ if for some $i \leq y$, }f(\bar{x},i) = 0
\\
1 & \textit{ otherwise}
\end{array}\right.
\]
is also recursive, and that $f$ is primitive recursive then $h$ is
primitive recursive.  Be careful: do not assume that any functions are
recursive other than the initial functions and those shown to be
recursive in Section 13.2.3 (or earlier in this problem set).  When
you obtain a function by composition or primitive recursion, make sure
it is clear from what functions it is obtained.

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