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\section*{Computability and Logic Homework 6}

\noindent
\textbf{Due: } Thursday, October 27, 2005

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

\noindent
Exercises 13.1.1 and 13.1.3 from the textbook

\subsection*{Problems to Hand In}

\bigskip\noindent 

\noindent
For Problems 1 and 2 (only), you should explicitly show the states and
transitions of the Turing machines you construct.

\paragraph{1.} (This is Exercise 13.1.2 on p.~773 of the text.)
Construct a Turing machine that starts with the symbol \# in one cell
of its tape, with all other cells blank; the beginning position of the
read-write head is not specified.  The machine should halt with the
head pointing at the cell containing the \#, with all other cells
blank.  

\paragraph{2.} Construct a Turing machine to accept exactly
those strings over $\{a,b\}$ that are palindromes (\emph{i.e.}, $w =
w^R$).  Make sure your machine works properly regardless of whether its
input is of even or odd length.

\paragraph{3.} Given any Turing machine $M$, show how to construct a
machine $M'$ such that:
\begin{enumerate}
\item $M'$ accepts exactly the same language as $M$, and 
\item Whenever $M'$ enters the halt state, it does so with the tape
  completely blank.
\end{enumerate}

You do not have to explicitly list the states and transitions of $M'$,
but make sure you describe its operation completely.

\paragraph{4.} There are many, many variations on the definition of a
Turing machine used in different textbooks.  Naturally, they are all
equivalent.
\begin{itemize}
\item[a.] Our Turing machines can write a symbol on the tape \emph{and}
  move left or right in a single step; in other definitions, a machine
  can \emph{either} write a symbol \emph{or} move the head, but not
  both at the same time.  Explain why our machines are no more
  powerful than machines with this restriction.
\item[b.] Our Turing machines have a ``2-way infinite'' tape, that is,
  their tape heads can move arbitrarily far in either direction.  Many
  books define a Turing machine to have a tape that is only infinite
  in one direction, that is, a tape with a left end but no right end.
  Explain why our machines are no more powerful than these;
  specifically, describe how any Turing machine with a 2-way infinite
  tape can be simulated by one with a 1-way infinite tape.  (Hint:
  Exercise 13.1.3 is helpful here.)
\end{itemize}

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