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\handout{ps7}{Homework Assignment 7}{March 20, 1992}
\due{Friday, April 3}

\medskip

Remember to write up problems 1 through 5 in such a way that each of
these problems can be turned in separately.  Put your name on each
problem, with your last name clearly written in CAPITAL LETTERS, and
with your section and the NAME of your recitation instructor  (e.g.
RIVEST/LYNCH, W1) also in capital letters. Try to always provide
clear (this is not in contradiction with short!) justifications to
your constructions.

Here are the statistics of the last two problems:\\

%\begin{itemize}
%\item PS4 :  Average 65.77, \ Standard deviation:  26.24,
%\item PS4 :  Average 75.20 \ Standard deviation:  20.23
%\end{itemize}

\begin{tabular}{lll}
 \mbox{ }	& PS4 &	PS5 \\
Average		& 65.77 & 75.20 \\
Standard dev.	& 26.24	 & 20.23 
\end{tabular}


\paragraph{Problems.}

\begin{enumerate}



\item[1.] (20 points)
     Write a short (two-paragraph to one page) essay describing the most 
     ``natural'' and  ``important'' problem you can think of that would be 
     great to have a computer program for, but for which you can't have such
     a program because the problem is undecidable.  You don't need to
     prove that it is undecidable.  
	Also, the problem you select does NOT need to be one that
	was described or proved undecidable in class.  Any 
	undecidable problem is OK.

     You should focus on making sure the
     problem is clearly described and that the justifications for its 
     importance are clearly articulated.  Argue directly for the importance 
     of the problem, not indirectly (e.g., don't say ``the halting problem is
     important because many other undecidable problems are reducible to
     the halting problem''; say ``many hours of CPU time are wasted with
     programs going into endless loops'' or the like...)  Have fun.
     Be imaginative.  Grammar, spelling, organization, CLARITY and style count.




\item[2.](25 points)
 Is each of the following recursive or not recursive? If it is recursive, sketch a
decision procedure. If  not, give a proof that it is not.

\begin{enumerate}
  \item (5 points) $\{ M : M$ is a TM with input alphabet $\{0,1\}$ and such that
        $M$ halts on input 00 but not on input 11$\}$

  \item (5 points) $\{ <M_1, M_2> : $ a pair of TM's with the same input 
         alphabet and such that $L(M_1) = \overline{L(M_2)}\}$

  \item (5 points) $\{ <M_1, M_2> : $ a pair of DFA's with the same 
         alphabet and such that $L(M_1) = \overline{L(M_2)}\}$

  \item (5 points) $\{ <M,x,k> : $ a triple consisting of a TM, a string over 
         its input alphabet and a natural number such that $M$ halts on 
          input $x$ within $k$ steps $\}$

  \item (5 points) $\{ <M,x,q> : $ a triple consisting of a TM, a string over 
         its input alphabet and a state of the machine such that $M$ on input
         $x$ reaches state $q \}$
\end{enumerate}

\item[3.] (15 points)
Let $A$ be any Turing acceptable (r.e) set. Prove that  $A \leq_m K_0$.

\item[4.] (20 points)
If $f$ is a strictly increasing total recursive function
then range($f$) is a recursive set.

\item[5.] (20 points)  Martin 20.9

\end{enumerate}


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