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\handout{ps6}{Homework Assignment 6}{March 13, 1992}
\due{Friday, March 20}

\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).


\paragraph{Problems.}

\begin{enumerate}

\item[1.] (14 points)
\begin{enumerate}
  \item (7 points) 17.12
  \item (7 points) Show that the intersection of a CFL and a Turing-acceptable
     language is Turing-acceptable.
\end{enumerate}

\item[2.] (23 points)
\begin{enumerate}
  \item (15 points) Sketch  a proof that 
  $\{ <M_1, M_2> : M_1$ and  $M_2$  are DFA's with the same input
    alphabet such that $ L(M_1) \subseteq L(M_2) \}$ is a recursive set.
  \item (4 points) 18.4 : prove your answer.
  \item (4 points) Explain why the following argument  is not correct. 
``Assume that each language $L_i; i=1,2, \ldots$ is recursively enumerable and accepted
by some Turing machine $M_i$. Define the Turing machine $M$ that acts as follows on
any input $x$: it begins by giving two steps to machine $M_1$ working on input $x$, then 
two steps to  $M_2$ (on input $x$), then one step to $M_3$ (on input $x$),
then gives  an additional 
step to $M_1$, $M_2$, $M_3$, then gives one step to $M_4$ (on input $x$), then 
gives one additional step to $M_1, M_2, M_3, M_4$, 
then gives one step to $M_5$ (on input $x$)... In short, as soon as $M$ has given
a step to a new machine $M_{i}$ (working on input $x$) it gives
another step to $M_1,\ldots,M_{i}$ and then gives a step to 
the new machine $M_{i+1}$. $M$ halts as soon and only when one
of the machines $M_i$ halts in this simulation. If $x$ is in one of the 
languages $L_i$ $M$ will then halt on $x$. This proves that $\cup_{i=1}^{\infty} L_i$
is recursively enumerable.''
\end{enumerate}

\item[3.] (15 points) 18.17

\item[4.] (20 points) 18.2

\item[5.] (28 points: 7 points for each of the four parts)  20.2

\end{enumerate}


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