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\handout{ps3}{Homework Assignment 3}{February 14, 1992}
\due{Friday, Feb. 21}
%Homework Assignment 1
%Given out: Friday, January 31
%Due: Friday, Feb. 7

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All problems are from Martin.  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.

%\reading{Martin Chs. }

\paragraph{Problems.}

\begin{enumerate}
\item[1.] {5.3 (a), (c), (e); 5.5}

\item[2.] {5.10; 5.17 (d)}

\item[3.] {6.3; 6.6 (a); 6.8 (b)}

\item[4.] {8.5; 8.7 (a), (b), (d), (f).  Recall that $N_a(x)$ is the number of $a$s in the string $x$.}

\item[5.] 8.8 Note that an arithmetic progression need not be infinite.
For instance, $S = \{n, n+p, n+2p, \ldots, n+kp \}$ (for some
non-negative integer $k$) is considered to be an arithmetic
progression.  Hence, if we have regular language $A = \{aaa, aaaaa\}$,
then $S = \{|x| \, | x \in A \} = \{3,5\}$ is an arithmetic
progression where $n = 3$, $p = 2$, and $k = 1$.  Hint for part (b):
using the Pumping Lemma, argue that $S$ is a union of a finite number
of finite arithmetic progressions and some number of infinite
arithmetic progressions, where the additive terms $p$ of the infinite
progressions is bounded above by $N$.  Then argue that there can be at
most $p$ infinite progressions with additive term $p$, so that the
total number of progressions in $S$ is finite.

\end{enumerate}


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