CS 390 Test

July 20, 2002

1. Which of the following statements are true and which are false ?

(a) $(L^{*})^{+} = L^{+}$.
(b) $(0^{*} 1^{*})^{*} = (0^{*} + 1)^{*}$.
(c) $(L_{1}L_{2} \cup L_{3})^{*}$ is regular if $L_{1}$, $L_{2}$, and $L_{3}$ ar e regular.
(d) The string $101$ is represented neither by $(0^{*} + 1^{*})$ nor by $(01^{*} + 10^{*} + 1^{*}0 + (0^{*}1)^{*})$ .
(e) Suppose that A and B are sets, that S and T are subsets of A and that $f$ is a function from A to B. Then $f(S \cup T) \subseteq ( f(S) \cup f(T) )$.

2 (a) Find an $NFA$ that recognizes the same language as the following $NFA-\Lambda$:

State $q$	$a$	$b$	$\Lambda$
$1$	$\emptyset$	{$4$}	{$2$}
$2$	$\emptyset$	{$2$}	{$3$}
$3$	$\emptyset$	$\emptyset$	$\emptyset$
$4$	{$3$}	$\emptyset$	{$1$}

The initial state is $1$ and the accepting state is $3$.
(b) Find $\Lambda(\{4\})$.
(c) Find $\delta^{*}(1,bb)$.

3. Find a DFA that accepts the same language as the following NFA:

$q$	$\sigma$	$\delta(q, \sigma)$	$q$	$\sigma$	$\delta(q, \sigma)$
$q_{0}$	$a$	{$q_{3}$}	$q_{2}$	$a$	{$q_{2}$}
$q_{0}$	$b$	{$q_{1}, q_{2}$}	$q_{2}$	$b$	{$q_{3}$}
$q_{1}$	$a$	{$q_{3}$}	$q_{3}$	$a$	$\emptyset$
$q_{1}$	$b$	$\emptyset$	$q_{3}$	$b$	$\emptyset$

4. Find a regular expression for each of the following languages over the alphabet {0, 1}:

(a) The language of all strings containing at least two 0's.
(b) The language of all strings not containing the string 00 as their substring.

5. Prove that $\mid x \mid$ = $\mid x^{r} \mid$ for all strings $x$ over the alphabet {a, b} by structural (general) induction, where $x^{r}$ is the reversal of string $x$ (i.e. $x$ spelled backward) and $\mid x \mid$ denotes the length of string $x$.