CS 390 Test II

July 25, 2000

1. Find an NFA without $\Lambda$ that accepts the same language as the following NFA-$\Lambda$: [26]

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

The initial state is $q_{0}$ and the accepting state is $q_{2}$.
The transitions not given in the table are to the empty set.

2. Find an FA that accepts the same language as the following NFA: [26]

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

The initial state is $q_{0}$ and the accepting state is $q_{3}$.

3. Let $S$ and $T$ be sets of states of an NFA-$\Lambda$.
Prove that $\Lambda(S \cup T) \subseteq \Lambda (S) \cup \Lambda (T)$ by structural induction. [20]

4. For the FA given below answer the following questions:
(a) Find $R(1,3,3)$ using the recursive formula given by Kleene's Theorem. [10]
(b) Find the language accepted by this FA. You may answer this by inspection. [18]

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

The initial state is $q_{0}$ and the accepting state is $q_{3}$.