1. Part Two : Nondeterministic Finite Automata

2. Example 1 A C B accept all strings of 0 and 1 ending in 11
Given the word "1111", for the first '1' do we stay in state A or go to state B? A machine cannot determine that.
A string is accepted by an NFA if there is any sequence of transitions that ends in a favorable state. A C B 1
Start with a few examples that are accepted:
1111 11
Easy fail example:
Two different ways to fail :

3. DFA v. NFA Both: one start state one or more accepting states DFA:
every state indicates how to process each character of the input alphabet in exactly one way
NFA:
states may have more than one possible transition for an input
just try them all
states may not indicate how to process a particular input
indicates the word is not accepted

4. Example 2 - Epsilon Transitions
ε - change state without consuming an input character
This NFA accepts words of 0 and 1 that contain either 11 or 101.
Example words:
> A A A A - rejected
> A B C e E - accepted
> several possible transitions, none reach a favorable state A B C E D ε 1
0,1

5. FA Theory Two FAs are equivalent if they accept the same language.
For each NFA, there is an equivalent DFA.
We will skip the boring proof.

6. NFA to DFA algorithm Start the DFA with the start state of the NFA.
For each new state in the DFA
for each character in the alphabet
if there is more than one choice of transitions, create a new state that is the union of those two states
if that character does not lead out of this state, create transition to a new dead state
Favorable states are any states that reference a favorable state from the NFA.

7. NFA to DFA Example 1
Again, accept all words of 0 and 1 that end with 11.
NFA in table format:
> A {A} {A,B}
B ф {C}
C ф ф A C B 1
new DFA - step one
> A A AB
state A is already
defined
state AB is new
new DFA - step two
> A A AB
AB A ABC
state AB is the union
of states A and B
0: {A} U ф = A
1: {A,B} U {C} = ABC
Final DFA
> A A AB
AB A ABC
ABC A ABC

8. NFA to DFA Example 1 continued
Accept all words of 0 and 1 that end with 11.
Final DFA
> A A AB
AB A ABC
ABC A ABC A C B 1 1 1 1 A AB
ABC

9. NFA to DFA example 2 accept all binary words that start with zero A B
NFA in Table Format:
0 1
>A B ф
B B B A B
0,1
new DFA:
0 1
>A B C
B B B
C C C
C is a trap state. A B
0,1 1 C
0,1

10. Practice Draw an NFA that accepts strings of {0,1} that contain "11".
Convert that NFA to a DFA.

11. Epsilon NFA to DFA example
Again, this NFA accepts words of 0 and 1 that contain
either 11 or 101. 1 1 A B C ε
0,1 1
0,1
e-NFA: e*
>A A A,B A
B D C B
C C,E
D E D
E E E E D E
states reachable by using
zero or more epsilon moves
These cells are pointers to the
correct row. Use the e* values
from those rows.

12. e-NFA to DFA example continued
>A A A,B A
B D C B
C C,E
D E D
E E E E
DFA:
>A A AB
AB AD ABCE
AD A ABE
ABCE ADE ABCE
ADE AE ABE
ABE ADE ABE
AE AE ABE

13. e-NFA to DFA continued 0 1 >A A AB AB AD ABCE AD A ABE
>A A AB
AB AD ABCE
AD A ABE
ABCE ADE ABCE
ADE AE ABE
ABE ADE ABE
AE AE ABE
e-NFA to DFA continued
accept words of 0 and 1 that contain
either 11 or 101 1 A AB 1 AD
ABE 1 1 1 1
ABCE
ADE AE 1

14. Practice
The following eNFA accepts strings of {a,b} that start with a single 'a', followed by zero or more 'b's, and ending with a single 'a'.
ab*a
b b
a e a a A B C D D