This document introduces non-deterministic finite automata (NFA). It defines an NFA as a 5-tuple (Q, Σ, δ, q0, F) where Q is a finite set of states, Σ is a finite input alphabet, q0 is the initial state, F is a set of accepting states, and δ is a transition function that returns a set of states rather than a single state. This allows an NFA to be in multiple states simultaneously and accept inputs if any path reaches an accepting state. The document provides examples of strings being accepted and rejected by NFAs, and explains how epsilon transitions allow the read head to remain in the same position. It concludes by distinguishing NFAs

1. WELCOME TO A
JOURNEY TO
CS419
Dr. Hussien Sharaf
Computer Science Department
dr.sharaf@from-masr.com

2. Dr. Hussien M. Sharaf
NON DETERMINISTIC FINITE AUTOMATA NFA

There is a fixed number of states but we can be in multiple states at one
time.
NFA = “a 5-tuple “ (Q, Σ, , q0, F)
Q A finite set of states
Σ
A finite input alphabet
q0 The initial/starting state, q0 is in Q
F
A set of final/accepting states, which is a subset of Q
δ
A transition function, which is a total function from Q x Σ to 2Q ,
this function:

Takes a state and input symbol as arguments.

Returns a set of states instead a single state as in DFA.
δ: (Q x Σ) –> 2Q -2Q is the power set of Q, the set of all subsets of Q
δ(q,s) is a function from Q x S to 2Q (but not to Q)
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3. NFA
A finite automaton is deterministic if
 It has no edges/transitions labeled with
epsilon/lamda.
 For each state and for each symbol in the
alphabet, there is exactly one edge labeled
with that symbol.

4. Dr. Hussien M. Sharaf
NFA

NFA travels all possible paths, and so it remains in many states at once.
As long as at least one of the paths results in an accepting state, the NFA
accepts the input.
Alphabet = {a }
a
q0
q1
a
q2
a
q3
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5. Dr. Hussien M. Sharaf
NFA
Alphabet =
{a }
Two choices
a
q0
q1
a
q2
a
q3
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6. Dr. Hussien M. Sharaf
NFA
Alphabet =
{a }
Two choices
a
q0
q1
a
q2
No transition
a
q3
No transition
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7. Dr. Hussien M. Sharaf
NFA
An NFA accepts a string:
if there is a computation of the NFA
that accepts the string
i.e., all the input string is processed and
the automaton is in an accepting state
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8. Dr. Hussien M. Sharaf
Acceptance Example 1
NFA
a
a
a
q0
q1
a
q2
a
q3
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9. Dr. Hussien M. Sharaf
First Choice
NFA
a
a
a
q0
q1
a
q2
a
q3
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10. Dr. Hussien M. Sharaf
First Choice
NFA
a
a
All input is consumed
a
q0
q1
a
q3
a
q2
“accept”

11. Dr. Hussien M. Sharaf
Second Choice
NFA
a
a
a
q0
q1
a
q3
a
q2

12. Dr. Hussien M. Sharaf
Second Choice
NFA
a
a
Input cannot be consumed
a
q1
a
q2
Automaton Halts
q0
a
q 3 “reject”

13. Dr. Hussien M. Sharaf
NFA
aa is accepted by the NFA:
“accept”
a
q0
q1
q2
a
q0
a
q3
because this
computation
accepts aa
a
q1
a
q2
a
q3
“reject”
this computation
is ignored
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14. Dr. Hussien M. Sharaf
NFA
An NFA rejects a string:
if there is no computation of the NFA
that accepts the string.
For each computation:
• All the input is consumed and the
automaton is in a non final state
OR
• The input cannot be consumed
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15. Dr. Hussien M. Sharaf
NFA
a
is rejected by the NFA:
“reject”
a
q0
q1
a
q2
a
q0
a
q 3 “reject”
q1
a
q2
a
q3
All possible computations lead to rejection
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16. Dr. Hussien M. Sharaf
NFA
aaa is rejected by the NFA:
“reject”
a
q0
q1
a
q2
a
q0
a
q3
q1
a
q2
a
q3
“reject”
All possible computations lead to rejection
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17. Dr. Hussien M. Sharaf
LAMBDA TRANSITIONS
q0
a
q1

q2
a
q3
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18. Dr. Hussien M. Sharaf
LAMBDA TRANSITIONS
Acceptance Example 2
a
a
q0
a
q1

q2
a
q3
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19. Dr. Hussien M. Sharaf
LAMBDA TRANSITIONS
a
a
q0
a
q1

q2
a
q3
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20. Dr. Hussien M. Sharaf
LAMBDA TRANSITIONS
input tape head does not move
a
a
q0
a
q1

q2
a
q3

21. Dr. Hussien M. Sharaf
LAMBDA TRANSITIONS
all input is consumed
a
a
“accept”
q0
a
q1

q2
a
q3
String aa is accepted
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22. Dr. Hussien M. Sharaf
LAMBDA TRANSITIONS
Rejection Example 3
a
a
a
q0
a
q1

q2
a
q3
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23. Dr. Hussien M. Sharaf
LAMBDA TRANSITIONS
a
a
a
q0
a
q1

q2
a
q3
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24. Dr. Hussien M. Sharaf
LAMBDA TRANSITIONS
(read head doesn’t move)
a
a
a
q0
a
q1

q2
a
q3
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25. Dr. Hussien M. Sharaf
LAMBDA TRANSITIONS
Input cannot be consumed
a
a
a
Automaton halts
“reject”
q0
a
q1

q2
a
q3
String aaa is rejected
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26. Dr. Hussien M. Sharaf
LAMBDA TRANSITIONS
Language accepted:
q0
a
q1

L  {aa }
q2
a
q3
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27. Dr. Hussien M. Sharaf
Example 4
q0
a
b
q1
q2

q3

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28. Dr. Hussien M. Sharaf
a b
q0
a
b
q1
q2

q3

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29. Dr. Hussien M. Sharaf
a b
q0
a
b
q1
q2

q3

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30. Dr. Hussien M. Sharaf
a b
“accept”
q0
a
b
q1
q2

q3

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31. Dr. Hussien M. Sharaf
Another String
a b a b
q0
a
b
q1
q2

q3

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32. Dr. Hussien M. Sharaf
a b a b
q0
a
b
q1
q2

q3

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33. Dr. Hussien M. Sharaf
a b a b
q0
a
b
q1
q2

q3

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34. Dr. Hussien M. Sharaf
a b a b
q0
a
b
q1
q2

q3

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35. Dr. Hussien M. Sharaf
a b a b
q0
a
b
q1
q2

q3

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36. Dr. Hussien M. Sharaf
a b a b
q0
a
b
q1
q2

q3

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37. Dr. Hussien M. Sharaf
a b a b
“accept”
q0
a
b
q1
q2

q3

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38. Dr. Hussien M. Sharaf
Language accepted
L  ab , abab , ababab , ... 
 ab 
q0
a

b
q1
q2

q3

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39. Dr. Hussien M. Sharaf
EXAMPLE 5
0
q0
1
q1
0, 1
q2

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40. Dr. Hussien M. Sharaf
Language accepted
L ( M ) = {λ , 10 , 1010 , 101010 , ... }
= {10 } *
0
q0
1
q1
0, 1
q2
(redundant
state)

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41. DETERMINISTIC AND
NONDETERMINISTIC AUTOMATA

Deterministic Finite Automata (DFA)
 One
transition per input per state
 No -moves

Nondeterministic Finite Automata (NFA)
 Can
have multiple transitions for one input in a
given state
 Can have -moves
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42. THANK YOU
Dr. Hussien M. Sharaf
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