Deterministic
Finite Automata
And Regular Languages
2nd Sem 2017
Lecture 3
1
Deterministic Finite Automaton (DFA)
Input Tape
String
Finite
Automaton
2nd Sem 2017
Lecture 3
Output
“Accept”
or
“Reject”
2
Transition Graph
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
initial
state
state
2nd Sem 2017
transition
Lecture 3
a, b
q4
accepting
state
3
Alphabet
  {a , b }
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
a, b
q4
For every state, there is a transition
for every symbol in the alphabet
2nd Sem 2017
Lecture 3
4
head
Initial Configuration
a b b a
Input Tape
Input String
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
a, b
q4
Initial state
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Lecture 3
5
Scanning the Input
a b b a
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
6
a b b a
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
7
a b b a
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
8
Input finished
a b b a
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
a, b
q4
accept
2nd Sem 2017
Lecture 3
9
A Rejection Case
a b a
Input String
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
10
a b a
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
11
a b a
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
12
Input finished
a b a
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
reject
a, b
q4
13
Another Rejection Case
Tape is empty
( )
Input Finished
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
a, b
q4
reject
2nd Sem 2017
Lecture 3
14
Language Accepted:
L  abba 
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
15
To accept a string:
all the input string is scanned
and the last state is accepting
To reject a string:
all the input string is scanned
and the last state is non-accepting
2nd Sem 2017
Lecture 3
16
Another Example
L  , ab , abba 
a, b
q5
b
q0 a
Accept
state
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Accept
state
Lecture 3
a, b
q4
Accept
state
17
Empty Tape
( )
Input Finished
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
a, b
q4
accept
2nd Sem 2017
Lecture 3
18
Another Example
a, b
a
q0
b
q1
Accept
state
2nd Sem 2017
Lecture 3
a, b
q2
trap state
19
a a b
Input String
a, b
a
q0
2nd Sem 2017
b
q1
Lecture 3
a, b
q2
20
a a b
a, b
a
q0
2nd Sem 2017
b
q1
Lecture 3
a, b
q2
21
a a b
a, b
a
q0
2nd Sem 2017
b
q1
Lecture 3
a, b
q2
22
Input finished
a a b
a
q0
2nd Sem 2017
a, b
accept
b
q1
Lecture 3
a, b
q2
23
A rejection case
b a b
Input String
a, b
a
q0
2nd Sem 2017
b
q1
Lecture 3
a, b
q2
24
b a b
a, b
a
q0
2nd Sem 2017
b
q1
Lecture 3
a, b
q2
25
b a b
a, b
a
q0
2nd Sem 2017
b
q1
Lecture 3
a, b
q2
26
Input finished
b a b
a, b
a
q0
b
q1
a, b
q2
reject
2nd Sem 2017
Lecture 3
27
Language Accepted:
L  {a b : n  0}
n
a, b
a
q0
2nd Sem 2017
b
q1
Lecture 3
a, b
q2
28
Another Example
Alphabet:
  {1}
1
q0
q1
1
Language Accepted:
EVEN  {x : x   and x is even}
*
 {, 11, 1111, 111111, }
2nd Sem 2017
Lecture 3
29
Formal Definition
Deterministic Finite Automaton (DFA)
M  Q, ,  , q0 , F 
Q


q0
: set of states
: input alphabet
 
: transition function
: initial state
F : set of accepting states
2nd Sem 2017
Lecture 3
30
Set of States Q
Example
Q  q0 , q1, q2 , q3 , q4 , q5 
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
31
Input Alphabet 
 
:the input alphabet never contains
Example
  a, b
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3

a, b
q4
32
Initial State q0
Example
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
33
Set of Accepting States F  Q
Example
F  q4 
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
34
Transition Function
 :Q  Q
 (q , x )  q 
q
x
q
Describes the result of a transition
from state q with symbol x
2nd Sem 2017
Lecture 3
35
Example:
 q0 , a   q1
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
36
 q0 , b   q5
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
37
 q2 , b   q3
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
38
Transition Table for
states

symbols
q0
a
q1
q2
q5
q1
q3
q4
q5
2nd Sem 2017

q5
q4
q5
q5
b
q5
q2
q3
q5
q5
q5
a, b
q5
b
q0 a
Lecture 3
a
a
b
q1 b q2 b q3 a
a, b
q4
39
Extended Transition Function
 :Q   Q
*
*
 (q ,w )  q 
*
Describes the resulting state
after scanning string w from state
2nd Sem 2017
Lecture 3
q
40
Example:
 q0 , ab   q2
*
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
41
 q0 , abbbaa   q5
*
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
42
 q1 , bba   q4
*
a, b
q5
b
q0 a
2nd Sem 2017
a
a
b
q1 b q2 b q3 a
Lecture 3
a, b
q4
43
Special case:
for any state q
 q ,    q
*
2nd Sem 2017
Lecture 3
44
In general:
 q ,w   q 
*
implies that there is a walk of transitions
w   1 2  k
q
1
2
k
q
states may be repeated
q
2nd Sem 2017
w
Lecture 3
q
45
Language Accepted by DFA
Language of DFA
M:
it is denoted as L M  and contains
all the strings accepted by M
We say that a language L
is accepted (or recognized)
by DFA M if L M  L
 
2nd Sem 2017
Lecture 3
46
For a DFA
M  Q, ,  , q0 , F 
Language accepted by
M:

L M   w   :  q0 ,w   F
q0
2nd Sem 2017
*
w
Lecture 3
*
q

q  F
47
Language rejected by

M:
LM   w   :  q0 ,w   F
q0
2nd Sem 2017
*
w
*
q
Lecture 3

q  F
48
More DFA Examples
  {a , b }
a, b
a, b
q0
q0
L(M )  { }
L( M )  
Empty language
All strings
2nd Sem 2017
Lecture 3
*
49
  {a , b }
a, b
q0
a, b
q0
L(M )  { }
Language of the empty string
2nd Sem 2017
Lecture 3
50
  {a , b }
LM = { all strings with prefix ab }
a, b
q0
a
q1
b
a
q3
2nd Sem 2017
Lecture 3
b
q2
accept
a, b
51
LM  = { all binary strings containing
substring 001 }
0,1
0
1
1

0
0
00
1
001
0
2nd Sem 2017
Lecture 3
52
LM  = { all binary strings without
substring 001 }
0
1
0,1
1

0
0
00
1
001
0
2nd Sem 2017
Lecture 3
53

L(M )  awa : w  a , b 
*
b

a
b
q0
a
q2
q3
a
b
q4
a, b
2nd Sem 2017
Lecture 3
54
Regular Languages
Definition:
A language L is regular if there is
a DFA M that accepts it ( L(M )  L )
The languages accepted by all DFAs
form the family of regular languages
2nd Sem 2017
Lecture 3
55
Example regular languages:
abba  , ab, abba
n
*
{a b : n  0} awa : w  a , b  
{ all strings in {a,b}* with prefix
ab }
{ all binary strings without substring 001}
{x : x  {1} and x is even}
*
{ } { } {a , b }
*
There exist automata that accept these
languages (see previous slides).
2nd Sem 2017
Lecture 3
56
There exist languages which are not Regular:
L {a b : n  0}
n n
ADDITION  {x  y  z : x  1 , y  1 , z  1 ,
n
m
k
nm k}
There is no DFA that accepts these languages
(we will prove this in a later class)
2nd Sem 2017
Lecture 3
57