Transcript Languages and Finite Automata

Deterministic
Finite Automata
And Regular Languages
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Deterministic Finite Automaton (DFA)
Input Tape
String
Finite
Automaton
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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
transition
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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
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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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Scanning the Input
a b b a
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
6
a b b a
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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a b b a
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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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
Last state determines the outcome
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A Rejection Case
a b a
Input String
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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a b a
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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a b a
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
12
Input finished
a b a
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
reject
a, b
q4
Last state determines the outcome
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Another Rejection Case
Tape is empty
( )
Input Finished (no symbol read)
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
a, b
q4
reject
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This automaton accepts only one string
Language Accepted:
L  abba 
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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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
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Another Example
L  , ab , abba 
a, b
q5
b
q0 a
Accept
state
a
a
b
q1 b q2 b q3 a
Accept
state
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a, b
q4
Accept
state
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Empty Tape
( )
Input Finished
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
a, b
q4
accept
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Another Example
a, b
a
q0
b
q1
Accept
state
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a, b
q2
trap state
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a a b
Input String
a, b
a
q0
b
q1
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a, b
q2
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a a b
a, b
a
q0
b
q1
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a, b
q2
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a a b
a, b
a
q0
b
q1
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a, b
q2
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Input finished
a a b
a
q0
a, b
accept
b
q1
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a, b
q2
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A rejection case
b a b
Input String
a, b
a
q0
b
q1
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a, b
q2
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b a b
a, b
a
q0
b
q1
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a, b
q2
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b a b
a, b
a
q0
b
q1
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a, b
q2
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Input finished
b a b
a, b
a
q0
b
q1
a, b
q2
reject
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Language Accepted:
L  {a b : n  0}
n
a, b
a
q0
b
q1
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a, b
q2
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Another Example
Alphabet:
  {1}
1
q0
q1
1
Language Accepted:
EVEN  {x : x   and x is even}
*
 {, 11, 1111, 111111, }
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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
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Set of States Q
Example
Q  q0 , q1, q2 , q3 , q4 , q5 
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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Input Alphabet 
 
:the input alphabet never contains
Example
  a, b
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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
a, b
q4
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Initial State q0
Example
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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Set of Accepting States F  Q
Example
F  q4 
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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Transition Function
 :Q  Q
 (q , x )  q 
q
x
q
Describes the result of a transition
from state q with symbol x
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Example:
 q0 , a   q1
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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 q0 , b   q5
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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 q2 , b   q3
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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Transition Table for
states

symbols
q0
a
q1
q2
q5
q1
q3
q4
q5

q5
q4
q5
q5
b
q5
q2
q3
q5
q5
q5
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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Extended Transition Function
 :Q   Q
*
*
 (q ,w )  q 
*
Describes the resulting state
after scanning string w from state
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q
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Example:
 q0 , ab   q2
*
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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 q0 , abbbaa   q5
*
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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 q1 , bba   q4
*
a, b
q5
b
q0 a
a
a
b
q1 b q2 b q3 a
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a, b
q4
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Special case:
for any state q
 q ,    q
*
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In general:
 q ,w   q 
*
implies that there is a walk of transitions
w   1 2  k
q
1
k
2
q
states may be repeated
q
w
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q
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Language Accepted by DFA
Language accepted by DFA
M:
it is denoted as L M  and contains
all the strings accepted by M
We also say that
M
recognizes L M 
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For a DFA
M  Q, ,  , q0 , F 
Language accepted by

M:
L M   w   :  q0 ,w   F
q0
*
w
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*
q

q  F
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Language rejected by

M:
LM   w   :  q0 ,w   F
q0
*
w
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*
q

q  F
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More DFA Examples
  {a , b }
a, b
a, b
q0
q0
L(M )  { }
L( M )  
Empty language
All strings
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*
49
  {a , b }
a, b
q0
a, b
q1
L(M )  { }
Language of the empty string
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  {a , b }
LM = { all strings with prefix ab }
a, b
q0
a
q1
b
a
q3
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b
q2
accept
a, b
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LM  = { all binary strings containing
substring 001 }
0,1
0
1
1

0
0
00
1
001
0
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LM  = { all binary strings without
substring 001 }
0
1
0,1
1

0
0
00
1
001
0
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
L(M )  awa : w  a , b 
*
b

a
b
q0
a
q2
q3
a
b
q1
a, b
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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
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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 DFAs that accept these
languages (see previous slides).
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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 are no DFAs that accept these languages
(we will prove this in a later class)
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