Standard Representations
of Regular Languages
Regular Languages
DFAs
NFAs
Regular
Grammars
Regular
Expressions
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When we say:
We mean:
We are given
a Regular Language
L
Language L is in a standard
representation
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Elementary Questions
about
Regular Languages
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Membership Question
Question:
Answer:
Given regular language L
and string w
how can we check if w 
L?
Take the DFA that accepts L
and check if w is accepted
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DFA
w
w L
DFA
w
w L
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Question:
Answer:
Given regular language L
how can we check
if L is empty: ( L  ) ?
Take the DFA that accepts
L
Check if there is any path from
the initial state to a final state
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DFA
L
DFA
L
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Question:
Given regular language
how can we check
if L is finite?
Answer: Take the DFA that accepts
L
L
Check if there is a walk with cycle
from the initial state to a final state
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DFA
L is infinite
DFA
L is finite
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Question: Given regular languages L1 and
how can we check if L1  L2 ?
Answer:
Find if
L2
( L1  L2 )  ( L1  L2 )  
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( L1  L2 )  ( L1  L2 )  
and
L1  L2  
L1
L1  L2  
L2
L2 L
2
L1  L2
L1 L1
L2  L1
L1  L2
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( L1  L2 )  ( L1  L2 )  
L1  L2  
L1
or
L1  L2  
L2
L2
L1  L2
L1
L2  L1
L1  L2
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Non-regular languages
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{a b : n  0}
n n
Non-regular languages
{vv : v {a, b}*}
R
Regular languages
a *b
b*c  a
b  c ( a  b) *
etc...
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How can we prove that a language
is not regular?
L
Prove that there is no DFA that accepts
L
Problem: this is not easy to prove
Solution: the Pumping Lemma !!!
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The Pigeonhole Principle
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4 pigeons
3 pigeonholes
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A pigeonhole must
contain at least two pigeons
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n pigeons
...........
m
pigeonholes
nm
...........
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The Pigeonhole Principle
n pigeons
m
pigeonholes
nm
There is a pigeonhole
with at least 2 pigeons
...........
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The Pigeonhole Principle
and
DFAs
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DFA with
b
q1
4
states
b
b
a
q2
b
q3
a
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q4
a
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In walks of strings:
no state
is repeated
a
aa
aab
b
q1
b
b
a
q2
a
q3
a
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q4
a
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In walks of strings:
a state
is repeated
aabb
bbaa
abbabb
abbbabbabb...
b
q1
b
b
a
q2
a
q3
a
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q4
a
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If string
w has length | w |  4 :
Then the transitions of string w
are more than the states of the DFA
Thus, a state must be repeated
b
q1
b
b
a
q2
a
q3
a
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q4
a
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In general, for any DFA:
String
A state
w has length  number of states
q
walk of
must be repeated in the walk of
w
w
......
q
......
Repeated
state
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In other words for a string
w:
a
transitions are pigeons
q
states are pigeonholes
walk of
w
......
q
......
Repeated state
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