This document discusses regular expressions and regular languages. It begins with definitions of regular expressions and examples. It then defines the language of a regular expression as the set of strings it describes. It proves that languages described by regular expressions are precisely the regular languages by showing regular expression languages are regular and any regular language can be described by a regular expression. It provides examples and discusses converting between regular expressions and non-deterministic finite automata.

1. © 2020 Wael Badawy
1
Regular Expressions
1
DISCLAIMER:
This video is optimized for HD large display using
patented and patent-pending “Nile Codec”, the first
Egyptian Video Codec for more information,
PLEASE check https://NileCodec.com
Also available as a PodCast

2. © 2020 Wael Badawy
Copyright © 2020 Wael Badawy. All rights reserved
n This video is subject to copyright owned by Wael Badawy “WB”. Any
reproduction or republication of all or part of this video is expressly prohibited
unless WB has explicitly granted its prior written consent. All other rights
reserved.
n This video is intended for education and information only and is offered AS IS,
without any warranty of the accuracy or the quality of the content. Any other
use is strictly prohibited. The viewer is fully responsible to verify the accuracy
of the contents received without any claims of costs or liability arising .
n The names, trademarks service marked as logos of WB or the sponsors
appearing in this video may not be used in any any product or service,
without prior express written permission from WB and the video sponsors
n Neither WB nor any party involved in creating, producing or delivering
information and material via this video shall be liable for any direction,
incidental, consequential, indirect of punitive damages arising out of access
to, use or inability to use this content or any errors or omissions in the
content thereof.
n If you will continue to watch this video, you agree to the terms above and
other terms that may be available on http://nu.edu.eg & https://caiwave.net
2

3. © 2020 Wael Badawy

4. © 2020 Wael Badawy
Regular Expressions
Regular expressions
describe regular languages
Example:
describes the language
4
*
)( cba ×+
{ } { },...,,,,,,
*
bcaabcaabcabca e=

5. © 2020 Wael Badawy
Recursive Definition
5
ae,,Æ
( )1
*
1
21
21
r
r
rr
rr
×
+
Are regular expressions
Primitive regular expressions:
2r1rGiven regular expressions and

6. © 2020 Wael Badawy
Examples
6
( ) )(
*
Æ+××+ ccbaA regular expression:
( )++ baNot a regular expression:

7. © 2020 Wael Badawy
Languages of Regular
Expressions
: language of regular expression
Example
7
( )rL r
( ) { },...,,,,,)( *
bcaabcaabcacbaL e=×+

8. © 2020 Wael Badawy
Definition
For primitive regular expressions:
8
( )
( ) { }
( ) { }aaL
L
L
=
=
Æ=Æ
ee

9. © 2020 Wael Badawy
Definition (continued)
For regular expressions and
9
1r 2r
( ) ( ) ( )2121 rLrLrrL È=+
( ) ( ) ( )2121 rLrLrrL =×
( ) ( )( )*
1
*
1 rLrL =
( )( ) ( )11 rLrL =

10. © 2020 Wael Badawy
Example
Regular expression:
10
( ) *
aba ×+
( )( )*abaL ×+ ( )( ) ( )*
aLbaL +=
( ) ( )*
aLbaL +=
( ) ( )( ) ( )( )*
aLbLaL È=
{ } { }( ) { }( )*
aba È=
{ }{ },...,,,, aaaaaaba e=
{ },...,,,...,,, baababaaaaaa=

11. © 2020 Wael Badawy
Example
Regular expression
11
( ) ( )bbabar ++=
*
( ) { },...,,,,, bbbbaabbaabbarL =

12. © 2020 Wael Badawy
Example
Regular expression
12
( ) ( ) bbbaar
**
=
( ) }0,:{ 22
³= mnbbarL mn

13. © 2020 Wael Badawy
Example
Regular expression
13
**
)10(00)10( ++=r
)(rL = { all strings containing substring 00 }

14. © 2020 Wael Badawy
Example
Regular expression
14
)0()011( *
e++=r
)(rL = { all strings without substring 00 }

15. © 2020 Wael Badawy
Equivalent Regular Expressions
Definition:
Regular expressions and
are equivalent if
15
1r 2r
)()( 21 rLrL =

16. © 2020 Wael Badawy
Example
16
L = { all strings without substring 00 }
)0()011( *
1 e++=r
)0(1)0()0111( ****
2 ee +++=r
LrLrL == )()( 21
1r 2rand
are equivalent
regular expressions

17. 17
Regular Expressions
and
Regular Languages
17

18. © 2020 Wael Badawy
Theorem
18
Languages
Generated by
Regular Expressions
Regular
Languages=

19. © 2020 Wael Badawy
19
Languages
Generated by
Regular Expressions
Regular
Languages
Í
Languages
Generated by
Regular Expressions
Regular
Languages
Ê
Proof:

20. © 2020 Wael Badawy
20
Proof - Part 1
r
)(rL
For any regular expression
the language is regular
Languages
Generated by
Regular Expressions
Regular
Languages
Í
Proof by induction on the size of r

21. © 2020 Wael Badawy
Induction Basis
Primitive Regular Expressions:
21
ae,,Æ
Corresponding
NFAs
)()( 1 Æ=Æ= LML
)(}{)( 2 ee LML ==
)(}{)( 3 aLaML ==
regular
languages
a

22. © 2020 Wael Badawy
Inductive Hypothesis
Suppose
that for regular expressions and ,
and are regular languages
22
1r 2r
)( 1rL )( 2rL

23. © 2020 Wael Badawy
Inductive Step
We will prove:
23
( )
( )
( )
( )( )1
*
1
21
21
rL
rL
rrL
rrL
×
+
Are regular
Languages

24. © 2020 Wael Badawy
By definition of regular expressions:
24
( ) ( ) ( )
( ) ( ) ( )
( ) ( )( )
( )( ) ( )11
*
1
*
1
2121
2121
rLrL
rLrL
rLrLrrL
rLrLrrL
=
=
=×
È=+

25. © 2020 Wael Badawy
25
)( 1rL )( 2rL
By inductive hypothesis we know:
and are regular languages
Regular languages are closed under:
( ) ( )
( ) ( )
( )( )*
1
21
21
rL
rLrL
rLrL ÈUnion
Concatenation
Star
We also know:

26. © 2020 Wael Badawy
Therefore:
26
( ) ( ) ( )
( ) ( ) ( )
( ) ( )( )*
1
*
1
2121
2121
rLrL
rLrLrrL
rLrLrrL
=
=×
È=+
Are regular
languages
)())(( 11 rLrL = is trivially a regular language
(by induction hypothesis)
End of Proof-Part 1

27. © 2020 Wael Badawy
27
Using the regular closure of operations,
we can construct recursively the NFA
that accepts
M
)()( rLML =
Example: 21 rrr +=
)()( 11 rLML =
)()( 22 rLML =
)()( rLML =
e
e

28. © 2020 Wael Badawy
28
For any regular language there is
a regular expression with
Proof - Part 2
Languages
Generated by
Regular Expressions
Regular
Languages
Ê
L
r LrL =)(
We will convert an NFA that accepts
to a regular expression
L

29. © 2020 Wael Badawy
Since is regular, there is a
NFA that accepts it
29
L
M
LML =)(
Take it with a single accept state

30. © 2020 Wael Badawy
From construct the equivalent
Generalized Transition Graph
in which transition labels are regular expressions
30
M
Example:
a
ba,
c
M
a
ba +
c
Corresponding
Generalized transition graph

31. © 2020 Wael Badawy
Another Example:
31
ba +
a
b
b
0q 1q 2q
ba,
a
b
b
0q 1q 2q
b
bTransition labels
are regular
expressions

32. © 2020 Wael Badawy
Reducing the states:
32
ba +
a
b
b
0q 1q 2q
b
0q 2q
babb*
)(*
babb +
Transition labels
are regular
expressions

33. © 2020 Wael Badawy
Resulting Regular Expression:
33
0q 2q
babb*
)(*
babb +
****
)()( bbabbabbr +=
LMLrL == )()(

34. © 2020 Wael Badawy
In General
Removing a state:
34
iq q jq
a b
cd
e
iq jq
dae*
bce*
dce*
bae*
2-neighbors

35. © 2020 Wael Badawy
35
iq jq
dae*
bce*
dce*
bae*
iq q jq
a b
cd
e
kq
f g
kq
fge*
dge*
fae*
bge*
fce*
This can be generalized
to arbitrary number
of neighbors to q
3-neighbors

36. © 2020 Wael Badawy
36
0q fq
1r
2r
3r
4r
*
2
*
1342
*
1 )( rrrrrrr +=
LMLrL == )()(
The resulting regular expression:
By repeating the process until
two states are left, the resulting graph is
Initial graph Resulting graph
End of Proof-Part 2

37. © 2020 Wael Badawy
Standard Representations
of Regular Languages
37
Regular Languages
DFAs
NFAs
Regular
Expressions

38. © 2020 Wael Badawy
38
When we say: We are given
a Regular Language
We mean:
L
Language is in a standard
representation
L
(DFA, NFA, or Regular Expression)