38
Courtesy: Costas/Ullman 1 Regular Expressions
Courtesy: Costas/Ullman 1
Regular Expressions
2
RE’s: Introduction
• Regular expressions describe languages by an algebra.
• They describe exactly the regular languages.
• If E is a regular expression, then L(E) is the language it defines.
• We’ll describe RE’s and their languages recursively.
Courtesy: Costas/Ullman
3
Operations on Languages• RE’s use three operations: union,
concatenation, and Kleene star.
• The union of languages is the usual thing, since languages are sets.
• Example: {01,111,10}{00, 01} = {01,111,10,00}.
Courtesy: Costas/Ullman
4
Concatenation
• The concatenation of languages L and M is denoted LM.
• It contains every string wx such that w is in L and x is in M.
• Example: {01,111,10}{00, 01} = {0100, 0101, 11100, 11101, 1000, 1001}.
Courtesy: Costas/Ullman
5
Kleene Star• If L is a language, then L*, the Kleene
star or just “star,” is the set of strings formed by concatenating zero or more strings from L, in any order.
• L* = {ε} L LL LLL …
• Example: {0,10}* = {ε, 0, 10, 00, 010, 100, 1010,…}
Courtesy: Costas/Ullman
6
RE’s: Definition
• Basis 1: If a is any symbol, then a is a RE, and L(a) = {a}.– Note: {a} is the language containing one
string, and that string is of length 1.
• Basis 2: ε is a RE, and L(ε) = {ε}.
• Basis 3: ∅ is a RE, and L(∅) = ∅.
Courtesy: Costas/Ullman
7
RE’s: Definition – (2)• Induction 1: If E1 and E2 are regular
expressions, then E1+E2 is a regular expression, and L(E1+E2) = L(E1)L(E2).
• Induction 2: If E1 and E2 are regular expressions, then E1E2 is a regular expression, and L(E1E2) = L(E1)L(E2).
• Induction 3: If E is a RE, then E* is a RE, and L(E*) = (L(E))*.Courtesy: Costas/Ullman
Courtesy: Costas/Ullman 8
Definition (continued)
For regular expressions and
1r 2r
2121 rLrLrrL
2121 rLrLrrL
** 11 rLrL
11 rLrL
9
Precedence of Operators
• Parentheses may be used wherever needed to influence the grouping of operators.
• Order of precedence is * (highest), then concatenation, then + (lowest).
Courtesy: Costas/Ullman
Courtesy: Costas/Ullman 10
ExampleRegular expression: *aba
*abaL *aLbaL *aLbaL *aLbLaL
*aba ,...,,,, aaaaaaba
,...,,,...,,, baababaaaaaa
Courtesy: Costas/Ullman 11
Example
Regular expression bbabar *
,...,,,,, bbbbaabbaabbarL
Courtesy: Costas/Ullman 12
Example
Regular expression bbbaar **
}0,:{ 22 mnbbarL mn
Courtesy: Costas/Ullman 13
Example
Regular expression *)10(00*)10( r
)(rL = { all strings containing substring 00 }
Courtesy: Costas/Ullman 14
Example
Regular expression )0(*)011( r
)(rL = { all strings without substring 00 }
Courtesy: Costas/Ullman 15
Equivalent Regular Expressions
Definition:
Regular expressions and
are equivalent if
1r 2r
)()( 21 rLrL
Courtesy: Costas/Ullman 16
Example L= { all strings without substring 00 }
)0(*)011(1 r
)0(*1)0(**)011*1(2 r
LrLrL )()( 211r 2rand
are equivalentregular expressions
Courtesy: Costas/Ullman 17
Regular Expressionsand
Regular Languages
Courtesy: Costas/Ullman 18
Theorem
LanguagesGenerated byRegular Expressions
RegularLanguages
Courtesy: Costas/Ullman 19
LanguagesGenerated byRegular Expressions
RegularLanguages
LanguagesGenerated byRegular Expressions
RegularLanguages
Proof:
Courtesy: Costas/Ullman 20
Proof - Part 1
r)(rL
For any regular expression the language is regular
LanguagesGenerated byRegular Expressions
RegularLanguages
Proof by induction on the size of r
Courtesy: Costas/Ullman 21
Induction BasisPrimitive Regular Expressions: ,,Corresponding NFAs
)()( 1 LML
)(}{)( 2 LML
)(}{)( 3 aLaML
regularlanguages
a
Courtesy: Costas/Ullman 22
Inductive Hypothesis Suppose that for regular expressions and , and are regular languages
1r 2r)( 1rL )( 2rL
Courtesy: Costas/Ullman 23
Inductive StepWe will prove:
1
1
21
21
*
rL
rL
rrL
rrL
Are regular Languages
Courtesy: Costas/Ullman 24
By definition of regular expressions:
11
11
2121
2121
**
rLrL
rLrL
rLrLrrL
rLrLrrL
Courtesy: Costas/Ullman 25
)( 1rL )( 2rLBy inductive hypothesis we know: and are regular languages
Regular languages are closed under: *1
21
21
rLrLrLrLrL Union
Concatenation
Star
We also know:
Courtesy: Costas/Ullman 26
Therefore:
** 11
2121
2121
rLrL
rLrLrrL
rLrLrrL
Are regularlanguages
)())(( 11 rLrL is trivially a regular language(by induction hypothesis)
End of Proof-Part 1
Courtesy: Costas/Ullman 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
Courtesy: Costas/Ullman 28
For any regular language there is a regular expression with
Proof - Part 2
LanguagesGenerated byRegular Expressions
RegularLanguages
Lr LrL )(
We will convert an NFA that accepts to a regular expression
L
Courtesy: Costas/Ullman 29
Since is regular, there is aNFA that accepts it
LM
LML )(
Take it with a single accept state
Courtesy: Costas/Ullman 30
From construct the equivalentGeneralized Transition Graphin which transition labels are regular
expressions
M
Example:
a
ba,
cM
a
ba
c
CorrespondingGeneralized transition graph
Courtesy: Costas/Ullman 31
Another Example:
ba a
b
b0q 1q 2q
ba,a
b
b0q 1q 2q
b
bTransition labels are regular expressions
Courtesy: Costas/Ullman 32
Reducing the states:ba
ab
b0q 1q 2q
b
0q 2q
babb*
)(* babb
Transition labels are regular expressions
Courtesy: Costas/Ullman 33
Resulting Regular Expression:
0q 2q
babb*
)(* babb
*)(**)*( bbabbabbr
LMLrL )()(
Courtesy: Costas/Ullman 34
In GeneralRemoving a state:
iq q jqa b
cde
iq jq
dae* bce*dce*
bae*
2-neighbors
Courtesy: Costas/Ullman 35
iq jq
dae* bce*dce*
bae*
iq q jqa b
cde
kq
f g
kq
fge*
dge*
fae*
bge*fce*
This can be generalized to arbitrary number of neighbors to q
3-neighbors
Courtesy: Costas/Ullman 36
0q fq
1r
2r
3r4r
*)*(* 213421 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
Courtesy: Costas/Ullman 37
Standard Representations of Regular Languages
Regular Languages
DFAs
NFAsRegularExpressions
Courtesy: Costas/Ullman 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)
