Regular Expressions
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Regular Expressions
Regular expressions
describe regular languages
Example:
( a  b  c) *
describes the language
a, bc*   , a, bc, aa, abc, bca,...
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Recursive Definition
Primitive regular expressions:
Given regular expressions
r1
,  , 
and r2
r1  r2
r1  r2
r1 *
Are regular expressions
r1 
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Examples
A regular expression:
a  b  c  * (c  )
Not a regular expression:
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a  b  
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Languages of Regular Expressions
L r  :
language of regular expression
r
Example
L(a  b  c) *   , a, bc, aa, abc, bca,...
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Definition
For primitive regular expressions:
L   
L    
La   a
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Definition (continued)
For regular expressions r1 and r2
Lr1  r2   Lr1   Lr2 
Lr1  r2   Lr1  Lr2 
Lr1 *  Lr1  *
Lr1   Lr1 
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Example
Regular expression: a  b   a *
La  b   a *  La  b La *
 La  b La *
 La   Lb La  *
 a b a *
 a, b , a, aa, aaa,...
 a, aa, aaa,..., b, ba, baa,...
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Example
Regular expression
r  a  b  * a  bb
Lr   a, bb, aa, abb, ba, bbb,...
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Example
r  aa  * bb * b
Regular expression
Lr   {a b
2n 2m
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b : n, m  0}
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Example
Regular expression
r  (0  1) * 00 (0  1) *
L(r ) = { all strings containing substring 00 }
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Example
Regular expression
r  (1  01) * (0   )
L(r ) = { all strings without substring 00 }
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Equivalent Regular Expressions
Definition:
Regular expressions
are equivalent if
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r1
and
r2
L(r1)  L(r2 )
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Example
L = { all strings without substring 00 }
r1  (1  01) * (0   )
r2  (1* 011*) * (0   )  1* (0   )
and r2
are equivalent
regular expressions
r1
L(r1)  L(r2 )  L
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Regular Expressions
and
Regular Languages
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Theorem
Languages
Generated by
Regular Expressions
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
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Regular
Languages
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Proof:
Languages
Generated by
Regular Expressions

Regular
Languages
Languages
Generated by
Regular Expressions

Regular
Languages
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Proof - Part 1
Languages
Generated by
Regular Expressions

Regular
Languages
For any regular expression
the language
r
L(r ) is regular
Proof by induction on the size of
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r
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Induction Basis
Primitive Regular Expressions:
Corresponding
NFAs
,  , 
L( M1 )    L()
L( M 2 )  {}  L( )
a
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regular
languages
L( M 3 )  {a}  L(a)
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Inductive Hypothesis
Suppose
that for regular expressions r1 and r2 ,
L(r1) and L(r2 ) are regular languages
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Inductive Step
We will prove:
Lr1  r2 
Lr1  r2 
Lr1 *
Are regular
Languages
Lr1 
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By definition of regular expressions:
Lr1  r2   Lr1   Lr2 
Lr1  r2   Lr1  Lr2 
Lr1 *   Lr1  *
Lr1   Lr1 
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By inductive hypothesis we know:
L(r1) and L(r2 ) are regular languages
We also know:
Regular languages are closed under:
Union
Lr1   Lr2 
Concatenation Lr1  Lr2 
Star
 Lr1  *
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Therefore:
Lr1  r2   Lr1   Lr2 
Lr1  r2   Lr1  Lr2 
Are regular
languages
Lr1 *   Lr1  *
L((r1 ))  L(r1 )
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is trivially a regular language
(by induction hypothesis)
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End of Proof-Part 1
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Using the regular closure of these operations,
we can construct recursively the NFA M
that accepts L(M )  L(r )
Example: r  r1  r2
L(M1 )  L(r1 )
L(M )  L(r )


L(M2 )  L(r2 )
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Proof - Part 2
Languages
Generated by
Regular Expressions

Regular
Languages
For any regular language L there is
a regular expression r with L(r ) 
We will convert an NFA that accepts
to
a
regular
expression
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L
L
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Since L is regular, there is a
NFA M that accepts it
L( M )  L
Take it with a single final state
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From M construct the equivalent
Generalized Transition Graph
in which transition labels are regular expressions
Example:
Corresponding
Generalized transition graph
M
a
c
a
ab
a, b
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c
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Another Example:
a
q0
b
b
q1 a, b
q2
b
b
b
Transition labels
are regular
expressions
a
q0
q1 a  b q2
b
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Reducing the states:
b
a
q0
b
q1 a  b q2
b
Transition labels
are regular
expressions
bb * a
q0
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b
bb * (a  b)
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q2
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Resulting Regular Expression:
bb * a
q0
b
bb * (a  b)
q2
r  (bb * a) * bb * (a  b)b *
L( r )  L( M )  L
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In General
e
Removing a state:
d
qi
c
qj
q
a
b
ae * d
ce * b
ce * d
qj
qi
ae * b
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By repeating the process until
two states are left, the resulting graph is
Initial graph
Resulting graph
r1
r4
r3
q0
r2
qf
The resulting regular expression:
r  r1 * r2 (r4  r3r1 * r2 ) *
L( r )  L( M )  L
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End of Proof-Part 2
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Standard Representations
of Regular Languages
Regular Languages
DFAs
NFAs
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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
(DFA, NFA, or Regular Expression)
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Languages and Finite Automata