1

A Single Final Statefor Finite Accepters

2

ObservationAny Finite Accepter (NFA or DFA)

can be converted to an equivalent NFA

with a single final state

3

Example

a

b

b

aNFA

Equivalent NFA

a

b

b

a

4

In GeneralNFA

Equivalent NFA

Singlefinal state

5

Extreme Case

NFA without final state

Add a final state

6

Properties of Regular Languages

7

PropertiesTake any regular languages andWe will prove:

1L 2L

21 LL

21LL

*1L

Union:

Concatenation:

Star:Are regularLanguages

Complement:

Intersection: 21 LL

1L

8

We SayRegular Languages are closedclosed:

– Under union:

– Under concatenation:

– Under the star operation:

– Under complement:

– Under intersection:

21 LL

21LL

*1L

21 LL

1L

9

For regular languages and take NFAs and with

1M 2M

22

11

LML

LML

1M 2M

Single final state

1L 2L

10

Example

baL n1

baL 2

a

b

ab

1M

2M

11

UnionNFA for

1M

2M

21 LL

12

Example

a

b

ab

baL n1

baL 2

abbaLL n ,21 NFA for

13

ConcatenationNFA for 21LL

1M 2M

14

Example NFA for

a

b ab

baL n1 baL 2

bbaababaLL nn 21

15

Star OperationNFA for *1L

1M

16

ExampleNFA for *1* baL n

a

b

baL n1

17

ComplementFor the complement of regular language :

Take the DFA that accepts

Construct such that: – Each final state of is nonfinal in nonfinal final

We have:

LL

LM

M M M

LMLML

18

Example

a

b ba,

ba, baL n

a

b ba,

ba, baL n

19

IntersectionFor regular languages and : 1L 2L

21

2121

LL

LLLL

1Lregular

1L

regular

2L 2L

21 LL

regular

21 LL

21 LL regular

regular

20

ExampleRegular languages:

lkbaL 1 babL m2

bbLL m 21The language

0,, mlk

is regular

21

Regular Expressions

22

Regular ExpressionsRegular expressionsare another way of expressing regular languages

Example:

Stands for the language

*)( cba

,...,,,,,*, bcaabcaabcabca

23

Recursive DefinitionRegular Expressions:

• Primitive regular expressions:

• Given regular expressions and

,,

2r1r

11

21

21

*

r

r

rr

rr

Are regular expressions

24

ExamplesA regular expression

Not a regular expression

)(* ccba

ba

25

Languages of Regular Expressions : language of regular expression

Example

rL r

,...,,,,,*)( bcaabcaabcacbaL

26

Definition

For primitive regular expressions:

aaL

L

L

27

Definition (continued)For regular expressions and

1r 2r

11

11

2121

2121

**

rLrL

rLrL

rLrLrrL

rLrLrrL

28

ExampleRegular expression: *aba

*abaL

,...,,,...,,,

,...,,,,

*

*

*

*

baababaaaaaa

aaaaaaba

aba

aLbLaL

aLbaL

aLbaL

29

Example

Regular expression bbabar *

,...,,,,, bbbbaabbaabbarL

30

Example

Regular expression bbbaar **

0,:22 mnbbarL mn

31

Example

Regular expression *)10(00*)10( r

)(rL = { all strings with at least two consecutive 0 }

32

Example

Regular expression )0(*)011( r

)(rL = { all strings without two consecutive 0 }

33

Equivalent Regular Expressions

Definition:

Regular expressions and

are equivalent if

1r 2r

)()( 21 rLrL

34

Example L= { all strings with at least

two consecutive 0 }

)0(*)011(1 r

)0(*1)0(**)011*1(2 r

LrLrL )()( 211r 2rand

are equivalentregular expr.

35

Regular Expressionsand

Regular Languages

36

Theorem

The class of languages described by Regular expressionsis identical to the Regular languages

37

In Other Words

• For any regular expression the language is regular

• For any regular language there is a regular expression with

r

)(rL

L

r LrL )(

38

ProofFirst we prove:

• For any regular expression the language is regular

r)(rL

39

Induction BasisPrimitive Regular Expressions: ,,

NFAs

)()( 1 LML

)(}{)( 2 LML

)(}{)( 3 aLaML

regularlanguages

40

Inductive Hypothesis

Assume for regular expressions andthat and are regular languages

1r 2r

)( 1rL )( 2rL

41

Inductive StepWe will prove that:

1

1

21

21

*

rL

rL

rrL

rrL

Are regular Languages

42

By definition of regular expressions:

11

11

2121

2121

**

rLrL

rLrL

rLrLrrL

rLrLrrL

43

By inductive hypothesis and are regular languages

We know:

)( 1rL )( 2rL

Regular languagesare closed under union concatenation star operation

*1

21

21

rL

rLrL

rLrL

44

Therefore:

** 11

2121

2121

rLrL

rLrLrrL

rLrLrrL

Are regularlanguages

45

And trivially:

))(( 1rL is a regular language

46

Proof - Second PartNow we want to prove:

• For any regular language there is a regular expression with

L

r LrL )(

47

Since is regular take the NFA that accepts it

LM

LML )(

Single final state

48

From Construct the equivalentGeneralized Transition Graph

labels of transitions are regular expressions

M

Example:

a

ba,

cM

a

ba

c

49

Another Example:

ba a

b

b

0q 1q 2q

ba,a

b

b

0q 1q 2q

b

b

50

Reducing the states:

ba a

b

b

0q 1q 2q

b

0q 2q

babb*

)(* babb

51

Resulting Regular Expression:

0q 2q

babb*

)(* babb

*)(**)*( bbabbabbr

LMLrL )()(

52

In GeneralRemoving states:

iq q jqa b

cde

iq jq

dae* bce*dce*

bae*

53

Obtaining the final regular expression:

0q fq

1r

2r

3r4r

*)*(* 213421 rrrrrrr

LMLrL )()(