Automata theory ppt

7/24/2019 Automata theory ppt

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CS205

Language Theory



2

Time Table

Day Time Class Room

Monday 12:00-12:55 L201Tuesday 10:00-10:55 CR101

Wednesday 11:00-11:55 CR101

Friday 11:00-11:55 CR101



3

Course Contents

• Introduction

•Finite Automata and regular languages

•Push down Automata and Contet !ree languages

•Push down Automata

•Turing "achines and Com#utability

•$ecidability% undecidability and reducibility

•Com#utational Com#leity & 'P(Com#leteness



4

)oo*s

Text Book: +o#cro!t% ,ohn -./ "otwani% a1ee & 3llman% ,e!!rey $.% Introduction to

Automata Theory% Languages% and Com#utation% Third -dition% Pearson -ducation Inc.%

'ew $elhi% 2004.

Reference Book: Si#ser% "ichael% Introduction to the Theory o! Com#utation% Second

-dition% Cengage Learning India Pt. Ltd.% 'ew $elhi% 2004



5

"ar*s $istribution Quiz 1: 10%

Mid Semester: 25%

Quiz 2: 10%

*End Semester: 50%

+!"ss #erform"nce: 5%

*End Semester co$ers &o!e s'!!"(us st"rtin) from

tod"' itse!f

+!"ss #erform"nce is e$"!u"ted ("sed on "ttend"nce

"nd "ssi)nments



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Language Processing System

Source Program

"odi!ied source #rogram

Target assembly #rogram

elocatable machine code

Target machine code

re#rocessor

om#i!er

ssem(!er

,inker-,o"der

-#ands macros

library !iles

relocatable ob1ects



7

e will show later in class

• +ow to build com#ilers !or #rogramming languages

• Some com#utational #roblems cannot be soled

• Some #roblems are hard to sole



8

"athematical Preliminaries



9

"athematical Preliminaries

• Sets

• elations

• 6ra#hs

• Proo! Techni7ues



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}321!= A

A set is a collection o! elements

S-TS

}! airplanebicyclebustrain B =

e write

A∈1

B ship∉



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Set e#resentations

C 8 9 a% b% c% d% e% !% g% h% i% 1% * :C 8 9 a% b% ;% * :

S 8 9 2% <% =% ; :

S 8 9 1 > 1 ? 0% and 1 8 2* !or some *?0 :

S 8 9 1 > 1 is nonnegatie and een :

!inite set

in!inite set



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A 8 9 @% 2% % <% 5 :

3niersal Set> All #ossible elements

3 8 9 @ % ; % @0 :

@ 2

< 5

A

3

=

4

B

@0



13

Set D#erations

A 8 9 @% 2% : ) 8 9 2% % <% 5:

• 3nion

A 3 ) 8 9 @% 2% % <% 5 :

• Intersection

A ) 8 9 2% :

• $i!!erence

A ( ) 8 9 @ :

) ( A 8 9 <% 5 :

3

A )

A()



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• Com#lement

3niersal set 8 9@% ;% 4:

A 8 9 @% 2% : A 8 9 <% 5% =% 4:

@2

<

5

=

4

A A

A 8 A



15

02

<

=

@

5

4

een

9 een integers : 8 9 odd integers :

odd

Integers



16

$e"organEs Laws

A 3 ) 8 A ) 3

A ) 8 A 3 ) 3



17

-m#ty% 'ull Set>

8 9 :

S 3 8 S

S 8

S ( 8 S

( S 8

38 3niersal Set



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Subset

A 8 9 @% 2% : ) 8 9 @% 2% % <% 5 :

A ) 3

Pro#er Subset> A ) 3

A

)



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$is1oint Sets

A 8 9 @% 2% : ) 8 9 5% =:

A ) 8 3

A )



20

Set Cardinality

• For !inite sets

A 8 9 2% 5% 4 :

A 8



21

Powersets

A #owerset is a set o! sets

Powerset o! S 8 the set o! all the subsets o! S

S 8 9 a% b% c :

2S 8 9 % 9a:% 9b:% 9c:% 9a% b:% 9a% c:% 9b% c:% 9a% b% c: :

Dbseration> 2S 8 2S G B 8 2 H

C P d



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Cartesian Product

A 8 9 2% < : ) 8 9 2% % 5 :

A ) 8 9 G2% 2H% G2% H% G2% 5H% G <% 2H% G<% H% G<% <H :



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-LATID'S

8 9G@% y@H% G2% y2H% G% yH% ;:

i yi



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-7uialence elations

• e!leie>

• Symmetric> y y

• Transitie> J and y K K

-am#le> 8 8

• 8

• 8 y y 8

• 8 y and y 8 K 8 K



25

6AP+SA directed gra#h

• 'odes GMerticesH

M 8 9 a% b% c% d% e :

• -dges

- 8 9 Ga% bH% Gb% cH% Gc% aH% Gb% dH% Gd% cH% Ge% dH :

e

a

b

c

dnode

e d g e



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Labeled 6ra#h

a

b

c

d

e

@

5 =

2=



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al*

a

b

c

d

e

al* is a se7uence o! ad1acent edges Ge% dH% Gd% cH% Gc% aH



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Path

a

b

c

d

e

Path is a wal* where no edge is re#eated

Sim#le #ath> no node is re#eated

l



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Cycle

a

b

c

d

e

@2

Cycle> a wal* !rom a node GbaseH to itsel!

Sim#le cycle> only the base node is re#eated

base

- l T



30

-uler Tour

a

b

c

d

e @

2

<5

=

4

B base

A cycle that contains each edge once



31

+amiltonian Cycle

a

b

cd

e@

2

<

5base

A sim#le cycle that contains all nodes



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Finding All Sim#le Paths

a

b

cd

e!

St @



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a

b

c

d

e

Gc% aH

Gc% eH

!

Ste# @



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a

b

c

d

e

Gc% aH

Gc% aH% Ga% bH

Gc% eH

Gc% eH% Ge% bH

Gc% eH% Ge% dH

Ste# 2

!

Ste#



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Ste#

ab

c

d

e

!

Gc% aHGc% aH% Ga% bH

Gc% eH

Gc% eH% Ge% bH

Gc% eH% Ge% dH

Gc% eH% Ge% dH% Gd% !H

e#eat the same

!or each starting node

Trees



36

Treesroot

lea!

#arent

child

Trees hae no cycles



37

root

lea!

Leel 0

Leel @

Leel 2

Leel

+eight

)i T



38

)inary Trees



39

PDDF T-C+'IN3-S

• Proo! by induction

• Proo! by contradiction



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Induction

e hae statements P@% P2% P% ;

I! we *now

• !or some * that P@% P2% ;% P* are true• !or any n ?8 * that

P@% P2% ;% Pn im#ly PnO@

Then

-ery Pi is true



41

Proo! by Induction• Inductie basis

Find P@% P2% ;% P* which are true

• Inductie hy#othesisLetEs assume P@% P2% ;% Pn are true%

!or any n ?8 *

• Inductie ste#

Show that PnO@ is true



42

-am#le

Theorem> A binary tree o! height n

has at most 2n leaes.

Proo!>

let lGiH be the number o! leaes at leel i

lG0H 8 @

lGH 8 B



43

e want to show> lGiH 8 2i

• Inductie basis

lG0H 8 @ Gthe root nodeH

• Inductie hy#othesis

LetEs assume lGiH 8 2i !or all i 8 0% @% ;% n

• Induction ste#

we need to show that lGn O @H 8 2nO@

I d ti St



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Induction Ste#

hy#othesis> lGnH 8 2n

Leel

n

nO@

I d ti St



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hy#othesis> lGnH 8 2n

Leel

n

nO@

lGnO@H 8 2 Q lGnH 8 2 Q 2n 8 2nO@

Induction Ste#

Proo! by Contradiction



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Proo! by Contradiction

e want to #roe that a statement P is true

• we assume that P is !alse

• then we arrie at an incorrect conclusion

• there!ore% statement P must be true

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Automata theory ppt