introduction to toc

8/12/2019 Introduction to toc

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Theory of Computation

Subject code: 160704


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Text and Reference book

Text Book

Introduction To Languages And Theory Of Computation ByJohn C !artin" Third #dition" T!$

Reference Books

% Automata Theory" Languages and Computation" $opcroft"!ot&ani" '((man" )earson #ducation

* Theory of automata" Langusges and computation" +umar"

!c,ra$i((- The Theory of Computation" !oret" )earson #ducation

. Introduction to Computer Theory" Cohen" /i(ey0India


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Computation

C)' memory


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C)'

input memory

output memory

)rogram memory

temporary memory


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C)'

input memory

output memory

)rogram memory

temporary memory

3)( xxf =

compute xx

compute xx 2

#xamp(e1


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C)'

input memory

output memory

)rogram memory

temporary memory

3)( xxf =

compute xx

compute xx 2

2=x


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C)'

input memory

output memory

)rogram memory

temporary memory 3)( xxf =

compute xx

compute xx 2

2=x

42*2 ==z

82*)( ==zxf


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C)'

input memory

output memory)rogram memory

temporary memory 3)( xxf =

compute xx

compute xx 2

2=x

42*2 ==z

82*)( ==zxf

8)( =xf


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Automaton

C)'

input memory

output memory

)rogram memory

temporary memory

Automaton


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2ifferent +inds of Automata

Automata are distinguished by the temporary memory

Finite Automata1 no temporary memory

Pushdown Automata1 stack

Turing Machines1 random access memory


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input memory

output memory

temporary memory

3inite

Automaton

3inite Automaton

4ending !achines 5sma(( computing po&er6


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input memory

output memory

Stack

)ushdo&n

Automaton

)ushdo&n Automaton

)rogramming Languages 5medium computing po&er6

)ush" )op


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input memory

output memory

Random Access !emory

Turing

!achine

Turing !achine

A(gorithms 5highest computing po&er6


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3inite

Automata

)ushdo&n

Automata

Turing

!achine

)o&er of Automata


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/e &i(( sho& (ater in c(ass

$o& to bui(d compi(ers for programming (anguages

7ome computationa( prob(ems cannot be so(8ed

7ome prob(ems are hard to so(8e


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!athematica( )re(iminaries


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!athematica( )re(iminaries

7ets

3unctions

Re(ations

)roof Techni9ues

#


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}3,2,1{=A

A set is a co((ection of e(ements

7#T7

},,,{ airplanebicyclebustrainB =

/e &rite

A1

Bship


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7et Representations

C : ; a" b" c" d" e" f" g" h" i"


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A : ; %" *" -" ." =

'ni8ersa( 7et1 A(( possib(e e(ements

' : ; % " > " % =

% * -

.

A

'

?

D

E%


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7et Operations

A : ; %" *" - = B : ; *" -" ." =

'nion

A ' B : ; %" *" -" ." =

Intersection

A B : ; *" - =

2ifference

A 0 B : ; % =

B 0 A : ; ." =

'

A B

A0B


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Comp(ement

'ni8ersa( set : ;%" >" =

A : ; %" *" - = A : ; ." " ?" =

%*

-.

?

A A

A : A


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*

.

?

%

-

e8en

; e8en integers = : ; odd integers =

odd

Integers


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2e!organFs La&s

A ' B : A B'

A B : A ' B'


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#mpty" Gu(( 7et1

: ; =

7 ' : 7

7 :

7 0 : 7

0 7 :

': 'ni8ersa( 7et


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7ubset

A : ; %" *" -= B : ; %" *" -" ." =

A B'

)roper 7ubset1 A B'

A

B


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2is


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7et Cardina(ity

3or finite sets

A : ; *" " =

HAH : -


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)o&ersets

A po&erset is a set of sets

)o&erset of 7 : the set of a(( the subsets of 7

7 : ; a" b" c =

*7: ; " ;a=" ;b=" ;c=" ;a" b=" ;a" c=" ;b" c=" ;a" b" c= =

Obser8ation1H *7H : *H7H 5 D : *- 6

C t i ) d t


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Cartesian )roduct

A : ; *" . = B : ; *" -" =

A B : ; 5*" *6" 5*" -6" 5*" 6" 5 ." *6" 5." -6" 5." .6 =

HA BH : HAH HBH

,enera(ies to more than t&o sets

A B > K


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3'GCTIOG7domain

%*

-

a

bc

range

f 1 A 0@ B

A B

If A : domainthen f is a tota( function

other&ise f is a partia( function

f5%6 : a


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R#LATIOG7

R : ;5x%" y%6" 5x*" y*6" 5x-" y-6" >=

xiR yi

e g ifR : @F1 * @ %" - @ *" - @ %

In re(ations xican be repeated


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#9ui8a(ence Re(ations

Ref(exi8e1 x R x

7ymmetric1 x R y y R x

Transiti8e1 x R M and y R x R

#xamp(e1R : :

x : x

x : y y : x

x : y andy : x :


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#9ui8a(ence C(asses3or e9ui8a(ence re(ationR

e9ui8a(ence c(ass ofx : ;y 1 x R y=

#xamp(e1R : ; 5%" %6" 5*" *6" 5%" *6" 5*" %6"

5-" -6" 5." .6" 5-" .6" 5." -6 =

#9ui8a(ence c(ass of% : ;%" *=

#9ui8a(ence c(ass of- : ;-" .=


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)ROO3 T#C$GIN'#7

)roof by induction

)roof by contradiction


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Induction

/e ha8e statements)%" )*" )-" >

If &e kno&

for some k that )%" )*" >" )kare true for any n @: k that

)%" )*" >" )n imp(y )n%

Then

#8ery )i is true


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)roof by InductionInducti8e basis

3ind )%" )*" >" )k&hich are true

Inducti8e hypothesisLetFs assume )%" )*" >" )nare true"

for any n @: k

Inducti8e step

7ho& that )n%is true


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#xamp(e

Theorem1 A binary tree of height n

has at most *n (ea8es

)roof1

(et (5i6be the number of (ea8es at (e8e( i

(56 : %

(5-6 : D


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/e &ant to sho&1 (5i6 P: *i

Inducti8e basis

(56 : % 5the root node6

Inducti8e hypothesis

LetFs assume (5i6 P: *ifor a(( i : " %" >" n

Induction step

&e need to sho& that (5n %6 P: *n%

I d i


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Induction 7tep

hypothesis1(5n6 P: *nLe8e(

n

n%

I d i 7


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hypothesis1(5n6 P: *nLe8e(

n

n%

(5n%6 P: * Q (5n6 P: * Q *n : *n%

Induction 7tep

R k


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Remark

Recursion is another thing

#xamp(e of recursi8e function1

f5n6 : f5n0%6 f5n0*6

f56 : %" f5%6 : %

)roof by Contradiction


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)roof by Contradiction

/e &ant to pro8e that a statement ) is true

&e assume that ) is fa(se

then &e arri8e at an incorrect conc(usion

therefore" statement ) must be true


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#xamp(e

Theorem1 is not rationa(

)roof1

Assume by contradiction that it is rationa( : nm

n and m ha8e no common factors

/e &i(( sho& that this is impossib(e

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2

nm * m* n*2


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: nm * m*: n*

Therefore" n* is e8enn is e8en

n : * k

* m*: .k* m*: *k*m is e8en

m : * p

Thus" m and n ha8e common factor *

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introduction to toc

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