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Theory of

ComputationLecture # 1-2-3

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BookIntroduction to Theory of Computation by Anil

Maheshwari Michiel Smid, 2014

Introduction to computer theory by Daniel I.A. Cohen Second

Edition

Introduction to the Theory of Computation

second!third edition"by Michael Sipser

Introduction to Languages and Theory of Computation by !. C.

"artin "c#ra$ %ill &oo' Co. ())* Second Edition

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Main contentsRE

NFA

FA!F"

A

NS

$M%$he !hurch-$urin& $hesis '

eci(able an( )n(eci(able Lan&ua&es

N !o*pleteness !o*ple+ity $heory

Re(ucibility , ntractability

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Lan&ua&eNatural Lan&ua&esEn&lish. !hinese. French. )r(u etc

ro&ra**in& Lan&ua&esBasic. Fortran. !. !//. 0aa etc

Mathe*atics

State (ia&ra*

etc

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!o*ponents o Lan&ua&eAlphabetsBasic ele*ents

Set o letters or characters

Rules"ra**ar$ells you that or(s belon&s to a lan&ua&e

Synta+

Meanin&Se*antics

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e4nitionLan&ua&eSet o strin&s o characters ro* the alphabets

5or(A set o characters belon&s to the lan&ua&e

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En&lish 5or(sAlphabetsa b c667 A B ! 6689 punctuation *arks.

blank space etc

5or(sAll or(s in stan(ar( (ictionary

5hat is on the e+a*$he :uick bron o+ ;u*pe( oer the la7y (o&

E8& conte+t-ree. re&ular66

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!-Lan&ua&eAlphabetsAS! characters

5or(sro&ra*s

#inclu(e

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!onentions

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5hat (oes $heory o auto*ata

*eanD$he or( ?$heory *eans that this sub;ect is

a *ore *athe*atical sub;ect an( lesspractical8

Auto*ata is the plural o the or( Auto*atonhich *eans ?sel-actin&

n &eneral. this sub;ect ocuses on thetheoretical aspects o co*puter science8

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Theory of Automata

Applications$his sub;ect plays a *a;or role in

e4nin& co*puter lan&ua&es

!o*piler !onstruction

arsin&

etc

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Types of languages$here are to types o lan&ua&esFor*al Lan&ua&es are use( as a basis or

(e4nin& co*puter lan&ua&es A pre(e4ne( set o sy*bols an( strin& For*al lan&ua&e theory stu(ies purely syntactical

aspects o a lan&ua&e %e8&8. or( abcd'

nor*al Lan&ua&es such as En&lish has *any(ierent ersions

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Basic Element of a Formal

Language Alphabetse4nitionA 4nite non-e*pty set o sy*bols %letters'. is

calle( an alphabet8 t is (enote( by "reek lettersi&*a G8

E+a*ple

GH>1.2.3CGH>I.1C JJBinary (i&its

GH>i.;.kC

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Example Computer Languages! Lan&ua&e

!//

0aa

Kb8net

!#8net

etc

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What are StringsA Strin& is or*e( by co*binin& arious sy*bols

ro* an alphabet8

E+a*ple

GH >1.IC then so*e sa*ple strin&s areI. 1. 11II11. 688

Si*ilarly. GH >a. bC then so*e sa*ple strin&sarea. b. abbbbbb. aaaabbbbb. 688

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5hat is an EM$ or N)LL Strin&A strin& ith no sy*bol is (enote( by %Small

"reek letter La*b(a' or %Capital "reekletter La*b(a' 8 t is calle( an e*pty strin&or null strin&8

lease (ont conuse it ith lo&ical operator

Oan(8

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What are Words5or(s are strin&s that belon& to so*e speci4c

lan&ua&e8

E+a*ple GH >aC then a lan&ua&e L can be (eine( as

LH>a.aa.aaa.68C here L is a set o or(s o

the lan&ua&e (e4ne by &ien set o alphabets8Also a. aa.6 are the or(s o L but not ab8

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!e"ning Alphabets #uidelines$he olloin& are three i*portant rules or (e4nin&

Alphabets or a lan&ua&e

Shoul( not contain e*pty sy*bol

Shoul( be 4nite8 $hus. the nu*ber o sy*bols are4nite

Shoul( not be a*bi&uous E+a*ple an alphabet *ay contain letters consistin&

o &roup o sy*bols or e+a*ple G1H >A. aA. bab. (C8

All startin& letters are uni:ue

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!e"ning Alphabets #uidelinesNo consi(er an alphabet

G2H >A. Aa. bab. (C an( a strin& AababA8

$his strin& can be actore( in to (ierent ays%Aa'. %bab'. %A'

%A'. %abab'. %A'

5hich shos that the secon( &roup cannot be

i(entiie( as a strin&. (eine( oer G H >a. bC8$his is (ue to a*bi&uity in the (e4ne( alphabet

G2

%Because o A an( Aa'

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Ambiguity ExamplesG1H >A. aA. bab. (C

G2H >A. Aa. bab. (C

$%is a &alidalphabet hile $'is an in(&alidalphabet8

Si*ilarly.

G1H >a. ab. acC

G2H >a. ba. caC

n this case. $% is a in&alidalphabet hile $' is a

&alid alphabet8

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Length of Stringse4nition

$he len&th o strin& s. (enote( by PsP. is thenu*ber o lettersJsy*bols in the strin&8

E+a*ple

GH>a.bC

sHaaabb

PsPHQ

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5or( Len&th E+a*pleNu*ber o or(s in a strin&

E+a*pleGH >A. aA. bab. (C

sHAaAbabA(

Factorin&H%A'. %aA'. %bab'. %A'. %('

PsPHQ

ne i*portant point to note here is that aA has alen&th 1 an( not 28

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Length of strings o&er n

alphabetsFormula) Nu*ber o strin&s o length *m+ (e4ne(

oer alphabet o *n+ lettersis nm

E+a*ples$he lan&ua&e o strin&s o length '. (e4ne( oer

GH>a.bCis LH>aa. ab. ba. bbC i#e#nu*ber o strin&s H22

$he lan&ua&e o strin&s o len&th 3. (e4ne( oerGH>a.bC is LH>aaa. aab. aba. baa. abb. bab. bba. bbbC

i#e#nu*ber o strin&s H 23

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,e&erse of a String$he reerse o a strin& s (enote( by ,e&-s. or sr. is

obtaine( by ritin& the letters o s in reerse or(er8

E+a*ple

sHabc is a strin& (eine( oer GH>a.b.cC

then Re%s' or srH cba

GH >A. aA. bab. (C

sHAaAbabA(

Re%s'H(AbabAaA

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ALNRME+,alindrome e-amples !K!. MAAM. RAAR. S$A$S. $ALLA$. R$A$R/

$he lan&ua&e consistin& o an( the strin&s s(eine( oer G such that Re%s'Hs8

t is to be (enote( that the or(s o ALNRMEare calle( palin(ro*es8

E+a*pleFor GH>a. bC

ALNRMEH> . a. b. aa. bb. aaa. aba. bab.bbb. 888C

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Een EenAll strin&s that contains een nu*ber o as

an( een nu*ber o bs

E8&

. aa. bb. aabb. abab. abba. baab etc

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Tleene Star closuret is (enote( by GUan( represent all collection

o strin&s (eine( oer G inclu(in& Null strin&8

$he lan&ua&e pro(uce( by Tleene closure isin4nite8 t contains in4nite or(s. hoeereach or( has 4nite len&th8

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Tleene Star closureExamples

G H >+C

$hen GU H >. +. ++. +++. ++++. 68C

G H >I.1C

$hen GU H >. I. 1. II. I1. 1I. 11. 68C

G H >aaB. cC

$hen GU H > .aaB. c. aaBaaB. aaBc. caaB. cc.68C

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Tleene Star closureConsider the language S/0 1here S 2 3a b45

6o1 many 1ords does this language ha&e oflength '7 of length 87 of length n7

Nu*ber o or(s H n* %n is nu*ber o sy*bols an( * is len&th'

Len&th 2 22H V

Len&th 3 23H W

Len&th n 2n

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9lus :peration5ith lus peration. co*bination o (ierent

letters are or*e(8 @oeer. Null Strin& is notpart o the &enerate( lan&ua&e8

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9lus :perationExamples

G H >+C

$hen G/

H > +. ++. +++. ++++. 68C G H >I.1C

$hen G/H > I. 1. II. I1. 1I. 11. 68C

G H >aaB. cC

$hen G/H >aaB. c. aaBaaB. aaBc. caaB. cc.68C

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e4nin& lan&ua&eFour (ierent ays. in hich a lan&ua&e can be (e4ne($here are our ays that e ill stu(y in this course

escriptie ay

Recursie ay

Re&ular E+pression

Finite Auto*ata

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escriptie ay an( its e+a*ples$he lan&ua&e an( its associate( con(itions are (e4ne( in plain

En&lish8

Example) $he lan&ua&e L o strin&s o e&en length. (eine( oer GH>bC. can

be ritten as

LH>bb. bbbb. 688C

Example) $he lan&ua&e L o strin&s that does not start 1ith a. (e4ne( oer

GH>a.b.cC. can be ritten as

LH>b. c. ba. bb. bc. ca. cb. cc. 6C

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escriptie ay an( its e+a*plesExample)$he lan&ua&e L o strings of length 8. (eine( oer GH>I.1.2C. can be

ritten as

LH>III. I12. I22.1I1. 1I1.12I.6C

Example)$he lan&ua&e L o strings ending in %. (eine( oer G H>I.1C. can be

ritten as

LH>1.II1.1I1.III1.I1I1.1II1.11I1.6C

Example)$he lan&ua&e E;a.bC. can be ritten as

LH> .ab.aabb.abab.baba.abba.6C

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escriptie ay an( its e+a*plesExample) $he lan&ua&e E>E?(E>E?. o strin&s ith een nu*ber o as

an( een nu*ber o bs. (eine( oer GH>a.bC. can be ritten as

LH>. aa. bb. aaaa. aabb.abab. abba. baab. baba. bbaa. bbbb.

6C

Example) $he lan&ua&e @?TE#E,. o strin&s (e4ne( oer

GH>-.I.1.2.3.V.Q.X.Y.W.ZC. can be ritten as

N$E"ER H >6.-2.-1.I.1.2.6C

Example) $he lan&ua&e E>E?. o stin&s (e4ne( oer

GH>-.I.1.2.3.V.Q.X.Y.W.ZC. can be ritten as

EKEN H > 6.-V.-2.I.2.V.6C

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escriptie ay an( its e+a*plesExample) $he lan&ua&e >anbnC. o strin&s (e4ne( oer

GH>a.bC. as

>an bn nH1.2.3.6C. can be ritten asLH >ab. aabb. aaabbb.aaaabbbb.6C

Example) $he lan&ua&e >anbncnC. o strin&s (e4ne( oer

GH>a.b.cC. as >anbncn nH1.2.3.6C. can be rittenas

LH >abc. aabbcc. aaabbbccc.aaaabbbbcccc.6C