lesson 1uaf

8/9/2019 Lesson 1UAF

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Welcome to ! Lecture # 1

Theory Of Automata


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Text and Reference Material

1. Introductionto Computer Theory, by DanielI. Cohen, John Wiley and Sons, Inc., 1991,Second Edition

2. Introduction to Languages and Theory ofComputation, by J. C. Martin, McGraw HillBook Co., 1997, Second Edition


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Grading

There will be One mid exam and one final

exam. The final exam will be comprehensive.These will contribute the followingpercentages to the final grade:

Mid-Term Exams. (12 Marks)

Assignments (04 Marks)

Final Exams. (24 Marks)

Practical Exam (20 Marks)


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What does automata mean?

It is the plural of automaton, and itmeanssomethingthatworks automatically


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Introduction to languages

There aretwotypes of languages

Formal Languages (Syntactic languages)

Informal Languages (Semantic

languages)


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Alphabets

Definition:

A finitenon-empty setofsymbols (letters), is

called an alphabet. Itis denoted by ( Greekletter sigma).

Example:

={a,b}

={0,1} //importantas this is the language//whichthecomputerunderstands.

={i,j,k}


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Strings

Definition:

Concatenationoffinite symbols fromthe

alphabetis called a string. Example:

If={a,b} then

a, abab, aaabb, ababababababababab


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NOTE:

EMPTYSTRING orNULL STRING

Sometimes a string withno symbol atall is

used, denoted by (Small Greek letterLambda) or (Capital Greek letterLambda) , is calledan empty stringornull string.

Thecap

ital lambda w

ill mostly beusedtodenotethe empty string, infurtherdiscussion.


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Words

Definition:

Words are strings belongingto some

language.Example:

If={x} then a languageLcan bedefined as

L={xn: n=1,2,3,..} orL={x,xx,xxx,.}

Here x,xx, arethe words ofL


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NOTE:

All words are strings, butnotall stringsare words.


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Valid/In-valid alphabets

Whiledefining an alphabet, an alphabetmaycontain letters consistingofgroupofsymbols

for example1={B, aB, bab, d}.

Now consider an alphabet2={B, Ba, bab, d} and a string BababB.


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This stringcan betokenizedintwodifferentways

(Ba), (bab), (B) (B), (abab), (B)

Which shows thatthe secondgroupcannot

beidentified as a string, definedover ={a, b}.


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As whenthis stringis scanned bythecompiler (Lexical Analyzer), firstsymbol B is

identified as a letter belongingto, whileforthe second letterthe lexical analyzer wouldnotbe abletoidentify, so whiledefining analphabetitshould be keptinmindthat

ambiguity shouldnotbecreated.


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Remarks:

Whiledefining an alphabetof lettersconsistingofmorethanone symbols, no

letter should be started withthe letterofthesame alphabeti.e. one letter shouldnotbetheprefix ofanother. However, a lettermaybe endedinthe letterofsame alphabeti.e.

one lettermay bethe suffix ofanother.


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Conclusion

1={B, aB, bab, d}

2={B, Ba, bab, d}

1is a valid alphabetwhile2is anin-validalphabet.


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Length ofStrings

Definition:

The lengthofstring s, denoted by |s|, is the

numberof letters inthe string. Example:

={a,b}

s=ababa|s|=5


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Example:

={B, aB, bab, d}

s=BaBbabBd

Tokenizing=(B), (aB), (bab), (B), (d)

|s|=5


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Reverse of a String

Definition:

Thereverseofa string s denoted byRev(s)

or sr, is obtained by writingthe letters ofsinreverseorder.

Example:

Ifs=abcis a stringdefinedover={a,b,c}thenRev(s) or sr=cba


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Example:

={B, aB, bab, d}

s=BaBbabBd

Rev(s)=dBbabaBB


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Defining Languages

The languages can bedefinedindifferentways , such as Descriptivedefinition,Recurs

ivedef

inition, us

ing

RegularExpressions(RE) andusing Finite

Automaton(FA) etc.

Descriptive definition of language:The languageis defined, describingtheconditions imposedonits words.


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Example:

The language Lofstrings ofodd length,

definedover

={a},

can be wr

itten asL={a, aaa, aaaaa,..}

Example:

The languageLofstrings thatdoes notstart

with a, definedover={a,b,c}, can be writtenas

L={b, c, ba, bb, bc, ca, cb, cc, }


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Example:

The languageLofstrings of length2,

definedover={0,1,2}, can be written asL={00, 01, 02,10, 11,12,20,21,22}

Example:

The languageLofstrings endingin 0,definedover ={0,1}, can be written as

L={0,00,10,000,010,100,110,}


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Example: The languageEQUAL, ofstrings withnumberofas equal tonumberofbs, definedover={a,b}, can be written as

{ ,ab,aabb,abab,baba,abba,}

Example: The languageEVEN-EVEN, ofstringswith evennumberofas and evennumberof

bs, defi

nedover={

a,b},c

an be wri

tten as{, aa, bb, aaaa,aabb,abab, abba, baab, baba,bbaa, bbbb,}


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Example: The languageINTEGER, ofstringsdefinedover={-,0,1,2,3,4,5,6,7,8,9}, can

be written asINTEGER={,-2,-1,0,1,2,}

Example: The languageEVEN, ofstingsdefin

edove

r={

-,0,1,2,3,4,

5,6,7,8,9},

c

anbe written as

EVEN={ ,-4,-2,0,2,4,}


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Example: The language{anbn}, ofstringsdefinedover={a,b}, as

{a

n

b

n

: n=1,2,3,},can be wr

itten as{ab, aabb, aaabbb,aaaabbbb,}

Example: The language{anbnan}, ofstringsdefinedover={a,b}, as

{anbnan: n=1,2,3,}, can be written as

{aba, aabbaa, aaabbbaaa,aaaabbbbaaaa,}


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Example: The languagefactorial, ofstringsdefinedover={1,2,3,4,5,6,7,8,9} i.e.

{1,2,6,24,120,} Example: The languageFACTORIAL, of

strings definedover={a}, as

{an!: n=1,2,3,}, can be written as

{a,aa,aaaaaa,}. Itis to benotedthatthelanguage FACTORIALcan bedefinedoverany single letter alphabet.


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Example: The languageDOUBLEFACTORIAL,ofstrings definedover={a, b}, as

{an!

bn!

: n=1,2,3,}, can be written as{ab, aabb, aaaaaabbbbbb,}

Example: The languageSQUARE, ofstringsdefinedover={a}, as

{an2

: n=1,2,3,}, can be written as

{a, aaaa, aaaaaaaaa,}


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Example: The languageDOUBLESQUARE, ofstrings defined

over={a,b}, as{an

2

bn2

: n=1,2,3,}, can be written as

{ab, aaaabbbb, aaaaaaaaabbbbbbbbb,}


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Example: The languagePRIME, ofstringsdefinedover={a}, as

{ap: pis prime}, can be written as

{aa,aaa,aaaaa,aaaaaaa,aaaaaaaaaaa}


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An Important language

PALINDROME:

The languageconsistingof andthe

strings s definedover suchthatRev(s)=s.

Itis to bedenotedthatthe words ofPALINDROME arecalledpalindromes.

Example:For={a,b},PALINDROME={ , a, b, aa, bb, aaa, aba,bab, bbb, ...}


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Remark

There are as manypalindromes of length2nas there areof length2n-1.

Toprovethe aboveremark, thefollowingis

to benoted:


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Note

Numberofstrings of length mdefinedoveralphabetof n letters is nm.

Examples: The languageofstrings of length2, definedover={a,b} is L={aa, ab, ba, bb} i.e.numberofstrings =22

The languageofstrings of length 3, definedover={a,b} is L={aaa, aab, aba, baa, abb,bab, bba, bbb} i.e. numberofstrings =23


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Tocalculatethenumberofpalindromes oflength(2n), considerthefollowingdiagram,


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which shows thatthere are as manypalindromes of length2n as there arethe

strings of lengt

hn

i.e. t

herequ

irednumberofpalindromes are2n.


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Tocalculatethenumberofpalindromes oflength (2n-1) with a as themiddle letter,

considerthefollowingdiagram,


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which shows thatthere are as manypalindromes of length2n-1 as there arethe

strings of lengt

hn-1

i.e. t

herequ

irednumberofpalindromes are2n-1.

Similarlythenumberofpalindromes of length2n-1, with b as middle letter, will be2n-1as

well. Hencethetotal numberofpalindromes oflength2n-1 will be2n-1+ 2n-1=2 (2n-1)=2n.


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Exercise

Q) Provethatthere are as manypalindromes

of length2n, definedover ={a,b,c}, asthere areof length2n-1. Determinethenumberofpalindromes of length2ndefinedoverthe same alphabetas well.


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SummingUp Lecture-1

Introductiontothecoursetitle, Formal andIn-

formal languages, Alphabets, Str

ings, Null str

ing,Words, Valid andIn-valid alphabets, lengthofa

string, Reverseofa string, Defining languages,Descriptivedefinitionof languages, EQUAL,EVEN-EVEN, INTEGER, EVEN, { anbn}, { anbnan

}, factorial, FACTORIAL, DOUBLEFACTORIAL,SQUARE, DOUBLESQUARE, PRIME,PALINDROME.

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