CS25 Lecture Presentation 4

8/20/2019 CS25 Lecture Presentation 4

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CS 25Automata Theory & Formal Languages

Lecture 4


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EQUIVALENCE OF FINITE AUTOMATA WITHREGULAR EXPRESSIONS:

● Regular express!"s a"# $"%e au%!&a%a are e'u(ale"% " %)er#es*rp%(e p!+er,

● A la"guage s regular $ a"# !"l- $ s!&e regular express!"#es*r.es %,


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GENERALI/E0 NON0ETERMINISTIC FINITE AUTOMATA 1GNFA2

● NFA wherein transition arrows may have any regular expressions as labels.

● Special Form:

3, T)ere are %ra"s%!"s g!"g $r!& %)e "%al s%a%e %! all !%)er s%a%es4 a"# %)ereare "! %ra"s%!"s "%! %)e "%al s%a%e,

5, T)ere s a s"gle a**ep% s%a%e %)a% )as !"l- %ra"s%!"s *!&"g "%! % 1a"# "!!u%g!"g %ra"s%!"s2,

6, T)e a**ep% s%a%e s #s%"*% $r!& %)e "%al s%a%e,

7, Ex*ep% $!r %)e "%al a"# a**ep%"g s%a%es4 all !%)er s%a%es are *!""e*%e# %! all!%)er s%a%es (a a %ra"s%!", I" par%*ular4 ea*) s%a%e )as a %ra"s%!" %! %sel$,


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CONVERTING GNFA TO SPECIAL FORM

3, A## "e+ s%ar% s%a%e +%) a" 8 arr!+ %! %)e !l# s%ar% s%a%e,

5, A## "e+ a**ep% s%a%e +%) 8 arr!+s p!"%"g $r!& %)e !l# a**ep%s%a%es,

6, I$ %)ere are &ul%ple arr!+s .e%+ee" 5 s%a%es a"# " sa&e

#re*%!"4 +e repla*e ea*) +%) a s"gle arr!+ +)!se la.el s %)eu"!" !$ %)e pre(!us la.els,

7, A## arr!+s la.ele# 9 .e%+ee" s%a%es %)a% )a# "! arr!+s,


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FORMAL 0EFINITION OF A GNFA

;%uple 1Q4 Σ4 δ4 's%ar%4 'a**ep%24 +)ere

3, Q s a $"%e se% *alle# states

2. Σ s %)e input alphabet

3. δ: 1Q;


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CONVERTING 0FA TO REGULAR EXPRESSION

1.Convert DFA to a GNFA, adding new initial andfinal states.

2.Remove all states one-by-one, until we ave

only te initial and final states.!."ut#ut regular e$#ression is te label on te

%single& transition left in te GNFA.


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CONVERTING GNFA TO REGULAR EXPRESSION

● I$ > 1"u&.er !$ s%a%es2 ? 54 *!"s%ru*% a" e'u(ale"% GNFA $!r& +%)>;3 s%a%es@ repea% u"%l >5,

● I$ >54 %)ere s !"l- a s"gle arr!+ $r!& %)e s%ar% %! %)e a**ep% s%a%e,T)e la.el !" %)s arr!+ s %)e regular express!",


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CONVERTING GNFA TO REGULAR EXPRESSION

Algorithm for ripping a single state:

For e(er- s%a%e qrip do

For e(er- "*!&"g s%a%e qi do For e(er- !u%g!"g s%a%e q j do

Re&!(e all %ra"s%!" pa%)s $r!& qi %! q j (a qrip .-

*rea%"g a #re*% %ra"s%!" .e%+ee" qi a"# q j ,


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EXAMPLE OF CONVERTING 0FABNFA TO REGULAREXPRESSION

1. The original 3-state DFA.


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2. Convert to 5-state GNFA %null transitions are unre#resented&.

∪


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3. Convert to 4-state GNFA by eliminating state A.

∪

∪


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4. Convert to 3-state GNFA by eliminating state B.

∪

∪

∪


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5. Convert to 2-state GNFA by eliminating state C.

∪

∪

∪∪

∪

∪

∪

'us, te automata is e(uivalent to te regular e$#ression %ab)a * b&%a * b&).


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EXERCISE:

C!"(er% %)e g(e" 0FA .el!+ %! a" e'u(ale"% Regular Express!",

1 2 !a a

b

b

a, b

1 2


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Alge.ra* La+s $!r REs

● *nion and +on+atenation beave sort of lie

addition and multi#li+ation.

– *nion is +ommutative and asso+iative +on+atenation

is asso+iative.

– Con+atenation distributes over union.

– $+e#tion/ Con+atenation is not +ommutative.


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0dentities and Anniilators

● ∅ is te identity for∪.

– R∪∅ R.

● ε is te identity for +on+atenation.

– εR Rε R.

● ∅ is te anniilator for +on+atenation.

– ∅R R∅ ∅.

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CS25 Lecture Presentation 4