Presentation on Regular Languages

8/18/2019 Presentation on Regular Languages

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Regular Languages,Regular OperationsSeptember 11, 2001


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Agenda

TodayRegular languages

Finite languages are regular

Regular operations on languagesUnion ( ∪ )Concatenation ( • )

leene star (!)

For next time:Read 1"# and $andout on minimi%ation

Thursday, 9/20 ( revised ) : &'1 collected


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e nition o* RegularLanguage

Recall t$e de nition o* a regular language+F+ -$e language accepted by an FA M is t$e set o* all strings .$ic$ are acceptedb/ M and is denoted b/ L (M)"

'ould li e to understand .$at t/pes o*languages are regular" Languages o* t$ist/pe are amenable to super *astrecognition o* t$eir elements

'ould be nice to no. *or e ample, .$ic$o* t$e *ollo.ing are regular+


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Language amples

Unar/ prime numbers+3 11, 111, 11111, 1111111, 11111111111, 4 56 31 2 , 1 # , 1 7 , 1 8 , 1 11 , 1 1# , 4 5

6 3 1 p

9 p

is a prime number 5Unar/ s:uares+3 ε , 1, 1 ; , 1 alindromic bit strings+

3 ε , 0, 1, 00, 11, 000, 010, 101, 111, 456 3 x ∈ 30,15! 9 x 6 x R 5 o

'ill e plore .$et$er or not t$ese are regularin *uture"


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Finite Languages

All t$e pre?ious e amples $ad t$e *ollo.ingpropert/ in common+ infnite cardinalit/

@O- + -$e strings .$ic$ made up t$elanguage .ere nite (as t$e/ al.a/s .illbe in t$is course) $o.e?er, t$e collectiono* suc$ strings .as in nite"

Be*ore loo ing at in nite languages, s$ouldde nitel/ loo at nite languages"


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Languages o* Cardinalit/ 1

+ Ds t$e singleton languagecontaining one string regularE For

e ample, is3 banana 5

regularE


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A+ es"

+ '$atGs, .rong .it$ t$ise ampleE


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A+ @ot$ing, reall/" -$is an e ample o* anondeterministic FA" -$is turns out to bet$e most concise .a/ to encapsulate t$elanguage 3 banana 5

But .e .ill deal .it$ nondeterminism incoming lectures" So+

+ Ds t$ere a .a/ o* ing t$is and ma ing itdeterministicE


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Languages o* Cardinalit/ 1A+ es, Hust add a fail state :8 D"e", put

a state t$at suc s in all strings diIerent*rom JbananaK *or all eternit/ unlesst$e/ $appen to be t$e JbananaK

pre xes 3 ε , b, ba, ban, bana, banan5"


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-.o Strings

+ &o. about t.o stringsE Fore ample

3 banana, nab 5 E


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-.o Strings

A+ Must add anot$er route+


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Arbitrar/ Finite @umber o*Strings

1+ &o. about moreE For e ample3 banana, nab, ban, babba 5 E

2+ Or less (t$e empt/ set)+N 6 35 E


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Arbitrar/ Finite @umber o*StringsA1+


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Arbitrar/ Finite @umber o*Strings+ mpt/ Language

A2+ Build a 1 state automaton.$ose accept states set F is

empty!


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Arbitrar/ Finite @umber o*Strings

-& + All nite languages are regular"Proo + Can al.a/s construct a tree .$ose lea?es

are .ord ending" Dn our e ample t$e tree is+

@o. ma e .ord endings into accept states, add a

*ail sin state and add lin s to t$e *ail state tonis$ t$e construction"

ba a

b

a

n b

a

n

b

an


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Dn nite Cardinalit/

+ Are all regular languages niteE


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Dn nite Cardinalit/

A+ @oP an/ in nite languages are regular"Common ista e 1+ -$e strings o* regular

languages are nite, t$ere*ore t$e regularlanguages must be nite"

Common ista e 2+ Regular languages areb/ de nition accepted b/ fnite

automata, t$ere*ore regular languages arenite"

+ Qi?e an e ample o* a infnite but regularlanguage"


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Dn nite Cardinalit/bit strings .it$ an e?en number o* bGs

Simplest e ample is Σ∗

man/, man/ more&ome e ercise+ t$in o* a criterion *or

non niteness


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Regular Operations ou ma/ $a?e come across t$e regular

operations .$en doing ad?anced searc$esutili%ing programs suc$ as emacs , egrep , perl , python , etc" -$ere are t$ree basic

operations .e .ill .or .it$+1" Union2" Concatenation#" leene star

And a *ourt$ de nable in terms o* t$e pre?ious+;" leene plus


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Regular Operations Summari%ing -able

Operation

S/mbol

U@D?ersion

eaning

Union ∪ 9 matc$ one o*t$e patterns

Concatenation •

implicit inU@D

matc$patterns inse:uence

leenestar ! !

atc$ pattern0 or more

times

leene atc$ pattern1 or more


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Regular operations Union

U@D + to searc$ *or all lines containing?o.els in a te t one could use t$e

commandegrep -i

`

a|e|i|o|u’

&ere t$e pattern J vowel K is matc$ed

b/ an/ line containing one o* a, e, i,o or u"+ '$at is a string patternE


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String >atterns

A+ A good .a/ to de ne a pattern isas a set o* strings, i"e" a language"

-$e language *or a gi?en pattern ist$e set o* all strings satis*/ing t$epredicate o* t$e pattern"Q+ vowel-pattern 63 t$e set o* strings .$ic$contain at least one o*+ a e i o u 5


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U@D patterns ?s"Computabilit/ patterns

Dn U@D , a pattern is implicitl/assumed to occur as a substring o*

t$e matc$ed strings"Dn our course, $o.e?er, a pattern

needs to speci*/ t$e whole string,

and not Hust a substring"


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Regular operations Union

Computabilit/+ union is e actl/ .$at.e e pect" D* /ou $a?e patterns

A 6 3aard?ar 5, 6 3bobcat5,6 3c$impan%ee5

union t$e patterns toget$er to get A∪ ∪ 6 3aard?ar , bobcat,

c$impan%ee5


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Regular operations Concatenation

U@D + to searc$ *or all consecuti?edouble occurrences o* ?o.els, use+egrep -i

`

(a|e|i|o|u)(a|e|i|o|u)’&ere t$e pattern J vowel K $as been

repeated" >arent$eses $a?e been

introduced to speci*/ .$ere e actl/in t$e pattern t$e concatenation isoccurring"


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Computabilit/" Consider t$epre?ious result+

L 6 3aard?ar , bobcat, c$impan%ee5

+ '$at language results .$en .econcatenate L .it$ itsel* obtaining

L• L E


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A+L• L 63aard?ar , bobcat, c$impan%ee5 • 3aard?ar , bobcat, c$impan%ee5

63aard?ar aard?ar , aard?ar bobcat, aard?ar c$impan%ee, bobcataard?ar , bobcatbobcat, bobcatc$impan%ee, c$impan%eeaard?ar , c$impan%eebobcat,

c$impan%eec$impan%ee5

1+ '$at is L•{ε} E2+ '$at is L• N E


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Algebra o* LanguagesA1+ L•{ε} 6 L" Dn general, {ε} is t$e identity in

t$e JalgebraK o* languages" D"e", i* .e t$in o*concatenation as being li e multiplication,{ε} acts li e t$e number 1"

A2+ L• N 6 N" Opposite to {ε} , N acts li e t$enumber %ero obliterating e?er/t$ing it isconcatenated .it$"

@ote+ 'e can carr/ on t$e analog/ bet.eennumbers and languages" Addition becomesunion, multiplication becomes concatenation"

-$is *orms a so called JalgebraK"


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Regular operations leene !

U@D + searc$ *or lines consisting purel/ o*?o.els (including t$e empt/ line)+

egrep -i `^(a|e|i|o|u)*$’

@O- + ^ and $ are special s/mbols in U@Dregular e pressions .$ic$ respecti?el/anc$or t$e pattern at t$e beginning and

end o* a line" -$e tric abo?e can be usedto con?ert an/ Computabilit/ regulare pression into an e:ui?alent U@D *orm"


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Computabilit/+ Suppose .e $a?e alanguage

6 3 ba, na 5

+ '$at is t$e language ! E


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A+! 6 3 ba, na 5!6

3 ε ,ba, na,

baba, bana, naba, nana,

bababa, babana, banaba, banana,nababa, nabana, nanaba, nanana,babababa, bababana, 4 5


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Regular operations leene

leene is Hust li e leene ! e cept t$at t$epattern is *orced to occur at least once "

U@D + searc$ *or lines consisting purel/ o* ?o.els(not including t$e empt/ line)+

egrep -i `^(a|e|i|o|u)+$’

Computabilit/+ 6 3 ba, na 5 63 ba, na, baba, bana, naba, nana, bababa, babana, banaba, banana,

nababa, nabana, nanaba, nanana,babababa, bababana, 4 5


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Qenerating t$e Regular

Languages -$e real reason t$at regular languages arecalled regular is t$e *ollo.ing+

-& + -$e regular languages are all t$oselanguages .$ic$ can be generatedstarting *rom t$e nite languages b/appl/ing t$e regular operations"

-$is .ill be pro?ed in t$e coming lectures"+ Can .e start .it$ e?en more basiclanguages t$an arbitrar/ nitelanguagesE


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Qenerating t$e Regular

LanguagesA+ es" 'e can start .it$ languages consistingo* single strings .$ic$ are t$emsel?es Hust asingle c$aracter" -$ese are t$e JatomicKregular languages"

Q+ -o generate t$e nite languageL 6 3 banana, nab 5.e can start .it$ t$e atomic languages

A 6 3a5, 6 3b5, " 6 3n5"

-$en .e can e press L as+L 6 ( • A • " • A • " • A ) ∪ ( " • A • )


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Blac board ercises

press t$e FA patterns *rom t$epre?ious board e ercises using

regular operations in bot$ U@Dst/le and Computabilit/ st/le"

Presentation on Regular Languages

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