REGULAR CUBIC LANGUAGE AND REGULAR CUBIC EXPRESSION

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1 Advnes n Fuzzy ets nd ystems 05 Pushp Pulshng House Allhd nd Pulshed Onlne: Novemer 05 Volume 0 Numer 05 Pges 97-3 N: X REGUAR CUBC ANGUAGE AND REGUAR CUBC EXPREON Thr Mhmood leem Adullh nd Qsr Khn Deprtment of Mthemts nterntonl slm Unversty slmd Pkstn e-ml: thrkht@yhoo.om qsrkhn4@gml.om Deprtment of Mthemts Qud--Azm Unversty slmd Pkstn e-ml: sleemdullh8@yhoo.om Astrt n ths pper usng u set we ntrodue u lnguge regulr u grmmr regulr u lnguge fnte stte utomt wth u trnstons fnte stte utomt wth u fnl sttes Myhll- Nerode theorem for u lnguges nd defne n lgorthm for mnmzton of fnte stte utomt wth u fnl stte nd progrmme n Mtl.. ntroduton n 965 Zdeh ntrodued the onept of fuzzy set [7] whh s the generlzton of mthemtl log. Fuzzy set s new mthemtl tool to Reeved: Jnury 3 05; Revsed: Mrh 0 05; Aepted: Aprl Mthemts ujet Clssfton: 68Q70 68Q45 03D05 65Y04 08A70. Keywords nd phrses: u lnguge regulr u grmmr fnte stte utomt regulr u expresson Myhll-Nerode theorem for u lnguge.

2 98 Thr Mhmood leem Adullh nd Qsr Khn desre the unertnty. n [5 6] Zdeh mde n extenson of the onept of fuzzy sets y n ntervl-vlued fuzzy set.e. fuzzy set wth n ntervlvlued memershp funton. n trdtonl fuzzy log to represent e.g. the expert s degree of ertnty n dfferent sttements numers from the ntervl [ 0 ] re used. After ths Atnssov ntrodued the ntutonst fuzzy set n 986 [ ] n whh he gve the onept of memershp funton nd nonmemershp funton to represent the unertnty. And then Gu nd Buehrer ntrodued the onept of vgue set n 993 nd Bustne nd Burllo proved tht vgue set s n ntutonst fuzzy set [9 3]. t s often dffult for n expert to extly quntfy hs or her ertnty; therefore nsted of rel numer t s more dequte to represent ths degree of ertnty y n ntervl or even y fuzzy set. n the frst se we get n ntervl-vlued fuzzy set. n the seond se we get seond order fuzzy set. ntervl-vlued fuzzy sets hve een tvely used n rel-lfe pplton. n 0 Jun et l. [0] ntrodued the onept of u set. Cu set s n ordered pr of ntervl-vlued fuzzy set nd fuzzy set. These ll re mthemtl modelng to solve the prolems n our dly lfe. These tools hve ts own nherent prolems to solve these types of unertnty whle the u set s more nformtve tool to solve ths unertnty. Fuzzy grmmr nd fuzzy lnguges were ntrodued y ee nd Zdeh n 969 []. ne Wee [4] n 967 ntrodued the onept of fuzzy utomt followng ee nd Zdeh [] fuzzy utomt theory hs een developed y mny reserhers. After the ntroduton of fuzzy lnguge nd fuzzy utomt there re mny generlztons of fuzzy lnguge nd fuzzy utomt. ntervl-vlued fuzzy vgue nd ntutonst fuzzy regulr lnguges were ntrodued y Chouey nd Rv [4 6 7]. Mteesu et l. ntrodued fnte fuzzy utomt wth fuzzy trnston nd fnte fuzzy utomt wth fuzzy fnl stte nd fuzzy regulr expresson n 995 [3]. n 009 Chouey nd Rv ntrodued ntutonst fuzzy utomt nd ntutonst fuzzy regulr expressons [5]. n 03 Chouey nd Rv ntrodued the onept of mnmzng fnte stte utomt wth vgue fnl stte nd ntutonst fuzzy fnl sttes [8].

3 Regulr Cu nguge nd Regulr Cu Expresson 99 The rest of the pper s dvded n the followng setons. Prelmnres re dsussed n eton. n eton 3 u regulr lnguge u grmmr some lger operton on u regulr lnguge nd relted results re dsussed. n eton 4 we dsussed fnte utomt wth u trnston fnte utomt wth u fnl stte u regulr expresson nd relted results. n eton 5 we gve Myhll-Nerode theorem for u lnguges nd defned n lgorthm for mnmzton of fnte stte utomt wth u fnl stte nd progrmme n Mtl. Also the lst seton onssts of onluson nd referenes.. Prelmnres Defnton. [3]. et U e n lphet set nd ω ( ) : U [ 0 ] s funton. Then the set of the form = { ω ( ) U } s lled fuzzy lnguge over U nd ω ( ) the memershp funton of. Defnton. [3]. et e fuzzy lnguge over U the fnte lphet set wth ω ( ) : U [ 0 ] s ts memershp funton. Then s lled fuzzy regulr lnguge f; () the set { [ 0 ] M ( ) } s fnte nd () for eh [ 0 ] the strng M ( ) s regulr. Defnton.3 [0]. et X e non-empty set. Then the set of the form A = { x A( x) λ( x) } s u set n whh A ( x) s ntervl-vlued fuzzy set nd λ ( x) s fuzzy set n X. For smplty we wrte A = A λ nsted of A = { x A( x) λ( x) x X}. Defnton.4 [0]. For ny A = { A ( x) λ ( x) x X} we defne P-unon P-nterseton R-unon nd R-nterseton s follows:

4 00 Thr Mhmood leem Adullh nd Qsr Khn A B { x ( A )( x) ( λ ) ( x) x X} (P-unon) p = Λ Λ A B { x ( A )( x) ( λ ) ( x) x X} (p-nterseton) p = Λ Λ A B { x ( A )( x) ( λ ) ( x) x X} (R-unon) R = Λ Λ A B { x ( A )( x) ( λ ) ( x) x X} (R-nterseton). R = Λ Λ 3. Cu Regulr nguge Defnton 3.. et U e non-empty set of lphets. A u lnguge over U s n ojet of the form n whh A ( x) : U [] = {( x A ( x) λ ( x) x U )} nd λ ( x) : U represent respetvely ntervl-vlued fuzzy nd fuzzy memershp funtons of for ll x U. Here [] represents the set of ll losed su-ntervl n [ 0 ] nd represents the losed ntervl [ 0 ]. A ( x) ssgns ntervl-vlue nd λ ( x) ssgns rel numers n the losed ntervl [ 0 ] to eh strng of the lnguge. et e u lnguge over the lphet U nd A ( x) : U [] nd λ ( x) : U re respetvely ntervl-vlued fuzzy nd fuzzy memershp funtons of. Then for eh [ ] [ ] denoted y ([ ] ) the set ([ ] ) = { x x U nd A ( x) = [ ]} nd for eh the set { x x U nd λ ( x) = } s denoted y ( ). Note tht s n nverse funton of memershp funtons. Exmple 3.. et U = { x y} nd A ( x) : U []. Then nd λ ( x ): U

5 Regulr Cu nguge nd Regulr Cu Expresson 0 = {( x [ ] 0.4)( x y [ ] 0.5) x x y U } represent u lnguge. Cu lnguge s generted y u grmmr. The defnton of u grmmr s gven elow. Defnton 3.3. A four-tuple G = ( UT U N P ) s lled u grmmr where () The elements of U T represent termnl symols. U N () The elements of =. U N represent non-termnl symols nd U T () The elements of P re lled produtons. s the strtng symol nd U N. The elements of P re expressons of the form α ( β γ) = ([ ϑ ρ] δ) for [ ϑ ρ] [ ] nd δ where β nd γ re strngs n ( U ) nd T U N [ ϑ ρ] δ represents the grde of memershps of γ for gven β. For smplty the ove expresson n lso e wrtten s α ( β γ) = ([ ϑ ρ] δ) to [ ϑ ρ] \ δ β γ or β γ. The produtons of P re of the form C [ ϑ ρ] \ δ D or C [ ϑ ρ] \ δ C D U N U T s llowed n u regulr grmmr nd lso the produton Λ s llowed wth memershp vlues nd 0 respetvely. Cu grmmr s generlzton of fuzzy grmmr. For termnl strng x s sd to e n ( G) f nd only f x s dervle from strtng symol.

6 0 Thr Mhmood leem Adullh nd Qsr Khn Defnton 3.4. Two u lnguges nd re sd to e equl nd wrtten s A ( x) = A ( x) f nd only f λ ( x) = λ( x) for ll x U. Defnton 3.5. A u lnguge s suset of u lnguge denoted y f nd only f A ( x) A ( x) nd λ ( x) λ( x) for ll x U. n the ove defnton A ( x) A ( x) nd λ ( x) λ ( x) represent respetvely the ntervl-vlued fuzzy nd fuzzy memershp funtons of nd. These lnguges re onstruted over n lphet U ome lger opertons over u lnguges et nd e two u lnguges over n lphet U. et A A nd λ λ e the memershps funtons of nd respetvely. Then the followng s opertons suh s unon omplement nterseton ontenton str on the lnguges nd wll hold. () Unon. The unon of the lnguges nd s denoted nd defned s = = {( x { A ( x) A ( x) } { λ ( x) λ ( x) } x U )}. () nterseton. The nterseton of the lnguges nd s denoted nd defned s = = {( x { A ( x) A ( x) } { λ ( x) λ ( x) } x U )}. () Complement. The omplement of the lnguges s denoted nd defned s = {( A ( x) λ ( x)) x U } x U = {( x [ A ( x) A ( x)] λ ( x)) x U }.

7 Regulr Cu nguge nd Regulr Cu Expresson 03 (d) Contenton. The ontenton of the lnguges nd s denoted nd defned s = = {( x { ( A ( ) A ( ) )} { ( λ ( ) λ ( ) )} x = U nd x U )}. s (e) tr. The str operton of the lnguges s denoted nd defned = ( x { ( A ( ) A ( )... A ( ))} { ( λ ( ) λ ( )... λ ( ))}) x = (f)... n... n U n n 0} x U. + denotes the lnguge for the operton + on nd defned s = ( x { ( A ( ) A ( )... A ( n ))} { ( λ( ) λ( )... λ( n ))}) x =... n... n U n } x U. Defnton 3.7. et e u lnguge over the lphet U nd A ( x) : U [] nd λ ( x) : U re respetvely ntervl-vlued fuzzy nd fuzzy memershp funtons of. Then we ll u regulr lnguge f () the sets {[ ] [ ] [ ] nd ([ ] ) φ} nd { nd ( ) φ} re fnte nd () for eh [ ] [ ] ([ ] ) nd ( ) s regulr. Exmple 3.8. et e u lnguge over the lphet U = { x y} nd the memershp funtons A nd λ re gven elow: n

8 04 Thr Mhmood leem Adullh nd Qsr Khn f x 0 f x A ( x) = [ ] f x yx nd λ ( x) = 0.5 f x yx 0 otherwse otherwse. Then = {( x 0) ( x yx [ ] 0.5) x x yx U } s u regulr lnguge s ts strngs re regulr wth memershp funtons re fnte vlues. Theorem 3.9. Cu regulr lnguges re losed under nterseton unon omplement ontenton nd str opertons. Proof. et nd e two u lnguges over n lphet U. et A : U [] A : U [] nd λ : U λ : U e the ntervl-vlued fuzzy nd fuzzy memershp funtons of nd respetvely. et e the resultng u lnguge fter the opertons (unon nterseton str) wth A U : [] nd λ : U s ts ntervlvlued fuzzy nd fuzzy memershp funtons respetvely. Then lerly [] [] [] nd (n the se of unon nterseton ontenton nd str opertons) or [ ] = {[ ] [ ] [ ] } nd = { } (n se of omplement) re fnte for whh the orrespondng strngs re regulr. et [ ] nd e fnte vlues n [ ] nd respetvely. Then ([ ] ) nd ( ) re defned s follows: Unon [ ] [ ] [ ] [ ] [ ] [ ] f > [ ] [ ] [ ] [ ] [ ] [ ] [ ] f > [ ] ([ ] ) = ( [ ] [ ]) [ ] [ ] > [ ] [ ] [ ] [ ] [ ] [ ] f > [ ]

9 Regulr Cu nguge nd Regulr Cu Expresson 05 ( ) ( ) ( ) ( ) ( ) (( ( ) ( )) ( ) ( ) ( ) = < < < <. f f f nterseton [ ] ( ) [ ] [ ] [ ] [ ] [ ] [] [] [ ] [ ] [ ] [ ] [ ] [] [] (( [ ] [ ]) [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] = < < < < f f f nd ( ) ( ) ( ) ( ) ( ) (( ( ) ( )) ( ) ( ) ( ) = > > > >. f f f Complementton [] [ ][ ] [ ] { } = nd { } = [ ] ( ) [ ] ( ) = nd ( ) ( ). = Contenton [ ] ( ) [ ] [ ] ( ) [ ] [ ] [ ] [ ] [ ] [ ] ( ) [ ] ( ) = = [ ] [ ] ( ) [ ] [ ] [ ] [ ] [ ] [ ] ( ) [ ] ( ) >

10 06 Thr Mhmood leem Adullh nd Qsr Khn nd ([ ] ) = ( ) ( ) ( ) ( ) tr Assumng tht 3 4. ( ) = ( ) < [] = {[ ][ ]... [ k ]} k [ ] [ ] > [ ] > > [ k ] [ 0 0] k {... } 0 < < < = k k ([ ] ) = ( ([ ] ) f [ ] = [ ] ([ ] ) = { Λ} + ([ ]) = ( ([ ]) { Λ} f [ ] [ ] ([ m m ]) = ([ ]) ([ ] ) { Λ} n m n n l< m l + f m k nd ( ) = ( ( ) f = 0 ( 0) = { Λ} nd + ( ) = ( ) { Λ} f 0 + ( m ) = ( ) ( ) { Λ} n m n l > m l f m k. Therefore Kleene losure s stsfed.

11 4.. Defnton Regulr Cu nguge nd Regulr Cu Expresson Fnte Automt wth Cu Trnstons A nondetermnst fnte utomt wth u trnstons s 5-tuple M = ( M U A s F ) where the elements of M re fnte set of sttes. The elements of U re fnte set of nput symols A = A λ s u set n M U M A : M U M nd λ : M U M re ntervl-vlued fuzzy nd fuzzy memershp funtons of M respetvely s denotes the ntl stte nd F M s the set of fnl stte. tht s [ ] For U nd r s M defne the exted memershp funtons of stte trnstons s 0 f = Λ nd r s f = Λ nd r = s A ( r s) = u M { A ( r u) A( u s)} = for ll U U otherwse nd 0 λ ( r s) = u M { λ ( r u) λ( u r)} = for ll U U f = Λ nd f = Λ nd otherwse. r r = s s Then we sy tht d[ ] M ( ) nd d M ( ) where d U s reognzed y M wth degrees of memershp [ ] M ( ) = { A ( s u) u F} nd d M ( ) = { λ ( s u) u F}. We denote the lnguge epted y M s ( M) nd s gven to e the set ( M) = {( d[ ] ( ) d ( )) X }. M M

12 08 Thr Mhmood leem Adullh nd Qsr Khn 4.. Exmple et M = ( M U A s F ) e nondetermnst fnte utomt wth u trnston s shown n the fgure elow where M = { s r u} U = { x y} ntl stte = s fnl stte = {u}. Also u trnstons re defned s follows: A ( s x r) = [ ] λ( s x r) = 0.3 A( s x u) = [ ] λ( s x u) = 0.4 A ( r y r) = [ ] λ( r y r) = 0.5 A( r x u) = [ ] λ( r x u) = 0.6 A ( u y u) = [ ] λ( u y u) = 0.7. The u regulr lnguge epted y the non-determnst fnte utomt wth u trnstons s ( M) = {( [ ] 0.7) xy xy } {( [ ] 0.7) xy }. The trnston dgrm of the ove non-determnst fnte utomt wth u trnston s gven y Defnton 4.3. A determnst fnte utomt wth u trnstons s non-determnst fnte utomt wth u trnston wth the ondton tht for eh r u M nd for eh x U f A( r x u) > 0 λ( r x u) < nd A( r x v) > 0 λ( r x v) < then u = v. Theorem 4.4. e u regulr lnguge f nd only f t s reognzed y non-determnst fnte utomton wth u trnston wth the reognton of Λ the empty word.

13 Regulr Cu nguge nd Regulr Cu Expresson 09 Proof. et e u regulr lnguge over n lphet U. et A : U [] nd λ : U e ntervl-vlued fuzzy nd fuzzy memershp funtons of respetvely. et [] = {... k } nd = {... k } e the fnte sets of ntervl-vlued fuzzy nd fuzzy memershp funtons of respetvely. As s u regulr lnguge so for eh [ ] we hve ( ) nd ( ) re regulr where k. ( ) ( ) = nd ( ) ( ) = for j sne re funtons. nd Also note tht A nd et M = ( M U A s F ) e determnst fnte utomton wth u trnston (or non-determnst fnte utomton wth u trnston) suh tht ( ) = ( M ) nd ( ) = ( M ) where k. Now we onstrut M ι = ( M U A s F ) where λ A ( r x u) = 0 f ( r x u) A otherwse nd ( ) λ λ f r x u ( r x u) = otherwse represent the ntervl-vlued fuzzy nd fuzzy memershp funtons of respetvely. We suppose tht ( M U A s F ) suh tht M M j = for j. et us defne M =

14 0 Thr Mhmood leem Adullh nd Qsr Khn M = { s} s M M M M M M n n F = F F Fn A( r x u) Aι ( r x u) A ( s x u) 0 = ι f r u M k f r = s u M k otherwse λι( r x u) f r u M k λ( r x u) = λι( s x u) f r = s u M k otherwse. Hene M reognzes wth the exepton of Λ the empty strng. Conversely let M = ( M U A s F ) e non-determnst fnte utomt wth u trnston. et us defne u lnguge wth A ( x) = d[ ] ( x) nd λ ( x) = d M ( x) s ts ntervl-vlued fuzzy nd fuzzy memershp funtons respetvely ( A ( Λ) = 0 nd λ ( Λ) = ). M Now we hve to prove tht s u regulr lnguge. et [ ]= {[ ] A( r x u) = [ ] r u M x U } nd = { λ( r x u) = r u M x U }. Then [] nd re fnte. Now suppose tht [ ] = {... k } wth > > > k k nd = {... k } wth < < < k k. et us defne M = ( M U A s F ) e non-determnst fnte utomton where A = {( r x u) A( r x u) } nd λ = {( r x u) λ( r x u) }. Now defne the lnguges of s for k for the nresng sequene

15 Regulr Cu nguge nd Regulr Cu Expresson = ( M ) = ( M ) ( ) M = ( M ) ( ( M ) ( )) 3 3 M ( ) = M ( M ). j= j Therefore ( ) = nd s u regulr lnguge for every k. Theorem 4.5. et e u regulr lnguge. Then s reognzed y determnst fnte utomton wth u trnston f nd only f t + + stsfes the followng xoms for eh U U = nd A ( ) > 0 λ( ) < mply tht A( ) A( ) nd λ ( ) λ( ). Proof. et M = ( M U A s F ) e determnst fnte utomton wth u trnstons whh reognzed the u regulr lnguge. We prove tht stsfes the ove ondton. + et U + U suh tht =. f d[ ] ( ) = 0 M nd ( ) = then A( ) A( ) nd λ ( ) λ( ) re true otherwse lso A( ) = d ( ) = { A ( r) A( r u) r u F} M A ( r) = d[ ] ( ) = A( ) M λ( ) = d ( ) = { λ ( r) λ( r u) r u F} M λ ( r) = d ( ) = λ( ). M d M

16 Thr Mhmood leem Adullh nd Qsr Khn λ Conversely let e u regulr lnguge nd A : U [] nd : U e ntervl-vlued fuzzy nd fuzzy memershp funtons of respetvely stsfyng the gven ondton. uppose tht [ ] = {... k } nd = {... k }. Construt determnst fnte utomton M = ( M U A s F ) suh tht ( ) = ( M ) nd ( ) = ( M ) where k. ( M ) ( M ) = ( ) ( ) = nd ( M ) ( M ) ( ) ( ) = for every j. j j = j Now we defne determnst fnte utomton M = ( M U A s F ) where M = M M M s = { s s... s } A : M U M s defned y k k A( ( p p... p ) x) ( A ( p x) A ( p x)... A ( p x)) k = k k j nd λ : M U M s defned y λ (( p p... p ) x) = ( λ ( p x) λ ( p x)... ( p )) k λ k k x nd F = F F F where k Fk = {( p p... p ) ( p p... p ) k k M p F nd p j Fj for j} j = n. Note tht F F = for j nd j ( ) = { x x U nd A ( s x) F } nd ( ) = { x x U nd λ ( s x) F }.

17 Regulr Cu nguge nd Regulr Cu Expresson 3 Bsed on the ove determnst fnte utomton M we n defne determnst fnte utomt wth u trnston M s M = ( M U A s F ) suh tht A ( r x u) = 0 f A( r x) = u F f A ( r x) = u F otherwse nd λ( r x u) = 0 f λ( r x) = u F f λ ( r x) = u F otherwse. We re now only to prove tht [ ] ( x) = A ( x) nd ( x) = λ ( x) + for ll x U. d M et us show tht M hs the followng property. d M For ll + x U wth x = for eh U U d [ ] ( x) = > 0 ff A ( s r) nd A( r u) = M for some u F k nd d ( x) = < ff λ ( s r) nd λ( r u) = M for some u F k. (E) Then f prt n e proved esly. For the only f prt t holds trvlly when = Λ. For Λ we re ssumng the ontrry tht s A ( s r) = nd A ( r u) = j > for k j. Also λ ( s r) = nd λ ( r u) = j < for k j. Then there exsts deomposton of = def for ll suh tht d f U nd e U

18 4 Thr Mhmood leem Adullh nd Qsr Khn A ( d v) A( v e z) = nd A( z f r) lso λ ( d v) λ( v e z) = nd λ ( z f r). By the defnton of M we know tht z nd u. F F j Thus we hve A ( de) = nd A ( x) = j lso λ ( de ) = nd λ ( x ) = j. ne we ssume tht gven ondton. Hene (E) holds. > nd < ths ontrdton to the j j Agn (E) mples ( ) tht s A ( x) = nd ( ) tht s λ ( x ) =. Hene the theorem s proved. Fnte utomt wth u (fnl) sttes Defnton 4.6. A 7-tuple M = ( M U A λ q F [ ] F ) s lled fnte utomt wth u fnl sttes where the elements of U re fnte set of sttes the elements of re fnte set of nput letters nd M A λ : M U re the trnston funtons q s the u ntl stte nd F [ ] : M [] M M M F M : M re the ntervl-vlued fuzzy nd fuzzy memershp funtons of the u (fnl) sttes set respetvely. Defne d [ ] ( ) = { F [ ] ( q r) A } M M nd d ( ) = { F ( q r) λ } M M M where A λ : M U respetvely denotes the reflexve nd trnstve losure A of λ. The strng s epted y M wth the ntervlvlued fuzzy nd fuzzy degrees of memershp d [ ] M ( ) nd d M ( ).

19 Regulr Cu nguge nd Regulr Cu Expresson 5 The u regulr lnguge epted y M s denoted y ( M ) defned s ( ) {( [ ] ( ) M = d d ( )) U }. M M nd Defnton 4.7. A determnst fnte utomton wth u fnl sttes s 7-tuple M = ( M U A λ q F [ ] F ) M M s non-determnst utomton wth u fnl stte wth A λ : M U M s funtons. For eh U d [ ] ( ) = F [ ] ( r ) where r = A ( q ) M M nd ( ) = F ( ) where r = λ ( q 0 ). d r M M Defne d [ ] M ( ) = 0 nd d M ( ) = f A ( q0 ) not defned. nd λ ( q ) re 0 Theorem 4.8. et e u lnguge. Then s u regulr lnguge f nd only f t s reognzed y determnst fnte utomton wth u fnl stte. A Proof. et e u regulr lnguge over n lphet U wth : U [] nd λ : U s ts ntervl-vlued fuzzy nd fuzzy memershp funtons where [ ] represents the set of ll losed su-ntervls n the losed ntervl [ 0 ] nd represents the losed ntervl [ 0 ]. ne s u regulr lnguge so for every [ ] nd the sets ( ) nd ( ) re regulr. uppose tht = {... k } nd = {... k } where k. Now we defne determnst fnte utomton wth u fnl stte M = ( M U A λ s F [ ] M F ) ι M suh ι tht ( M ) = ( ) nd ( M ) ( ). =

20 6 Thr Mhmood leem Adullh nd Qsr Khn Now we onstrut determnst fnte utomton wth u fnl sttes M = ( M U A λ s F [ ] F ) M M to e the ross produt of M k wth F [ ] ( p ( ) p ( )... p ( ) ) M k () ( ) j p F [ ] for some k nd p F = M M 0 j j otherwse nd F ( p () p ( )... p ( ) ) M k () ( ) j p F for some k nd p F j = M M j otherwse. k Now f ( p () p ( )... p ( ) ) s rehle from ( s s... s k ) for every k n M then t s not possle to get ( p ) ( F [ ] F ) nd M M ( j p ) ( F [ ] F ) for j euse ( M ) ( ) = for j k. Mj M j M j Hene M reognzes. Conversely let M = ( M U A λ s F [ ] F ) e determnst fnte utomt wth u fnl stte. Defne nd M M [] = { F [ ] ( p) = for some p M} M = { F ( p) = for some p M}. M Thus [ ] nd re fnte sets. For eh [ ] nd defne M = ( M U A λ s F F )

21 Regulr Cu nguge nd Regulr Cu Expresson 7 where F = { p F [ ] ( p) } M nd F = { p F ( p ) = }. = M et = ( M) tht s A = d [ nd λ = d. ]M M Hene for eh [] the set ( ) nd the set ( ) re regulr. Therefore s u regulr lnguge. Theorem 4.9. A u regulr lnguge s reognzed y nondetermnst fnte utomton wth u fnl stte f nd only f t s reognzed y determnst fnte utomton wth u fnl stte. Proof. et M e u lnguge. Here we hve to prove f M s non-determnst fnte utomt wth u fnl stte nd = ( M) then = ( M ) where M s determnst fnte utomt wth u fnl stte. et M = ( M U A λ s F [ ] F ) represent non-determnst M M fnte utomt wth u fnl stte. We defne determnst fnte utomton wth u fnl stte M = ( M U A λ s F [ ] M F M ) usng the method of stndrd suset onstruton nd for eh Q M ( Q M ) defne nd F [ ] ( Q) = { F [ ] ( p) = p Q} mx M M ( Q) = mn { F ( p) = p } F Q M M where nd represent respetvely the ntervl-vlued fuzzy nd fuzzy memershp vlues of the strng n the lnguge. Therefore = ( M ).

22 8 Thr Mhmood leem Adullh nd Qsr Khn Cu regulr expresson Eh strng n u regulr lnguge hs fnte ntervl-vlued fuzzy nd fuzzy memershp vlues. The set of fnte words ssoted wth these vlues forms regulr lnguge. Therefore u regulr lnguge my e represented y modfed u regulr expresson. et U = { x y} e n lphet. f { Λ x xx xxx...} [ ] 0.5 { y xy yx xyx...} [ ] 0. 4 represents u regulr lnguge then we n represent t y u regulr expresson s x [ ] x yx [ ] 0.4. The forml defnton of u regulr expresson s defned elow s. Defnton 4.0. et U e n lphet nd [ ] e fnte sets of ntervlvlued fuzzy nd fuzzy vlues n [ 0 ]. et e regulr expresson over U nd [ d] [ ] nd. Then we ll = [ d] n u regulr expresson where [ d] nd respetvely represent ntervl-vlued fuzzy nd fuzzy vlues of. et nd e two u regulr expressons over n lphet U. Then the followng hold: () elongs to u regulr expresson wth memershp vlues nd 0 () Λ elongs to u regulr expresson wth memershp vlues nd 0 (3) elongs to u regulr expresson wth ntervl-vlued fuzzy nd fuzzy memershp vlues n [ 0 ] for ll X (4) for ll nd elong to u regulr expressons ( + ) elongs to u regulr expressons ( ) elongs to u regulr expressons ( ) elongs to u regulr expressons.

23 Regulr Cu nguge nd Regulr Cu Expresson 9 By pplyng the ove mentoned steps (() nd ()) fnte numer of tmes we otned u regulr expresson. Defnton 4.. et e u regulr expresson over n lphet U. Then the orrespondng lnguge tht s u regulr lnguge ( ) s defned y ( ) = {( [ d e] h) ( )}. Here ( ) represents the lnguge for regulr expresson. f ( ) ( ) ( ) ( = + = = ) then ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( = = = )) respetvely. Defnton 4.. A u regulr expresson over n lphet U s sd to e normlzed u regulr expresson f t s of the form... k k k where... k re regulr expressons over U nd... k nd... k re ntervl-vlued fuzzy nd fuzzy memershp vlues n [ 0 ]. Note tht f = nd = 0 then n smply e wrtten s. We ssumed nd hs hgher prortes thn /. o ertn prs of prenthess n e omtted. Exmple 4.3. The followng re ll vld u regulr expressons: () x y [ ] x yx [ ] () xy [ ] x [ ] (3) ( x + y) xx( x + y) [ ] 0. 6 (4) ( y + Λ)( x + xy) [ ] 0. 47

24 0 Thr Mhmood leem Adullh nd Qsr Khn (5) ( x + xy) [ ] (6) ( y xy [ ] 0.6) ( x yx [ ] 0.59) + x + y. The ove from () to (5) re normlzed nd (6) s not normlzed. The followng re not vld u regulr expressons: () x y [ ] 0.49 [ ] zy [ ] 0. 3 () ( x [ ] 0.46) [ ] x yx [ ] 0. Defnton 4.4. A u regulr expresson over n lphet s strtly normlzed u regulr expresson f t s normlzed: =... k k k nd for ny j nd j ( ) ( ) =. j Exmple 4.5. () xy [ ] xz [ ] yz [ ] 0.59 () (( yy + yyx) [ ] ({ x y} { yx}} [ ] 0.6 () x y [ ] ( x + y) x( x + y) y [ ] 0.7 (v) x x [ ] ( x + y) y( x + ) x [ ] Myhll-Nerode Theorem For Cu Regulr nguge Theorem 5.. The followng three sttements re equvlent to one nother: () A u regulr lnguge over n lphet U s epted y some determnst fnte utomton wth u (fnl) sttes. () s the unon of some of the equvlene lsses of rght nvrnt equvlene relton of fnte ndex.

25 A Regulr Cu nguge nd Regulr Cu Expresson (3) The relton U U E s defned y E ff for ll U ( ) = A ( ) nd λ ( ) = λ ( ) then of fnte ndex. E s n equvlene relton Proof. () () et e u regulr lnguge over n lphet U. uppose tht s reognzed y some determnst fnte utomton wth u fnl stte M = ( M U A λ s F [ ] M F M ). et E M e the equvlene relton E M f nd only f A ( ) = A( ) nd λ ( ) = λ( ). ne E M s rght nvrnt so for ny A ( ) = A( ) f A ( ) = A( ) nd λ ( ) = λ( ) f λ ( ) = λ( ). Then the ndex of E M s fnte sne the ndex s t most the numer of sttes n M. Furthermore s the unon of those equvlene lsses hvng strng suh tht A ( ) nd λ ( ) re respetvely n F [ ]M nd F M. Tht s the equvlene lsses orrespondng to the fnl sttes. () () We prove tht ny equvlene relton R stsfyng () s refnement of E ; tht s some equvlene lsses of E wll e superset of every equvlene lss R. Thus the ndex of E nnot e greter thn the ndex of R nd so s fnte. Assume tht R for eh X R nd thus ( ) = ( ) sne R s rght nvrnt. Hene E. We onlude tht eh equvlene lss of R s the suset of some equvlene lsses of E. () () To prove tht E s rght nvrnt ssume E nd let U we must show tht E ; tht s for ny d ( d ) = ( d ). ne E for ny e ( e) = ( e) (y the defnton of E ). uppose e = d to show E s rght nvrnt.

26 Thr Mhmood leem Adullh nd Qsr Khn Now we present the mnmzed determnst fnte utomt wth u fnl stte y defnng equvlene lsses of E. et M e fnte set of equvlene lsses of ontnng. E nd M Defne A ( h) = h nd λ ( h) = h. Ths defnton s onsstent s E s rght nvrnt. f we hoose nsted of from then we wll hve A ( h) = h nd λ ( ) = h. But E so ( ) = ( d ). n prtulr f d = hd ( h d ) = ( hd ) so he h nd h = h. et s = Λ F [ ] = { } nd F = { }. M M The fnte utomton M = ( M U A λ s F [ ] F ) epts therefore A ( s ) = A ( Λ ) = nd λ ( s ) = λ ( Λ ) =. Hene ( M ) = ( M). 5.. Algorthm for mnmzng determnst fnte stte utomton wth u fnl stte et M = ( M U A λ s F [ ] F ) e determnst fnte utomton M M wt u fnl sttes. uppose tht M = { s m m... mk } k 0 nd let P = {( m m j ) m m j M nd 0 < j k}. egn tep : for eh pr ( m m j ) P nd F [ ] M( m ) F[ ] M( m j ) or ( m ) F ( m ) FM M j M M

27 Regulr Cu nguge nd Regulr Cu Expresson 3 do mrk ( m m j ); tep : for eh unmrked pr ( m m ) P do j f for some U ( A( m ) A( m j )) nd ( λ ( m ) λ ( m j )) s mrked then tep.: mrk ( m m j ); tep.. reursvely mrked ll unmrked prs on the lst of ( m m j ) nd the lst of other prs tht re mrked on ths step. else tep.3: for ll nput symols do put ( m m j ) on the lst for ( A( m ) A( m j )) nd ( λ ( m ) λ( m )) unless ( A( m ) A( m )) nd j ( λ ( m ) λ( m )) j j tep 3: Equvlene lsses of M re onstruted s follows; For = 0 to k do For j = + to k do tep 4: Defne mnmum determnst fnte stte utomton wth u fnl stte M = ( M U A λ s F F ) s follows;. M M M = { m m M} A ( m x) = A( m x) nd λ ( m x) = λ( m x) nd F ( m ) = F ( m ). A A s F [ ] A( m ) = F [ ] A( m ) s =

28 4 Thr Mhmood leem Adullh nd Qsr Khn Exmple 5.3. et M = ( M U A λ s F[ ] F ) e determnst M M fnte utomton wth u fnl sttes. M = { s u v w y z} U = { } u ntl stte s s wth ntervl-vlued fuzzy memershp vlue F[ ] M ( s) = [ ] nd fuzzy memershp vlue F M ( s) = 0.3 A λ : M U M re the trnston funtons gven elow: A ( s ) = v A( s ) = u A( u ) = w A( u ) = s A( v ) = z A( v ) = y A ( w ) = z A( w ) = y A( y ) = z A( y ) = y A( z ) = z A( z ) = z λ ( s ) = v λ( s ) = u λ( u ) = w λ( u ) = s λ( v ) = z λ( v ) = y λ ( w ) = z λ( w ) = y λ( y ) = z λ( y ) = y λ( z ) = z λ( z ) = z nd FM ( u) = [ ] FM ( u) = 0.4 F[ ] M( v) = [ ] FM ( v) = 0.5 F[ ] M ( w) = [ ] FM ( w) = 0.5 F[ ] M( y) = [ ] FM ( v) = 0.6 F[ ] M ( z) = [ ] FM ( z) = 0. 7 show the ntervl-vlued fuzzy nd fuzzy memershp funtons of { u } { v } { w } { y} { z} respetvely. The trnston dgrm of determnst fnte stte utomton s gven elow:

29 Regulr Cu nguge nd Regulr Cu Expresson 5 The trnston dgrm of the mnmzed determnst fnte stte utomton wth u fnl stte: The lnguge reognzed y the ove determnst fnte utomton wth u fnl stte nd mnmzed fnte utomton wth u fnl stte s gven elow: ( M) = {( ) \[ ] 0.3 ( ) \[ ] 0.4 [ ] \[ ] 0.6 ( + ) [ ] 0.7 ( + ) \[ ] 0.7} Progrmme n Mtl for mnmzng determnst fnte utomton wth u fnl stte noofsttes=nput( Enter the numer of sttes of DFA-CF... ); sttes=[noofsttes]; lower=[noofsttes]; upper=[noofsttes]; fuzzy=[noofsttes]; sttes()=nput('enter the ntl stte nme...''s'); lower()=nput('enter lower ound of ntervl for ntl stte...');

30 6 Thr Mhmood leem Adullh nd Qsr Khn upper()=nput('enter upper ound of ntervl for ntl stte...'); fuzzy()=nput('enter fuzzy vlue for ntl stte...'); for k=:noofsttes sttes(k)=nput(strt('enter the stte nme...'strt(numstr(k)'...'))'s'); lower(k)=nput(strt('enter lower ound of ntervl for stte...'strt(numstr(k)'...'))); upper(k)=nput(strt('enter upper ound of ntervl for stte...'strt(numstr(k)'...'))); fuzzy(k)=nput(strt('enter fuzzy vlue for stte...'strt(numstr(k)'...'))); noofnputs=nput('enter the numer of nput symols of DFA-CF...'); nputsymols=[noofnputs]; for k=:noofnputs nputsymols(k)=nput(strt('enter the nme of nput symol...' strt (numstr(k)'...'))'s'); trnstons=[noofsttesnoofnputs]; for k=:noofsttes for p=:noofnputs trnstons(kp)=nput(strt(strt(strt('enter trget stte for stte...' sttes(k))'... nd for nput symol...')strt(nputsymols(p)'...'))'s'); for =:noofsttes f (sttes()==trnstons(kp)) trnstons(kp)=;

31 Regulr Cu nguge nd Regulr Cu Expresson 7 ps=[noofsttesnoofsttes]; lst=[noofsttesnoofsttesnoofsttesnoofsttes]; for k=:noofsttes for j=:noofsttes for l=:noofsttes for m=:noofsttes lst(kjlm)=0; for k=:noofsttes for j=:noofsttes ps(kj)=3; f (k<j) ps(kj)=0; f(lower(k)=lower(j)&&upper(k)=upper(j)&&fuzzy(k)=fuzzy(j)) ps(kj)=;

32 8 Thr Mhmood leem Adullh nd Qsr Khn for k=:noofsttes for j=:noofsttes f (ps(kj)==0) temp=0; for =:noofnputs f (ps(trnstons(k)trnstons(j))==) ps(kj)=; lstout=[noofsttes]; [lstout ur]=lstmrk(kjlstnoofstteslstout); for m=:ur- ps(lstout(m)lstout(m))=; temp=; f temp==0 for =:noofnputs f trnstons(k)=trnstons(j)&&k<j lst(kjtrnstons(k)trnstons(j))=;

33 Regulr Cu nguge nd Regulr Cu Expresson 9 eqlsses=[noofsttes*noofsttes]; urr=; for k=:noofsttes- for j=k+:noofsttes f(ps(kj)==0) eqlsses(urr)=k; eqlsses(urr)=j; urr=urr+; fprntf('the nput symols re...\n') for k=:noofnputs fprntf('%...'nputsymols(k)); fprntf('\nthe output sttes re...\n') for k=:noofsttes for l=:urr- f k=eqlsses(l) fprntf('%...%f...%f...%f...'sttes(k)lower(k)upper(k)fuzzy(k)); fprntf('\nthe trget sttes re...\n') for k=:noofsttes

34 30 Thr Mhmood leem Adullh nd Qsr Khn for l=:urr- f k=eqlsses(l) for j=:noofnputs f trnstons(kj)==eqlsses(l) fprntf('for the stte...%...nd nput...%...trget stte s...%...'sttes(k) nputsymols(j)eqlsses(l)); else fprntf('for the stte...%...nd nput...%...trget stte s...%...' sttes(k) nputsymols(j)trnstons(kj)); eond progrm funton [lstouturrent ] = lstmrk(kjlststteslstouturrent) for =:sttes for l=:sttes f(lst(kjl)==) lstout(urrent)=; lstout(urrent)=l; urrent=urrent+; lstmrk(llststteslstouturrent);

35 Regulr Cu nguge nd Regulr Cu Expresson 3 6. Conluson We ntrodued u lnguge whh s the generlzton of fuzzy lnguge. We lso ntrodue u grmmr nd u regulr lnguge nd show tht u regulr lnguge s losed under unon nterseton ontenton nd str opertons. We ntrodue fnte stte utomton wth u trnstons nd fnte stte utomton wth u fnl sttes. We lso gve Myhll-Nerode theorem for u regulr lnguge. We lso ntrodue lgorthm for mnmzton of fnte stte utomton wth u fnl sttes nd onstrut progrmme n Mtl. Referenes [] K. T. Atnssov ntutonst fuzzy sets Fuzzy ets nd ystems 0() (986) [] K. T. Atnssov More on ntutonst fuzzy sets Fuzzy ets nd ystems 33() (989) [3] H. Bustne nd P. Burllo Vgue sets re ntutonst fuzzy sets Fuzzy ets nd ystems 79(3) (996) [4] A. Chouey nd K. M. Rv ntutonst fuzzy regulr lnguge Proeedngs of nterntonl Conferene on Modellng nd multon CTCOM pp [5] A. Chouey nd K. M. Rv ntutonst fuzzy utomt nd ntutonst fuzzy regulr expressons J. Appl. Mth. nformts 7(-) (009) [6] A. Chouey nd K. M. Rv Vgue regulr lnguge Advnes n Fuzzy Mthemts 4() (009) [7] A. Chouey ntervl-vlued fuzzy regulr lnguge J. Appl. Mth. nformts 8(3-4) (00) [8] A. Chouey nd K. M. Rv Mnmzton of determnst fnte utomt wth vgue fnl stte nd ntutonst fuzzy fnl stte rnn Journl of Fuzzy ystems 0() (03) [9] W.. Gu nd D. J. Buehrer Vgue sets EEE Trnstons on ystems Mn nd Cyernets 3() (993) [0] Y. B. Jun C.. Km nd K. O. Yng Cu sets Ann. Fuzzy Mth. nform. 4 (0)

36 3 Thr Mhmood leem Adullh nd Qsr Khn [] Edwrd T. ee nd otf A. Zdeh Note on fuzzy lnguges nform.. 4 (969) [] Y. nd W. Pedryz Fuzzy fnte utomt nd fuzzy regulr expressons wth memershp vlues n ltte-ordered monods Fuzzy ets nd ystems 56() (005) [3] A. Mteesu A. lom K. lom nd. Yu exl nlyss wth smple fnte-fuzzy-utomton model The Journl of Unversl Computer ene (995) 9-3. [4] Wllm Go Wee On Generlztons of Adptve Algorthms nd Applton of the Fuzzy ets Conept to Pttern Clssfton Pulsher Purdue Unversty West fyette UA pp. [5] otf A. Zdeh Fuzzy sets nformton nd Control 8(3) (965) [6] otf A. Zdeh The onept of lngust vrle nd ts pplton to pproxmte resonng- nform.. 8(3) (975) [7] otf A. Zdeh The onept of lngust vrle nd ts pplton to pproxmte resonng- nform.. 8(4) (975)