Content. Languages, Alphabets and Strings. Operations on Strings. a ab abba baba. aaabbbaaba b 5. Languages. A language is a set of strings

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1 CD5560 FABER Forml guges, Automt d Models of Computtio ecture Mälrdle Uiversity 006 Cotet guges, Alphets d Strigs Strigs & Strig Opertios guges & guge Opertios Regulr Expressios Fiite Automt, FA Determiistic Fiite Automt, DFA guges guges, Alphets d Strigs A lguge is set of strigs A Strig is sequece of letters defied over lphet: {,, c, K, z} A lphet is set of symols 3 4 Alphets d Strigs We will use smll lphets: { } Strigs u = v = w = Opertios o Strigs 5 6

2 w = v = Strig Opertios m wv = x = y = Coctetio (smmfogig) m xy = 7 w = w R Reverse (reverserig) = Exmple: ogest odd legth plidrome i turl lguge: sippukuppis (Fiish: sop silsm) 8 w = egth: Strig egth w = Exmples: = 4 = = Recursive Defiitio of egth For y letter: For y strig : Exmple: = w w = w + = + = + + = = = Exmple: egth of Coctetio u =, uv = u + v =, v u = 3 v = 5 uv = = 8 uv = u + v = = 8 Clim: Proof of Coctetio egth uv = u + Proof: By iductio o the legth Iductio sis: v v = From defiitio of legth: v uv = u + = u + v

3 Iductive hypothesis: for v v = + uv = u + uv = u + Iductive step: we will prove for v v Iductive Step v = w w =, = Write, where From defiitio of legth: From iductive hypothesis: Thus: uv = uw = uw + w = w + uw = u + w uv = u + w + = u + w = u + v 3 END OF PROOF 4 Empty Strig A strig with o letters: (Also deoted s ε) λ Sustrig (delsträg) Sustrig of strig: susequece of cosecutive chrcters Strig Sustrig Oservtios: λ = 0 λw = wλ = w λ = λ = = {} { λ} 5 6 Prefixes λ Prefix d Suffix Suffixes λ prefix w = uv suffix 7 w = Exmple: Defiitio: ww... w } Repetitio ( ) = 0 w = λ (Strig w 0 ( ) = λ repeted times) 8 3

4 Σ* Σ* = The * (Kleee str) Opertio the set of ll possile strigs from lphet {, } { λ,,,,,,,,,k} [Kleee is proouced "cly-kee ] Σ 9 Σ The + Opertio + : the set of ll possile strigs from lphet except Σ { } { λ } Σ* =,,,,,,,,,K Σ + = Σ* λ Σ + = λ {,,,,,,,,K} 0 { oj, fy } {, usch Exmple Σ* = λ, oj, fy, usch, ojoj, fyfy,uschusch, ojfy, ojusch K} Σ + = Σ* λ Σ + = { oj, fy, usch, ojoj, fyfy,uschusch, ojfy, ojusch K} Opertios o guges guge A lguge is y suset of Exmple: guges: Σ* = Σ * {, } { λ } {} λ { },,,,,,,,K,, { λ,,,,, } 3 Exmple A ifiite lguge = { : 0} λ 4 4

5 Opertios o guges The usul set opertios {,, } {, } = {,, {,, } {, } = { } {,, } {, } = {, } Complemet:, } * = { λ,,,,,,,,,k} {, } = { λ,,,,,,k} Σ = Σ* Defiitio: Exmples: R Reverse R = { w : w } R {,, } = {,, } = { R = { : 0} : 0} 5 6 Coctetio Defiitio: = { xy : x y } Exmple, {,, }{, } = {,,,,, } Defiitio: Specil cse: Repet = 3 3 {, } = {, }{, }{, } = {,,,,,,, } 0 = {} λ 0 {,, } = { λ} 7 8 Exmple Str-Closure (Kleee *) = { : 0} Defiitio: 0 * = m m = { :, m 0} 9 Exmple: {, } λ,,, * =,,,,,,,,K 30 5

6 Defiitio {, } + Positive Closure + = U = * {} λ U,, =,,,,,,,,K Regulr Expressios 3 3 r + r r r r * ( r ) Regulr Expressios: Recursive Defiitio, Primitive regulr expressios: Give regulr expressios r d r λ, α re Regulr Expressios 33 A regulr expressio: Exmples Not regulr expressio: ( + c) * ( c + ) ( + + ) 34 Buildig Regulr Expressios {,, c} Zero or more. * mes "zero or more 's." To sy "zero or more 's," tht is, {,,,,...}, you eed to sy ()*. * deotes {,,,,,...}. Buildig Regulr Expressios {,, c} Oe or more. Sice * mes "zero or more 's", you c use * (or equivletly, *) to me "oe or more 's. Similrly, to descrie "oe or more 's," tht is, {,,,...}, you c use ()*

7 Buildig Regulr Expressios {,, c} Ay strig t ll. To descrie y strig t ll (with = {,, c}), you c use (++c)*. Ay oempty strig. This c e writte s y chrcter from followed y y strig t ll: (++c)(++c)*. Buildig Regulr Expressios {,, c} Ay strig ot cotiig... To descrie y strig t ll tht does't coti (with = {,, c}), you c use (+c)*. Ay strig cotiig exctly oe... To descrie y strig tht cotis exctly oe, put "y strig ot cotiig," o either side of the, like this: (+c)*(+c)* guges of Regulr Expressios ( r) lguge of regulr expressio Exmple (( + c)* ) = { λ,, c,, c, c,... } r Defiitio For primitive regulr expressios: ( ) = ( λ) = { λ} ( ) = { } Defiitio (cotiued) For regulr expressios r d r ( r + r ) = ( r ) ( r ) ( r r ) ( r ) ( r ) ( r ) ( ( ))* = = * r (( r )) ( ) = r 4 Exmple Regulr expressio: ( + ) * (( + ) *) = (( + ) ) ( *) = ( + ) ( *) = ( ( ) ( ) ) ( ( ) )* = ({ } { } ) ({ } )* = {, }{ λ,,,,... } = {,,,...,,,,... } 4 7

8 Regulr expressio Exmple ( + ) ( ) r = * + ( r) {,,,,,,... } = Regulr expressio Exmple ( ) *( ) r = * m ( r) = { :, m 0} { 0,} Exmple { 0,} Exmple (r) = { ll strigs with t lest two cosecutive 0 } Regulr expressio r = ( 0 + )*00 (0 + )* (r) = { ll strigs without two cosecutive 0 } Regulr expressio r = ( + 0) * (0 + λ) (cosists of repetig s d 0 s) Exmple = { ll strigs without two cosecutive 0 } Equivlet solutio: r = (*0*)*(0 + λ ) + *(0 + λ) Equivlet Regulr Expressios Defiitio: Regulr expressios r d r (I order ot to get 00 i strig, fter ech 0 there must e, which mes tht strigs of the form re repeted. Tht is the first prethesis. To tke ito ccout strigs tht ed with 0, d those cosistig of s solely, the rest of the expressio is dded.) re equivlet if ( r ) = ( r )

9 Exmple = { ll strigs without two cosecutive 0 } r = ( + 0) * (0 + λ) r = (*0*)*(0 + λ ) + *(0 + λ) ( r d r re equivlet regulr expressios. r ) = ( r ) = 49 Fiite Automt FA 50 There is o forml defiitio for "utomto". Isted, there re vrious kids of utomt, ech with it's ow forml defiitio. Geerlly, utomto hs some form of iput hs some form of output hs iterl sttes, my or my ot hve some form of storge is hrd-wired rther th progrmmle Iput Strig Fiite Automto Fiite Automto Output Strig 5 5 Fiite Accepter FA s Directed Grph Iput Strig Fiite Automto Output Accept or Reject 53 Nodes = Sttes Edges = Trsitios q 0 A edge with severl symols is short-hd for severl edges: q 0 q q 0 q q 54 9

10 DFA Determiistic Fiite Automt DFA Determiistic there is o elemet of choice Fiite oly fiite umer of sttes d rcs Acceptors produce oly yes/o swer iitil stte Trsitio Grph -Fiite Acceptor Alphet = {, } stte trsitio fil stte ccept 57 Iitil Cofigurtio Iput Strig q 0 58 Redig the Iput

11 6 6 Rejectio q 0 Output: ccept

12 Output: reject Aother Exmple,, 69 70,, 7 7

13 Rejectio Output: ccept,, 73 74,, 75 76,, Output: reject

14 Forml defiitios Iput Aplhet Σ Determiistic Fiite Accepter (DFA) ( Q Σ, δ, q F ) M =,, 0 { } Q Σ δ : trsitio fuctio q 0 : iitil stte F : set of sttes : iput lphet : set of fil sttes Set of Sttes Q Iitil Stte Q = { q, q, q, q, q q } 0 3 4, 5 q Set of Fil Sttes F Trsitio Fuctio δ F = { q 4 } δ : Q Σ Q q

15 δ ( q 0, ) = q δ ( q 0, ) = q q δ ( q, ) = q 3 δ q 0 q5 q5 q q q 5 3 q 3 q5 q q 4 q5 5 Trsitio Fuctio δ Exteded Trsitio Fuctio δ * δ *: Q Σ* Q ( q 0, ) = δ * q q

16 ( q 0, ) = 4 δ * q ( q 0, ) = 5 δ * q q Oservtio: There is wlk from with lel ( q 0, ) = 5 δ * q q 0 to δ * δ * Recursive Defiitio ( q, λ) = q ( q, w) = δ ( δ *( q, w), ) q δ *(, ) = δ ( δ *(, ), ) = δ ( δ ( δ *(, λ), ), ) δ ( δ (, ), ) = δ ( ) = q, = guges Accepted y DFAs M Tke DFA Defiitio: The lguge ( M ) cotis ll iput strigs ccepted y M 95 ( M ) M = { strigs tht drive to fil stte} 96 6

17 ( M ) { } Exmple = M Aother Exmple ( M ) { λ, } =, M Alphet = {, } Alphet = {, } ccept ccept ccept ccept For DFA Formlly ( Q Σ, δ, q F ) M =,, 0 M guge ccepted y : lphet ( M ) = { w Σ : δ *( q, w) F} trsitio fuctio * 0 iitil stte fil sttes 99 guge ccepted y Oservtio M ( M ) = { w Σ : δ *( q, w) F} guge rejected y * 0 M ( M ) = { w Σ : δ *( q, w) F} * 0 00 More Exmples ( M ) = { : 0} ( M ) = { ll strigs with prefix }, ccept Alphet = {, } ccept trp stte 0 Alphet = {, } q 3 0 7

18 ( M ) = { ll strigs without sustrig 00 } λ , A lguge DFA Regulr guges is regulr if there is M such tht = ( M ) All regulr lguges form lguge fmily {0,} Alphet = The lguge = Exmple { w : w {, } *} is regulr q 4 Alphet = {, } 05 8