We will use small alphabets: Strings. Reverse (reversering) ababaaabbb. bbbaaababa

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1 CD556 FABER Forml guges, Automt d odels of Computtio ecture älrdle Uiversity 7 Cotet guges, Alphets d Strigs Strigs & Strig Opertios guges & guge Opertios Regulr Expressios Fiite Automt, FA Determiistic Fiite Automt, DFA guges, Alphets d Strigs 3 guges A lguge is set of strigs. A Strig is sequece of letters defied over lphet: {,, c,k} A lphet is set of symols. Alphets d Strigs We will use smll lphets: Strigs u { } v w Opertios o Strigs 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 9

2 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 Proof of Coctetio egth (iductive) Clim: uv u + Proof: By iductio o the legth Iductio sis: v v From defiitio of legth: v uv u + u + v 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 Empty Strig A strig with o letters: (Also deoted s ε) Oservtios: λ λ λw wλ w λ λ {} { λ} 3 END OF PROOF 4 5 Sustrig (delsträg) Sustrig of strig: susequece of cosecutive chrcters Strig Sustrig 6 Prefixes λ Prefix d Suffix Suffixes λ prefix w uv suffix 7 w Exmple: Defiitio: ww... w Repetitio ( ) w λ ( ) λ (Strig w repeted times) 8

3 Σ* Σ* The * (Kleee str) Opertio the set of ll possile strigs from lphet {, } { λ,,,,,,,,,k} [Kleee is proouced "cly-kee ] Σ Σ The + Opertio + : the set of ll possile strigs from lphet except Σ { } { λ } Σ*,,,,,,,,,K Σ + Σ* λ Σ + λ {,,,,,,,,K} { 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} 9 Opertios o guges guge A lguge is y suset of Exmple: guges: Σ* Σ* {, } { λ } {} λ { },,,,,,,,K,, { λ,,,,, } 3 Exmple A ifiite lguge { : } λ 4 Opertios o guges The usul set opertios {,, } {, } {,, {,, } {, } { } {,, } {, } {, } Complemet:, } * { λ,,,,,,,,,k} {, } { λ,,,,,,k} Σ Σ* Defiitio: Exmples: R Reverse R { w : w } R {,, } {,, } { R { : } : } Coctetio Defiitio: { xy : x y } Exmple, {,, }{, } {,,,,, } 5 6 7

4 Defiitio: Specil cse: Repet 3 3 {, } {, }{, }{, } {,,,,,,, } {} λ {,, } { λ} 8 Exmple { : } m m { :, m } 9 Defiitio: Exmple: {, } Str-Closure (Kleee *) * λ,,, *,,,,,,,,K 3 Defiitio {, } + Positive Closure + U * {} λ U,,,,,,,,,,K Regulr Expressios r + r r r r * ( r ) Regulr Expressios: Recursive Defiitio, Primitive regulr expressios: Give regulr expressios r d r λ, α re Regulr Expressios A regulr expressio: Exmples Not regulr expressio: ( + c) * ( c + ) ( + + ) 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 ()*

5 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 Defiitio (cotiued) Exmple Regulr expressio: ( + ) * For primitive regulr expressios: ( ) ( λ) { λ} ( ) { } For regulr expressios r d r ( r + r ) ( r ) ( r ) ( r r ) ( r ) ( r ) ( r ) ( ( ))* * r (( r )) ( ) r (( + ) *) (( + ) ) ( *) ( + ) ( *) ( ( ) ( ) ) ( ( ) )* ({ } { } ) ({ } )* {, }{ λ,,,,... } {,,,...,,,,... } Regulr expressio Exmple ( + ) ( ) r * + ( r) {,,,,,,... } Regulr expressio Exmple ( ) *( ) r * m ( r) { :, m } {,} (r) Exmple { ll strigs with t lest two cosecutive } Regulr expressio r ( + )* ( + )*

6 {,} (r) Regulr expressio Exmple { ll strigs without two cosecutive } r ( + ) * ( + λ) (cosists of repetig s d s). Exmple { ll strigs without two cosecutive } Equivlet solutio: r (**)*( + λ ) + *( + λ) (I order ot to get i strig, fter ech 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, d those cosistig of s solely, the rest of the expressio is dded.) Equivlet Regulr Expressios Defiitio: Regulr expressios r d r re equivlet if ( r ) ( r ) Exmple { ll strigs without two cosecutive } r ( + ) *( + λ) r (**)*( + λ ) + *( + λ) Fiite Automt FA 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, ( r d r re equivlet regulr expressios. r ) ( r ) 49 5 my or my ot hve some form of storge is hrd-wired rther th progrmmle 5 Fiite Automto Fiite Accepter FA s Directed Grph Iput Strig Fiite Automto Output Strig 5 Iput Strig Fiite Automto Output Accept or Reject 53 Nodes Sttes Edges Trsitios A edge with severl symols is short-hd for severl edges: q q q 54

7 Determiistic Fiite Automt DFA 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 Iitil Cofigurtio Iput Strig Redig the Iput Output: ccept 63

8 Rejectio Aother Exmple Output: reject, ,,, 7 7 7

9 Rejectio Output: ccept,,, ,,, Output: reject Forml defiitios Iput Aplhet Σ Set of Sttes Q Determiistic Fiite Accepter (DFA) ( Q Σ, δ, q F ),, { } Q { q, q, q, q, q q } 3 4, 5 Q Σ δ : trsitio fuctio : iitil stte F : set of sttes : iput lphet : set of fil sttes

10 Iitil Stte Set of Fil Sttes F Trsitio Fuctio δ F { q 4 } δ : Q Σ Q q δ (, ) q δ (, ) δ ( q, ) q 3 q δ 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 (, ) δ * q q

11 (, ) 4 δ * q (, ) 5 δ * q Oservtio: There is wlk from to q5 with lel (, ) 5 δ * q δ * δ * Recursive Defiitio ( q, λ) q ( q, w) δ ( δ *( q, w), ) δ *(, ) δ ( δ *(, ), ) δ ( δ ( δ *(, λ), ), ) δ ( δ ( q ) ),, δ ( ) q, guges Accepted y DFAs Tke DFA Defiitio: The lguge ( ) cotis ll iput strigs ccepted y ( ) { strigs tht drive to fil stte} 96 ( ) { } Exmple Alphet {, } ccept 97 Aother Exmple ( ) { λ, }, Alphet {, } ccept ccept ccept 98 For DFA Formlly ( Q Σ, δ, q F ),, guge ccepted y : lphet ( ) { w Σ : δ *( q, w) F} trsitio fuctio * iitil stte fil sttes 99

12 guge ccepted y Oservtio ( ) { w Σ : δ *( q, w) F} guge rejected y * ( ) { w Σ : δ *( q, w) F} * ore Exmples ( ) { : }, ccept trp stte Alphet {, } ( ) { ll strigs with prefix } Alphet {, } q 3 ccept ( ) { ll strigs without sustrig } λ, Regulr guges A lguge is regulr if there is ( ) DFA such tht All regulr lguges form lguge fmily The lguge Exmple { w : w {, } *} is regulr q 4 {,} Alphet 3 4 Alphet {, } 5

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...