Review Languages and Grammars. CS Lecture 5 Regular Grammars, Regular Languages, and Properties of Regular Languages

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1 CS 0 - Lecture 5 Regulr Grmmrs, Regulr Lguges, d Properties of Regulr Lguges Fll 008 Review Lguges d Grmmrs Alphets, strigs, lguges Regulr Lguges Determiistic Fiite Automt Nodetermiistic Fiite Automt Euivlece of NFA d DFA Regulr Expressios Tody: Regulr Grmmrs d Regulr Lguges Properties of Regulr Lguges Grmmrs Grmmrs express lguges Exmple: setece the Eglish lguge ou _ phrse predicte Grmmr Nottio Productio Rules ou ct ou dog ou _ phrse rticle ou Vrile Termil predicte ver

2 Some Termil Rules rticle rticle the ou ct ou dog ver rus ver wlks A Resultig Setece setece ou _ phrse predicte ou _ phrse ver rticle ou ver the ou ver the dog ver the dog wlks The Resultig Lguge L = { ct rus, ct wlks, the ct rus, the ct wlks, dog rus, dog wlks, the dog rus, the dog wlks } Defiitio of Grmmr G = ( V, T, S, P) V : T : S : P : Set of vriles Set of termil symols Strt vrile Set of Productio rules

3 Grmmr: A Simple Grmmr S S S Derivtio of setece : S S S S S Exmple Grmmr Nottio S S S ( V, T, S P) G =, V = {S} T = {, } P = { S S, S } Derivig Strigs i the Grmmr Grmmr: S S S S S S Derivtio of setece : Setetil Form A setece tht cotis vriles d termils S S S S S S S Setetil Forms setece

4 Geerl Nottio for Derivtios I geerl we write: w * w If: w w w w Why Nottio Is Useful We c ow write: S * It is lwys the cse tht: w * w Isted of: S S S S Lguge of Grmmr Grmmr c produce some set of strigs Set of strigs over lphet is lguge Lguge of grmmr is ll strigs produced y the grmmr L( G) = { w: S w} Strig of termils Exmple Lguge S S S Cosider the set of ll strigs tht c derived from this grmmr.. S S S S S S S S S Wht lguge is eig descried? 4

5 The Resultig Lguge S S S Alwys dd o d o ech side resultig i: s t the left s t the right eul umer of s d s The imge cot e displyed. Your computer my ot hve eough memory to ope the imge, Lier Grmmrs Grmmrs with t most oe vrile t the right side of productio Exmples: S S S A No-Lier Grmmr Grmmr G : S SS S S S S S L( G) = { w: ( w) ( w)} = Aother Lier Grmmr G Grmmr : S A A B B A L( G) = { : 0} Numer of i strig 5

6 Right-Lier Grmmrs All productios hve form: Exmple: S S S A xb or A x strig of termils Left-Lier Grmmrs All productios hve form: Exmple: S A A A B B A Bx or A x strig of termils Regulr Grmmrs Regulr Grmmrs A regulr grmmr is y right-lier or left-lier grmmr Exmples: S S S S A A A B B Wht lguges re geerted y these grmmrs? 6

7 Lguges d Grmmrs S S S L ( G ) = ( ) * S A A A B B L ( G) = ( )* Note oth these lguges re regulr we hve regulr expressios for these lguges (ove) we c covert regulr expressio ito NFA (how?) we c covert NFA ito DFA (how?) we c covert DFA ito regulr expressio (how?) Do regulr grmmrs lso descrie regulr lguges?? Regulr Grmmrs Geerte Regulr Lguges Lguges Geerted y Regulr Grmmrs Theorem = Regulr Lguges Lguges Geerted y Regulr Grmmrs Theorem - Prt Regulr Lguges Ay regulr grmmr geertes regulr lguge 7

8 Lguges Geerted y Regulr Grmmrs Proof Prt L(G) The lguge y regulr grmmr Regulr Lguges geerted y is regulr G Let The cse of Right-Lier Grmmrs G We will prove: e right-lier grmmr L(G) is regulr Proof ide: We will costruct NFA usig the grmmr trsitios Exmple Give right lier grmmr: Step : Crete Sttes for Ech Vrile M Costruct NFA such tht every stte is grmmr vrile: 8

9 Step.: Edges for Productios Productios of the form result i Step.: Edges for Productios Productios of the form re oly slightly hrder. Crete row of sttes tht derive w d ed i Step.: Edges for Productios Productios of the form Crete row of sttes tht derive w d ed i fil stte I Geerl Give y right-lier grmmr, the previous procedure produces NFA We sketched proof y costructio Result is oth proof d lgorithm Why does t this work for o lier grmmr? Sice we hve NFA for the lguge, the right-lier grmmr produces regulr lguge 9

10 Regulr Lguges Proof - Prt Lguges Geerted y Regulr Grmmrs Ay regulr lguge is geerted y some regulr grmmr L G Ay regulr lguge is geerted y some regulr grmmr G Proof ide: Let e the NFA with. Costruct from such tht L M L = L(M ) M L ( M ) = L( G) regulr grmmr G NFA to Grmmr Exmple Sice L is regulr there is NFA This trsitio i the NFA Looks lot like productio rule 0 Step : Covert Edges to Productios 0 M 0 0

11 0 Step : Edges d Fil Sttes M 0 0 Step : Edges d Fil Sttes M 0 If is fil stte, dd I Geerl Give y NFA, the previous procedure produces right lier grmmr We sketched proof y costructio Result is oth proof d lgorithm Every regulr lguge hs NFA C covert tht NFA ito right lier grmmr Thus every regulr lguge hs right lier grmmr Comied with Prt, we hve show right lier grmmrs re yet other wy to descrie regulr lguges But Wht Aout Left-Lier Grmmrs Wht hppes if we reverse left lier grmmr s follows: Reverses to Reverses to The result is right lier grmmr. If the left lier grmmr produced L, the wht does the resultig right lier grmmr produce?

12 But Wht Aout Left-Lier Grmmrs The previous slide reversed the lguge! Reverses to Reverses to If the left lier grmmr produced lguge, the the resultig right lier grmmr produces Clim we just proved left lier grmmrs produce regulr lguges? Why? Left-Lier Grmmrs Produce Regulr Lguges Strt with Left Lier grmmr tht produces wt to show is regulr C produce right lier grmmr tht produces All right lier grmmrs produce regulr lguges so is regulr lguge The reverse of regulr lguge is regulr so is regulr lguge! For regulr lguges we will prove tht: L d L We sy: Regulr lguges re closed uder Uio: L L Uio: L L Coctetio: L L Coctetio: L L Str: Reversl: L * R L Are regulr Lguges Str: Reversl: L * R L Complemet: L Complemet: L Itersectio: L L Itersectio: L L

13 Regulr lguge L ( M ) = L L NFA M NFA M Sigle fil stte Regulr lguge L ( M ) = L Sigle fil stte L L = { } 0 L = { } Exmple M M Uio M NFA for L L M Exmple NFA for L L = { } { } L = { } L = { }

14 NFA for L L Coctetio M M NFA for Exmple L L = { }{ } = { } L = { } L = { } Str Opertio Exmple NFA for L * M L * NFA for L * = { }* L = { } w = w w w w L i k 4

15 L M Reverse NFA for R L M L = { } Exmple M. Reverse ll trsitios. Mke iitil stte fil stte d vice vers R L = { } M M Complemet L L M L = { } Exmple, M,. Tke the DFA tht ccepts L. Mke fil sttes o-fil, d vice-vers L = {, }* { }, M, 5

16 Itersectio DeMorg s Lw: L L = L L L, L L, L L L L L L L regulr regulr regulr regulr regulr Red Wht s Next Liz Chpter,.,..., (skip.4),, d Chpter 4 JFLAP Strtup, Chpter,., (skip.),, 4 Next Lecture Topics from Chpter 4. d 4. Properties of regulr lguges The pumpig lemm (for regulr lguges) Quiz i Recittio o Wedesdy 9/7 Covers Liz.,.,.,.,., d JFLAP,. Closed ook, ut you my rig oe sheet of 8.5 x ich pper with y otes you like. Quiz will tke the full hour o Wedesdy Homework Homework Due Thursdy 6