2 Unit-J (3) Insodtuction To Automata Thed\y. (a Non- Determinstic Finite Automata (NDA) (8) Deteministic Finite Automata C BFA) + Intrcctuction “Th Auwomata Teony? + Alphabet: t id a sya to reprerent a Language denoted wit Aigma " By". — Chinite set) &: B= Ao}, da,bh * + Going? ie the raquence of pyrrbols over Ye alphabet - Cintinite et ) &: S =4{00, ot, lo, i, hows. of S~ dab aa, ab, ba, bb-- x5 4 B+ Ka, bh Gop eceS el aaa? ee » danguage: It is depined OD she not of atnings accepted by finite cutomata - / Note: Reguiax Exprersions: &: 0% = de, 00,000: }. a® = fe, 0, 00,000, 0000 ---$ ; oe f a « (61, wine 5 ; ~yer se see ntiaae Coby = {0€, ab, tix, abab, Wensa, tai) + 9 ermydy set re ae 5 ‘ anon y indude at = da, Aa, aaa-- | Cot)* = for, 0!, O1olol-- = } e Corts’ = LE, 0,1, 00,01, 10, 1-777 s (AGS TWA cj ynin feb, ba, Bh aoar Ia Si = 4a€, cab, cabab, aababab : -+ b by L= 4a files with aubastring abl (orl 9 L{ aerting vith any me. oF coreeiqgtl o's g a's and amding with a} stloalas i Finite Automatas x Shudwal Ragmertation of TIS Svs efefefeleL [ELT s | te sae! Automation Model 3 Fe fine awaraia a a ae veod herd ard prile contrat: Gnput Tope: Ho 0 bres tape having ‘, each input aymelnot i placed in each feo ‘ae ins Tape feade): H reads the cell © text to night a oe inpue nypebt oh a , ’ % Finite Control: It decides the rext one top from the input tape. # Jaq ed Aefinision of Auternata Mathematically , the 4inite re taper . @: Finite set of SatOD Bi: Finite net of symbols (alphabet &: Tramsntion function Goi Initial stale f: Feo) sede ixe stase Tramition Ragan: . are state Aaansition — diagiar © ured to tepreent FA whe -e ataieon ae reprerented by o 8 gramitiono are represent ad by =e &: 0) B= 40,1} L= 40,0, 1,10, ot} in Adder to accept the language the sate tamition diagary > tepreneniad by: 2 440, sae 8 Br donth Fo + {Ao} f=} @ B=dorh * f L=4{000, 111, tot, (00, Col, 10,01, E10} Ge) Oo) @) o,1 2)" @ 40, 1 Fa, Fay B= do,1h 8: gp 2 Ageh F {4shNFA to DFA: slob 3 DFA refers to detenministic finlke automata te, a finite cutomata Ww Adaid to he dotormimtic tf cdheeponding 4d an fapul symbol thee tb a aingle Yesuttant stote (only, one taansition)- Dit & a eet of 5 taples: M= 48, 2,8, 4%, FS wee Qe ik in a Mon-empy sinile pek of control + f E= i a non-empty tine oak of input oymbdo- B= ik © a tranaitton -qunetion thot takes 2 cuqurents, A 2kode & an inp Sgral, ib yekuano a ingle state Qo & b&b the aarting stale, ore <2 sho sialO in ® PS Tues of Riglte Automata: a) Determinstic finlke Automata - DPA @ Non-Determinstic inte Automata — NFA stakes in the inile > NFA tefers to mon- determinstic finite automata ie, a 4{nie automata i maid to be ron detorminntic if there are mde’ than one povible -ramition from ore sate on the same input symeoo!: lt w & pa of 5 tuples: M=4@,Z 8, qo, FY where Q= ik BW a Non -empy toile set of states in tHe sinite corbol, ik in a non-emply. tinile ack of input rymbols &= ik ® a tranottion Wurction Ahok takes & state from inpul signal § a nubs of G. Jez Ub wo ne atorting Mode, ore of tre axcde 108. Sa an rekunnegro. non-empty set Fe ib th the # non. g, ah yinlle states dy q alates from the Feit bo ob finite states dy accor prakes fom RH cat belongs 0, Q. to 8 > DEA cannot une empty String | NFA carn une emply €) Ramition. transition, > DFA can be understood ap |-5 NFA can be ore machine multiple Atle 2 sting at the pame mae "i St ib equ, dificult ' 4 4 a Ae tt in earter to comtw Conntaurck 3 AU DFA oe NFA -) Nok all NFA’ are -> &GxE 4g 3 8: @x(e0ej ae 5 DPA teauires moe space. ? on ate Examples of DEA? : a —————— G) To derign a tinike automata , F = 40,1} ‘ which ptarto wilh 4 & onde wilh me hon 107 100, NO, \N09, 1010, 1110, 1000+ F + 1. @ ear, qa} z= {0,14 = aF= 4rae Design finite cuttomata == do, 1} accepto even no. of @ zeroon A RED KO, Of crm. ©0, W, C000, ILI, COt!, 1190, L001, O10, O101, 1010 9= {92,5 A, 434 T= doh %o = I =o oe ae a tinile automata wilh z= Xo, th pera only tre ee bk O Exe ze eh % > Yo Ys = ) Comtruuct a finite automata *& F-4o,I) f% 3 connecutie O94, 0001, 1000, \CC0!, 4 a ee ° ale ey AAs) pe bro | \ Pants @ = 44e.4» 43,43 % 1 \ rx dor ry %. 7 fe Y= 4o 1 FE %3 :Alor ju # deorminctic File Adoroaas €gu): Comtauct BFA oen on alphatbety =={o,ih larquoge = atevits with 200" 4 9219, dah es z= do,15 % = Vo €Q F> WER nel (2) Comtauct DPA over an cdphabet == {a,b Yorguage sow wih b a endo with: a- Q~ {a0 A, 991935 r= for} qe = Go 9 Fy EQ@ Combuct DFA RA an alphabet Z- {0,15 fy L={oo;10, 11, ols. u “OF O$_@GH® awe = 19, WW, MD Bo z= {o,1} 42° % WF = Ia { Pe (8) Comtrick DPA f& E=day, LAGa,an,aaa-.-} R =aaloalay : = NFA => ton Determinstic finite Aemeter 7 wy ta i ; pevhamiive text t g: 2+ fo,1f Pearls daacking L= aning end wlth one: 4 FS thin language ovey = construct NFA § WAVO, OOl, 0, (Ol eee 5 State aoe atte NEA: ag es an @ > =: lout Fa, =% €8 le Pe Rig @ conmmuck NEA {By the language aoting sans & ord wif} -b- NEA =(3) Conatauct NFA f& Are atring BP 9 length =-fo,1f w= 400,10, 01, 11} & on Gee ©) 69> 4%, VW} Re © ©) z=4o,1s Q =" EW €= 9a Gg NOTE > Bey NEA b a DFA ie, We CON comet NFA's #0 DFAS became NFA machines dasa not exint in real tit ite pepo tw, thee one eany to comic 9 it will be wed in back tacking, oxhountite pooch techniques. but eveuy DFA to nok oO NFA- { % Cryesion of NFA tQ DEA: “a & convert the Ruowing NFP Oe. 8 = K40, 25 w a) 3 Q—+@) "Zoo, by Yo = So Fem into BRAYNeco -vansttion table 48) BFA‘ danguage : String eis ed ending with ab - tre NFA DPA occept> the language dhe. endo, with ab: : + fg ci ) as tes OF @ g= {90,495 z= Loh 7) Sp. S07 Rca rNeo xansition table fh DPA: ot 1 (ofry*olofiyi0 (fy ° \ Qo, DEA: aS CS tangange : The NEA dy DRA acceptin the tanguage with in any mo. of Os & 4'S tw) @= {0,1,2,3} = =4a, by ae = {05 F=fons &. ‘i 70 \ Ana | Zh. 5 {ia} } dias dosh adray| 425 Corguage 4° aa*(bay 3 S= 4a, a0b,ab, a0.) -- i Stun 9 seviting eith oOoS|o0 [oul : « Epsilon © ~ neh: NEA witty ermpty ip (ataing) aymbol 0 1 * O5H56 Ce 5: input oyrnbolo m= 4g, 2,8, %,F } 3 Sox s ufey 3 29 Covet the Busshg epsilon, neh +o, NFA , oa: és Ceti j W Qs , of } a7 4 © | Z (ite tnaee qn —€ le 4 ~92- >)the #AItion function S' 4& NFA ia: ais aN es ‘ =e fa0.41.49) 41%} % \ fash don ae) 82%, VW 142 4% “| Lay} far} arom 90! Ww Fp 1042 ,WUB We fo.) [Lon ar} *Laaidrs iaeande) | {ar.q0) {ain} | Mars {4.92} * 42 WM ' \ 0,1 Qo, Ve V2 } 8 fear Gare iae- CoH 92, (09) 2 fo, 7 8: al = fo, \5 Qe: He O eq > 18,3 ioes ° “fi ¢ 0,1 d ‘Seas me! € - closure (A) — 4n,8,D} €-closwe (8) — 48,D4 E-closwe Ce) - 4c} €- close (2) oy x ae ((pe — has es & Be —. CO ee c—c—?¢ * EC eo ee The tnarsition Tundion =& NPA DSEe is aud SA \{n.a,cdh {of ‘ ee a 8: ALH, B,C, df Z doth {oh {4,D} Qo: A Fs D, Ned, BD, cD Dd \ {dy {vj > NPA ": 25% sje) a\aaoy s4ps0o} {anacado} spe) gs h, ABCD, D, MBCDS, DB, x to} any ae a e Lol s4faccpey | frac} soe5 a A e: {necd, Dy AncDd, DB, ‘py de, db) x{oay {pc}ZT Aah} whore [w]e 2) Constauct DFA over ‘ W DY Grn f= 4a,by Us| med = Saing: fle 9 =O , |W 2 =o y language: fan, ab, ba, bb, aaca/bbbb ,abab |, \wlpp 320% faa, albb, bbb, bab, OS OHO trates (A i nlmod2=0 =) Ob { yy ? - Ob : Oe +; *S) covinick “BPA Ghat ooels 6 bs both we een. > GosS product method | ab UW a Oe fas we | | CO e cee pS? nal a » OOF 1 « see wen b a : [4 GJ.) YD 1a i he | oO | 7 on» Unita * y Finite Automata ond Regwar Grammar Requiox Explesions: These ue mathensatical expressions descnhing a language which » accepted by finite automata and the language & called an regula, language - lek & he 0n alphabet then tre RE over td — a vequiar expresaion that desenbes an empty Set 5 defined as folly: @ E -wabo a RE that descnbeo enul stning sels i) A- ib a RE Oler & than thet describes tre sat witha. @ 4 a © ae 2 langiges ae Gi (iy let 'A' and 's" te ae tre RE described as: shts © equivalent to GUL + NS bb equitalent to Lidle CG) le en® io equivalent to ut &:a) wite tre RE fd the tang accepbing all combinations of o's over tre ==Lak w= 4E, 4, 00,000, aaaa.... & Re = aX @ waite te RE fd the lang accepting strings any —™: of a's followed by ony ro. of Bs followed by any 0, of CS Over the z=4a,b,ch Ww= £E, abc, oabbce , aabc, --- J Re = fay? Coy we unite a ER the lang accepting set of al stvrgo which containo “he dd chuvacter hom the ight ond Of Fhe Shur % aku @! rey E=AI DY L =4aanbab, apa... + Re = fatby’ ala rbyatb)ep eye tree Rr the lang, accepting Saws wilh g erdy | Mie Gi, ARG fe) W = dab, aab, abb pe - A (atby* 6 tro nare of = the state arts with RE fh sna letter « 4he a follows wilh ©) waite capital tether ee = [A-z3[o-2]" sdoogunge Assodated wth Mailer peso: DTre language which regular languoge- te the RE, ne(ea)* tren what» tre lang of thin RE: w- £€, ba, baba, babala - a: of substury of (Ba) + % spgetbedl ley ties GS Ce &. 0 > Lla) = any no. RE = arta we {aa,aba, abba... \ gaing starting wih o Fllowed hy any ro. F bls ard ending with a. ® 2 = c(Re) = Po - any ro, of IS ending with o © ~ sinicg Starting wih o Q enditg with any @) oo* of o's * © (tot) » (10 (0.c0,000---)) oS w= AE, 100, loo0, 10000, - L = Alning Startting with 1 fotlowod by oY ro of os.of — Finite Awotncda check whether these 2 fats we equidlert or Mot + = Gu (34) (Hn) (2,2') (23) | 6) 6,8) fivie Sep: tre above «2 Feal automata’s are equivalent in the ready arred transition ‘able, wo we geting the paU2s of states: os. (zF, 1F) 7 WWF, NF) Slep-2: Suppose if wo gek t2 paias ao combination of (Fine) cB) (NF, F) cB) CHE) BILEL) @ GF, me) ete we cot equivalent 2 “$$ Omm &§ ooh ecm ard si F(-99)|( 2,43) [o.95) “to the pain (qa 43) 9 (43) Car.93y | tas ay) A camanodion of QAP) tren trese 2 a @is4y) 2,94) Vana) Bis ©. ridcniuet308194) (864) (@>,8¢) | (8s) (@,Q) (82,3) | (Gs, ¢) (82 ,@3) (@>, @)\_ C@@x) (Q2,Q5) (81, Qu) sofas oe oe A Tre abowe 2FAS we equuivctlent let BST ae regular expressiay> then @) (Rts) +7 = R+ (S47) © Ra R=aK @ Rep = +R =R HW) R= OR = > (empty set) O Res =S4R @Re = eR= QR @ R(ST) = @)T @) &ls4) = es + RT ( B+T)R = SRR () o* = Le ay at e* are (ey (2) RR = RR =R* i + (13) R+s)* = (Ge st) = (Rss) + CM} (&)* = (RR 3) -Us) .ee. oF RES! en ye Equivatans a converting of ® comating of FA to RE RE to FA ¢ oonyertiog RE fo = NEA, & hae) = S then FA r= € C accepting ompty string ) ompky ne nok accepting stning fees wed a sving wie Ae, Reo) then eA OD (AP) (Cr s 628. Wen 5 =Ap-8 QHDailol? 4 convert MwA tho flousing %e into PA a alo conchud 4 x (ab VU aba) a *) (ab + aba) a i) a w= da, Aba, aaa ---- 4 CanbaetOn NGA : e-dosine of (46) 2440, W. 43, %.4a} i) > 44 Aas (2) 9 {92,93,4.,44,49 4 () ‘wy aa) =f @) » Jan) (4) = 40, 9 649 93-44, COR e Gti gy (48) 2446» 92143449, 40} Gy) » {tJ | Se (a) aeyCiregce ee anes AM oe Pataca? —ip oo Vo QW Sem weer ae ee. 4% —8— 4s —4s er a = Qe ais & a & 2 6 & G —C —Gy — Yr Te Dacian sa a —4s me US ee Owe ae. Ao ES Tse % — ®— W—WW Vien ee ee? & anes ep & & W UH Ue — Ys wy ayo -$ LT So ee ho eg Cie» & I—%-o—-b eee ee ; Wea 4 < Quy Ar} > ome wee %—DdD —. 4 — ay G% —D —$ ty me —% eeee oO e e € is PT) tee ae Ghetto G-w- o+- : Bee ie. att eesti te eta a eae Cx, ' ef chieg® = 4- WT faorA, -q— — > a4 9a, Iy os aa : 3 eo Chop fb 9 woes ote — Eb alte 4s a an — Ou ty 7 + epic? es x re et € an—9 —P— > ayy == $ a 444 49 cu} fs any 4, ivf Jee a. — 3@ {au 19aqunry > fay, 4a.qu} 5 dass $ § ri 4394 46% fas ac} om 2 lar aeqaqureqa tay fie 9s ananaes os : %3 400 sq on Fa Aa Io} das Ge qu j ‘ me Neca” faz Ge anany | Lash Hinas wsancetot} \ as} ay Jauqrasew 9699) {a PI A446 qa f 495 444 J i $ | am. Is Fo + 45% te P4994 V649 44} v diy - ol (o-1)* ‘c3) Sa i cuca €-nFA +t DFA: a 0) Co, 0 2? © Bee f(y), ~ oe aye A (a) = {aaa} (a2) = Age} é ° x - a Yo ao) di) Sl) 540 = hy —> & ¢ go) a) %» — 92 —42 Y —3 © et oe son => se!) —a) — 4) % —9, —@a)c RP —W Ean iron. e ! Slo) -—Q2 — 4.-@) slo -42 — & -@+ tomerting” FA to_8E' a eae Fa converting PA to RE ve we Adens Thederm, ¥ xx, He Dandents “ThedeM) : tet PA Q be, 2 RE over the input alphabet & » The RE ty giten 9 R= Q4RP | which hoo a wnique solution R= QP*- j a Conversion KAM | G) tee g, ve on intial state. @ ree WE AQidaqu-.. gn we no. & states. The tina], Stade rau be ROR «Qs Where YAN. @ tek ty be qj sansitlon fom qi two qy Hale. W) Coloulate 43 = % =%M+tE FY R=4%; G=& P=0 Rage 9 Yo = Eo% ®» q- 14 + 19, to* +14, R= 4, @ = Io ) ps) R= gp =) ag = “ lotr? 2ae ® > w= Of +O +H1Q0 * oot 4 ost 142 OM 4 0g + 142 = oF + (ott) 4 Rego, gzoti* 'P=o+) R "1 fee tT Ot) = of ort). tee) fh the andorra he RE o Cont sos pumpin leoma RA Rguer lonauate? sit lo Wed to pole that a nguage nob regulon: - > Buk cannot be ured to do thot a language io Megilar SIE IA a requiow language they A has a pumpirg length spl puch) that . any ating ‘Ss! coher isi zp may be diided ito 9 parts s=ay2 oud that the fuow'ng conditions must bo tue: a) aye ¢ A fe avery 120 ®% wl>o @ Wy)

Sz dod = ol = & © equbr wxt=2 >) S= alam 0 yore “KRW rok a nequlay Aanguoge,lomma— prole 0 ae pumpryy tes {wil ye (0, yt t ip rot a requlay, lang 0 | ii: fio = oo Y= did) \ mee Yq al ale ( J tat p27 = codecs (o000000 | 4 nw mab 2 if i ceayla + (8) D1 of} wt 10 9 B= SF MJol OF = Foolol = S107 € sequlan, “The gien language in not a regular language ¥ Qpet hee Grammar (CAG) | > Tre context fee gamma, G v Acted od 4 tuples: S| S151, 28) l where V5 variable 6) ron fe Cpa git ) a care letters T=) terminals Gepesented with tower care itis ) P=) productions $3 sta symbo) (1% aymbd fo LHS) 3 Wee ae too types. of denivations: - ) WY most denituation (Und) ® Right most donivation (RMD)RmD ‘ Sat The preduction i of the iM 3 The production is of the fy thei ’ A—f\a then ik Wb lied od Aaa 4 n ik is Called 4 night Urea, fm lek Unecn, fern 62 can derive the Shing by Yepacing left side symbol ?-e, the vorable can be placed with ot tonmingl 1 7 cal a Dervation : 35k © a sequence of production Aviles which v was t get the inpuk staring T° srese — production. > ie we of 2 +yped @ le most Porivation (umd) ; ) Right frost Derivation ( RMD) 3 In UMD, the input ‘© acanred g replaced wih te Production ule Rom lebt to aignt 3 In RMD, tre inp © manned a repared vith he Production nw omy night to (ott - : 0) €= €4+€ ee eg heh o> Artbigilous ramon € = alb Ve = a-b4+a Fee ApaY und 9 ND from the obre producqT -ab, a © (emd) Gaz EXE (und) E— E-€ unp> = =s a-F OE a Cer Get ‘D, a-b+e uM. gq —b+o AGE o@)8 agbe ab® Gs-c) abSb 28 abbov Eg(s)1 Dose tre Staring oolol ® the context fee grammar given by: S— A168 Ve A + of}le Me : SAB Suey dae ® +> o6]se}e Bese j >) ae end => ALB <=> AB os CAABR BOR A 08 soy ond OOALB a eV ae Rony ; ZS oe 18 SON A 404 UR =>) 9048) Lady Alol um, Shh = ‘00108 x fey oto} ; an z Ss 004046 SS SCOAlo} um 4d a Ane SS ooloje a COE lo} ud re * = ooio) rs oro => N- qromnax may _ produce more than ore Kf mos anivatican oy mde than ore night TWost daxivation Then Beat Cealucl GS GbR Ne Gaerne Posse Tree. . on) to called * Toxivatton . hee: ©v > Wwe gqartical legit of the dlerivatt a denvation Mée &’ parse se- oulouloy —_ > The pase “nee oO be RCO oo uy Make a “yoot Mode a ptait symried (ity Intonidy P02 Ge vorvinblwo 4 NON. varlables «Oe mnt ire diy Anda leaf nodes are tonminals fq uy Constauck = parse tree fc the following e— exe / exe fid 2 ip > id+ id xid € ——— root a CA) E 4 E- intend oF aree a Preeseere Geass ce O\ Shetagbecause the highest operat precedence © fox thew the Soot node: Clery os GSS ee Ret grammar v9 S— bSaS ambiquous 4 the lop me aly ‘ababmp ae S = aShs Se imp a ee = > abSas bS Bz aebS | wate = aboShs => abeasbsS ie cS und => aha €bs =) abaebS is ay) = ababe = > abab& eo und = abab = > abab ‘a we Be grammar jo ambiguous gramme’: % Chomsky texarchy + CEL PDA FSMIFA TYPES < Wed < Type) < Type 0 RG > Requioy Gramma Fem =) Finike: Stato Mochire [FA > Finite’ Autoncta CEG > Conkaxt Free Grammar CAL =) context Free lang | PDA » push down adore CES % Context Sersitive Gramma, LBA 9 Uner Bounded Automata.« Unie « ° ad & Intacctuction) * 25 Aish down adomata w» mde pouorful thon FA > In FA tree ib a fienited memsy but in PDA & ho me mendiy because & Kas tre Sa dota Suckot b ued to Ade the inputs - PDA = FSM + Slack FSIN-Finido State ffachicg wilh In PDR woruse context fiez granynar rorre lane regular qrarnm a in PA 5A stack Bb A Woy eth tro tOp. OF =the stack: to waarge the erento one » he 2 opaations son the ..Stack are! PusH a PoP. PUSH = ew element io adldecl on dep of the tack PoP -) the Peete te vice fod 8 lead % Model of PDA:~ PPA has 4 comporerilo! 0) toput cell Wy read] wotite head Gd ite tontig) unit AY push & pop 9 Fe wathemotical def of PDA: = COSI R) teles re) r S > dinkte sot of sates Z> finkke set of inpuk Symbols TE) TS Yoke & Non-empy set-of stack Symbdls. S > vansition junction which is det” an 8 @xtzve} xy gx a* gr 2 arr % >» initial state €9 Zo D Stat symigol of the stack € F> fiool tote €Q os\oupy Fy Design a PDA fdr a tang t= 4 ]n2i} 5 Vp s8Ng = cont ©, 6/025 ' “Wy OJoa Se & 2 S +, 000% €Refe< ee Metrerti]e SHG = OocoNI Ne &(90,E, 2) = S(Fe 20) &(4, ©, ©) = & (%, 02%) &(q0,0,0) = § (41 0070) Slqq, 0,032 §[41 0002) 8p,20) = § (411 000024) gla, = § 2, 000%) §(42, 1,€) = 82,0020) sq2,1,€) 2 §(%, 02.) §(%, 1, €) = SC 4,2) $ (42, &,€) = S(qs,€) Ip stiog = conse S(4o, €, 2) = §(%, 2) s(Qp, 0,0) = § (4), O20) &(q1, 0,0) = &(%, 1, €) = §(%) 0%) &(%,1,€) = § (92, %) blp,e,€) = S99 © eR °, o[o% Q (E €,20)% t pee €, €] to 9) fe) @) 1,9/020 © § (4, 0020)™Inrit =~ Ro): Design a PA fH a lang {oi >) ‘p String =) OO\l S$, €, 25) = § (40520) S(9o 0,0) = § (qr, 02) S[4:,0,0) = §(q), coz) ; Scie ae