“+ ene Unit ~ |; Trwee_cenlaal omens + the theory af computation cere, — ‘ 1D) Avtomata al Compoiasifiy 7 - D||__Complertiy — a — T a — DI com lexth The conte cpsatn oy _semplext theory Ts = | ‘What make ‘some problem cpio hoard ond others ) ena _ — a H eotarkabty, we dant know the omswer to mane severah j partion she ws sone ees that a tn be L a 7 17 By sndérstanding which ospéit-o} the. rcblem i 2 is Sik the rook of — > aites fe 90 that. the pankiem & snost— — ey ‘solvable. 7 os le. tocettie fore lew than a pertert sotutin. P fo the puoblem, Gn’ cesjafn cases find Fog solutfons a __approfimate the perfest one & velasfluy eauy . 4) Some problems axe hard_oniy fn the worst stvatfon bot emsy most o} the me Depending on the applfcatfon consider, altewnatfve Hme 24 computation syen as sondoriee _ computation that con et up sestain take. Computatittty Th = The theories compatatltiy ral lex “iy ey {rrolased. + = plexity plerliy theory the_objestive Ts to claithy belo os ee Gnd _herd_one,whereas’ in _Compataailf a pro_chasf{featfon of problem ave those —terroiputatFomatty solvable or not. : =7 Adonai TReony.- 8 deat vith the deffnatfor aod | Property} smathemaltal modete 0} compotatfen - naires cased the thite autometan'tn tert | re Complex defen and hardwore datyn. | fe sed fi MPU cadted taent trex. Gorm (Ce) : L m0. h | Programing lngoagea cnd_cutonraia, tresrg "© Alphabstons oy a oo is | Computer cde Poestnnae bale bling bck. We delfne an Alphabet toben ron ole — ya las oce 2 hte alphabet re the_symbel. We ganevatug —~—}— te srepterent the get 0} albhabel . = Bea ae 2 Atiing over alphabet Ra. ffite seqpenta o} cyto ae —}vsvalty unten ext to one another. and not separated by commas 4} oe ste Pysnt te thi'ch_ fe_the nor of symbole Fe contakrs, __|— 7 “The sting Of ten: a sepreaented by i | Revewe ©} vy "f usittien ox wh, fe the sning obtained | by unting wy fn opposite onder. an (fig 2 aobeeg o| wi If = oppenss eenaaquetfueny | usin we. apt cael the empty svg ond { ite seat = Soa Sen | Lanppoge A_lemgage 1. he stot of sng| Fite state automata+ fe fae ats a. He Wwple ‘model oe 2.451 %0.F)- — 4o- Snifal_ state ‘e |_F + Seto} final state | Qs scojamstae € = Alphobet & “Fasitfn foretion (axe 0) ele -—| > aan cue $0 mput- Cocentic civde. ‘represerlt Kinal state. i ‘for thie_poobiem %{ flesh inpot 6 4 only then shins is auspies _|o we ga to 4, (dead state) 4 never ceme out ol Be. _ | Q | prow a srite state “audometal over alphabet 0,1 whidacepe QL the amet | even length.—|__veelty. for tis * VULVA VIN 2) eee cris -|ivad State, e no inpeks TT bids evan, i tif sate te nal state, | |G) means one a frre, Se for lect Aramitfen clther 1 oF 4, length =) 80 dent stale, "reek ert length? go even ¢ so on | Doaw finfie etate awtomelal vthfch aunepts au the (Jergtha) | Steioge. Greatter than 4 ond fs moMilpie of 2,. Here 7 418 given so nial ahote eam!) be {nat Bos. stir moltipleo} 2 ts 000,94 11 oF omy by 8 lergih greater Yam 1 hnffakeern't be ffrakoa raater than 4. Oru finkiectare aylomeiah ever 0.) alphabel vshte | aseptic O1 and. 40 ow a xing | Ax fieetone ether © of 4, then chek for nest | Remar be © and 4 $0. chesk state auordinigiy T He foaea aera] aver GT, 10, TH will be shang (Ike 040, 000% pe a Cut mot atreptable so_goa fe 9q..Q Draw finite state avtometal over 0! alphabet which artepts siting contain a Angle 1. &) which contafns ak leat one 4. | for raul, one := me mater how “many Hime © appeorrs, Te { As not aueplable so sc loop, a4ie gesting. single 4, Qu other = pe after mot asroptable. o | + S_ ooo _ b) |4, || 2 }—9x}9x-} 9 o ee a — 7 | b : Dyan rte sake otto over 0:1 alphabet wh ance 0, an0 sting _ as _ __ - ~ Or 4,14 PRES Sr Ore once 001 oct, ait su¥fng ore auytabie ajter wat @) ela Nee) _ &, |Deow ffnthe state auomelehover 0, Nh, Gcceps all stving_ tn Te ane in. stig I fllorel by ne + Galetat ones) : 7 js _ : o A | — HQ" eS fay tas 4 ae @ |a,- HF Drow or @ finite stoke cutometal oves ,) atpna sarong atk even no: fn bina many i's alway be odd, “no tromsiion Dons o {fotte. stole automaton binary, seapts oh au S when Yoterpreted as binowy amormber 1 agate PG form has 0 ak 1S6- yo meet steht sags whith ave de by = —— eo a?[fist asi aer Het 2 A) as. 4 tn Ne EOP OT © | Construct a OFA that crept. aU the stings who's short |_and_end over alphabet (a,b)« CStamt 1 end acre’) qcctb Ww 4a @-— 1% @ 2 1% qe ‘Construd a DEA that _atrepts _ushos tts} amd lost atane are not same. ee Constret o DEA that cuepts ans stan with even o- of gre. amd even mo:of onels. over aUphoabot oO. — Set a ee ° ers . ce . : = i“ = L- Af ae (Ga a suns Constrock the DFA over the alphabet ©,) Ah whieh orerepls lh of the. strings cons Gg _even no: 0} gem and odd no* oper: >} - _smait shefing $$ so intios atage Te no} frat state cyating sol aueaptt (AL the. aiing axe emp | 0)" Gy" at contains af least 3 ome By enopls ALL the staings Un é sie 8 ee 4o aA = y_ __— «| aecepls att Ane statnge vaheh- who! s_length | unatis 24 symbol i 4p that conjakn_ the Cond o DIA uitth ab the _shfing - 3 cpbatioy O10 | cobatig 018Qonstroct a OFA which cumepis uu staing Wh whiun decom’ tensho: contain the subsning 440. | SOLOS” | make au final 40 mtnnfinak , and fion-fmat to fined f complement | first drow the avtemeta. whith autepls thak string ord se W fsany ekaing except il and 114. ‘Or "Oe AGS f ae ee | emeSifan Fonction er give maui. = OED on ——o<yo a © NEA Non. dete A non. determinl CME, G19Q10 Fe set of Aina late Yo = Smilial shote G2 Q%Z-P(Q) FQ) 22% Ze > Allphabet- Q> cA of slate We flinthe tate Aude Me stare Autorn ©) Draw a NFA which start with ayb over alphabet 9\b *) Contain ay I _ i) | - I Conctruct a NFA over 0,) ustifch CLecepts siring Wwowhere | We bp, comvest_therm into. OFA- Re ne encng nyt: Gr) Lee - - : = NEA o five tuple mode! | | |— © |, Concho _o NFA over O,] whith axepls sing WL where ( 4 W =, convert them Into OFA, E> accept emp y siting cow, oO stdtng Ye aoa _a sting) ont = - . DEA _ Tati fen. D a at ma os Ge —S401945)-K 46} — — a = : F Now by sony fonefon, = BFA Ccomestel) _ All sta tod conteuin News oe of WEA A is Yr {ge 9 oa here Consuct NFA_over 0) “use dutepts Ut the al A howe = those ard symbo} foom abé last “BL convert Ty into DEA.nih G5)" )"&) ° o 8 4.5 Mord 4eq2Aele | Get) — ets tots e | _€Gets) 40+ etre 4oide ds —— Jott i qo4sts—— Aaa 4,97 "shir DFAL [commas NFA ever alpinbel bc which will cuept at the vo is i js those ore Sec uth ab be,"ca and convert this into Se Ves ——4e4s 499) Aes — 9.9594 a AeA) Fast ag#Gonetrod an NEA when wl asiegs ati sringy Ove asphatet Rieiey A shea has efther even noo} Als, even NO Des | or even mo’ o} c's, ve us) = = O-med 3 | ae be mo ttiple a) = [Rm (w= : Omed (no: of b shoud be smott ple} & |“ ever alphabet La by both condffins shovld be ther __rmeans both “jo%g Shedd be final.4), Na Cur) SO med 3 ‘er’ nj(us)= 0 mod 3 ___|| 4 omy OMe OF 49 OF 4, Gs there jhe state fs made final thad!s th 1s the ony dBHerence. | —__9 || Conshud o DFA which c.epts aul strings Wi Over Gbhabel eo Koy where w= on | n7-0 tr, MEY a i ©) || Conshuct on NFA which auept att the shing w on alphabet ; I 4° \ aha uber =Xom riak am phy 20} — = Ge os 8 Comsirucl a NFA oveY alphabet: aby hitch cuseple out the — StRegp se stoi ancy semis are aftferent 24 7 7 Ca Oysen Aer tem » BEA Ke cstornadidin, <4 ry have ei EO = = ewe. are, ceed mean Oe atcepts aut the binary by Sand s_ = | © | Construct an NFA over 2 (0;!), whieh ces _j_Stsings_wltch when interpseted ci fible a [Mate © OFA Jo both the ashime amd take eros predoct — ____f on is asa taifa protien] yeni OQ Geax ee Ga) io Sooo eG: 6 & ey) waD- trom [po] = txtetowwse [ $ 4.19)0 $@,.2) PE 4 yo) ) 1 = E-cgsnre r4a0 v6] = S-aosvre [4g] =A a] D-tansfA, 1] = € -~cocore 8 C1p1 PSC 110 6a! = -dewe [4 UU = €-docove (4, ] lad on th = . ge navded 09 he <4, 92-4 = 2 Ca Remand s endeSTe Mater ineeded Deavare [A2] = €-clowore ( §[q,,2)US (aa?) S22) = €-cosure [ $0 > 0421 | = €-closore [4,} tt : _= £4,422 = tesuse [804 3) vs. Gigs o) DUS 2D] Tea} © -ctowwre (8 OH, = e-cesvre 9.) 7 Sco = : . { poe (= = ches ore (OJ Cais) : { =D = . 4 s 1 Doms [ 11) = = chesere Cod 9) 1 | a | Detronsl oo) = e-dowre L912) 2 Oy p2) Fe = = \ [L_Datwene [2,8] = D—Themal pi) = D—Tome | PA L=P~~ [Bat there ome nd +€ -dasare from amg stake a then 4S dave Kay nee] -* =— © ctesose CH)> G omy] Se -ctoane (©) have & ony nase iat few to whith ue = | exve mmoving throxgh lenty from 0.Ctn ‘beluizen m0 tse) — | &-dosuse (6) = Lo,t 12 FY a | O-trans (Aaj = € wetowrre (20,8) 0 Scales (%a)u i oe rsrs—SC + closure (h 0 UZ bUs) : €-cosure (2,2) Crom Ste whew it ace 2 K1j243 46, 3y =e 3 BT ea Bie [A= 6 -dere Cendeduswh) al a) = _= €~ dosove (5) : 5 § = KU AMSG FY SC ay = ——~ 5 — : =J Fe oy May TE wall same 0 befere. Co NE i phe _ ag B1oo comet eeeaes ug co mwen vid _anake minimized © 1c Se. aT® i" 4 — Aan, aay A 95,4443) 4 - abiaing bo fst IOS Meu __Co a day day es 4934 | Slvidlng C3. mow | WE ACS De Wb LEI LL I IPS II PI EPS L Cot Gy Cy Kast Are) 93 Laa.agy© Craw a PSESPIS AL the by. I and minimize. ry shring the ‘Sea. Z GS oe —ey- SBicting St forthe _ _ ee { X94 Aantal < a __ aiid hi or (011) wht firite state automaton ove E G/e= 1 20. goer on ot °F on * £3 = 3xuPeaxyt > B44 =q ys =F goa 499, Z same for ash. Gorsitfon . {sa & tuple model t state or © | Melay moshine:—A mle Mashing | CQ, #141814, 40)- - : —+ 9m metas mashine the output depen on curser’ symbol a | | olf ° 4 ° __ s@ [8a aba) — __s(surba) S343 ,a4)— 844 4)- os ¢' snot coor (ab © cConctrvel « metay anashine over €(AtP) |, Member of a's im the aking ‘mod + =) 0/4, 2 Uilll be ovtput (mod 2) “a | Canstrod_ 0 amashime ahh _oucept a | ding ai’ fnpot” ond genersts Bs Ais complen = O => Ake Se —— ent. tN — ° | Conatret ame i | Sopot and _peedvees binom _ | Os ootpet._ Q” =e sete : : oft : C00 3904 =e) : ao)! ood me oso ote ott £00—> toe vqaahne_wshtch ates binary 70° s no. ‘megemented b+ £ : t. i to enesion) iis isdue 40 carry genevotl ee complemented a=JAIL ASE dddd dd ddd ® % a : | 4, 4 ie eech 7. I HU d ee () f Seat Ee BLA | & a Poe | biel een | se ct$ a Ole b of _ Convess?. 5 Senwenion Of metas) to -moove leo >| b | ove —}- 42} +4 | 9 a . Geet lege oeleqs ° 7 WV 4 {| 9, — - see which States hove some cote ego! Gnd ~ vahfcb_ do nor 7 q,_43 hose change io_olpok_both_ 44 a rat abe have two outputs = = Yo y= == Let give © er Lay eons or re 7|] tu, t}——%»— io 2 2G5 | oy is ile state, a» — i “8 Bo eigen Spee, ~yogelar expression. — t : _ "D) Letial Aratyser vserthe ian expen — : “9 on to Te evs —— a — al Node tee: Used do esrvar}_o Ir _ caued postiifonin ‘qnetnod_tor saint fan O} | steps: — ———_—_— oo 4.) Create the pales o} aul the states volved in the given + 4 4 4 = = pA - 8. Mork. alt the pars ‘such that one state Final _ “duke and_one state fe non- firal state. — : | f thove fe_cmy,_ommmantied paie(Ga.,Qb.)suh that SQ gy), SEQ 2) Te mated the Taek |e Qp_then. 24.0 tye _eignal- _ «| Repent Bie step onfll_, mp menting com bedigre. 42 Combine_ay the wmmarked pales and make them Single ete