Proof of Pumping Lemma. PL Use. Example. Since there are only n different states, two of q 0, must be the same say q i

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1 COSC 2B 22 Summr Proo o Pumpin Lmm Sin w lim L is rulr, tr must DFA A su tt L = L(A) Lt A v n stts; oos tis n or t pumpin lmm Lt w strin o lnt n in L, sy w = 2 m, wr m n Lt i t stt A is in tr rin t irst i symols o w = strt stt, = δ(, ), 2 = ˆ δ (, 2 ), t. 22 Summr COSC 2 Sin tr r only n irnt stts, two o,, n must t sm sy i = j, wr i < j n Lt x = i y = i+ j z = j+ m Tn y rptin loop rom i to j wit ll i+ j k tims, w n sow tt xy k z is pt y A y x j i+ j i 22 Summr COSC 2 2 i z j + m Pumpin Lmm us strin y is "pump" Not: ntur o FA's mns nnot ontrol numr o tims pump So, rulr lnu wit strins o lnt n is lwys ininit PL only intrstin or ininit lnus ut works or init lnus, wi r lwys rulr in tis s n is lrr tn t lonst strin so notin n pump T PL is n pplition o t "pion-ol prinipl" PL Us W us t PL to sow lnu L is not rulr Strt y ssumin L is rulr Tn tr must som n tt srvs s t PL onstnt W my not know wt n is, ut w n work t rst o t "m" wit n s prmtr W oos som w tt is known to in L Typilly, w pns on n 22 Summr COSC Summr COSC 2 4 Exmpl Applyin t PL, w know w n rokn into xyz, stisyin t PL proprtis Ain, w my not know ow to rk w, so w us x, y, z s prmtrs W riv ontrition y pikin i (wi mit pn on n, x, y, n/or z) su tt xy i z is not in L Consir t st o strins o 's wos lnt is prt sur; ormlly L = { i i is sur} W lim L is not rulr Suppos L is rulr. Tn tr is onstnt n stisyin t PL onitions Consirw = n2, wi is surly in L Tn w = xyz, wr xy n n y ε 22 Summr COSC Summr COSC 2 6

2 COSC 2B 22 Summr T PL "m" By PL,xyyz is in L. But t lnt o xyyz is rtr tn n 2 n no rtr tn n 2 + n (Wy?) Howvr, t nxt prt sur tr n 2 is (n+ ) 2 = n 2 + 2n + Tus,xyyz is not o sur lnt n is not in L Sin w v riv ontrition, t only unprov ssumption tt L is rulr must t ult, n w v "proo y ontrition" tt L is not rulr Gol: win t PL m inst our opponnt y stlisin ontrition o t PL, wil t opponnt tris to oil us. 22 Summr COSC Summr COSC 2 8 Four Stps. Numr o stts in utomton is n Not: w on't v to know wt n is, sin w us n to in our strin 2. Givn n, w pik strin w in L o lnt ul to or rtr tn n W r r to oos ny w, sujt to w L n w n W usully in t strin in trms o n Four Stps (II) 3. Our opponnt ooss t omposition xyz, sujt to xy n y 4. W try to pik i (t powr tor in xy i z) in su wy tt t pump strin w i is not in L. I w n o so, w win t m 22 Summr COSC Summr COSC 2 Exmpl Σ ={,}; onsir L = {ww R w Σ*}. Wtvr n t opponnt ooss in stp, w n lwys oos w s ollows: n n n n x y z Bus o tis oi n t ruirmnt tt xy n, t opponnt is rstrit in stp 3 to oosin y tt onsists ntirly o 's. In stp 4, w us i=2.t strin xy 2 z s mor 's on t lt tn on t rit, so it nnot o orm ww R. So L is not rulr. 22 Summr COSC 2 22 Summr COSC 2 2 2

3 COSC 2B 22 Summr Exmpl 2 Consir L = { i i i } Givn n, w oos t strin n n or our rumnt. I t lnu is rulr, our opponnt n rk tis strin into xyz wr or ny j xy j z is in L Bus xy n n y >, t strin y s to onsist o s only. So pumpin y s to t numr o s n n tr r mor s tn s 22 Summr COSC 2 3 Exmpl 3 Consir L = {w w s n ul numr o 's n 's} Givn n, w oos t strin () n W n to sow splittin tis strin into xyz wr xy i z is in L is impossil But it is possil! I x = ε, y =, n z = () n-, xy i z is in L or vry vlu o i. Ar w out o luk? 22 Summr COSC 2 4 First lw o PL us I your strin os not su, try notr! Lt's try n n. Ain, w n to sow splittin tis strin into xyz wr xy i z is in L is impossil But it is possil! I x n y r t mpty strin n y is n n, tn xy i z lwys s n ul numr o 's n 's. Ar w still in troul? Not tis tim t PL sys tt our strin s to ivi so tt xy n n y > I xy Qtn y must onsist only o 's, so xyyz L Contrition! W win! 22 Summr COSC Summr COSC 2 6 Exmpl 4 Consir L = {ww w Σ*} W oos t strin n n, wr n is t numr o stts in t FA. W now sow tt tr is no omposition o tis strin into xyz wr or ny j xy j z is in L. Ain, it is ruil tt t PL insists tt xy n, us witout it w oul oul pump t strin i w lt x n z t mpty strin. Wit tis onition, it's sy to sow tt t PL won't pply us y must onsist only o 's, so xyyz is not in L. As or, t oi o strin is ritil: w osn n n (wi is mmr o L) inst o n n, it wouln't work us it n pump n still stisy t PL. MORAL Coos your strins wisly. 22 Summr COSC Summr COSC 2 8 3

4 COSC 2B 22 Summr Exmpl 5 "Pumpin own" L = { i j i > j} Givn n, oos s = n+ n. Split into xyz t. Bus y t PL, xy n, y onsists only o 's. Is xyyz in L? T PL stts tt xy i z is in L vn wn i = So, onsir t strin xy z Rmovin strin y rss t numr o 's in s s s only on mor tn Tror, xz nnot v mor 's tn 's, n is not mmr o L. Contrition! 22 Summr COSC Summr COSC 2 2 Rmmr You n to in only on strin or wi t PL os not ol to prov lnu is not rulr But you must sow tt or ny omposition into xyz t PL ols Tis somtims mns onsirin svrl irnt ss Exmpl L = { 3 i i-3 i > 3} Assum L is rulr, PL ols. Coos w = 3 n n-3 wit n s t PL onstnt. Tr wys to prtition w into xyz:. y ontins only s 2. y ontins only s 3. y ontins s n s Hv to sow tt o ts prtitions ls to ontrition, i.. tt tr is no possil wy to ivi w into xyz so tt t PL ols 22 Summr COSC Summr COSC 2 22 Cs: y ontins only s Tn x ontins to 2 s, y ontins to 3 s, n z ontins to 2 s ontnt onto t rst o t strin n n-3, su tt tr r xtly 3 s. So t prtition is x = k y = j z = 3-k-j n n-3 wr k, j >, n k+j 3. It soul tru tt xy i z L or ll i. xy 2 z = (x)(y)(y)(z) = ( k )( j )( j )( 3-j-k n n-3 ) = 3+j n n-3 L sin j >, tr r too mny s CONTRADICTION! Cs 2: y ontins only s Tn x ontins 3 s ollow y or mor s, y ontins to n-3 s, n z ontins 3 to n-3 s ontnt onto t rst o t strin n-3. So t prtition is x = 3 k y = j z = n-k-j n-3 wr k, j >, n k+j Q-3. It soul tru tt xy i z L or ll i. xy z = 3 n-j n-3 L sin j >, tr r too w s CONTRADICTION! 22 Summr COSC Summr COSC

5 COSC 2B 22 Summr Cs 3: y ontins s n s Tn x ontins to 2 s, y ontins to 3 s n to n-3 s, n z ontins 3 to n- s ontnt onto t rst o t strin n-3. So t prtition is x = 3-k y = k j z = n-j n-3 wr 3 N, n n-3 M>. It soul tru tt xy i z L or ll i. xy 2 z = 3 j k n n-3 L sin j, k >, n tr r s or s Tr is no prtition o w L is not rulr! 22 Summr COSC 2 25 Closur Proprtis Crtin oprtions on rulr lnus r urnt to prou rulr lnus Exmpl: t union o rulr lnus is rulr: strt wit RE's, n pply + to t n RE or t union. 22 Summr COSC 2 26 Asi: Applitions FST: Finit Stt Trnsur Implmntin FSTs in Jv, C, Solvin prtil prolms wit FSTs FST: Finit Stt Trnsur Lik DFA, ut rs n writs Ars sow input n output: "in/out" I/O n omplx: "x in Σ-{}/x" / Strt x in Σ/x /ε 22 Summr COSC Summr COSC 2 28 FST Implmnttion Clmpin FST N to r/writ, trk stt: stt = ; Strt x in Σ/x until EOF: in = r() swit on stt: /ε s : writ(in); stt = ; n s s : i in=='' tn stt=; ls writ(''); stt=; n i n s n swit n until / riv rom FA: / strt / / /,/ 22 Summr COSC Summr COSC 2 3 5

6 COSC 2B 22 Summr Clmpin FST in Jv int stt = ; r inr, outr; wil (!in.o() ) { outr = inr = in.r(); swit( stt ) { s : swit( inr ) s '': stt = ; rk; s '': stt = ; rk; ult: // rror Solvin prtil prolms RE's us in: rp, rp lx lxil nlysr nrtor } rk; //... } print(outr); } 22 Summr COSC 2 32 initions ruls usr o Lx/Flx %{ #inlu %} %% [-9]+ print("int %s\n", yytxt); [-+*/] print("op %s\n", yytxt); [ \t\n]+ /* t up wit sp */. print("unk: %s\n", yytxt); %% int min(int r, r *rv[]) { yylx(); } Lx/Flx (II) nrts u il o tls lik: stti yyonst sort int yy_pt[2] = {,,, 6, 4, 3, 3, 2,, 3,, }; plus untions lik yylx, wi mults n utomton tls riv trnsitions m y yylx 22 Summr COSC Summr COSC 2 34 Sustitution o RLs into RL is Rulr Swp rulr lnus or vry symol in t lpt o rulr lnu, n t rsultin lnu is rulr: Σ (or in Σ), w in sustitution s: s() = L wr L is RL s( 2 n )= s( )s( 2 ) s( n ) xtn to strins = L L 2 L n s(m) = w in M s(w) xtn to lnus Tn s(l) is rulr 22 Summr COSC 2 35 Closur Unr Rvrsl Rvrs o strin w = 2... n is n... 2 Dnot w R Not ε R = ε Rvrs o lnu L is st ontinin rvrs o strin in L Dnot L R I L is rulr, so is L R Proo: us RE's, rursiv rvrsl s in txt 22 Summr COSC

7 COSC 2B 22 Summr Dision Proprtis o RLs Givn rprsnttion (.., RE, FA) o rulr lnu L, wt n w tll out L? Hv loritms to onvrt twn rprsnttions, so n oos rprsnttion tt mks tst sist Mmrsip: Is strin w in rulr lnu L? Us DFA rprsnttion or L & simult DFA on input w. Emptinss: Is L = φ? Us DFA rprsnttion + rp-rility loritm to tst i t lst on ptin stt is rl rom strt stt 22 Summr COSC 2 37 Finitnss Is L init lnu? Not vry init lnu is rulr (wy?), ut rulr lnu is not nssrily init DFA mto: Givn DFA or L, limint ll stts tt r not rl rom t strt stt n ll stts tt o not r n ptin stt Tst i tr r ny yls in t rminin DFA; i so, L is ininit, i not, tn L is init 22 Summr COSC 2 38 Finit L? RE Mto * in RE mns ininit lnu (lmost) Wt out ε* or *φ? Follow stps:. () Fin n () limint su-xprssions uivlnt to φ 2. Fin su-xprssions uivlnt to ε 3. L(R) is ininit i tr is non-ε su-xprssion E* Fin n Elimint φ ) Fin REs uivlnt to φ: E+F i E=F=φ EF i E=φ or F=φ But nvr: ε or or E* ) Rpl su-xprssions uivlnt to φ: E+φ E φ+e E φ* ε 22 Summr COSC Summr COSC 2 4 Look or non-ε E* Su-xprssions uivlnt to ε : E+F i E=F=ε EF i E=F=ε E* i E=ε Now, i L(R) s su-xprssion E* su tt E is not uivlnt to ε, tn L is ininit. 22 Summr COSC 2 4 Exmpl Consir ( + φ)* + φ* ) φ (twi) n φ r su-xprssions uivlnt to φ )Rmov ts su-xprssions ( + φ)* ( + φ)* * n φ* ε * + ε rmins 2) only su-xprssion ε is uivlnt to ε * + rmins Sin is strr, lnu is ininit. 22 Summr COSC

8 COSC 2B 22 Summr Minimiztion o Stts Rl ol is tstin uivln o (rprsnttions o) two rulr lnus. Intrstin t: DFA's v uniu (up to stt nms) minimum-stt uivlnts. Distinuisl Stts Ky i: in stts p n tt r istinuisl us tr is som input w tt tks xtly on o p n to n ptin stt. Bsis: ny non-ptin stt is istinuisl rom ny ptin stt (w = ε). Inution: p n r istinuisl i tr is som input symol su tt δ(p, ) is istinuisl rom δ(, ). All otr pirs o stts r inistinuisl, n n mr into on stt. 22 Summr COSC Summr COSC 2 44 Exmpl (vry simpl) Cn w istinuis rom r? Consir FA p is istinuisl rom n r y sis p r No strin innin wit works. ot stts o to p, n tror ny strin o t orm x tks n r to t sm stt. No strin innin wit works. Strtin in itr or r, s lon s w v input, w r in on o t ptin stts. Wn is r, w o to t sm stt (p) n tn rrlss o input, t sm stts orvr tr. p r 22 Summr COSC Summr COSC 2 46 Construtin Minimum-Stt DFA For roup o inistinuisl stts, pik "rprsnttiv." Not roup n lr,..,, 2,, k, i ll pirs r inistinuisl. Inistinuisility is trnsitiv (wy?) so inistinuisility prtitions stts. I p is rprsnttiv, n δ(p, ) =, in minimum-stt DFA t trnsition rom p on is t rprsnttiv o 's roup (or to itsl, i is itr lon in roup or rprsnttiv). 22 Summr COSC 2 47 Strt stt is rprsnttiv o t oriinl strt stt. Aptin stts r rprsnttivs o roups o ptin stts. Noti w oul not v "mix" (ptin + non-ptin) roup (wy?). Dlt ny stt tt is not rl rom t strt stt. 22 Summr COSC

9 COSC 2B 22 Summr Exmpl For t DFA ivn rlir, p is in roup y itsl; {, r} is t otr roup. p r p, Wy Cn't Bt Tis Minimiztion Proo y ontrition: suppos w minimiz DFA A, onstrutin DFA M suppos N is notr DFA tt pts sm lnu s A n M, yt s wr stts tn M Run stt-istinuisility pross on stts o M n N totr Strt stts o M n N r inistinuisl us L(M) = L(N) 22 Summr COSC Summr COSC 2 5 I {p,} r inistinuisl, tn tir sussors on ny on input symol r lso inistinuisl sin nitr M nor N oul v n inssil stt, vry stt o M is inistinuisl rom t lst on stt o N Sin N s wr stts tn M, tr r two stts o M tt r inistinuisl rom t sm stt o N, n tror inistinuisl rom otr But M ws sin so tt ll its stts r istinuisl rom otr Contrition! so ssumption tt N xists is wron, n M in t s s w stts s ny uivlnt DFA or A In t, tr must - orrsponn twn t stts o ny otr minimum-stt N n t DFA M, sowin tt t minimum-stt DFA or A is uniu up to rnmin o t stts. 22 Summr COSC Summr COSC 2 52 DFA Minimiztion: T I Bkroun: Hv sn tt sinl RL s mny DFAs. Hv sn tt n simpliy DFAs, REs, t., y uivlns. Qustions: Is tr uniu simplst DFA or RL? I so, n w onstrut it? DFA Minimiztion: T I Ys, tr is uniu miniml DFA & w n onstrut it. Miniml: Miniml numr o stts. Uniu: Uniu up to rnmin o stts. I.., s sm sp. Isomorpi. 22 Summr COSC Summr COSC

10 COSC 2B 22 Summr Aloritm I Intiy & ollps stts vin sm vior. Buil uivln rltion on stts: p z, δˆ (p,z) F δˆ (,z) F I.., i or vry strin z, on o t ollowin is tru: p z p z or z z 22 Summr COSC 2 55 DFA Minimiztion: Aloritm Buil tl to ompr unorr pir o istint stts p,. E tl ntry s mrk s to wtr p n r known to not uivlnt, n list o ntris, rorin pnnis: I tis ntry is ltr mrk, lso mrk ts. 22 Summr COSC 2 56 DFA Minimiztion: Aloritm DFA Minimiztion: Exmpl. Initiliz ll ntris s unmrk n wit no pnnis. 2. Mrk ll pirs o inl n non-inl stt. 3. For unmrk pir p, n input symol :. Lt r=δ(p,), s=δ(,). 2. I (r,s) unmrk, (p,) to (r,s) s pnnis, 3. Otrwis mrk (p,), n rursivly mrk ll pnnis o nwly-mrk ntris. 4. Cols unmrk pirs o stts. 5. Dlt inssil stts.. Initiliz tl ntris: Unmrk, mpty list 22 Summr COSC Summr COSC 2 58 DFA Minimiztion: Exmpl DFA Minimiztion: Exmpl δ(,)? δ(,) δ(,)? δ(,) 2. Mrk pirs o inl & non-inl stts 3. For unmrk pir & symol, 22 Summr COSC Summr COSC 2 6

11 COSC 2B 22 Summr DFA Minimiztion: Exmpl DFA Minimiztion: Exmpl?? My. No! 3. For unmrk pir & symol, 3. For unmrk pir & symol, 22 Summr COSC Summr COSC 2 62 DFA Minimiztion: Exmpl DFA Minimiztion: Exmpl δ(,)? δ(,) δ(,)? δ(,)?? My. Ys. 3. For unmrk pir & symol, 3. For unmrk pir & symol, (,) 22 Summr COSC Summr COSC 2 64 DFA Minimiztion: Exmpl DFA Minimiztion: Exmpl N to mrk. So, mrk (,) lso. 3. For unmrk pir & symol, (,) (,) 3. For unmrk pir & symol, (,) (,) (,) (,) 22 Summr COSC Summr COSC 2 66

12 COSC 2B 22 Summr DFA Minimiztion: Exmpl DFA Minimiztion: Exmpl 3. For unmrk pir & symol, (,) 22 Summr COSC 2 67 (,) 4. Cols unmrk pirs o stts. (,) 22 Summr COSC 2 68 DFA Minimiztion: Exmpl 5. Dlt unrl stts. Non. (,) 22 Summr COSC 2 69 Contxt-Fr Grmmrs Nottion or rursiv sription o lnus. Exmpl: Roll <ROLL> Clss Stus </ROLL> Clss <CLASS> Txt </CLASS> Txt Cr Txt Txt Cr Cr (otr rs) Stus Stu Stus Stus ε Stu <STUD> Txt </STUD> 22 Summr COSC 2 7 Exmpl (II) Gnrts "oumnts" su s: <ROLL><CLASS>s24</CLASS> <STUD>Slly</STUD> <STUD>Fr</STUD> </ROLL> Exmpl (III) Vrils (.., Stus) rprsnt sts o strins (i.., lnus). In snsil rmmrs, ts strins sr som ommon rtristi or roll. Trminls (.., or <ROLL>) = symols o wi strins r ompos <ROLL> oul onsir itr sinl trminl or t ontntion o 6 trminls 22 Summr COSC Summr COSC

13 COSC 2B 22 Summr CFG Nottion Proutions = ruls o t orm oy is vril. oy is strin o zro or mor vrils n/or trminls. Strt Symol = vril tt rprsnts "t lnu." Nottion: G = (V, Σ, P, S) V = vrils Σ = trminls P = proutions S = strt symol 22 Summr COSC 2 73 Exmpl A simpl xmpl nrts strins o 's n 's su tt lok o 's is ollow y t lst s mny 's S AS ε A A A Not vrtil r sprts irnt ois or t sm. 22 Summr COSC 2 74 Drivtions αaβ αγβ wnvr tr is proution A γ (Susript wit nm o rmmr,.., G, i n.) Exmpl: AS AS α * β mns strin n om β in zro or mor rivtion stps. Exmpls: AS * AS (zro stps) AS * AS (on stp) * AS (tr stps) Lnu o CFG L(G) = st o trminl strins w su tt S G w, wr S is t strt symol. Nottion,, r trminls; y, z r strins o trminls. Grk lttrs r strins o vrils n/or trminls, otn ll sntntil orms. A,B, r vrils., Y, Z r vrils or trminls. S is typilly t strt symol. 22 Summr COSC Summr COSC 2 76 Ltmost/Ritmost Drivtions oi o vril to rpl t stp Drivtions my ppr irnt only us w mk t sm rplmnts in irnt orr. To voi su irns, w my rstrit t oi. A ltmost rivtion lwys rpls t ltmost vril in sntntil orm. Yils lt-sntntil orms. Ritmost rivtion in nloously.,, t., us to init rivtions r lm rm ltmost or ritmost. Exmpl Ltmost rivtion: S AS AS S AS AS S Ritmost rivtion: S AS AAS AA AA A A Not w riv t sm strin 22 Summr COSC Summr COSC