DFA Minimization. DFA minimization: the idea. Not in Sipser. Background: Questions: Assignments: Previously: Today: Then:

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1 Assinmnts: DFA Minimiztion CMPU 24 Lnu Tory n Computtion Fll 28 Assinmnt 3 out toy. Prviously: Computtionl mols or t rulr lnus: DFAs, NFAs, rulr xprssions. Toy: How o w in t miniml DFA or lnu? Tis is t lst topi in t irst prt o t ours! Tn: Tusy w ll rviw or Exm. Not in Sipsr Rin rom Hoprot t l. is on t ours wsit. Usr nm n psswor r t sm s or ssinmnt solutions. Wy r w in it? DFA minimiztion: t i Bkroun: A rulr lnu n pt y mny DFAs Qustions: Is tr uniqu simplst DFA or rulr lnu? I so, n w onstrut it? Tin is lik iliustr

2 T i Ys, tr is uniqu miniml DFA n w n onstrut it! Miniml? Miniml numr o stts. Uniqu? Uniqu up to rnmin o stts. I.., s sm sp. Isomorpi. Distinuisl strins Lt L ny lnu in Σ*. For ny two strins x n y Σ*, w n pply t inistinuisility rltion I L, su tt x I L y i n only i or ny z Σ*, itr ot xz n yz r in L, or ot xz n yz r in L (t omplmnt o L). Stts n strins W n tink o stt in n utomton s rprsntin rtin miniml mount o inormtion w n to rmmr out t strin w r prossin. E stt n intii y t st o strins tt us us to rmmr xtly tt inormtion, I.., t st o strins tt us us to t tt stt in t ronition loritm A stt is n strtion nywy, so wy not tink o stt s st o strins? Distinuisl stts I w s stts s sts o strins, it s sy to s tt istinuisility pplis to stts s wll. Stts p n q r inistinuisl i or ll strins z, itr ot δ (p,z) F n δ (q,z) F or δ (p,z) F n δ (q,z) F p q z z or p q z z

3 Exmpl (vry simpl) Consir: p p is istinuisl rom q n r us p is non-ptin n q n r r ptin. So ppnin t mpty strin to t sts o strins t stts rprsnt ls to non-ptin stt in on s n n ptin stt in t otr. q r Cn w istinuis q rom r? No strin innin wit works, us ot stts o to p, n tror ny strin o t orm x tks q n r to t sm stt. No strin innin wit works. Tnilly, δ(q, ) = r n δ(r, ) = q r not istinuisl. Wt ppns is tt, strtin in itr q or r, s lon s w v inputs, w r in on o t ptin stts, n wn is r, w o to t sm stt (p) n tn rrlss o input, t sm stts orvr tr. p q r Equivln lsss For strins: Σ* n prtition into sts o inistinuisl strins For stts: T stts o n utomton Q n prtition into sts o inistinuisl stts Tis ins quivln lsss ovr Σ* or Q W will tlk out quivln lsss in nrl, sin stts n viw s sts o strins. Myill Nro Torm L is rulr lnu i n only i t st o quivln lsss is init Exmpl: L = {x {,}* x ns wit } Consir 3 strins: ε,, n Any two o ts strins r istinuisl wit rspt to L T strin ε istinuiss ε n n lso n T strin istinuiss ε n

4 So t tr quivln lsss [ε], [], n [] r istint Any strin y is inistinuisl rom on o ts strins I y ns in it is inistinuisl rom I y ns in it is inistinuisl rom Otrwis (i y = ε, y =, or y ns wit ), y is inistinuisl rom ε T ision m or ny input strin pns only on its lst two symols no n to istinuis twn on strin nin in n ny otr strin nin wit, or twn ny two strins nin in So t quivln lsss or tis utomton r ( ) * ( ) * ( ) * ε Bs on t n to rmmr only t inl two symols, oul uil t ollowin FA On t FA s sn two symols, it yls k n ort mon 4 stts rmmrin t lst two symols it sw Cn tis FA simplii? Consir stts n Nitr is n ptin stt Bot n t lst on mor symol to v strin in L Rows in t tl look xtly lik Consir strins tt us t FA to in stts n rsptivly Do not n to istinuis now, us nitr stt is ptin Cnnot istinuis on symol ltr, us t FA is in t sm stt Tror ts two stts r not istinuisl T two n mr stt input ε

5 Mrin stts Stt A: Mr n Stt B: Mr,, Nw FA wit 4 stts ε A B stt input ε Ol FA wit 7 stts n r now A,, n r now B In t nw tl, rows or ε n A r t sm (i.., n ) Nitr is ptin Inistinuisl Mr ε into A No urtr rutions possil so tis is t miniml FA ε A ε B A A B B B A B A A B B B A B A B Construtin t minimum-stt DFA For roup o inistinuisl stts, pik rprsnttiv. Not roup n lr,.., q, q 2,, q k, i ll pirs r inistinuisl. Inistinuisility is trnsitiv (wy?), so inistinuisility prtitions stts. I p is rprsnttiv, n δ(p, ) = q, in minimum-stt DFA t trnsition rom p on is t rprsnttiv o q s roup (or to q itsl, i q is itr lon in roup or rprsnttiv). Tos svn stts wr just tr stts in trn ot (wit room or n).

6 Exmpl 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 For t DFA ivn rlir, p is in roup y itsl; {q, r} is t otr roup. p q r p, q Forml loritm Aloritm Buil tl to ompr unorr pir o istint stts p, q. E tl ntry s mrk s to wtr p n q r known to not quivlnt, n list o ntris, rorin pnnis: I tis ntry is ltr mrk, lso mrk ts. Ky i: in stts p n q tt r istinuisl us tr is som input w tt tks xtly on o p n q to n ptin stt. Bsis: ny non-ptin stt is istinuisl rom ny ptin stt Inution: p n q r istinuisl i tr is som input symol su tt δ(p, ) is istinuisl rom δ(q, ). All otr pirs o stts r inistinuisl, n n mr into on stt.

7 Aloritm. Initiliz ll ntris s unmrk n wit no pnnis 2. Mrk ll pirs o inl n non-inl stt 3. For unmrk pir p,q n input symol : Lt r = δ(p, ), s = δ(q, ). I (r, s) unmrk, (p, q) to (r, s) s pnnis, Otrwis mrk (p, q), n rursivly mrk ll pnnis o nwly mrk ntris. 4. Cols unmrk pirs o stts 5. Dlt inssil stts Exmpl. Initiliz tl ntris: Unmrk, mpty list (,) δ(,)? δ(,) δ(,)? δ(,) 2. Mrk pirs o inl & non-inl stts 3. For unmrk pir & symol,

8 ? My 3. For unmrk pir & symol,? No! Mrk (,) s istinuisl 3. For unmrk pir & symol, 3. For unmrk pir & symol, (,) δ(,) is istinuisl rom δ(,) Mrk (,) δ(,)? δ(,) δ(,)? δ(,) 3. For unmrk pir & symol, (,)

9 3. For unmrk pir & symol,?? My Ys A (,) to (,) s pnnis 3. For unmrk pir & symol, (,) Distinuisl (,) (,) 3. For unmrk pir & symol, (,) My A (,) to (,) s n (,) s pnnis (,) 3. For unmrk pir & symol, (,) Distinuisl (,) (,) (,) (,) (,)

10 (,) δ(,) = (inl stt) δ(,) = (non-inl stt) Distinuisl (,) Distinuisl (,) Distinuisl 3. For unmrk pir & symol, (,) 3. For unmrk pir & symol, (,) (,) (,) (,) (,) 3. For unmrk pir & symol, (,) Distinuisl Mrk (,) Mrk (,) lso (,) (,)? (,) Distinuisl (,) (,) (,) (,)? Distinuisl Distinuisl Distinuisl (,) (,) (,) (,) (,)

11 (,) δ(,) = δ(,) = (,) istinuisl (,) istinuisl (,) istinuisl (,) istinuisl Distinuisl Mrk (,) (,) (,) (,) (,) (,) Finl Miniml FA 5. Dlt unrl stts 4. Cols unmrk pirs o stts Non. (,)

12 NFA minimiztion T minimiztion loritm osn t in uniqu miniml NFA. Mor importntly, in nrl, tr is no uniqu miniml NFA. Exmpl NFAs or + : Dision Proprtis o Rulr Lnus Givn ( rprsnttion,.., RE, FA, o) rulr lnu L, wt n w tll out L? Sin tr r loritms to onvrt twn ny two rprsnttions, w n oos t rprsnttion tt mks t tst sist. Mmrsip Is strin w in rulr lnu L? Coos DFA rprsnttion or L. Simult t DFA on input w. Emptinss Is L =? Us DFA rprsnttion. Us rp-rility loritm to tst i t lst on ptin stt is rl rom t strt stt. Bot miniml, ut not isomorpi. Finitnss Is L init lnu? Not vry init lnu is rulr, 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. RE mto Almost, w n look or * in t RE n sy its lnu is ininit i tr is on, init i not. Howvr, tr r xptions,.. ε* or *. Tus:. Fin su-xprssions quivlnt to y: (Bsis) is; ε n r not. (Inution) E F is i ot E n F r; EF is i itr E or F is; E* nvr is. Elimint su-xprssions quivlnt to y: Rpl E F or F E y F wnvr E is n F isn t. Rpl E* y ε wnvr E is quivlnt to.

13 3. Now, in su-xprssions tt r quivlnt to ε y: (Bsis) ε is; isn t. (Inution) E F is i ot E n F r; itto EF; E* is i E is. Now, w n tll i L(R) is ininit y lookin or su-xprssion E* su tt E is not quivlnt to ε. Appnix Nots on FA Minimiztion Aloritm Orr o sltin stt pirs ws ritrry. All orrs iv sm ultimt rsult. But, my ror mor or wr pnnis. Coosin stts y workin kwrs rom known nonquivlnt stts prous wst pnnis. Tis loritm: Humn (954), Moor (956). O(n 2 ) tim. Constnt work pr ntry: initil mrk tst n possily ltr sin o its pnnis. Mor iint loritms xist,.., Hoprot (97). Corrtnss o t Minimiztion Aloritm Wy is nw DFA no lrr tn ol DFA? Only rmovs stts, nvr introus nw stts. Ovious. Wy is nw DFA quivlnt to ol DFA? Only intiy stts tt provly v sm vior. Coul prov x L(M) x L(M ) y inutions on rivtions. Wy is miniml DFA uniqu (up to isomorpism)? Dpns on t uniqunss o miniml quivln lsss o strins in t lnu.

14 Wy t Minimiztion Aloritm Cn t B Btn Suppos w v DFA A, n w minimiz it to onstrut DFA M. But tr is notr DFA N tt pts t sm lnu s A n M, yt s wr stts tn M. Proo y ontrition tt tis n't ppn: Run t stt-istinuisility pross on t stts o M n N totr. Strt stts o M n N r inistinuisl us L(M) = L(N). I {p, q} r inistinuisl, tn tir sussors on ny on input symol r lso inistinuisl. Tus, 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. W v ontrition, so t ssumption tt N xists is wron, n M in t s s w stts s ny quivlnt DFA or A. In t (stronr), tr must on-to-on orrsponn twn t stts o ny otr minimum-stt N n t DFA M, sowin tt t minimum-stt DFA or A is uniqu up to rnmin o t stts.