Lecture 11: Safra s Determinization Construction

Transcription

1 Lece 11: Sf Deeminizion Concion In hi lece we hll ee concion de o S. Sf h n nondeeminiic Büchi omon of ize n ino n eivlen deeminiic Rin omon of ize 2 O(nlogn). A n immedie coolly we ee h we cn comlemen nondeeminiic Büchi omon ino deeminiic See omon of ize 2 O(nlogn). Sf concion i one of he meiece of hi e. Thee e nme of icle h y o demyify nd exlin he concion [2, 4]. They y nd exlin why imle ide do no wok nd led yo o Sf concion. A he ik of ofcing me I hll no ke hi oe nd ined y o ive Sf concion fom n nlyi of he n-ee of nondeeminiic Büchi om. The em hee i o exlin he ecil ce of he e ce conced omehing h ie nlly fom he ce of cceing n (he hn comliced ce oined y eence of efinemen of imle ide h do no wok). A nondeeminiic omon h mny n on n in wod. The deeminiic omon m imle ll hoe n (o ll he elevn one) in ingle one. The e of ll n of NBA A on wod w wold e n infinie ee in which ech h i n of A on w. (Thi i imil o he n of nivel lening omon, i.e. one in which ll he e e e. Of coe, he ccence cieion i diffeen. We only demnd h le one h hogh hi ee i cceing.) Hee i NBA A f nd i n on he wod...,,, We cold deeminiiclly imle hi omon level y level. Th i ey mch ll h we cn do in deeminiic imlion. B he nme of node in level i no onded. So, we need oxime nd minin ome onded mon of infomion. In he o clled owe-e concion, we oxime level y he e of e h lel ome veex he level. Th, we hve n omon of ize 2 n. Nex, we need o decide on he ccence condiion. Demnding h h infiniely mny level conin finl e i oo geneo. In he ove n, he cceing e F e in evey level in he n ee, howeve hee i no h in hi ee h vii F infiniely ofen (nd hence... i no cceed y he ove omon). Th he owee omon A = (2 Q, Σ,δ, {}, {X X F }) wih δ (X,) = { X. δ(,)} 1

2 cce moe wod hn A. The owe-e omon conced fom A f cce... hogh A doe no. Hee i n cceing n: Fo ny wo e X,Y of e fom Q nd wod, le wie X g Y if fo ech Y hee i X ch h hee i n fom o on h vii ome e in F. Fo exmle, in he omon A f, {,,} g {,}. We now decie diffeen ccence cieion fo he owee omon. Le S 1 0 S e he n of he owee omon on We cce ovided we cn x decomoe hi n S Si1 Si2... ch h fo ech S ik+1, hee i S ik ch h x k+1 g. We clim h ccence nde hi cieion gnee h A cce w. We cn ee hi follow: Conc DAG whoe node level k e he e in S ik. An edge i dwn fom node level i o node level i + 1 if nd only if x i+1 g. Lel hi edge y ch good n fom o. Thi finiely nching DAG h infiniely mny node echle fom he veice level 1. Th, hee m e n infinie h hogh hi DAG. The lel long he edge of hi h give n cceing n fo A on w. Cn we nle hi ccence condiion Büchi ccence condiion (y lowing he e ce if necey?) The ide i imil o he one ed o nle ABA ino NBA. Le A m = (Q m, Σ,δ m, ({}, ), {(X,X) X Q} whee Q m = {(X,Y ) Y X Q} δ m ((X,Y ),) = (δ (X,),δ (Y,) δ(x,) F) δ m ((X,X),) = (δ (X,),δ (X,) F) if X Y We imle he owee omon one level. In he ohe level, we kee ck of he e of e h hve een eched vi h h viied F. When hee wo e e he me, we ee he econd e nd oceed. Th we hve deeminiic omon A m wih L(A m ) L(A). In [2], he omon A m i clled he mked owee omon. Qie clely, L(A m ) will no in genel e he me L(A). Fo innce, he mked owee omon conced fom A f doe no cce... which i cceed y A f. 2

3 In genel, if hee i ded e d( noncceing e wih elf-loo on ll lee which i fhe no echle fom ny cceing e) i echle vi ome efix of w hen A m will no cce w. Thi i ece he e d e in evey level of he nee eyond finie efix on evey wod nd hee i NO h in he omon o ech d vi n cceing e. How do we ge ond hi? The nive ide wold e omehow modify he concion of A m o do ome elemen of Q fom he cen e (o h ny ded-e e doed.) B h i wh nondeeminiic omon doe! I imly do ll he e exce he one h e on he cceing h. So we need ee ide. I n o h wh we need o do i o he omon L(A m ) no igh he eginning, ined i fe ome iniil efix h een ed. In ome ene, we wi fo ll he ele h o nch off nd n he A m on coe of he nee. Of coe, we hve o do ll hi deeminiiclly. Conide he following fige. I decie nee of n NBA A on wod w = which i cceed y A. In he fige, we hve mked one cceing h in hi nee, nd indiced he e of F h e long hi h y dkened cicle. S 3 S S 0 S 2 y x 0 y 0 x 1 y 1 x 2 y 2 x 3 Hee 0, 1,... e he e fom F eched long he mked cceing n. We hve lced lice S 0,S 1,... on he -nee ooed 0, 1,... S 0 i he e of node eched 3

4 fom 0 on eding he egmen x 0, S 1 i he e of node eched fom 1 on eding he egmen x 1 nd o on. w = yx 0 y 0 x 1 y 1... (The egmen y i i he ffix oined when we do x i fom he wod h ke i o i+1.) y i x i+1 Noice h S i g S i+1. Thi migh led o elieve h we imly hve o he omon A m 0. Unfonely, S 1 need no e e of e viied y he omon A ed e 0 (on x 0 y 0 x 1 ), i.e. δ ( 0,x 0 y 0 x 1 ) cold e ic ee of S 1 = δ ( 1,x 1 ). A indiced in he fige elow, he e my e in δ ( 0,x 0 y 0 x 1 ) no in S 1. S 1 0 S 0 1 B hen i δ ( 1,x 1 y 1 x 2 ) he me δ ( 2,x 2 ) (i.e. cn we he A m omon 2?) Well, once gin δ ( 1,x 1 y 1 x 2 ) cold e ic ee of δ( 2,x 2 ). Noe h if δ ( i,x i y i ) = δ ( i+1,) fo ome hen δ ( i,x i y i v) = δ ( i+1,v) fo ll exenion v. i i+1 Now, le y h i nd i+1 mege if hee i n ch h δ ( i,x i y i ) = δ ( i+1,). We cn eily exend hi noion of meging o i nd j fo ny i of occnce of cceing e (long he cceing h nde conideion.) If i mege wih i+1 nd i+1 mege wih i+2 hen clely i nd i+2 lo mege. I i ey o check h mege i n eivlence elion h divide 0, 1... ino egmen ( 0,... i ), ( i+1... j ),... Alo noe h if i nd i+1 do no mege, hen fo ny exenion of x i y i, δ ( i,x i y i ) i ic ee of δ ( i+1,). B he e δ( i,) i e of Q nd hence finie. Th, mege i n eivlence elion of finie index. Th hee i n eivlence cl of mege coniing of { j j N 0 } fo ome N i N 0 w Soe, h w = w whee N0, hen we clim h he mked e omon ed e F lelling N0 cce he wod w. Th i, he omon A m = 4

5 (2 Q 2 Q, Σ,δ m, ({}, ), {(X,X) X Q} whee δ m i efoe, cce he wod w. Le (X 1,X 2 ) e he e eched fe ome efix v of w. Hence hee i ome j n h gone j no j+1. The e N0 nd j+1 m mege ome efix w 1 of w. A h oin he e of A m wold e (Y,Y ) fo ome Y. Th, he omon A m vii he cceing e infiniely ofen on w. Th, o deeminiic omon m (deeminiiclly!) ge h in he nee nd F lelled oiion in hi h (coeonding o P N0 ) nd n he mked e omon (which i deeminiic omon) A m fom hee on. Th, in mnne ie eminicen of he oof of McNghon heoem, we hve hown h nondeeminim i needed only in finie efix of he n. The eion i how o ge id of he iniil nondeeminim. A in he oof of McNghon heoem, he ide wold e o n ll oile coie ined of geing which coy o n nd mege coie wheneve oile. Th i, ech oin in he n if he wod ed o f led o ome finl e we fok off coy of A m hi oin. We lo mege coie h e he me e. Le exmine he coeondence wih he concion in Lece 8 cloely: 1. The imle owee omon A will ly he ole of A U. I ole i o idenify he efixe which coie of he deeminiic Büchi om e o e foked off. 2. The ole of he deeminiic Büchi omon A V will e lyed y collecion of deeminiic Büchi om {A m F }. Afe eding finie wod w, if he omon A h eched he e X hen, fo ech X F coy of A m i o e foked off. 3. Oeve h if hen A m nd A m diffe only in he e. Th i, he e ce of ll he A m i h of A m. Th he oeion of meging diffeen coie ill mke ene. In icl, if wo diffeen coie e he me e we mege hem ogehe. 4. The nme of e in A m i 2 2n. A ech e, we cold fok o F coie (deending on which e in F e in he cen e). To ene h if coy lo j mege (wih lowe nmeed coy) down hen lo j i emy (i.e. i cie he e ) in he nex e, i ffice o kee 2 2n + F lo. Thi ene h ny lo h ecome vcn (ece i h meged wih ome ohe coy) y vcn le fo one move. 5. A n i cceing eciely when hee i ome lo j which i only finiely ofen (o h he coy j imle ome A m on ome ffix of w) nd omon lo j vii he finl e infiniely ofen (i.e. he e in he jh lo i of he fom (X,X) infiniely ofen). Thi led o he following concion of he (dole exonenil ized) deeminiic 5

6 omon A d cceing L(A). Le K = 2 2n + F. Q d d = 2 Q (2 Q 2 Q { }) K = ({},,,... ) Le (S,W 1,W 2,...W K ) e e. To come δ d ((S,W 1,W 2...,W K ),), fi come he le V = (δ (S,),δ m (W 1,),δ m (W 2,)...δ m (W K,)) wih he ndending h δ m (,) =. Soe { 1, 2,..., l } = δ(s,) F. We need o fok coy of A i m fo ech i. Pick he l lowe nmeed lo in V h e occied y nd elce hem y ({ 1 }, ), ({ 2 }, )...({ l }, ) eecively. Now, if wo o moe lo e occied y he me e of A m, hen elce ll he lowe nmeed ch lo y. Thi V i δ d ((S,W 1,W 2,...W K ),). Finlly, he ccence condiion i Rin ccence condiion coniing of K i (F i,i i ). F i conin ll he le whee he ih coodine i. I i conin ll he le whee he ih coodine i n cceing e of A m, h i i i of he fom (X,X) fo ome X. I i ie ey o check, ing he me ide in he oof in Lece 8, h hi omon cce wod eciely when i i cceed y L(A). How o nfom hi dole exonenil ized omon o ingle exonenil ized omon i he oic of he nex ecion. 1 Sf Concion Sf concion incooe he ce of he nee ino he e ce of he deeminiic omon. We hll ille hi ing he following comlex omon which cce he e of ll wod wih infiniely mny :,,,, v, Hee i efix of he nee of hi omon on he in... 6

7 v v Thee e even occnce of cceing e in he n on he wod. A n when hee e e enconeed coie of A m i foked o. The e eched y hee even coie i mked nex o hee e. Moeove he e eched y comlee owee omon i mked nex o he oo. {,,v,,} ({},{}) ({,},{,}) ({,,v},{,}) ({},{}) ({},{}) v ({},{}) ({},{}) v In he mon A d hi wold hve een eeened y le ({,,v,,}, ({,}, {,}), ({}, ), ({,,v}, {,}), ({}, ), ({}, )) (wih lge nme of ineeed which we hve omied o void cle). Sf concion oe moe infomion. I ecod elionhi eween hee vio coie. Fo exmle, he hee coie coeonding o he level 2, nd he 7

8 level 3 nd 4 e woking on indeenden ee. Thee e wo coie woking in he ee coeonding o he coy ed w... he level 3 nd o on. I wold ecod hee deendencie nd oe he e ee. ({,,v,,},{,,v,}) ({,},{,}) ({,,v},{,}) ({},{}) ({},{}) ({},{}) ({},{}) ({},{}) I lo e coy of A m ined of A fo he omon h fok o deeminiic om. Thi llow fo nifom emen of ll he node in he ee. (We hve no elimined dlice coie hee. We will ge down o h oon.) In A d we minined he e le nd o he index i llow o efe o icl coy of A m. To do hi hee, we me h hee i n infinie odeed e of node n 1 < n 2... nd ech ime we dd node in he ee, we ick he nex node fom hi eence nd e i. Fo exmle, wih hi he ove ee wold look like: n 1 ({,,v,,},{,,v,}) ({,},{,}) n 3 ({,,v},{,}) n 4 n 2 ({},{}) ({},{}) n 5 n 7 n 6 n 8 ({},{}) ({},{}) ({},{}) Noice h he node e nmeed in he ode in which he node wee inodced ino he ee (We me ome fixed odeing on Q o when mlile childen e dded o node in ingle e, he ode i deemined y he odeing on Q). To ndend he ehvio of hi omon le ee wh hen o hi e when he nex in lee, y i ed: 1. Fi, we imle hi move in ech of he coie of A m (one e node in he ee). 2. Then, we need o inodce new coie of A m fo ech e in f F h i eched hi e. Whee do we inodce hee coie? Noice h ech ch f e long ome e of h ing he oo (ince he e of e lelling en of node i lwy ee of h lelling child). Fo ech ch h, find he highe level ch node on h h nd dd child lelled ({f}, ). In hi exmle, i n o h ll hee highe level node e leve nd o we dd childen o hee leve. B hi need no lwy e he ce. Fo he ove omon hi give: 8

9 n 1 ({,,v,,},{,,v,}) ({,,v},{,,v}) n 3 ({,,v},{,,v}) n 4 n 2 ({},{}) ({,v},{}) n 5 n 7 n 6 n 8 ({,,v},{}) ({,},{}) ({,,v},{}) ({},{}) n 9 n 10 n 11 n 12 ({},{}) ({},{}) ({},{}) 3. If he e lelling node nd i child e of he fom (X,Y 1 ) nd (X,Y 2 ) hi indice h he coie of A m ed he node nd i child hve meged nd o we imly delee he child nd nfe ll i childen o he en. In he ove exmle, doing hi we ge: n 1 ({,,v,,},{,,v,}) ({,,v},{,,v}) n 3 ({,,v},{,,v}) n 4 n 2 ({},{}) ({,v},{}) n 5 n 7 n 6 n 8 ({,,v},{}) ({,},{}) ({,,v},{}) ({},{}) n 9 n 10 n 11 n 12 ({},{}) ({},{}) ({},{}) Th, nlike in A d we do no cively mege ny i of coie h hve eched he me e only en-child i h hve eched he me e. Noice h hi ene h long ny h in he ee he nme of e in he fi comonen of he node lel decee icly. Th ny h in ch ee i of lengh onded y Q. On he ohe hnd, hee i no ond on he nme of childen of given node. (Oevion 1: Noe h wheneve child of n mege wih n, n m e lelled y e of he fom (X,X), i.e. n cceing e of A m. Thi i ece if node lelled (X,Y ) h childen wih lel (X 1,Y 1 )...(X k,y k ) hen Y = X 1 X 2...X k.) The iniil e of he omon i he ee wih ingle node n 1 wih he e ({}, ). The niion elion decied ove gnee h in ny echle e, if he node n i i he en of n j nd he lel of n i nd n j e (X i,y i ) nd (X j,y j ) eecively, hen 9

10 X j i ic e of X i. We ke o e of e o e ee ove n 1,n 2... lelled y e of A m h fhe ify hi oey. Th he ee coniing e hve deh onded y Q. Howeve, ince we hve n infinie e of lel he omon we hve conced h infiniely mny e. Finlly, we hve o define he ccence condiion. I cce wod w, if hee i node n i ch h long he nie infinie n on w n i e in ll he e in ome infinie ffix nd moeove n i i lelled y n cceing e of A m infiniely ofen. The coecne of hi concion i ie ovio: I imly minin he e of A d in moe comliced fom (nd wih ome edndncy) nd o i coecne follow fom he coecne of A d. So i eem like we hve ken e in he wong diecion: Fom imle dole exonenil omon o comlex infinie e omon. B one illin oevion of Sf ke ll he wy o 2 O(nlogn) concion. Hi oevion i he following: If Q e in he e lelling mlile h in he ee (i.e. e of he Sf omon), hen we cn do i fom ll one! (By hi we men h if node n i i no long he choen h fo nd i lel i (X,Y ) hen we my elce hi lel y (X \ {},Y \ {})). Thi choice of which h i o kee ck of h o e mde ceflly. Childen of ny node e odeed fom lef o igh in he ode in which hey wee ineed ino he ee. Given wo h ing he oo, he one h move lef he oin of hei nching i id o e o he lef of he ohe. Sf ide i o kee he e in he lef mo h fom he oo (long which e). Wih hi ide, he ove e ecome: n 1 ({,,v,,},{,,v,}) n 2 ({},{}) ({,,v},{,,v}) n 3 ({},{}) n 4 ({,v},{}) n 5 n 10 n 6 ({},{}) n 12 ({},{}) ({},{}) ({},{}) n 9 n 11 ({},{}) We cn delee node lelled ({}, {}) ince hey ly no fhe ole in he n. Th he e eched on wold clly look like: n 1 ({,,v,,},{,,v,}) n 2 ({},{}) ({,,v},{,,v}) ({,v},{}) n 5 n 3 ({},{}) n 9 The modified omon A, h i e of e ee ove he e node n 1,n 2... lelled y e of A m in ch wy fo ny Q, hee i he mo one h (ing 10

11 he oo) whoe lel conin. The niion elion i comed ove wih he following ddiionl e: 4. If e long lel long mlile h ing he oo, delee i fom ll he lef-mo h. The ccence cieion conine o e he me. Why doe hi modified omon A cce he me lngge A (nd A d )? Evey wod cceed y A i cceed y A. Thi i ece, doing ome e fom he e lelling he ee coeond o nning coie of A m in ch wy h occionlly we do ome e fom he e. Thi doe no incee he fmily of wod cceed hown y he following execie. Execie: Le A m e he omon defined follow: A m = (Q m, Σ,δ m, ({}, ), {(X,X) X Q} whee Q m = {(X,Y ) Y X Q} δ m((x,y ),) = {(δ (X,) \ Z,δ (Y,) δ(x,) F \ Z) Z Q} δ m((x,x),) = {(δ (X,) \ Z,δ (X,) F \ Z Z Q}) if X Y Show h L(A m) L(A m ). Hin: Simly how h if (X,X F) (Z,Z) in hi omon hen fo ech z Z hee i ome x X ch h x g z. So we e lef wih howing h ny wod cceed y A i lo cceed y he modified omon A. Le ρ = e n cceing n. Le T T1... e he coeonding n in A. We ocie eence of node N 1,N 2..., N i T i wih he e i of ρ. In T i, e N i o e he node he highe level nme conining i (Thi node need no e lef.) Noice h ech e one of he following hing my hen: 1. N i+1 i child of N i. Thi hen if i+1 F nd he h fom oo N i+1 i he lef-mo h wih i N i+1 i n nceo of N i. Thi hen of N i nd evel of i nceo hve he me lel N i+1 nd hey ll ge meged wih N i N i+1 i eched vi lef jm. Oevion 2: If he ocied node eche node n fom i decenden (y hi we men N i+1 = n nd i i eched ing ce 2 ove) hen n m e lelled y n cceing e of A m. Thi follow fom Oevion 1 nd he condiion fo meging e. Oevion 3: By Oevion 1, we do no need o kee he econd comonen of he 11

12 lel of he ee hi cn e comed fom he fi comonen of he childen. (Sf concion e hi oimizion.) Fi we oeve h if n 0 e N i fo infiniely mny i hen he node n 0 i lelled y n cceing e of A m infiniely ofen. Thi i ece, ech ime he h vii n j F, he node N j wold move o child of n 0. B i en o n 0 in he fe. B hi hen only when ome child of n 0 mege wih i. B, y Oevion 2, n 0 m e lelled y n cceing e wheneve hi hen. Ohewie, hee i oin j ch h N i n 0 fo ll i j. A he l oin, y N i, when he ocied node moved fom n 0 i m hve moved o child of n 0. Moeove hee e only finiely mny childen of he oo nd in icl only finiely mny o he lef of N i. So wheneve he ocied node en o level 2 (i.e. he level of childen of he oo) hee e only finiely mny choice (i cnno eve ech child of he oo h e o he igh of N i wiho viiing he oo, we hve ledy moved eyond he l vii o he oo). Th, he only wy o ge o ny of hee i eihe y mege fom elow o y lef jm. B lef jm o node level 2 cn hen only finie nme of ime (ince hee e only finiely mny of hee veice h lie o he lef of N i hi level). Th, hee i oin eyond which no veex level 2 i eched y lef jm. Beyond hi oin eihe ome veex level 2 i viied infiniely ofen vi mege fom elow, which (y Oevion 2) gnee h hi node i lelled y cceing e of A m infiniely ofen o ele hee i oin eyond which he n neve en o level 2. We my hen ee hi gmen wih veice level 3 nd o on. B ince he deh of he ee i onded y N i follow h ome node i eched infiniely ofen y mege fom elow which gnee h hi node i lelled y cceing e of A m infiniely ofen. Th he n i n cceing n of he omon A. Now we n hi ino finie e omon. Fi of ll oeve h he fi comonen of he lel of ny child of n i e of he fi comonen of he lel of n. Fhe he fi comonen of he childen e ll dijoin. Th if we foc j on he fi comonen, we he oo nd we go down, we kee ioning he e ino fine nd fine. Th, he ol nme of node in he ee i onded y 2 n whee Q = n. A ech e, we will dd he mo F n new node. Th, if we hve e of 3n + 1 node (following he ide in Lece 8 o he concion of A d ove) we e gneed h ny node h i doed doe no e in he following e. Wih hi we ge finie e omon, wih Rin cceing condiion wih 3n + 1 i. In he ih i (F i,i i ), F i coni of ll he ee whee n i doe no e nd I i coni of ll he ee whee n i i lelled y n cceing e of A m. Wh i he ize of hi omon? How mny ee of he kind decied ove cn e fomed of ize 2 n fom e of 3n + 1 diinc node. By Cyley heoem he nme of ee ove O(n) node i of he ode of 2 O(nlogn). We hve odeed ee (i.e. diffeen odeing of he childen of ny node yield diffene ee). Fom ech nodeed ee we my genee o n! diffeen lelled ee. Th he ol nme of odeed ee conine o e of he ode of 2 O(nlogn). Given ch ee how mny diffeen wy cn we lel he node wih e of Q ifying he ioning oey? Noe h ech Q e (if 12

13 ll) long one nie h. Th, if we deemine he il (i.e. he highe nmeed node) fo ech Q, he lelling of he ee i niely deemined. The nme of fncion fom Q o he e of node i n 3n+1. Th he ol nme of Sf ee i O(2 (O(nlogin) ). O eenion of Sf ee i omewh diffeen fom he one ed y Sf in hi e. Thi i ece of o em o ive fom n nlyi of he n-ee. We hll oline he oiginl concion iefly he eginning of he nex lece. Refeence [1] Chiof Löding: Oiml Bond fo Tnfomion of ω-om, Poceeding of he Inenionl Confeence on Fondion of Sofwe Techonology nd Theoeicl Come Science (FSTTCS) 1999, Singe Lece Noe in Come Science 1738, [2] Mdhvn Mknd: Finie Aom on Infinie Wod, Inenl Reo, SPIC Science Fondion, TCS-96-2, Aville h:// mdhvn/e//c gz [3] S. Sf: On he comlexiy of ω-om, Poceeding of he 29h FOCS, [4] Wolfgng Thom: Lngge, om, nd logic In he Hndook of Foml Lngge, volme III, ge Singe, New Yok,