Lower Bounds for Complementation of omega-Automata Via the Full Automata Technique

Logical Methods in Computer Science V ol. 4 (1:?) 2008, pp. 1–1–20 www .lmcs-online.org Submitted Jul. 25, 2007 Published Mar . ?? , 2008 LO WER BOUNDS F OR COMPLEMENT A TION OF ω -A UTOMA T A VIA THE FULL A UTOMA T A TECHNIQUE ∗ QIQI Y AN Departmen t of Computer Siene and Engineering, Shanghai Jiao T ong Univ ersit y , 200240, Shang- hai, P .R. China e-mail addr ess : on tatqiqiy an.om Abstra t. In this pap er, w e rst in tro due a lo w er b ound te hnique for the state om- plexit y of transformations of automata. Namely w e suggest rst onsidering the lass of full automata in lo w er b ound analysis, and later reduing the size of the large alphab et via alphab et substitutions. Then w e apply su h te hnique to the omplemen tation of non- deterministi ω -automata, and obtain sev eral lo w er b ound results. P artiularly , w e pro v e an Ω((0 . 76 n ) n ) lo w er b ound for Bü hi omplemen tation, whi h also holds for almost ev- ery omplemen tation or determinization transformation of nondeterministi ω -automata, and pro v e an optimal (Ω( nk )) n lo w er b ound for the omplemen tation of generalized Bü hi automata, whi h holds for Streett automata as w ell. 1. Intr odution The omplemen tation problem of nondeterministi ω -automata, i.e. nondeterministi automata o v er innite w ords, has v arious appliations in formal v eriation. F or example in automata-theoreti mo del  he king, in order to  he k whether a system represen ted b y automaton A 1 satises a prop ert y represen ted b y automaton A 2 , one  he ks that the in ter- setion of A 1 with an automaton that omplemen ts A 2 is an automaton aepting the empt y language [Kur94 , VW94℄. In su h a pro ess, sev eral t yp es of nondeterministi ω -automata are onerned, inluding Bü hi, generalized Bü hi, Rabin, Streett et., and the omplexit y of omplemen ting these automata has augh t great atten tion. The omplemen tation of Bü hi automata has b een in v estigated for o v er fort y y ears [V ar07 ℄. The rst eetiv e onstrution w as giv en in [ Bü62 ℄, and the rst exp onen tial onstrution w as giv en in [SVW85 ℄ with a 2 O ( n 2 ) state blo w-up ( n is the n um b er of states of the input automaton). Ev en b etter onstrutions with 2 O ( n log n ) state blo w-ups w ere giv en in [Saf88 , Kla91 , KV01℄, whi h mat h with Mi hel's n ! = 2 Ω( n log n ) lo w er b ound 2000 A CM Subje t Classi ation: F.4.1, F.4.3. Key wor ds and phr ases: full automata, state omplexit y , automata transformation, Bü hi omplemen ta- tion, ω -automata. ∗ A preliminary v ersion of this pap er app ears in the pro eedings of the 33rd In ternational Collo quium on Automata, Languages and Programming, 2006. Supp orted b y NSF C No. 60273050. LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.216 8/LMCS-4 (1:?) 2008 c  Q. Y an CC  Cre ative Comm ons 2 Q. Y AN [Mi88 ℄, and w ere th us onsidered optimal. Ho w ev er, a loser lo ok rev eals that the blo w-up of the onstrution in [KV01℄ is (6 n ) n , while Mi hel's lo w er b ound is only roughly ( n/e ) n = (0 . 36 n ) n , lea ving a big exp onen tial gap hiding in the asymptoti notation 1 . Motiv ated b y this omplexit y gap, the onstrution in [KV01 ℄ w as further rened in [FKV06 ℄ to (0 . 97 n ) n . On the other hand, Mi hel's lo w er b ound w as nev er impro v ed. F or generalized Bü hi, Rabin and Streett automata, the b est kno wn onstrutions are in [KV05b , KV05a ℄, whi h are 2 O ( n log nk ) , 2 O ( nk log n ) and 2 O ( nk log nk ) resp etiv ely . Here state blo w-ups are measured in terms of b oth n and k , where k is the index of the input automaton. Optimalit y problems of these onstrutions ha v e b een v astly op en, b eause only 2 Ω( n log n ) lo w er b ounds w ere kno wn b y v arian ts of Mi hel's pro of [ Löd99 ℄. What remains missing are stronger lo w er b ound results. Tigh ter lo w er b ounds usually lead us in to b etter understanding of the in triay of the omplemen tation of nondeterministi ω -automata, and are the main onern of this pap er. Su h understanding an suggest metho ds to further optimize the onstrutions, or to irum v en t those diult ases in pratie. T o understand wh y w e ha v e so few strong lo w er b ounds, w e observ e that at the ore of almost ev ery kno wn lo w er b ound is Mi hel's result, whi h w as obtained in the traditional w a y . That is, one rst onstruts a partiular family of automata ( A n ) n ≥ 1 , and then pro v es that omplemen ting ea h A n requires a large state blo w-up. The A n +1 of Mi hel's automata family is depited in Figure 1. Although ea h A n +1 has a simple struture, it is not straigh t- forw ard to see what language it aepts, and nor is it lear at all ho w w e an w ork with this automaton for lo w er b ound. s f s 1 s 2 s n 1 . . . n, ♯ 1 . . . n, ♯ 1 . . . n, ♯ 1 2 n Figure 1: Mi hel's Automata Class In man y ases, iden tifying su h an automata family is diult, and is the main obsta- le to w ards lo w er b ounds. In this pap er, w e prop ose a new te hnique to irum v en t this diult y . Namely , w e suggest rst onsidering the family of full automata in lo w er b ound analysis, and later reduing the size of the large alphab et via alphab et substitutions. A simple demonstration of su h te hnique is presen ted in Setion 3. With the help of full automata, w e tigh ten the state omplexit y B C ( n ) of Bü hi om- plemen tation from (0 . 36 n ) n ≤ B C ( n ) ≤ (0 . 97 n ) n to (0 . 76 n ) n ≤ B C ( n ) ≤ (0 . 97 n ) n . Sur- prisingly , this (0 . 76 n ) n lo w er b ound also holds for ev ery omplemen tation or determinization transformation onerning Bü hi, generalized Bü hi, Rabin, Streett, Muller, and parit y au- tomata. As to the omplemen tation of generalized Bü hi automata, w e pro v e an (Ω( nk )) n lo w er b ound, mat hing with the ( O ( nk )) n upp er b ound in [KV05b℄. This lo w er b ound also holds for the omplemen tation of Streett automata and the determinization of generalized 1 In on trast, for the omplemen tation of nondeterministi nite automata o v er nite w ords, the 2 n blo w- up of the subset onstrution [RS59 ℄ w as justied b y a tigh t lo w er b ound [SS78 ℄, whi h w orks ev en if the alphab et onerned is binary [Jir05 ℄. LO WER BOUNDS F OR COMPLEMENT A TION OF ω -A UTOMA T A 3 Bü hi automata in to Rabin automata. A summary of our lo w er b ounds is giv en in Setion 6. F ull Automata and Sak o da and Sipser's Languages. It turns out that the notion of full automata is similar to Sak o da and Sipser's languages in [SS78 ℄. Their language B n atually orresp onds to the ∆ -graphs of the w ords aepted b y some full automaton. Also as p oin ted to us b y Christos A. Kap outsis, the te hnique of alphab et substitution w as somewhat impliit in Sak o da and Sipser's pap er (but presen ted in a somewhat obsure w a y , refer to the paragraph b efore their Theorem 4.3.2). So the full automata te hnique is more lik e a new treatmen t of some te hniques in the Sak o da and Sipser's pap er, rather than a totally new in v en tion. Compared to Sak o da and Sipser's languages, the notion of full automata enjo ys a simple denition and is v ery handy to use. It is also more readily to b e extended to other kinds of automata lik e alternating automata. F or unlear reasons, Sak o da and Sipser's languages w ere rarely applied to elds other than 2-w a y automata after their pap er. W e hop e that our treatmen t will mak e a lear exp osition of the te hniques and demonstrate their usefulness in problems on automata o v er one-w a y inputs as w ell. 2. Basi Definitions A ( nondeterministi ) automaton is a tuple A = (Σ , S, I , ∆ , ∗ ) with alphab et Σ , nite state set S , initial state set I ⊆ S , transition relation ∆ ⊆ S × Σ × S and ∗ some extra omp onen ts. P artiularly A is deterministi if | I | = 1 and for all p ∈ S and a ∈ Σ , |{ q ∈ S | h p, a, q i ∈ ∆ }| ≤ 1 . F or a w ord w = a (0) a (1) . . . a ( l − 1) ∈ Σ ∗ with len g th ( w ) = l ≥ 0 , a nite run of A from state p to q o v er w is a nite state sequene ρ = ρ (0) ρ (1) . . . ρ ( l ) ∈ S ∗ su h that ρ (0) = p , ρ ( l ) = q and h ρ ( i ) , a ( i ) , ρ ( i + 1) i ∈ ∆ for all 0 ≤ i < l . W e sa y that ρ visits a state set T if ρ ( i ) ∈ T for some 0 ≤ i ≤ l . W e write p w − → q if a nite run from p to q o v er w exists, and p w − → T q if in addition the run visits T . A ( Nondeterministi ) Finite W or d A utomaton ( NFW for short) is an automaton A = (Σ , S, I , ∆ , F ) with nal state set F ⊆ S . A nite w ord w is aepted b y A if there is a nite run o v er w from an initial state to a nal state. The language aepted b y A , denoted b y L ( A ) , is the set of w ords aepted b y A , and its omplemen t Σ ∗ \L ( A ) is denoted b y L C ( A ) . F or an ω -w ord α = α (0) α (1) · · · ∈ Σ ω , i.e., an innite sequene of letters in Σ , a (innite) run of A o v er α is an innite state sequene ρ = ρ (0) ρ (1) · · · ∈ S ω su h that ρ (0) ∈ I and h ρ ( i ) , α ( i ) , ρ ( i + 1) i ∈ ∆ for all i ≥ 0 . W e let O cc ( ρ ) = { q ∈ S | ρ ( i ) = q for some i ∈ N } , I nf ( ρ ) = { q ∈ S | ρ ( i ) = q for innitely man y i ∈ N } , and write ρ [ l 1 , l 2 ] to denote the inx ρ ( l 1 ) ρ ( l 1 + 1) . . . ρ ( l 2 ) of ρ . An (nondeterministi) ω -automaton is an automaton A = (Σ , S, I , ∆ , Acc ) with aep- tane ondition Acc , whi h is used to deide if a run ρ of A is suessful. There are man y t yp es of ω -automata onsidered in the literature [Tho90 ℄. Here w e onsider six of the most ommon t yp es: • Bühi automaton , where Acc = F ⊆ S is a nal state set, and ρ is suessful if I nf ( ρ ) ∩ F 6 = ∅ . 4 Q. Y AN • gener alize d Bühi automaton , where Acc = { F 1 , . . . , F k } is a list of nal state sets, and ρ is suessful if I nf ( ρ ) ∩ F i 6 = ∅ for all 1 ≤ i ≤ k . • R abin automaton , where Acc = {h G 1 , B 1 i , . . . , h G k , B k i} is a list of pairs of state sets, and ρ is suessful if for some 1 ≤ i ≤ k , I nf ( ρ ) ∩ G i 6 = ∅ and I nf ( ρ ) ∩ B i = ∅ . • Str e ett automaton , where Acc = {h G 1 , B 1 i , . . . , h G k , B k i} is a list of pairs of state sets, and ρ is suessful if for all 1 ≤ i ≤ k , if I nf ( ρ ) ∩ B i 6 = ∅ , then I nf ( ρ ) ∩ G i 6 = ∅ . • Mul ler automaton , where Acc = F ⊆ P ow er set ( S ) is a set of state sets, and ρ is suessful if I nf ( ρ ) ∈ F . • p arity automaton , where Acc is a mapping c : S → { 0 . . . l } , and ρ is suessful if min { c ( q ) | q ∈ I nf ( ρ ) } is ev en. An ω -w ord α is a  epte d b y A if it has a suessful run. The ω -language aepted b y A , denoted b y L ( A ) , is the set of ω -w ords aepted b y A , and its omplemen t Σ ω \L ( A ) is denoted b y L C ( A ) . The n um b er k , if dened, is alled the index of A . W e refer to the ab o v e six t yp es of ω -automata as the  ommon typ es . F ollo wing the on v en tion in [KV05a ℄, w e will use aron yms lik e NBW , NGBW , NR W et. to refer to Nondeterministi Bü hi/generalized Bü hi/Rabin/et. W ord automata. T w o simple fats ab out these ommon t yp es of ω -automata are useful for us: fA t 2.1. [Löd99 ℄ (1) F or every NBW A and every  ommon typ e T , ther e exists an T automaton A ′ with the same numb er of states suh that A ′ is e quivalent to A . (2) F or every deterministi ω -automaton A of a  ommon typ e T whih is not Bühi nor gener alize d Bühi, ther e exists a deterministi ω -automaton A ′ of a  ommon typ e (not ne  essarily also T ) with the same numb er of states (and index, if appli able) suh that A ′  omplements A . T o visualize the b eha vior of automata o v er input w ords, w e in tro due the notion of ∆ - graphs. If A = (Σ , S, I , ∆ , ∗ ) is an automaton, then for a nite w ord w = a (0) a (1 ) . . . a ( l − 1) ∈ Σ ∗ of length l , or an ω -w ord w = a (0) a (1) · · · ∈ Σ ω of length l = ∞ , the ∆ -gr aph of w under A is the direted graph G A w = ( V A w , E A w ) with v ertex set V A w = {h p, i i | p ∈ S, 0 ≤ i ≤ l, i ∈ N } and edge set E A w dened as: for all p, q ∈ S and 0 ≤ i < l , hh p, i i , h q , i + 1 ii ∈ E A w i h p, a ( i ) , q i ∈ ∆ . F or a subset T of S , w e sa y that a v ertex h p, i i is a T -v ertex if p ∈ T . By denition p w − → q i there is a path (in the direted sense) in G A w from h p, 0 i to h q , leng th ( w ) i and p w − → T q if furthermore the path visits some T -v ertex. Finally w e dene the state  omplexity 2 funtions. Assume that T is either NFW or some ommon t yp e of ω -automata. Then for a T automaton A , C T ( A ) is dened as the minim um n um b er of states of a T automaton that omplemen ts A , i.e., aepts L C ( A ) . F or n ≥ 1 , C T ( n ) is the maxim um of C T ( A ) o v er all T automata with n states. If indies are dened for T , then C T ( n, k ) is the maxim um of C T ( A ) o v er all T automata with n states and index k . 2 In some literature, instead of merely oun ting the n um b er of states, sizes of transition relations et. are also tak en in to aoun t to b etter measure the sizes of automata. Here w e prefer state omplexit y b eause it is a measure easier to study , and its lo w er b ound results usually imply lo w er b ounds on size omplexit y , if the automata witnessing the lo w er b ound are o v er a not to o large alphab et. LO WER BOUNDS F OR COMPLEMENT A TION OF ω -A UTOMA T A 5 3. The Full A utoma t a Tehnique In the reen tly emerging area of state omplexit y (see [Y u05 ℄ for a surv ey) or in the theory of ω -automata, w e often onern pro ving theorems of su h a v or: Theorem 3.1. [Jir05 ℄ F or e ah n ≥ 1 , ther e exists an NFW A n with n states over { a, b } suh that C NFW ( A n ) ≥ 2 n . In other w ords, w e w an t to pro v e a lo w er b ound for the state omplexit y of a transfor- mation ( NFW omplemen tation in this ase, an b e determinization et.), and furthermore, w e hop e that the automata family witnessing the lo w er b ound ( ( A n ) n ≥ 1 in this ase) is o v er a xed small alphab et. Su h laims are usually diult to pro v e. The apparen tly easy Theorem 3.1 w as not pro v ed un til 2005 b y a v ery te hnial pro of in [ Jir05 ℄ 3 , after the eorts in [SS78 , Bir93 , HK02 ℄. T o understand the diult y in v olv ed, w e rst review the traditional approa h p eople attempt at su h results: Step I: Iden tify an automata family ( A n ) n ≥ 1 with ea h A n ha ving n states. Step I I: Pro v e that to transform ea h A n needs a large state blo w-up. Almost ev ery kno wn lo w er b ound w as obtained in this w a y , inluding Theorem 3.1 and the aforemen tioned Mi hel's lo w er b ound. In su h an approa h, Step I is w ell-kno wn to b e diult. Iden tifying the suitable family ( A n ) n ≥ 1 requires b oth ingen uit y and lu k. Ev en w orse, most automata families that p eople try are natural ones with simple strutures, while the ones witnessing the desired lo w er b ound ould b e highly unnatural and omplex. Finding the righ t family ( A n ) n ≥ 1 seems to b e a ma jor obstale to w ards lo w er b ound results. No w w e in tro due the notion of full automata to irum v en t this obstale. Denition 3.2. Giv en state set S , initial state set I , and extra omp onen ts ∗ , a ful l automa- ton A = (Σ , S, I , ∆ , ∗ ) is an automaton with alphab et Σ = P ow er set ( S × S ) and transition relation ∆ dened as: for all p, q ∈ S and a ∈ Σ , h p, a, q i ∈ ∆ i h p, q i ∈ a . By denition, the alphab et on tains ev ery binary relation o v er S , and therefore is of a big size of 2 | S | 2 . Due to su h ri h alphab ets, ev ery automaton has some em b edding in a full automaton with the same n um b er of states. It is then not diult to see that transforming an automaton an b e redued to transforming a full automaton, and full automata are the most diult automata to transform. T o b e sp ei, if w e onsider NFW omplemen tation, then: Theorem 3.3. F or al l n ≥ 1 , C NFW ( n ) = C NFW ( A ) for some ful l NFW A with n states. The theorem follo ws from the follo wing lemma. Lemma 3.4. If A 1 is an NFW with n states, then ther e is a ful l NFW A 2 with n states suh that C NFW ( A 2 ) ≥ C NFW ( A 1 ) . Pr o of. By denition of C NFW , it sues to sho w that for some full NFW A 2 with n states, if there is an NFW C A 2 that omplemen ts A 2 , then there is an NFW C A 1 omplemen ting A 1 with the same n um b er of states as C A 2 . Let A 1 = (Σ 1 , S 1 , I 1 , ∆ 1 , F 1 ) , and onsider the full NFW A 2 = (Σ 2 , S 1 , I 1 , ∆ 2 , F 1 ) with resp et to S 1 , I 1 and F 1 . F or ea h a 1 ∈ Σ 1 , dene letter ∆ 1 ( a 1 ) in Σ 2 = P ( S 1 × S 1 ) as: h p 1 , q 1 i ∈ ∆ 1 ( a 1 ) i h p 1 , a 1 , q 1 i ∈ ∆ 1 , for all p 1 , q 1 ∈ S 1 . By denition of full automata, 3 The result is atually sligh tly stronger in that his A n has only one initial state. (In some literature NFW s are not allo w ed to ha v e m ultiple initial states.) 6 Q. Y AN h p 1 , a 2 , q 1 i ∈ ∆ 2 i h p 1 , q 1 i ∈ a 2 , for all p 1 , q 1 ∈ S 1 , a 2 ∈ Σ 2 . So w e ha v e h p 1 , a 1 , q 1 i ∈ ∆ 1 i h p 1 , ∆ 1 ( a 1 ) , q 1 i ∈ ∆ 2 , for all a 1 ∈ Σ 1 , p 1 , q 1 ∈ S 1 . F or an arbitrary w ord α = a (0) a (1) . . . a ( l − 1) ∈ Σ ∗ 1 , onsider w ord α ′ = ∆ 1 ( a (0))∆ 1 ( a (1)) . . . ∆ 1 ( a ( l − 1)) ∈ Σ ∗ 2 . Then ev ery state sequene ρ 1 = ρ 1 (0) ρ 1 (1) . . . ρ 1 ( l ) ∈ S ∗ 1 is a run of A 1 o v er α i ρ 1 is a run of A 2 o v er α ′ . Sine A 1 and A 2 share the same initial and nal state sets, ρ 1 is suessful i ρ 2 is suessful. So α ∈ L ( A 1 ) i α ′ ∈ L ( A 2 ) . Let C A 2 = (Σ 2 , S C , I C , ∆ C , F C ) b e an NFW that omplemen ts L ( A 2 ) . So α ′ ∈ L ( A 2 ) i α ′ / ∈ L ( C A 2 ) . Dene C A 1 to b e the NFW (Σ 1 , S C , I C , ∆ ′ C , F C ) , where ∆ ′ C is dened as h p 2 , a 1 , q 2 i ∈ ∆ ′ C i h p 2 , ∆ 1 ( a 1 ) , q 2 i ∈ ∆ C , for all p 2 , q 2 ∈ S C and a 1 ∈ Σ 1 . Similarly ev ery state sequene ρ C = ρ C (0) ρ C (1) . . . ρ C ( l ) ∈ S ∗ C is a suessful run of C A 2 o v er α ′ i ρ C is a suessful run of C A 1 o v er α . So α ′ ∈ L ( C A 2 ) i α ∈ L ( C A 1 ) . No w for ev ery α ∈ Σ ∗ 1 , α ∈ L ( A 1 ) i α / ∈ L ( C A 1 ) . Therefore C A 1 with the same n um b er of states as C A 2 omplemen ts A 1 as required. Theorem 3.3 implies that to pro v e a lo w er b ound for NFW omplemen tation (without taking the size of the alphab et in to aoun t), w e an simply set ( A n ) n ≥ 1 to b e some family of full NFW s in Step I. Similarly , the same applies to NBW omplemen tation: Theorem 3.5. F or al l n ≥ 1 , C NBW ( n ) = C NBW ( A ) for some ful l NBW A with n states. No w w e apply full automata to obtain a simple pro of of Theorem 3.1 . Pr o of. (of Theorem 3.1 ) W e rst pro v e a 2 n lo w er b ound for C NFW ( n ) . F or ea h n ≥ 1 , let F A n = (Σ n , S n , I n , ∆ n , F n ) b e the full NFW with S n = I n = F n = { s 0 , . . . , s n − 1 } . It sues to pro v e that C NFW ( F A n ) ≥ 2 n . F or ea h subset T ⊆ S n , let I d ( T ) denote the letter {h q , q i | q ∈ T } and let u T = I d ( T ) , v T = I d ( S n \ T ) . Figure 2(a) depits one example of u T v T 's ∆ -graph. Sine all states in F A n are b oth initial and nal, a w ord w of length l is aepted b y F A n i there is a path from an h s i , 0 i v ertex to an h s j , l i v ertex in the ∆ -graph of w under F A n . In partiular u T v T is not aepted b y F A n . Supp ose that some NFW C A omplemen ts F A n . So for ea h T ⊆ S n , there is a state ˆ q T of C A su h that ˆ q I u T − → ˆ q T and ˆ q T v T − → ˆ q F for some initial state ˆ q I and nal state ˆ q F of C A . If w e pro v e that ˆ q T 1 6 = ˆ q T 2 whenev er T 1 6 = T 2 , then C A has at least 2 n states as required. Supp ose b y on tradition that ˆ q T 1 = ˆ q T 2 for some T 1 6 = T 2 . W.l.o.g. there is a state s of F A n in T 1 \ T 2 . Then s u T 1 − → s v T 2 − → s and hene u T 1 v T 2 ∈ L ( F A n ) . On the other hand, for some initial state ˆ q I and nal state ˆ q F of C A , ˆ q I u T 1 − → ˆ q T 1 = ˆ q T 2 v T 2 − → ˆ q F . So u T 1 v T 2 ∈ L ( C A ) , on tradition. The ab o v e pro of is not fully satisfying in that the automata family witnessing the lo w er b ound is o v er an exp onen tially gro wing alphab et. T o x a binary alphab et and pro v e Theorem 3.1, w e in tro due a Step I I I in whi h w e do alphab et substitution, as w e no w illustrate. W e rst rene the ab o v e pro of of C NFW ( F A n ) ≥ 2 n b y restriting the n um b er of dieren t letters in v olv ed. F or t w o w ords u, v ∈ Σ ∗ n , w e sa y that u is e quivalent to v with resp et to F A n , or simply u ∼ v , if for all p, q ∈ S n , p u → q i p v → q . A little though t sho ws that if w e substitute ea h I d ( T ) letter used in the ab o v e pro of b y some equiv alen t w ords, the pro of still w orks. First w e onsider the alphab et { c i } 0 ≤ i 1 to b e the full NBW (Σ n , S n , I n , ∆ n , F n ) with I n = { s 0 , . . . , s n − 2 } , F n = { s f } and S n = I n ∪ F n . W e also use S ′ n = I n to denote the main states. W e rst try to onstrut an ω -w ord α n not aepted b y F B n su h that a great n um b er of tigh t lev el rankings w ould ha v e to b e presen t in ev ery C-ranking for G F B n α n . Sine the n um b er of tigh t lev el rankings is the ma jor fator of the state blo w-up in Kupferman and V ardi's onstrution, this w ould pro due a hard ase for the onstrution. F or su h purp ose, w e 4 Our denitions of lev el ranking and tigh t lev el ranking here are sligh tly dieren t from [FKV06 ℄. LO WER BOUNDS F OR COMPLEMENT A TION OF ω -A UTOMA T A 9 onsider a sp eial lass of tigh t lev el rankings for F B n , Q -rankings. W e sa y that a TL ( m ) - ranking g for F B n is a Q ( m ) -r anking if g ( q ) is dened for ea h q ∈ S ′ n and is undened for q = s f . W e start dening our diult ω -w ord α n b y dening its omp osing segmen ts. Lemma 4.3. F or every p air of Q -r ankings ( f , g ) , ther e exists a wor d w f ,g suh that: (i): F or al l p, q ∈ S ′ n , p w f ,g − → q i ( f i ( p ) > f i +1 ( q ) or f i ( p ) = f i +1 ( q ) ∈ [2 m ] odd ). (ii): F or al l p, q ∈ S ′ n , p w f ,g − → F n q i f i ( p ) > f i +1 ( q ) . (iii): F or al l p, q ∈ S n , if p w f ,g − → q then p, q / ∈ F n . Pr o of. W e rst illustrate the onstrution using a t ypial example depited in Fig. 3. As in Fig. 3, the v erties of the ∆ -graph of w f ,g are separated b y the wider spae b elo w c ( f , g ) in to t w o parts. W e sa y that ea h ( s i , j ) v ertex in the left part is rank ed f ( s i ) b y f , and ea h ( s i , j ) v ertex in the righ t part is rank ed g ( s i ) b y g . So when one follo ws a path from a leftmost v ertex v 1 to a righ tmost v ertex v 2 , either one go es to a next v ertex with the same rank, or one visits a h s f , j i v ertex and then go es to a v ertex with a rank lo w er b y one. This explains the only if diretion of (ii). Also note that v 1 and v 2 annot ha v e the same ev en ranks b eause in the middle of this pro ess, one has to go to a v ertex with an o dd rank to pass c ( f , g ) . So the only if diretion in (i) holds to o. F or the if diretions of (i) and (ii), supp ose one w an ts to go from a leftmost v ertex v 1 with rank r to a righ tmost v ertex v 2 with rank r ′ and that either r > r ′ or r = r ′ ∈ [2 m ] odd . Let t b e an o dd rank su h that r ≥ t ≥ r ′ . Then b y the onstrution, one an go from v 1 to some v ertex with rank t in the left part, pass through c ( f , g ) with rank t , and then on tin ue to go to v 2 in the righ t part. Note that in the pro ess, if rank ev er dereases, then an h s f , j i v ertex m ust ha v e b een visited. So the if diretions of (i) and (ii) hold as w ell. Condition (iii) is ob viously true. f ( s ) − 3 2 3 1 s s f s 0 s 1 s 2 s 3 g ( s ) − 2 1 3 0 s s f s 0 s 1 s 2 s 3 d ( f , 3 , 2) d ( f , 2 , 1) c ( f , g ) d ( g , 3 , 2) d ( g , 2 , 1) d ( g , 1 , 0) Figure 3: ∆ -graph of w f ,g F or later purp oses, w e expliitly presen t our onstrution for w f ,g . F or a Q ( m ) -ranking h , w e dene the state sets Rank h ( r ) = { q ∈ S ′ n | r = h ( q ) } for r ∈ [2 m ] and O dd h to b e the union of Rank h ( r ) 's with r ∈ [2 m ] odd . Also for ea h T ⊆ S ′ n , dene letters in Σ n as I d ( T ) = {h q , q i | q ∈ T } , T toF ( T ) = I d ( S ′ n ) ∪ {h q , s f i | q ∈ T } , F toT ( T ) = I d ( S ′ n ) ∪ {h s f , q i | q ∈ T } and c ( f , g ) = {h p, q i | f ( p ) = g ( q ) ∈ [2 m ] odd , p, q ∈ S ′ n } . F or a Q ( m ) -ranking h and r , r ′ ∈ [2 m ] , w e write d ( h, r, r ′ ) to denote the w ord T toF ( R an k h ( r )) · F toT ( Ran k h ( r ′ )) . Then if r 1 , r 2 . . . , r k are the ranks in [2 m ] that are images of h in desending order, w e let u h = d ( h, r 1 , r 2 ) · d ( h, r 2 , r 3 ) · · · · · d ( h, r k − 1 , r k ) . Finally , w f ,g is dened to b e u f · c ( f , g ) · u g . 10 Q. Y AN Lemma 4.4. L et f 0 , f 1 , . . . , f l b e a list of Q ( m ) -r ankings with l > 0 , and let w b e the wor d w f 0 ,f 1 w f 1 ,f 2 . . . w f l − 1 f l . A lso let p, q ∈ S ′ n , then: (i) If f 0 ( p ) > f l ( q ) or f 0 ( p ) = f l ( q ) ∈ [2 m ] odd , then p w − → q . (ii) If f 0 ( p ) > f l ( q ) , then p w − → F n q . Pr o of. If l = 1 , then w = w f 0 ,f 1 , and the prop erties follo w from Theorem 4.3 trivially . So w e assume that l > 1 . Let t b e an o dd rank su h that f 0 ( p ) ≥ t ≥ f l ( q ) . By denition of Q ( m ) -ranking, there exists a state sequene q 1 , q 2 , . . . , q l − 1 su h that f i ( q i ) = t for all 1 ≤ i ≤ l − 1 . So q i w f i ,f i +1 − → q i +1 for all 1 ≤ i < l − 1 . Also b eause f 0 ( p ) ≥ t ≥ f l ( q ) , w e ha v e p w f 0 ,f 1 − → q 1 and q l − 1 w f l − 1 ,f l − → q . Conatenate these together, w e ha v e p w − → q , and (i) is satised. If f 0 ( p ) > f l ( q ) , then either f 0 ( p ) > t or t > f l ( q ) , and hene either p w f 0 ,f 1 − → F n q 1 or q l − 1 w f l − 1 ,f l − → F n q . So p w − → F n q , and (ii) is satised. Let L ( n, m ) b e the n um b er of dieren t Q ( m ) -rankings and let L ( n ) b e max 1 ≤ m f i +1 ( q i +1 ) for innitely man y i ∈ N . This is imp ossible sine f 0 ( q 0 ) is nite. Reall that Kupferman and V ardi's onstrution uses distint state sets to handle dier- en t TL ( m ) -rankings. It turns out that if a omplemen t automaton of F B n do es not ha v e as man y states as Q ( m ) -rankings, it w ould b e onfused b y α n together with another omplex ω -w ord α ′ deriv ed from α n . Lemma 4.6. F or e ah n > 1 and e ah ω -automaton C A with less than L states, if ρ is a run of C A over α n / ∈ L ( F B n ) , then ther e is a run ρ ′ of C A over some ω -wor d α ′ ∈ L ( F B n ) with O cc ( ρ ′ ) = Occ ( ρ ) and I nf ( ρ ′ ) = I nf ( ρ ) . Pr o of. Supp ose that C A = (Σ n , ˆ S , ˆ I , ˆ ∆ , Acc ) is an ω -automaton with less than L states and ρ = ρ (0) ρ (1) · · · ∈ ˆ S ω is a run of C A o v er α n . Let k 0 , k 1 , . . . b e a n um b er sequene su h that k 0 = 0 , k i +1 − k i = l eng th ( w i ) for all i ≥ 0 . So the k i 's mark the p ositions where the w i 's onatenate. Therefore ρ ( k i ) w i − → ρ ( k i +1 ) for all i ≥ 0 . Dene for ea h 0 ≤ i < L the nonempt y set: ˆ Q i = { ˆ q ∈ ˆ S | ρ ( k j L + i ) = ˆ q for innitely man y j ∈ N } . Sine C A has less than L states, there exists some state ˆ q in ˆ Q i ∩ ˆ Q j for some i 6 = j, 0 ≤ i, j < L . In partiular one has, b y denition, f i 6 = f j . W.l.o.g. there is a q ∈ S ′ n with f i ( q ) > f j ( q ) . By denitions of ˆ Q i and O cc ( ρ ) , there is a t 1 ∈ N suien tly large su h that LO WER BOUNDS F OR COMPLEMENT A TION OF ω -A UTOMA T A 11 ρ ( k t 1 L + i ) = ˆ q , ev ery state in O cc ( ρ ) o urs in ρ [0 , k t 1 L + i ] , and that ρ ( t ′ ) ∈ I nf ( ρ ) for all t ′ > k t 1 L + i . By denitions of I nf ( ρ ) and ˆ Q j , there is a suien tly large t 2 > t 1 su h that ρ ( k t 2 L + j ) = ˆ q and ev ery state in I nf ( ρ ) o urs in ρ [ k t 1 L + i , k t 2 L + j ] . Let u = w 0 . . . w t 1 L + i − 1 and v = w t 1 L + i . . . w t 2 L + j − 1 . Finally let α ′ b e uv ω . Let q I ∈ S ′ n b e su h that f 0 ( q I ) = 2 m − 1 ≥ f i ( q ) = f t 1 L + i ( q ) . By Lemma 4.4 , q I u − → q . Similarly , sine f t 1 L + i ( q ) = f i ( q ) > f j ( q ) = f t 2 L + j ( q ) , b y Lemma 4.4 w e ha v e q v − → F n q . T ogether w e ha v e q I u − → q v − → F n q v − → F n q . . . and α ′ is aepted b y F B n . Finally , note that ρ ′ = ρ [0 , k t 1 L + i ] · ( ρ [ k t 1 L + i + 1 , k t 2 L + j ]) ω is a run o v er α ′ , and w e ha v e guaran teed that O cc ( ρ ′ ) = O cc ( ρ ) and I nf ( ρ ′ ) = I nf ( ρ ) as required . Theorem 4.7. F or every n > 1 , L ( n ) ≤ C NBW ( F B n ) ≤ C NBW ( n ) , wher e L ( n ) = Θ((0 . 76 n ) n ) . Pr o of. By Lemma 4.6, ev ery NBW that omplemen ts F B n m ust ha v e at least L ( n ) states, otherwise b oth α n and α ′ n w ould b e aepted b y F B n , leading to on tradition. By a n umer- ial analysis of L ( n ) v ery similar to the one in [FKV06℄, w e ha v e that L ( n ) = Θ((0 . 76 n ) n ) . F or ompleteness, w e presen t the detail of the analysis in app endix. 4.3. Alphab et. F ollo wing the pro of of Theorem 4.7 , one onstruts full NBW s witnessing the lo w er b ound o v er a v ery large alphab et, whi h w e rarely onsider in pratie. In this subsetion, w e sho w that b y using alphab et substitutions lik e in the pro of of Theorem 3.1 , the NBW s witnessing the lo w er b ound an b e also o v er a xed alphab et. W e sa y t w o w ords u and v from Σ ∗ n are equiv alen t with resp et to F B n , or simply u ≈ v , if for all p, q ∈ S ′ n : (i) p u − → q i p v − → q , and, (ii) p u − → F n q i p v − → F n q . Then if one replaes ea h letter in v olv ed in the lo w er b ound pro of b y an equiv alen t w ord o v er some alphab et Γ , one sho ws that F B n ↾ Γ also witnesses the same L ( n ) lo w er b ound. Lemma 4.8. Ther e is an alphab et Γ of size 7 suh that for e ah p air h f , g i of Q ( m ) -r ankings for F B n , ther e is a wor d in Γ ∗ e quivalent to w f ,g . Pr o of. Let Γ b e the alphab et on taining the follo wing 7 letters: • r otate = {h s i +1 , s i i | 0 ≤ i < n − 2 } ∪ {h s 0 , s n − 2 i , h s f , s f i} , • cl ear 0 = I d ( S n \{ s 0 } ) , • s w ap 01 = ( I d ( S ′ n ) ∪ {h s 0 , s 1 i , h s 1 , s 0 i} ) \{h s 0 , s 0 i , h s 1 , s 1 i} , • copy 01 = I d ( S ′ n ) ∪ {h s 1 , s 0 i} , • 0 toF = I d ( S n ) ∪ {h s 0 , s f i} , • F to 0 = I d ( S n ) ∪ {h s f , s 0 i} , • cl ear F = I d ( S ′ n ) . Only three t yp es of letters are relev an t in the pro of of Theorem 4.7 : T toF ( T ) , F toT ( T ) and c ( f , g ) . F or ea h T ⊆ S ′ n , one an v erify that: • T toF ( T ) ≈ clear F · Y s i ∈ T ( r otate i · 0 toF · r otate n − 1 − i ) . • F toT ( T ) ≈ Y s i ∈ T ( r otate i · F to 0 · r otate n − 1 − i ) · clear F . 12 Q. Y AN As to c ( f , g ) , the task is a bit more ompliated, and let us view it in a dieren t w a y . F or a w ord w , dene set r j = { i | s i w − → s j , 0 ≤ i < n − 1 } for ev ery 0 ≤ j < n − 1 . Clearly for t w o w ords u and v , the follo wing are equiv alen t: • p u − → q i p v − → q for all p, q ∈ S ′ n . • r j ( u ) = r j ( v ) for all 0 ≤ j < n − 1 . So it is suien t to nd for ea h c ( f , g ) a w ord w o v er { r otate, clear 0 , sw ap 01 , copy 01 } su h that r j ( w ) = r j ( c ( f , g )) for all 0 ≤ j < n − 1 . App ending ea h letter a to the end of a w ord w  hanges the on ten t of the r i ( w ) 's. Consider these three t yp es of w ords in Γ ∗ : (1) sw ap i,j =            r otate i · swap 01 · r otate n − 1 − i if i + 1 = j ( sw ap i,i +1 · swap i +1 ,i +2 · · · · · s w ap j − 1 ,j ) · ( swap j − 2 ,j − 1 · sw ap j − 3 ,j − 2 · · · · · s w ap i,i +1 ) if i + 1 < j sw ap j,i if i > j the empt y w ord if i = j . (2) copy i,j =  sw ap 01 · copy 01 · sw ap 01 if i = 1 and j = 0 sw ap 0 ,i · sw ap 1 ,j · copy 01 · sw ap 1 ,j · swap 0 ,i otherwise . (3) clear i = swap 0 ,i · clear 0 · sw ap 0 ,i One an v erify that app ending a sw ap i,j to w ex hanges the on ten t of r i ( w ) and r j ( w ) , app ending a copy i,j sets r i ( w ) to b e r i ( w ) ∪ r j ( w ) , and app ending a clear i empties r i ( w ) . Ob viously these three op erations allo w one to rea h arbitrary ( r i ( w )) 0 ≤ i 1 , ther e exists an NBW B n with n states over a seven letters alphab et suh that L ( n ) ≤ C NBW ( B n ) . 4.4. Other T ransformations. Surprisingly , our lo w er b ound on Bü hi omplemen tation extends to almost ev ery omplemen tation or determinization transformation of nondeter- ministi ω -automata, via a redution making use of Lemma 4.6. Theorem 4.10. F or e ah n > 1 and e ah  ommon typ e T 1 of nondeterministi ω -automata, ther e exists a T 1 automaton A n with n states over a xe d alphab et suh that: (i): F or e ah  ommon typ e T 2 , every T 2 automaton that  omplements L ( A n ) has at le ast L ( n ) states. (ii): F or e ah  ommon typ e T 2 that is not Bühi nor gener alize d Bühi 5 , every deter- ministi T 2 automaton that a  epts L ( A n ) has at le ast L ( n ) states. Pr o of. F or ea h ommon t yp e T 1 , b y F at 2.1 , there is a T 1 automaton A n equiv alen t to NBW F B n with also n states [Löd99 ℄. (i) Supp ose that an automaton C A of a ommon t yp e aepts L C ( A n )= L C ( F B n ) . Sine aeptane of ω -automata of a ommon t yp e only dep ends on the I nf set of a run, the laim an b e obtained b y applying Lemma 4.6. (ii) If some deterministi T 2 automaton with less than L ( n ) states aepts L ( A n ) , and T 2 is not Bü hi or generalized Bü hi, then b y F at 2.1 there is a deterministi ω -automaton of 5 Deterministi Bü hi or generalized Bü hi automata are stritly w eak er in expressiv e p o w er than the other ommon t yp es of ω -automata. LO WER BOUNDS F OR COMPLEMENT A TION OF ω -A UTOMA T A 13 a ommon t yp e (not neessarily T 2 ) omplemen ting L ( A n ) with also less than L ( n ) states [Löd99 ℄, on trary to (i). Finally , the alphab et of A n an b e xed lik e in the pro of of Theorem 4.9 . F or the transformations in v olv ed in this theorem, less than half already had non trivial lo w er b ounds lik e n ! b y Mi hel's pro of or the bun h of pro ofs b y Lö ding [Löd99 ℄, while the others only ha v e trivial or w eak 2 Ω( n ) lo w er b ounds. These b ounds are summarized in Setion 6 . 5. Complement a tion of Generalized Bühi A utoma t a W e turn no w to NGBW omplemen tation. F or NGBW s, state omplexit y is prefer- ably measured in terms of b oth the n um b er of states n and index k , where index measures the size of the aeptane ondition. By applying full automata, doing a hard ase anal- ysis for the onstrution in [KV05b ℄ based on GC-ranking, and using a generalization of Mi hel's te hnique, w e pro v e an (Ω( nk )) n lo w er b ound, mat hing with the ( O ( nk )) n b ound in [KV05b℄. This lo w er b ound also extends to the omplemen tation of Streett automata and the determinization of generalized Bü hi automata in to Rabin automata. 5.1. Standard F ull Generalized Bü hi Automata F B n,k . W e rst dene full NGBW automata whi h w e will sho w to witness our desired lo w er b ound. W e sa y a generalized Bü hi aeptane ondition Acc = { F 1 , F 2 , . . . , F k } is minimal , if no F i , F j pair with i 6 = j satises that F i ⊆ F j . Note that if su h a pair exists, F j an b e remo v ed from Acc without altering the ω -language dened. So w e will only onsider minimal aeptane onditions. By the Sp erner's theorem in om binatoris [Lub66 ℄, if Acc is minimal, then k ≤  n ⌊ n/ 2 ⌋  . Denition 5.1. F or n > 1 and 1 < k ≤  n − 1 ⌊ ( n − 1) / 2 ⌋  , the standar d ful l NGBW F B n,k = (Σ n , S n , I n , ∆ n , Acc n,k ) is an NGBW with | S n | = n , I n = S n and a minimal aeptane ondition Acc n,k . Let s nf b e one of its state. W e denote S n \{ s nf } as S ′ n . Acc n,k is dened as an arbitrary xed set { F 1 , F 2 , . . . , F k } ⊆ P ( S ′ n ) su h that: (i) | F i | = ⌊ ( n − 1) / 2 ⌋ for ea h F i ∈ Acc n,k . (ii) F or ea h q ∈ S ′ n , the n um b er of F i 's in Acc n,k that do not on tain q is at least ⌊ k / 2 ⌋ . W e m ust sho w that there is really su h a minimal Acc n,k satisfying (i) and (ii). First let Acc n,k b e a olletion of arbitrary k distint subsets of S ′ n of ⌊ ( n − 1) / 2 ⌋ states and th us (i) is satised. Dene χ q for ea h q ∈ S ′ n as the n um b er of F i 's in Acc n,k that on tain q . By double oun ting, P q ∈ S ′ n χ q = k P i =1 | F i | . So if | χ p − χ q | ≤ 1 for all p, q ∈ S ′ n , then for all q ∈ S ′ n , χ q ≤ ⌈ k ⌊ ( n − 1) / 2 ⌋ n − 1 ⌉ ≤ ⌈ k / 2 ⌉ and (ii) is also satised. Supp ose χ p − χ q > 1 for some p, q ∈ S ′ n . A little though t sho ws that there is an F i ∈ Acc n,k su h that p ∈ F i and ( F i \{ p } ) ∪ { q } / ∈ Acc n,k . Replae F i in Acc n,k b y ( F i \{ p } ) ∪ { q } and w e mak e | χ p − χ q | stritly smaller. Rep eat this till | χ p − χ q | ≤ 1 for all p, q ∈ S ′ n . Then ondition (ii) is also satised. 14 Q. Y AN 5.2. A Generalization of Mi hel's T e hnique. W e generalize the te hnique used in Mi hel's pro of for Bü hi omplemen tation [Mi88℄ so that a tigh ter analysis of NGBW omplemen tation b eomes p ossible. Denition 5.2. A gener alize d  o-Bühi se gment (GC-segmen t for short) w of an NGBW B is a w ord su h that w ω / ∈ L ( B ) . T w o GC-segmen ts w 1 , w 2 of B  onit if all ω -w ords in the form w k 0 1 ( w k 1 1 w k 2 2 ) ω , k i > 0 are in L ( B ) . A set W of GC-segmen ts of B is a  onit set for B if ev ery t w o distint GC-segmen ts in W onit. Lemma 5.3. If W is a  onit set for NGBW B , then C NGBW ( B ) ≥ | W | . Pr o of. Supp ose that some NGBW C B = (Σ , ˆ S , ˆ I , ˆ ∆ , ˆ F ) omplemen ts B , then for ea h GC- segmen t w of B in W , C B aepts w ω . F or ev ery t w o distint GC-segmen ts w 1 , w 2 ∈ W , let l 1 = l eng th ( w 1 ) , l 2 = l eng th ( w 2 ) , and let ρ (0) ρ (1) . . . and ρ ′ (0) ρ ′ (1) . . . b e C B 's t w o suessful runs o v er w ω 1 and w ω 2 resp etiv ely . Dene ˆ Q 1 = { ˆ q ∈ ˆ S | ρ ( i · l 1 ) = ˆ q for innitely man y i ∈ N } and ˆ Q 2 = { ˆ q ∈ ˆ S | ρ ′ ( i · l 2 ) = ˆ q for innitely man y i ∈ N } . Clearly ˆ Q 1 and ˆ Q 2 are nonempt y . It sues to sho w that ˆ Q 1 ∩ ˆ Q 2 = ∅ , sine it implies that the n um b er of states of C B is no less than the n um b er of GC-segmen ts in W . Supp ose b y on tradition that some ˆ q is in ˆ Q 1 ∩ ˆ Q 2 . By denition of ˆ Q 1 , there is a suien tly large k 0 > 0 su h that ρ ( k 0 l 1 ) = ˆ q and for ea h i ≥ k 0 l 1 , ρ ( i ) ∈ I n f ( ρ ) . So ρ [0 , k 0 l 1 ] is a nite run o v er w k 0 1 from some initial state ˆ q I of C B to ˆ q , i.e., ˆ q I w k 0 1 − → ˆ q . By denitions of ˆ Q 1 and I nf ( ρ ) , there is a suien tly large k 1 > 0 su h that ρ (( k 0 + k 1 ) l 1 ) = ˆ q and in addition ρ [ k · l 1 , ( k 0 + k 1 ) l 1 ] is a nite run from ˆ q to ˆ q o v er w k 1 1 whi h visits ev ery state in I nf ( ρ ) . Similarly w e ha v e that for some k ′ 0 and k 2 > 0 , ρ ′ [ k ′ 0 l 2 , ( k ′ 0 + k 2 ) l 2 ] is a nite run from ˆ q to ˆ q o v er w k 2 2 whi h visits exatly ev ery state in I nf ( ρ ′ ) . W e onstrut a new run as follo ws: ρ new = ρ [0 , k 0 l 1 ] ·  ρ [ k 0 l 1 + 1 , ( k 0 + k 1 ) l 1 ] · ρ ′ [ k ′ 0 l 2 + 1 , ( k ′ 0 + k 2 ) l 2 ]  ω , whi h is a run o v er α = w k 0 1 ( w k 1 1 w k 2 2 ) ω with I nf ( ρ new ) = I nf ( ρ ) ∪ I n f ( ρ ′ ) . As ρ and ρ ′ are b oth suessful, ρ new is also suessful b y denition of generalized Bü hi automata. So α is aepted b y C B . Ho w ev er, as w 1 and w 2 onit, α is aepted b y B to o, on tradition. Corollary 5.4. If W is a  onit set for NGBW B , then every NSW (nondeterministi Str e ett automaton) that  omplements B has at le ast | W | states. Pr o of. Streett automata also satisfy that if ρ and ρ ′ are b oth suessful runs, then ev ery run ρ new satisfying I nf ( ρ new ) = I nf ( ρ ) ∪ I n f ( ρ ′ ) is also suessful. So the same pro of as of Lemma 5.3 applies here. 5.3. A Conit Set for F B n,k . It remains to dene a large onit set for F B n,k . The follo wing onept of pseudo generalized o-Bü hi lev el ranking is adapted from the onept of generalized o-Bü hi lev el ranking in the NGBW omplemen tation onstrution in [KV05b ℄. Denition 5.5. A pseudo gener alize d  o-Bühi level r anking (PGCL-ranking for short) for F B n,k is a pair h f , g i su h that f is a bijetion from S ′ n to { 1 , . . . , n − 1 } and g is a funtion from S ′ n to { 1 , 2 , . . . , k } su h that ea h q ∈ S ′ n is not on tained in F g ( q ) . LO WER BOUNDS F OR COMPLEMENT A TION OF ω -A UTOMA T A 15 By denition of F B n,k , there are at least ⌊ k / 2 ⌋  hoies for the v alue of g ( q ) for ea h q ∈ S ′ n . So there are at least ( n − 1)! × ( ⌊ k / 2 ⌋ ) n − 1 man y dieren t PGCL-rankings, whi h is (Ω( nk )) n b y Stirling's form ula. Let G b e a set of state sets. In the follo wing, w e use notations in the form p w − → G , ! B q to denote that there is a nite run o v er w from p to q su h that the run visits ev ery state set F in G , but it do es not visit B . Either G or B will b e omitted if is empt y . In the follo wing, w e set F = { F 1 , . . . , F k } . Lemma 5.6. F or e ah PGCL-r anking h f , g i , ther e exists a wor d seg f ,g with the pr op erties that for al l p, q ∈ S ′ n : (i): If p = q , i.e., f ( p ) = f ( q ) , then ther e is a unique nite run of F B n,k over seg f ,g fr om p to q , and it is in the form p seg f ,g − − − − − − − − − → F \ F g ( p ) , ! F g ( p ) q . (ii): If f ( p ) > f ( q ) , then ther e is a unique nite run of F B n,k over seg f ,g fr om p to q , and it is in the form p seg f ,g − → F q . (iii): If f ( p ) < f ( q ) , then ther e is no nite run of F B n,k fr om p to q over seg f ,g . Pr o of. F or notational on v eniene, w e use notation lik e  ⊕ p 1 → p 2 , ⊖ p 3 → p 4 , ⊖ p 5 → p 5  to denote letter {h q , q i | q ∈ S ′ n } ∪ {h p 1 , p 2 i}\{h p 3 , p 4 i , h p 5 , p 5 i} . W e also dene a  hoie funtion c ( i, p ) for ea h i ∈ { 1 , . . . , k } and state p ∈ S ′ n with g ( p ) 6 = i su h that c ( i, p ) equals to some arbitrary xed elemen t in F i \ F g ( p ) . F or ea h r ∈ { 1 , . . . , n − 1 } , let p ∈ S ′ n b e su h that f ( p ) = r , and dene: u r = Y i 6 = g ( p ) , 1 ≤ i ≤ k s = c ( i,p )  ⊕ p → s , ⊖ p → p , ⊕ s → s nf , ⊖ s → s   ⊕ s → p , ⊖ p → p , ⊕ s nf → s , ⊖ s → s  . (Reall that Π U means the onatenation of all w ords in U in lexiographial order.) Then for ea h q ∈ S ′ n , there is a unique nite run o v er u r from q to q , and it is in the form q u r − → F \ F g ( p ) , ! F g ( p ) q if p = q , or q u r − → ! F g ( p ) q otherwise. F or ea h r = { 2 , 3 , . . . , n − 1 } , let p, q , s ∈ S ′ n b e su h that f ( p ) = r , f ( q ) = r − 1 and s b e an arbitrary state in F g ( p ) . Dene: v r =  ⊕ p → s , ⊖ s → s , ⊕ s → s nf   ⊕ s → q , ⊖ s → s , ⊕ s nf → s  . Then there is a unique nite run o v er v r from p to q , and it is in the form p v r − → F g ( p ) q . Also for ev ery q ′ ∈ S ′ n , there is a unique nite run o v er v r from q ′ to q ′ , and it is in the form q ′ v r − → ! F g ( p ) q ′ . Finally let seg f ,g b e u n − 1 v n − 1 u n − 2 v n − 2 . . . v 2 u 1 . T o see that seg f ,g satises the required prop erties, rst note that for all p ∈ S ′ n , p u r − → ! F g ( p ) p and p v r − → ! F g ( p ) p . F or prop ert y (i), for ev ery p ∈ S ′ n with f ( p ) = r , there exists a unique nite run o v er seg f ,g , and it is in the form: p u n − 1 v n − 1 ...u r +1 v r +1 − − − − − − − − − − − − − → ! F g ( p ) p u r − − − − − − − − − → F \ F g ( p ) , ! F g ( p ) p v r u r − 1 ...v 2 u 1 − − − − − − − − → ! F g ( p ) p, 16 Q. Y AN that is, p seg f ,g − − − − − − − − − → F \ F g ( p ) , ! F g ( p ) p as required. F or prop ert y (ii), for ev ery p, q ∈ S ′ n with f ( p ) = r 1 > r 2 = f ( q ) , let s r ∈ S ′ n b e su h that f ( s r ) = r for ea h r 1 > r > r 2 . There is a unique nite run o v er seg f ,g , and it is in the form: p u n − 1 v n − 1 ...u r 1 +1 v r 1 +1 − − − − − − − − − − − − − − → p u r 1 − − − − − − − − − → F \ F g ( p ) , ! F g ( p ) p v r 1 − − − → F g ( p ) s r 1 − 1 u r 1 − 1 v r 1 − 1 − − − − − − − → s r 1 − 2 . . . s r 2 +1 u r 2 +1 v r 2 +1 − − − − − − − → q u r 2 ...v 2 u 1 − − − − − − → q , that is, p seg f ,g − → F q as required. Prop ert y (iii) is easy to v erify . Remark 5.7. F rom the pro of of the ab o v e lemma, it follo ws that an alphab et of size p olynomial in n is suien t to desrib e { seg f ,g | f , g are PGCL-rankings } . Lemma 5.8. F or e ah PGCL-r anking h f , g i for F B n,k , wor d seg f ,g is a GC-se gment of F B n,k . Pr o of. Let l = leng t h ( seg f ,g ) , and let ρ = ρ (0) ρ (1) . . . b e a run of F B n,k o v er seg ω f ,g in the form ρ (0) seg f ,g − → ρ ( l ) seg f ,g − → ρ (2 l ) . . . . Note that b y the onstrution of seg f ,g , ρ ( i · l ) ∈ S ′ n and f ( ρ ( i · l )) is dened for all i ≥ 0 . Then b y prop ert y (iii), f ( ρ (0)) ≥ f ( ρ ( l )) ≥ f ( ρ (2 l )) ≥ . . . and then for some t ∈ N , f ( ρ ( t ′ · l )) = f ( ρ ( t · l )) for all t ′ > t , that is ρ ( t ′ · l ) = ρ ( t · l ) for all t ′ > t sine f is a bijetion. Let j = g ( ρ ( t · l )) . By prop ert y (i), F j is not visited in ρ [ t ′ · l, ( t ′ + 1) · l ] for all t ′ ≥ t . So I nf ( ρ ) ∩ F j = ∅ and hene seg ω f ,g is not aepted b y F B n,k . Lemma 5.9. The set W = { seg f ,g | h f , g i is a PGCL-r anking for F B n,k } is a  onit set of size (Ω( nk )) n for F B n,k . Pr o of. Supp ose h f 1 , g 1 i and h f 2 , g 2 i are t w o distint PGCL-rankings. Let w 1 = seg f 1 ,g 1 and w 2 = seg f 2 ,g 2 . There are t w o ases. Case: I: f 1 and f 2 are t w o dieren t bijetions. So there exist p, q ∈ S ′ n su h that f 1 ( p ) > f 1 ( q ) and f 2 ( p ) < f 2 ( q ) . By prop ert y (i), p w 1 − → p , q w 2 − → q and so p w m − 1 1 − → p, q w m − 1 2 − → q for all m > 0 . By prop ert y (ii), p w 1 − → F q and q w 2 − → F p . So for all m > 0 , p w m 1 − → F q and q w m 2 − → F p . No w for ev ery ω -w ord α in the form w k 0 1 ( w k 1 1 w k 2 2 ) ω , k i > 0 , w e onstrut a suessful run o v er α as p w k 0 1 − → p w k 1 1 − → F q w k 2 2 − → F p w k 1 1 − → F q w k 2 2 − → F p . . . . So α is aepted b y F B n,k and w 1 onits with w 2 . Case: I I: f 1 = f 2 but g 1 6 = g 2 . Let p ∈ S ′ n b e su h that g 1 ( p ) 6 = g 2 ( p ) . By prop ert y (i), p w 1 − − − − − − − − − − → F \ F g 1 ( p ) , ! F g 1 ( p ) p and p w 2 − − − − − − − − − − → F \ F g 2 ( p ) , ! F g 2 ( p ) p . As g 1 ( p ) 6 = g 2 ( p ) , p w k 1 1 w k 2 2 − − − − − → F p for ev ery k 1 , k 2 > 0 . No w for ev ery ω -w ord α in the form w k 0 1 ( w k 1 1 w k 2 2 ) ω , k i > 0 , w e onstrut a suessful run o v er α as p w k 0 1 − → p w k 1 1 w k 2 2 − − − − − → F p w k 1 1 w k 2 2 − − − − − → F p . . . . So α is aepted b y F B n,k and w 1 onits with w 2 . Finally , the size of W is just the n um b er of dieren t PGCL-rankings for F B n,k , whi h is (Ω( nk )) n . LO WER BOUNDS F OR COMPLEMENT A TION OF ω -A UTOMA T A 17 5.4. Results. Theorem 5.10. F or n > 1 and 1 < k ≤  n − 1 ⌊ ( n − 1) / 2 ⌋  , C NGBW ( n, k ) = (Ω( nk )) n . Pr o of. The theorem follo ws from Lemma 5.3 and Lemma 5.9 diretly . This mat hes neatly 6 with the ( O ( nk )) n onstrution in [KV05b ℄, and th us settles the state omplexit y of NGBW omplemen tation. Lik e Mi hel's result, this lo w er b ound an b e extended to NSW omplemen tation and the determinization of NGBW in to DR W (state omplexit y denoted b y D NGBW → DR W ( n, k ) ): Theorem 5.11. F or al l n > 1 and 1 < k ≤  n − 1 ⌊ ( n − 1) / 2 ⌋  , C NSW ( n, k ) = (Ω( nk )) n and D NGBW → DR W ( n, k ) = (Ω( nk )) n . Pr o of. By F at 2.1 there is an NSW S n,k equiv alen t to ea h F B n,k with the same n um b er of states and the same index. By Corollary 5.4 and Lemma 5.9 , ev ery NSW that omplemen ts F B n,k has (Ω( nk )) n states. So C NSW ( S n,k ) = (Ω( nk )) n and C NSW ( n, k ) = (Ω( nk )) n . Supp ose b y on tradition that R is a DR W with less than | W | states that aepts L ( F B n,k ) , then b y F at 2.1 there is a DSW S omplemen ting F B n,k with the same n um b er of states as R , on trary to Corollary 5.4. So D NGBW → DR W ( n, k ) = (Ω( nk )) n . Remark 5.12. F or the ab o v e lo w er b ound, b y Remark 5.7, the alphab et in v olv ed in the pro of is of a size p olynomial in n . It seems diult to x a onstan t alphab et, but w e onjeture this to b e p ossible if w e aim at a w eak er b ound lik e 2 Ω( n log nk ) . 6. Summar y In the follo wing table, w e briey summarize our lo w er b ounds. Here An y means an y ommon t yp e of nondeterministi ω -automata (and the t w o An y's an b e dieren t). o. means omplemen tation and det. means determinization. L.B. /U.B. stands for lo w er/upp er b ound. W eak 2 Ω( n ) lo w er b ounds are onsidered trivial. # T ransformation Previous L.B. Our L.B. Kno wn U.B. 1 NBW o. − → NBW Ω((0 . 36 n ) n ) [Mi88 ℄ Ω((0 . 76 n ) n ) O ((0 . 97 n ) n ) [FKV06℄ 2 An y o. or det. − → An y trivial or n ! [Löd99 ℄ 2 Ω( n log n ) - 3 NBW det. − → DMW trivial 7 2 Ω( n log n ) 2 O ( n log n ) [Saf89℄ 4 NR W o. − → NR W trivial 8 2 Ω( n log n ) 2 O ( nk log n ) [KV05a ℄ 5 NGBW o. − → NGBW Ω(( n/e ) n ) [Mi88 ℄ (Ω( nk )) n ( O ( nk )) n [KV05b℄ 6 NSW o. − → NSW Ω(( n/e ) n ) [Löd99 ℄ (Ω( nk )) n 2 O ( nk log( nk )) [KV05a ℄ 7 NGBW det. − → DR W Ω(( n/e ) n ) [Löd99 ℄ (Ω( nk )) n 2 O ( nk log( nk )) [Saf89℄ In partiular, lo w er b ound #2 implies that the 2 Ω( n l og n ) blo w-up is inheren t in the omplemen tation and determinization of nondeterministi ω -automata, orresp onding to the 2 n blo w-up of nite automata. The sp eial ase #3 justies that Safra's onstrution is optimal in state omplexit y for the determinization of Bü hi automata in to Muller automata. 6 The gap hidden in the notation (Θ( nk )) n an b e at most c n for some c , while the gap hidden in the more widely used notation 2 Θ( n log nk ) an b e as large as ( nk ) n . 18 Q. Y AN W e single out this result b eause this determinization onstrution is tou hed in almost ev ery in tro dutory material on ω -automata, and its optimalit y problem w as expliitly left op en in [Löd99 ℄. F or man y of these transformations, it is still in teresting to try to narro w the omplexit y gap, and here w e disuss three of them. First, the omplexit y gap of Bü hi omplemen tation, although signian tly narro w ed, is still exp onen tial. By analyzing the dierene b et w een the lo w er and upp er b ounds, one an nd that the gap is mainly aused b y the use of the state omp onen t O in [FKV06 ℄ to main tain the states along paths that ha v e not visited an o dd v ertex sine the last time O has b een empt y . So w e should in v estigate ho w man y states are really neessary for su h a purp ose. Seond, for Streett omplemen tation, the gap is still quite large. W e feel that eorts should b e rst tak en to optimize the onstrution in [KV05a℄. Third, it is in teresting to see if an Ω( n n ) or similar lo w er b ound exists for the determinization of NBW s in to Muller or Rabin automata. Su h w ould imply that determinization is harder than omplemen tation for ω -automata, unlik e the ase of automata o v er nite w ords. Of ourse, one an also w ork on the rev erse diretion, trying to design ranking based onstrutions for determinization, whi h ould ha v e go o d omplexit y b ound as w ell as b etter appliabilit y to pratie. Finally , w e remark that the full automata te hnique has b een quite essen tial in obtaining our lo w er b ound results. It is also p ossible to extend the full automata te hnique to other kinds of automata, lik e alternating automata or tree automata. W e hop e that the full automata te hnique will stim ulate the diso v ery of new results in automata theory . A know le dgement. I thank Orna Kupferman and Moshe V ardi for the insigh tful disussion and the extremely v aluable suggestions. I thank Enshao Shen for his kind supp ort and guidane. I also thank the anon ymous referees for the detailed and useful ommen ts. Referenes [Bir93℄ J.C. Birget. P artial orders on w ords, minimal elemen ts of regular languages and state omplexit y (has online erratum). The or eti al Computer Sien e , 119(2):267291, 1993. [Bü62℄ J. R. Bü hi. On a deision metho d in restrited seond order arithmeti. In Pr o  e e dings of the In- ternational Congr ess on L o gi, Metho d, and Philosophy of Sien e , pages 112. Stanford Univ ersit y Press, 1962. [FKV06℄ E. F riedgut, O. Kupferman, and M.Y. V ardi. Bü hi omplemen tation made tigh ter. International Journal of F oundations of Computer Sien e , 17(4):851868, 2006. [HK02℄ M. Holzer and M. Kutrib. State omplexit y of basi op erations on nondeterministi nite automata. In Pr o  e e dings of 7th International Confer en e on Implementation and Appli ation of A utomata , v olume 2608 of L e tur e Notes in Computer Sien e , pages 148157, 2002. [Jir05℄ G. Jirásk o vá. State omplexit y of some op erations on binary regular languages. The or eti al Com- puter Sien e , 330(2):287298, 2005. [Kla91℄ N. Klarlund. Progress measures for omplemen tation of omega-automata with appliations to temp oral logi. In Pr o  e e dings of 32th IEEE Symp osium on F oundations of Computer Sien e , pages 358367, 1991. [Kur94℄ R.P . Kurshan. Computer-A ide d V eri ation of Co or dinating Pr o  esses: The A utomata-The or eti Appr o ah . Prineton Univ. Press, 1994. [KV01℄ O. Kupferman and M.Y. V ardi. W eak alternating automata are not that w eak. A CM T r ansations on Computational L o gi , 2(3):408429, 2001. [KV05a℄ O. Kupferman and M.Y. V ardi. Complemen tation onstrutions for nondeterministi automata on innite w ords. In Pr o  e e dings of 11th International Confer en e on T o ols and A lgorithms for the Constrution and A nalysis of Systems , v olume 3440 of L e tur e Notes in Computer Sien e , pages 206221, 2005. LO WER BOUNDS F OR COMPLEMENT A TION OF ω -A UTOMA T A 19 [KV05b℄ O. Kupferman and M.Y. V ardi. F rom omplemen tation to ertiation. The or eti al Computer Sien e , 345(1):83100, 2005. [Löd99℄ C. Lö ding. Optimal b ounds for transformations of omega-automata. In Pr o  e e dings of 19th Con- fer en e on F oundations of Softwar e T e hnolo gy and The or eti al Computer Sien e , v olume 1738 of L e tur e Notes in Computer Sien e , pages 97109, 1999. [Lub66℄ D. Lub ell. A short pro of of Sp erner's lemma. Journal of Combinatorial The ory , 1:299, 1966. [Mi88℄ M. Mi hel. Complemen tation is more diult with automata on innite w ords. CNET, Paris, 1988. [RS59℄ M.O. Rabin and D. Sott. Finite automata and their deision problems. IBM Journal of R ese ar h and Development , 3:114125, 1959. [Saf88℄ S. Safra. On the omplexit y of ω -automata. In Pr o  e e dings of 29th IEEE Symp osium on F ounda- tions of Computer Sien e , pages 319327, 1988. [Saf89℄ S. Safra. Complexity of automata on innite obje ts . PhD thesis, W eizmann Institute of Siene, 1989. [SS78℄ W.J. Sak o da and M. Sipser. Nondeterminism and the size of t w o w a y nite automata. In Pr o  e e d- ings of 10th A CM Symp osium on The ory of Computing , pages 275286, 1978. [SVW85℄ A.P . Sistla, M.Y. V ardi, and P . W olp er. The omplemen tation problem for Bü hi automata with appliations to temp oral logi (extended abstrat). In Pr o  e e dings of 12th International Col lo quium on A utomata, L anguages and Pr o gr amming , v olume 194 of L e tur e Notes in Computer Sien e , pages 465474, 1985. [T em93℄ N.M. T emme. Asymptoti estimates of stirling n um b ers. Studies in Applie d Mathematis , 89:233 243, 1993. [Tho90℄ W. Thomas. Automata on innite ob jets. In Jan v an Leeu w en, editor, Handb o ok of The or eti al Computer Sien e , v olume B, F ormal mo dels and seman tis, pages 133191. Elsevier, 1990. [V ar07℄ M.Y. V ardi. The Bü hi omplemen tation saga. In Pr o  e e dings of 23r d International Symp osium on The or eti al Asp e ts of Computer Sien e , v olume 4393 of L e tur e Notes in Computer Sien e , pages 1222, 2007. [VW94℄ M.Y. V ardi and P . W olp er. Reasoning ab out innite omputations. Information and Computation , 115(1):137, 1994. [Y u05℄ S. Y u. State omplexit y: Reen t results and op en problems. F undamenta Informati ae , 64:471480, 2005. Appendix A. Numerial Anal ysis of L ( n ) In this setion, w e pro v e that L ( n ) = Θ((0 . 76 n ) n ) . The analysis is v ery similar to the one in [FKV06℄, but w e still presen t it here for ompleteness. In the follo wing, w e write f ( n ) ≈ g ( n ) if t w o funtions dier b y only a p olynomial fator in n . F or example, b y Stirling's form ula, n ! ≈ ( n/e ) n . Let T ( n, m ) denote the n um b er of funtions from { 1 . . . n } on to { 1 . . . m } . The follo wing estimate of T ( n, m ) is impliit in T emme [ T em93 ℄: Lemma A.1. [T em93 ℄ F or 0 < β < 1 , let x b e the p ositive r e al numb er solving β x = 1 − e − x , and let a = − ln x + β ln( e x − 1) − (1 − β ) + (1 − β ) ln(1 /β − 1) . Then T ( n, ⌊ β n ⌋ ) ≈ ( M [ β ] n ) n , wher e M [ β ] = e a − β  β 1 − β  1 − β . T o pro v e a lo w er b ound for L ( n ) , w e rst express L ( n, m ) in the follo wing form: Lemma A.2. L ( n, m ) = P n − 1 t = m  n − 1 t  T ( t, m ) m n − 1 − t . Pr o of. T o oun t the n um b er of dieren t Q ( m ) -ranking, w e x t , whi h denotes the n um b er of states that ha v e o dd ranks. Then there are  n − 1 t  w a ys to  ho ose whi h t states ha v e o dd ranks, and there are T ( t, m ) w a ys to assign these t states the m dieren t o dd ranks. 20 Q. Y AN Moreo v er, for ea h of the other n − 1 − t states in S ′ n , there are m w a ys to  ho ose whi h ev en rank it is assigned. Theorem A.3. L ( n ) = Ω (( c l n ) n ) , wher e c l = 0 . 76 . Pr o of. By the previous lemma, L ( n ) = max m =1 ...n − 1 P n − 1 t = m  n − 1 t  T ( t, m ) m n − 1 − t . Sine w e do not are ab out p olynomial fators, P n − 1 t = m an b e replaed b y max t = m...n − 1 , and w e an replae m ! b y ( m/e ) m and  n − 1 t  b y n n t t ( n − t ) n − t as w ell. Also let γ = m /n and β = t/n , then w e ha v e: L ( n ) ≈ max 0 <γ ≤ β < 1 n n ( β n ) − β n ((1 − β ) n ) − (1 − β ) n · ( M [ γ /β ] β n ) β n · ( γ n ) n − 1 − β n ≈ max 0 <γ ≤ β < 1 ( h ( β , γ ) n ) n , where h ( β , γ ) = (1 − β ) β − 1 ( M [ γ /β ]) β γ 1 − β . Computed b y the Mathematia soft w are, h ( β , γ ) = 0 . 7645 when β = 0 . 7236 , γ = 0 . 574 4 . So (0 . 76 n ) n is an asymptoti lo w er b ound for L ( n ) . 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