How Deterministic are Good-For-Games Automata?

How Deterministic re Good-For-Gmes Automt? Udi Boker 1, Orn Kupfermn 2, nd Mich l Skrzypczk 3 1 Interdisciplinry Center, Herzliy, Isrel 2 The Herew University, Isrel 3 University of Wrsw, Polnd Astrct In good for gmes (GFG) utomt, it is possile to resolve nondeterminism in wy tht only depends on the pst nd still ccepts ll the words in the lnguge. The motivtion for GFG utomt comes from their dequcy for gmes nd synthesis, wherein generl nondeterminism is inpproprite. We continue the ongoing effort of studying the power of nondeterminism in GFG utomt. Initil indictions hve hinted tht every GFG utomton emodies deterministic one. Tody we know tht this is not the cse, nd in fct GFG utomt my e exponentilly more succinct thn deterministic ones. We focus on the typeness question, nmely the question of whether GFG utomton with certin cceptnce condition hs n equivlent GFG utomton with weker cceptnce condition on the sme structure. Beyond the theoreticl interest in studying typeness, its existence implies efficient trnsltions mong different cceptnce conditions. This prcticl issue is of specil interest in the context of gmes, where the Büchi nd co-büchi conditions dmit memoryless strtegies for oth plyers. Typeness is known to hold for deterministic utomt nd not to hold for generl nondeterministic utomt. We show tht GFG utomt enjoy the enefits of typeness, similrly to the cse of deterministic utomt. In prticulr, when Rin or Streett GFG utomt hve equivlent Büchi or co-büchi GFG utomt, respectively, then such equivlent utomt cn e defined on sustructure of the originl utomt. Using our typeness results, we further study the plce of GFG utomt in etween deterministic nd nondeterministic ones. Specificlly, considering utomt complementtion, we show tht GFG utomt len towrd nondeterministic ones, dmitting n exponentil stte low-up in the complementtion of Streett utomton into Rin utomton, s opposed to the constnt low-up in the deterministic cse. This reserch ws supported y the Isrel Science Foundtion, grnt no. 1373/16. This reserch hs received funding from the Europen Reserch Council under the EU s 7-th Frmework Progrmme (FP7/2007-2013) / ERC grnt greement no 278410. Supported y the Polish Ntionl Science Centre (decision UMO-2016/21/D/ST6/00491). 1

1 Introduction Nondeterminism is prime notion in theoreticl computer science. It llows computing mchine to exmine, in concurrent mnner, ll its possile runs on certin input. For utomt on finite words, nondeterminism does not increse the expressive power, yet it leds to n exponentil succinctness [15]. For utomt on infinite words, nondeterminism my increse the expressive power nd lso leds to n exponentil succinctness. For exmple, nondeterministic Büchi utomt re strictly more expressive thn their deterministic counterprt [11]. In the utomt-theoretic pproch to forml verifiction, we use utomt on infinite words in order to model systems nd their specifictions. In prticulr, temporl logic formuls re trnslted to nondeterministic word utomt [19]. In some pplictions, such s model checking, lgorithms cn proceed on the nondeterministic utomton, wheres in other pplictions, such s synthesis nd control, they cnnot. There, the dvntges of nondeterminism re lost, nd the lgorithms involve complicted determiniztion construction [16] or crotics for circumventing determiniztion [10]. Essentilly, the inherent difficulty of using nondeterminism in synthesis lies in the fct tht ech guess of the nondeterministic utomton should ccommodte ll possile futures. Some nondeterministic utomt re, however, good for gmes: in these utomt it is possile to resolve the nondeterminism in wy tht only depends on the pst while still ccepting ll the words in the lnguge. This notion, of good for gmes (GFG) utomt ws first introduced in [4]. 1 Formlly, nondeterministic utomton A over n lphet Σ is GFG if there is strtegy g tht mps ech finite word u Σ + to the trnsition to e tken fter u is red; nd following g results in ccepting ll the words in the lnguge of A. Note tht stte q of A my e rechle vi different words, nd g my suggest different trnsitions from q fter different words re red. Still, g depends only on the pst, nmely on the word red so fr. Oviously, there exist GFG utomt: deterministic ones, or nondeterministic ones tht re determinizle y pruning (DetByP); tht is, ones tht just dd trnsitions on top of deterministic utomton. In fct, the GFG utomt constructed in [4] re DetByP. 2 Our work continues series of works tht hve studied GFG utomt: their expressive power, succinctness, nd constructions for them, where the key chllenge is to understnd the power of nondeterminism in GFG utomt. Let us first survey the results known so fr. In terms of expressive power, it is shown in [8, 14] tht GFG utomt with n cceptnce condition of type γ (e.g., Büchi) re s expressive s deterministic γ utomt. 3 Thus, s fr s expressiveness is 1 GFGness is lso used in [3] in the frmework of cost functions under the nme historydeterminism. 2 As explined in [4], the fct tht the GFG utomt constructed there re DetByP does not contrdict their usefulness in prctice, s their trnsition reltion is simpler thn the one of the emodied deterministic utomton nd it cn e defined symoliclly. 3 The results in [8, 14] re given y mens of tree utomt for derived lnguges, yet, y [2], the results hold lso for GFG utomt. 2

concerned, GFG utomt ehve like deterministic ones. The picture in terms of succinctness is diverse. For utomt on finite words, GFG utomt re lwys DetByP [8, 12]. For utomt on infinite words, in prticulr Büchi nd co-büchi utomt 4, GFG utomt need not e DetByP [2]. Moreover, the est known determiniztion construction of GFG Büchi utomt is qudrtic, wheres determiniztion of GFG co-büchi utomt hs n exponentil low-up lower ound [6]. Thus, in terms of succinctness, GFG utomt on infinite words re more succinct (possily even exponentilly) thn deterministic ones. For deterministic utomt, where Büchi nd co-büchi utomt re less expressive thn Rin nd Streett ones, reserchers hve come up with the notion of n utomton eing type [5]. Consider deterministic utomton A with cceptnce condition of type γ nd ssume tht A recognizes lnguge tht cn e recognized y some deterministic utomton with n cceptnce condition of type β tht is weker thn γ. When deterministic γ utomt re β-type, it is gurnteed tht deterministic β-utomton for the lnguge of A cn e defined on top of the structure of A. For exmple, deterministic Rin utomt eing Büchi-type [5] mens tht if deterministic Rin utomton A recognizes lnguge tht cn e recognized y deterministic Büchi utomton, then A hs n equivlent deterministic Büchi utomton on the sme structure. Thus, the sic motivtion of typeness is to llow simplifictions of the cceptnce conditions of the considered utomt without complicting their structure. Applictions of this notion re much wider [5]. In prticulr, in the context of gmes, the Büchi nd co-büchi conditions dmit memoryless strtegies for oth plyers, which is not the cse for the Rin nd Streett conditions [18]. Thus, the study of typeness in the context of GFG utomt ddresses lso the question of simplifying the memory requirements of the plyers. In ddition, s we elorte in Section 7, it leds to new nd non-trivil ounds on the low-up of trnsformtions etween GFG utomt nd their complementtion. Recll tht deterministic Rin utomt re Büchi-type. Dully, deterministic Streett utomt re co-büchi-type. Typeness cn e defined lso with respect to nondeterministic utomt, yet it crucilly depends on the fct tht the utomton is deterministic. Indeed, nondeterministic Rin re not Büchi-type. Even with the co-büchi cceptnce condition, where nondeterministic co-büchi utomt recognize only suset of the ω-regulr lnguges, nondeterministic Streett utomt re not co-büchi-type [7]. We first show tht typeness is strongly relted with determinism even when slightly relxing the typeness notion to require the existence of n equivlent utomton on sustructure of the originl utomton, insted of on the exct originl structure, nd even when we restrict ttention to n unmiguous utomton, nmely one tht hs single ccepting run on ech word in its lnguge. We descrie n unmiguous prity utomton A, such tht its lnguge is recognized y deterministic Büchi utomton, yet it is impossile to define Büchi cceptnce condition on top of sustructure of A. We lso point to 4 See Section 2.1 for the full definition of the vrious cceptnce conditions. 3

dul result in [7], with respect to the co-büchi condition, nd oserve tht it pplies lso to the relxed typeness notion. We then show tht for GFG utomt, typeness, in its relxed form, does hold. Notice tht once considering GFG utomt with no redundnt trnsitions, which we cll tight, the two typeness notions coincide. Oviously, ll GFG utomt cn e tightened y removl of redundnt trnsitions (Lemm 2.4). In prticulr, we show tht the typeness picture in GFG utomt coincides with the one in deterministic utomt: Rin GFG utomt re Büchi type, Streett GFG utomt re co-büchi type, nd ll GFG utomt re type with respect to the wek cceptnce condition. Unlike the deterministic cse, however, the Rin cse is not simple duliztion of the Streett cse; it is much hrder to prove nd it requires stronger notion of tightness. We continue with using our typeness results for further studying the plce of GFG utomt in etween deterministic nd nondeterministic ones. We strt with showing tht ll GFG utomt tht recognize lnguges tht cn e defined y deterministic wek utomt re DetByP. This generlizes similr results out sfe nd co-sfe lnguges [7]. We then show tht ll unmiguous GFG utomt re lso DetByP. Considering complementtion, GFG utomt lie in etween the deterministic nd nondeterministic settings the complementtion of Büchi utomton into co-büchi utomton is polynomil, s is the cse with deterministic utomt, while the complementtion of co-büchi utomton into Büchi utomton s well s the complementtion of Streett utomton into Rin utomton is exponentil, s opposed to the constnt low-up in the deterministic cse. We conclude with proving douly-exponentil lower ound for the trnsltion of LTL into GFG utomt, s is the cse with deterministic utomt. The pper is structured s follows. In Section 2 we provide the relevnt notions out lnguges nd GFG utomt. Section 3 contins exmples showing tht typeness does not hold for the cse of unmiguous utomt. The next three sections, Sections 4, 5, nd 6, provide the min positive results of this work: co-büchi typeness for GFG-Streett; Büchi typeness for GFG-Rin; nd wek typeness for GFG-Büchi nd GFG-co-Büchi, respectively. Finlly, in Section 7 we continue to study the power of nondeterminism in GFG utomt, looking into utomt complementtion nd trnsltions of LTL formuls to GFG utomt. 2 Preliminries 2.1 Automt An utomton on infinite words is tuple A = Σ, Q, Q 0, δ, α, where Σ is n input lphet, Q is finite set of sttes, Q 0 Q is set of initil sttes, δ : Q Σ 2 Q is trnsition function tht mps stte nd letter to set of possile successors, nd α is n cceptnce condition. The first four elements, nmely Σ, Q, δ, Q 0, re the utomton s structure. We consider here the Büchi, 4

co-büchi, prity, Rin, nd Streett cceptnce conditions. (The wek condition is defined in Section 6.) In Büchi, nd co-büchi conditions, α Q is set of sttes. In prity condition, α: Q {0,..., k} is function mpping ech stte to its priority. In Rin nd Streett conditions, α 2 2Q 2 Q is set of pirs of sets of sttes. The index of Rin or Streett condition is the numer of pirs in it. For stte q of A, we denote y A q the utomton tht is derived from A y chnging the set of initil sttes to {q}. A trnsition of A is triple q,, q such tht q δ(q, ). We extend δ to sets of sttes nd to finite words in the expected wy. Thus, for set S Q, letter Σ, nd finite word u Σ, we hve tht δ(s, ɛ) = S, δ(s, ) = q S δ(q, ), nd δ(s, u ) = δ(δ(s, u), ). Then, we denote y A(u) the set of sttes tht A my rech when reding u. Thus, A(u) = δ(q 0, u). Since the set of initil sttes need not e singleton nd the trnsition function my specify severl successors for ech stte nd letter, the utomton A my e nondeterministic. If Q 0 = 1 nd δ(q, ) 1 for every q Q nd Σ, then A is deterministic. Given n input word w = 1 2 in Σ ω, run of A on w is n infinite sequence r = r 0, r 1, r 2,... Q ω such tht r 0 Q 0 nd for every i 0, we hve r i+1 δ(r i, i+1 ); i.e., the run strts in the initil stte nd oeys the trnsition function. For run r, let inf(r) denote the set of sttes tht r visits infinitely often. Tht is, inf(r) = {q Q for infinitely mny i 0, we hve r i = q}. A set of sttes S stisfies n cceptnce condition α (or is ccepting) iff S α, for Büchi condition. S α =, for co-büchi condition. min q inf(r) {α(q)} is even, for prity condition. There exists E, F α, such tht S E = nd S F for Rin condition. For ll E, F α, we hve S E = or S F for Streett condition. Notice tht Büchi nd co-büchi re dul, nd so re Rin nd Streett. Also note tht the Büchi nd co-büchi conditions re specil cses of prity, which is specil cse of Rin nd Streett. In the ltter conditions, we refer to the sets E nd F s the d nd good sets, respectively. Finlly, note tht Rin pir my hve n empty E component, while n empty F component mkes the pir redundnt (nd dully for Streett). A run r is ccepting if inf(r) stisfies α. An utomton A ccepts n input word w iff there exists n ccepting run of A on w. The lnguge of A, denoted y L(A), is the set of ll words in Σ ω tht A ccepts. A nondeterministic utomton A is unmiguous if for every word w L(A), there is single ccepting run of A on w. Thus, while A is nondeterministic nd my hve mny runs on ech input word, it hs only single ccepting run on words in its lnguge. 5

We denote the different utomt types y three-letter cronyms in the set {D, N} {B, C, P, R, S} {W}. The first letter stnds for the rnching mode of the utomton (deterministic or nondeterministic); the second for the cceptncecondition type (Büchi, co-büchi, prity, Rin, or Streett); nd the third indictes tht we consider utomt on words. For Rin nd Streett utomt, we sometimes lso indicte the index of the utomton. In this wy, for exmple, NBW re nondeterministic Büchi word utomt, nd DRW[1] re deterministic Rin utomt with index 1. For two utomt A nd A, we sy tht A nd A re equivlent if L(A) = L(A ). For n utomton type β (e.g., DBW) nd n utomton A, we sy tht A is β-relizle if there is β-utomton equivlent to A. Let A = A, Q, Q 0, δ, α e n utomton. For n cceptnce-condition clss γ (e.g., Büchi), we sy tht A is γ-type if A hs n equivlent γ utomton with the sme structure s A [5]. Tht is, there is n utomton A = Σ, Q, Q 0, δ, α such tht α is n cceptnce condition of the clss γ nd L(A ) = L(A). 2.2 Good-For-Gmes Automt An utomton A = Σ, Q, Q 0, δ, α is good for gmes (GFG, for short) if there is strtegy g : Σ Q, such tht for every word w = 1 2 Σ ω, the sequence g(w) = g(ɛ), g( 1 ), g( 1 2 ),... is run of A on w, nd whenever w L(A), then g(w) is ccepting. We then sy tht g witnesses A s GFGness. It is known [2] tht if A is GFG, then its GFGness cn e witnessed y finite-stte strtegy, thus one in which for every stte q Q, the set of words g 1 (q) is regulr. Finite-stte strtegies cn e modeled y trnsducers. Given sets I nd O of input nd output letters, n (I/O)-trnsducer is tuple T = I, O, M, m 0, ρ, τ, where M is finite set of sttes, to which we refer s memories, m 0 M is n initil memory, ρ: M I M is deterministic trnsition function, to which we refer s the memory updte function, nd τ : M O is n output function tht ssigns letter in O to ech memory. The trnsducer T genertes strtegy g T : I O, otined y following ρ nd τ in the expected wy: we first extend ρ to words in I y setting ρ(ɛ) = m 0 nd ρ(u ) = ρ(ρ(u), ), nd then define g T (u) = τ(ρ(u)). Consider GFG utomton A = Σ, Q, Q 0, δ, α, nd let g = Σ, Q, M, m 0, ρ, τ e finite-stte (Σ/Q)-trnsducer tht genertes strtegy g : Σ Q tht witnesses A s GFGness (we use nottions nd use g to denote oth the trnsducer nd the strtegy it genertes). Consider stte q Q. When τ(m) = q, we sy tht m is memory of q. We denote y A g the (deterministic) utomton tht models the opertion of A when it follows g. Thus, A g = Σ, M, m 0, ρ, α g, where the cceptnce condition α g is otined from α y replcing ech set F Q tht ppers in α (e.g. ccepting sttes, rejecting sttes, set in Rin or Streett pir, etc) y the set F g = {m τ(m) F }. Thus, F g M contins the memories of F s sttes. For stte q of A, pth π of A g is q-exclusive ccepting if π is ccepting, nd inf(π) \ {m m is memory of q} is not ccepting. 6

q 2 q 1 q 0 2 1 0 Figure 1: A wekly tight GFG-NPW A 0. The numers elow the sttes descrie their priorities. q 1 q 2 q 0 m 2 2 m 1 m 1 1 m 0 Figure 2: A strtegy witnessing the GFGness of the utomton A 0, depicted in Figure 1. 0 q 1 q 2 q 0 m 2 2 m 1 m 1 1 m 0 Figure 3: A strtegy witnessing the tightness of su-utomton of A 0. 0 Exmple 2.1. Consider the NPW A 0 ppering in Figure 1. We clim tht A 0 is GFG-NPW tht recognizes the lnguge L 0 = {w {, } ω there re infinitely mny s in w}. Indeed, if word w contins only finitely mny s then A 0 rejects w, s in ll the runs of A 0 on w, the lowest priority ppering infinitely often is 1. Therefore, L(A 0 ) L 0. We turn to descrie strtegy g : {, } Q with which A 0 ccepts ll words in L 0. The only nondeterminism in A 0 is when reding the letter in the stte q 1. Thus, we hve to descrie g only for words tht rech q 1 nd continue with n. In tht cse, the strtegy g moves to the stte q 2, if the previous stte is q 0, nd to the stte q 1, otherwise. Figure 2 descries (Σ/Q)-trnsducer tht genertes g. The rectngles denote the sttes of A 0, while the dots re their g-memories. The numers elow the rectngles descrie the priorities of the respective sttes of A 0. As L(A 0g ) L(A 0 ), it remins to formlly prove tht L 0 L(A 0g ). Consider word w L 0. Let r = r 0, r 1,... e the sequence of sttes of A 0 visited y A 0g on w. Assume y wy of contrdiction tht r is not ccepting. Thus, r visits q 1 infinitely mny times ut visits q 0 only finitely mny times. Let N e such tht r m q 0 for ll m N. Consider position k > N such tht r k = q 1. 7

Since w contins infinitely mny s, there is some miniml k k such tht the k -th letter in w is. Then, r k = r k+1 =... = r k = q 1 nd r k +1 = q 0, which contrdicts the choice of N. The following lemm generlizes known residul properties of GFG utomt (c.f., [6]). Lemm 2.2. Consider GFG utomton A = Σ, Q, Q 0, δ, α nd let g = Σ, Q, M, m 0, ρ, τ e strtegy witnessing its GFGness. (1) For every stte q Q nd memory m M of q tht is rechle in A g, we hve tht L(A m g ) = L(A q ). (2) For every memories m, m M tht re rechle in A g with τ(m) = τ(m ), we hve tht L(A m g ) = L(A m g ). Proof. We strt with the first clim. Oviously, L(A m g ) L(A q ). For the other direction, consider towrd contrdiction tht there is word w L(A q ) \ L(A m g ). Let u e finite word such tht A g (u) = m. Then, u w L(A g ). However, there is n ccepting run of A on u w: it follows the run of A g on u, nd continues with the ccepting run of A q on w. Hence, g does not witness A s GFGness, nd we hve reched contrdiction. The second clim is direct corollry of the first, s L(A m g ) = L(A τ(m) ) = L(A τ(m ) ) = L(A m g ). A finite pth π = q 0,..., q k in A is sequence of sttes such tht for i = 0,..., k 1 we hve q i+1 δ(q i, i ) for some i Σ. A pth is cycle if q 0 = q k. Ech pth π induces set sttes(π) = {q 0,..., q k } of sttes in Q. A set S of finite pths then induces the set sttes(s) = π S sttes(π). For set P of finite pths, comintion of pths from P is set sttes(s) for some nonempty S P. Consider strtegy g = Σ, Q, M, m 0, ρ, τ. We sy tht trnsition q,, q of A is used y g if there is word u Σ nd letter Σ such tht q = g(u) nd q = g(u ). Consider two memories m m M with τ(m) = τ(m ). Let P m m e the set of pths of A g from m to m. We sy tht m is replcele y m if P m m is empty or ll comintions of pths from P m m re ccepting. We sy tht A is tight with respect to g if ll the trnsitions of A re used in g, nd for ll memories m m M with τ(m) = τ(m ), we hve tht m is not replcele y m. Intuitively, the ltter condition implies tht oth m nd m re required in g, s n ttempt to merge them strictly reduces the lnguge of A g. When only the first condition holds, nmely when ll the trnsitions of A re used in g, we sy tht A is wekly tight with respect to g. When Rin utomton A is tight with respect to g, nd in ddition for every stte q tht ppers in some good set of A s cceptnce condition, there is q-exclusive ccepting cycle in A g, we sy tht A is strongly tight with respect to g. Then, A is (wekly, strongly) tight if it is (wekly, strongly) tight with respect to some strtegy. Exmple 2.3. The GFG-NPW A 0 from Exmple 2.1 is wekly tight nd is not tight with respect to the strtegy g. Indeed, while ll the trnsitions in A 0 re 8

used in g, the memory m 1 is replcele y m 1, s ll comintions of pths from m 1 to m 1 re ccepting. The following lemm formlizes the intuition tht every GFG utomton cn indeed e restricted to its tight prt, y removing redundnt trnsitions nd memories. Further, every tight Rin GFG utomton hs n equivlent strongly tight utomton over the sme structure. Lemm 2.4. For every GFG utomton A there exists n equivlent tight GFG utomton A. Moreover, A is defined on sustructure of A. Proof. Consider GFG utomton A = Σ, Q, Q 0, δ, α, nd let g = Σ, Q, M, m 0, ρ, τ e strtegy tht witnesses A s GFGness. We show how to mke A tight with respect to strtegy otined y merging memories in g. As long s A is not tight with respect to g, we proceed s follows. First, we remove from A ll the trnsitions tht re not used y g. Then, if there re two memories m, m M with τ(m) = τ(m ) such tht m is replcele y m, we remove m from g nd redirect trnsitions to m into m. Note tht the removl of m my cuse the otined strtegy not to use some trnsitions in A. We thus keep repeting oth steps s long s the otined utomton is not tight with respect to the otined strtegy. We prove tht oth steps do not chnge the lnguge of A nd its GFGness. First, clerly, removl of trnsitions tht re not used does not chnge the lnguge of A. Now, consider memories m m M with τ(m) = τ(m ) such tht m is replcele y m. Thus, P m m is empty or ll susets S P m m re such tht sttes(s) is ccepting. Let g e the strtegy otined y removing m from g nd redirecting trnsitions to m into m. Since L(A g ) L(A) = L(A g ) it is enough to prove tht L(A g ) L(A g ). We strt with the cse P m m is empty, thus there is no pth from m to m. Consider the ccepting run r of A g on some word w. If r does not include m, then the run of A g on w is identicl to r, nd is thus ccepting. Otherwise, let p e the first position of m in r, nd let w p+1 e the suffix of w from the position p + 1 onwrds. Since r is ccepting, w p+1 L(A m g ). Thus, y Lemm 2.2, we hve w p+1 L(A m g ). Now, since P m m is empty, the runs of A m g nd A m g re identicl on w p+1, nd re thus ccepting. Hence, A g ccepts w. We continue with the cse tht ll susets S P m m re such tht sttes(s) is ccepting. Consider word w A g, nd let r e the run of A g on w. The run r my use the memory m insted of m finitely or infinitely mny times. Consider first the cse tht r uses the memory m insted of m for k times. It is esy to prove, y n induction on k, tht r is ccepting. Indeed, the se cse is similr to the cse P m m is empty, nd the induction step chnges only finite prefix of the run. Consider now the cse tht the chnge is done infinitely mny times, in positions p 1, p 2,... of r. Every pth from p i to p i+1 is pth from m to m in A g. Hence, the set of sttes inf(r ) is sttes(s) for some nonempty S P m m, nd is thus ccepting. 9

Lemm 2.5. For every tight Rin GFG utomton, there exists n equivlent strongly tight Rin GFG utomton over the sme structure. Proof. Consider tight GFG Rin utomton A nd let g e strtegy tht witnesses A s GFGness nd with respect to which A is tight. We show tht the removl of redundnt sttes from the good sets of A s ccepting condition results in n utomton tht is equivlent to A nd strongly tight with respect to g. Consider stte q of A tht ppers in some good set G of A s cceptnce condition, nd for which there is no q-exclusive ccepting cycle in A g. We clim tht the utomton A tht is identicl to A, except for removing q from G, is GFG Rin utomton equivlent to A tht is tight w.r.t. g. Indeed: Regrding the lnguge equivlence, oviously, L(A ) L(A). As for the other direction, let r e the ccepting run of A g on some word w. Oserve tht r is lso n ccepting run of A g on w: If q does not pper infinitely often in r then clerly r is lso ccepting w.r.t. A. Now, if q does pper infinitely often in r, then since there is no q-exclusive ccepting cycle in A g, every cycle from q ck to q is ccepting w.r.t. A nd thus r is ccepting w.r.t. A. Regrding the GFGness of A, since L(A) = L(A g ) = L(A g) L(A ) L(A), we get tht g witnesses the GFGness of A. Regrding the tightness of A w.r.t. g, oserve tht A nd A hve the sme trnsitions, nd since A g hs no redundnt memories, neither does A g hve ones: Recll tht memory m is redundnt if exists memory m of the sme stte, such tht the set of pths of A g from m to m, which we denote y P m m, is empty or ll comintions of pths from P m m re ccepting. The set of pths of A g nd of A g from m to m re the sme, nd pth of A g cnnot e ccepting if it is not ccepting in A g. As there re finitely mny sttes in A, n itertive removl of sttes q s descried ove results in n utomton tht is strongly tight w.r.t. g. Exmple 2.6. In Figure 3 we descrie strtegy g tht witnesses the tightness of GFG-NPW on sustructure of the GFG-NPW A from Exmple 2.1. The strtegy g is otined from g y following the procedure descried in the proof of Lemm 2.4: ll the trnsitions to m 1 re redirected to m 1. This cuses the trnsition (q 1,, q 2 ) tht ws used y the memory m 1 not to e used, nd it is removed. A specil cse of GFG utomt re those who re determinizle y pruning (or shortly DetByP) there exists stte q 0 Q 0 nd function δ : Q Σ Q tht for every stte q nd letter stisfies δ (q, ) δ(q, ) such tht A = Σ, Q, q 0, δ, α is deterministic utomton recognizing the lnguge L(A). 10

q 00 q 01 2 1 p 0 p 1 p 2 q 10 q 11 1 1 1 1 0 Figure 4: A 1 : An unmiguous NPW tht is DBW-relizle yet is not Büchitype. 3 Typeness Does Not Hold for Unmiguous Automt As noted in [7], it is esy to see tht typeness does not hold for nondeterministic utomt: there exists n NRW tht recognizes n NBW-relizle lnguge, yet does not hve n equivlent NBW on the sme structure. Indeed, since ll ω- regulr lnguges re NBW-relizle, typeness in the nondeterministic setting would imply trnsltion of ll NRWs to NBWs on the sme structure, nd we know tht such trnsltion my involve low-up liner in the index of the NRW [17]. Even for Streett nd co-büchi utomt, where the restriction to NCW-relizle lnguges mounts to restriction to DCW-relizle lnguges, typeness does not hold. In this section we strengthen the reltion etween typeness nd determinism nd show tht typeness does not hold for nondeterministic utomt even when they recognize DBW-relizle lnguge nd, moreover, when they re unmiguous. Also, we prove the non-typeness results for NPWs, thus they pply to oth Rin nd Streett utomt. Proposition 3.1. Unmiguous NPWs re not Büchi-type with respect to DBWrelizle lnguges., Proof. Consider the utomton A 1 depicted in Figure 4. We will show tht A 1 is unmiguous nd recognizes DBW-relizle lnguge, yet A 1 is not Büchitype. Moreover, we cnnot prune trnsitions from A 1 nd otin n equivlent Büchi-type NPW. The NPW A 1 hs two components: the left component, consisting of the sttes q ij ; nd the right component, consisting of the sttes p 0, p 1, nd p 2. The right prt is deterministic, nd it recognizes the lnguge L 1,, = {w {, } ω there re infinitely mny s nd s in w}. is: We first prove tht the left component is unmiguous nd tht its lnguge L 1,, = {w {, } ω there is finite nd even numer of s in w }. 11

q 00 q 01 q 10 q 11 q 12 Figure 5: A DBW recognizing L 1. To see this, oserve tht fter reding finite word, the left component of A 1 cn rech stte of the form q ij iff i (w) (mod 2) (i.e. i is the prity of the numer of letters in w). The only ccepting runs of the left component re those tht get stuck in the stte q 00. This implies tht if w is ccepted y the left component, then w L 1,,. For the other direction, consider word w L 1,,. We show tht A 1 hs n (in fct, unique) ccepting run on w. We cn construct n ccepting run of the left component of A 1 on w y guessing whether the next lock of (i.e., su-word of the form + ) hs n even or odd length. If the guess is incorrect, the run is stuck reding in stte of the form q i1. If the guess is correct, the run reds the first fter the lock in stte of the form q i0. Thus, fter reding the lst lock of s, the constructed run reches the stte q 00, stys there forever, nd A 1 ccepts w in its left component. Further, ll other runs tht ttempt to ccept w in the left component re doomed to get stuck. Thus, the left component is unmiguous. Since L 1,, L 1,, =, the NPW A 1 is unmiguous nd its lnguge is L 1 = {w {, } ω w hs n infinite numer of s nd n infinite or even numer of s}. It is not hrd to see tht L 1 is DBW-relizle. An exmple of DBW tht recognizes L 1 is depicted in Figure 5. We prove tht A 1 is not Büchi-type. Assume y wy of contrdiction tht there exists suset α of A 1 s sttes such tht the utomton otined form A 1 y viewing it s n NBW with the cceptnce condition α recognizes L 1. If {q 00, q 11 } α, then the NBW ccepts the word ω, which is not in L 1. If {q 01, q 10 } α, then the NBW ccepts the word ω, which is lso not in L 1. Therefore, α {p 0, p 1, p 2 }. Clerly, p 1 / α, s otherwise the NBW ccepts ω. Similrly, if p 0 α, then the NBW ccepts ω, which is lso not in L 1. Thus, α = {p 2 } nd the NBW rejects ω, which is in L 1. Finlly, s A 1 is unmiguous nd ll its trnsitions re used in the ccepting run of some word, it cnnot e pruned to n equivlent NPW. The dul cse of unmiguous NPWs tht re not co-büchi-type with respect to DCW-relizle lnguges follows from the results of [7], nd we give it here for completeness, dding the oservtion tht the utomton descried there cnnot e pruned to n equivlent co-büchi-type NPW. 12

q 0 q 1 q 2 q 3 Figure 6: A 2 : An unmiguous NBW tht is DCW-relizle yet is not co-büchitype. Proposition 3.2. [7] Unmiguous NPWs (nd even NBWs) re not co-büchitype with respect to DCW-relizle lnguges. Proof. Consider the NBW A 2 depicted in Figure 6. We will show tht A 2 is unmiguous, nd recognizes DCW-relizle lnguge, yet A 2 is not co-büchitype. Moreover, we cnnot prune trnsitions from A 2 for otining n equivlent co-büchi-type NPW. Notice tht L(A 2 ) = {w {, } ω w contins letter }, which is DCWrelizle. Yet, there is no wy to define co-büchi cceptnce condition on top of A 2 nd otin n equivlent NCW. Moreover, s A 2 is unmiguous nd ll its trnsitions re used in n ccepting run of some word, it cnnot e pruned to n equivlent one. We conclude this section with the following rther simple proposition, showing tht utomt tht re oth unmiguous nd GFG re essentilly deterministic. Essentilly, it follows from the fct tht y restricting n unmiguous GFG utomton A to rechle nd nonempty sttes, we otin, y pruning, deterministic utomton, which is clerly equivlent to A. Proposition 3.3. Unmiguous GFG utomt re DetByP. Proof. Let A e n unmiguous GFG utomton, witnessed y strtegy g tht strts in stte q 0. Without loss of generlity, we cn ssume tht L(A). Let A e the restriction of A to rechle nd nonempty sttes (nmely to rechle sttes q, such tht L(A q ) ). It is cler tht A is otined from A y pruning nd tht L(A ) = L(A). We prove tht A is deterministic. Note first tht there is single nonempty initil stte. Indeed, ssume towrd contrdiction tht there is n initil stte q 0 q 0, from which A hs run ccepting some word w. Since A hs n ccepting run on w strting from q 0, s witnessed y g, we get contrdiction to its unmiguity. Next, we prove tht A is deterministic y showing tht for every finite word u over which A cn rech nonempty stte, we hve A (u) = 1. Let q e the stte tht A g reches when reding u nd ssume towrd contrdiction the 13

existence of stte q q, such tht q A (u). As q is nonempty, A q ccepts some word w. However, since uw L(A), we hve y the GFGness of A tht A q lso ccepts w. Hence, A hs two different ccepting runs on uw, contrdicting its unmiguity. 4 Co-Büchi Typeness for GFG-NSWs In this section we study typeness for GFG-NSWs nd show tht, s is the cse with deterministic utomt, tight GFG-NSWs re co-büchi-type. On more technicl level, the proof of Theorem 4.1 only requires the GFG utomt to e wekly tight (rther thn fully tight), implying tht Theorem 4.1 cn e strengthened in ccordnce. This fct is considered in Section 5, where the typeness of GFG-NRWs is shown to require full tightness. Theorem 4.1. Tight GFG-NSWs re co-büchi-type: Every tight GFG-NSW tht recognizes GFG-NCW-relizle lnguge hs n equivlent GFG-NCW on the sme structure. Proof. Consider GFG-NSW A = Σ, Q, Q 0, δ, α, with α = { E 1, F 1,..., E k, F k }. For 1 i k, we refer to the sets E i nd F i s the d nd good sets of α, respectively. Let g = Σ, Q, M, m 0, ρ, τ e strtegy tht witnesses A s GFGness nd such tht A is tight with respect to g. Formlly, the utomton A is defined s A with the co-büchi cceptnce condition α def = {q ll the cycles in A g tht go through g-memory of q re rejecting}. We prove tht L(A) = L(A ) nd tht A is GFG-NCW. Let Q = {q 1,..., q n }. We define sequence of NSWs A 0, A 1,..., A n nd prove tht: L(A) = L(A 0 ) = L(A 1 ) = = L(A n ); g witnesses the GFGness of A l for ll 0 l n; nd A n is essentilly the NCW A. For ll 0 l n, the NSW A l hs the sme structure s A. The cceptnce condition of A l is α l { α l, }, where α l nd α l re defined s follows. First, α 0 = α nd α 0 =. Thus, going form A to A 0 we only dd to α redundnt pir,. Clerly, L(A) = L(A 0 ) nd A 0 is GFG witnessed y g. For 1 l n, we otin α l nd α l from α l 1 nd α l 1 in the following wy. First, we remove q l from ll the d sets in α l 1. Then, if q l α, we dd it to α l. We now prove tht L(A l ) = L(A l 1 ) nd tht A l is GFG witnessed y g. We distinguish etween two cses. If q l α, the proof is not hrd: dding q l to α l forces it to e visited only finitely often regrdless of visits in the good sets. Thus, L(A l ) L(A l 1 ). In ddition, L(A l 1 ) L(A l ), nd g witnesses lso the GFGness of A l. Indeed, n ccepting run in L(A l 1 ) remins ccepting in L(A l ). To see this, ssume y wy of contrdiction tht there is run r tht stisfies α l 1 { α l 1, } yet does not stisfy α l { α l, }. Since α l is esier to stisfy thn α l 1, it must e tht r violtes the pir α l,. Since r stisfies 14

the pir α l 1,, it must visit q l infinitely often. Since, however, q l α, the ltter indictes tht r eventully trverses only rejecting cycles in A g nd is thus rejecting lso in A l. If q l α, we proceed s follows. Consider stte q tht hs memory with n ccepting cycle, nd let A e the NSW tht is derived from A y tking q out of the d sets. The chnge cn oviously only enlrge the utomton s lnguge. Assume towrd contrdiction tht there is word w L(A ) \ L(A). Since L(A ) \ L(A) is n ω-regulr lnguge, we my ssume tht w is lsso word, nmely of the form w = uv ω. As the only difference etween A nd A is the removl of q from d sets, it follows tht n ccepting run r of A on w visits q infinitely often. Hence, there re positions i nd j of w, such tht: I) r visits q in oth i nd j, II) the inner position within v is the sme in positions i nd j, nd III) the cycle C r tht r goes through etween positions i nd j is ccepting. Let x e the prefix of w up to position i nd y the infix of w etween positions i nd j. Notice tht xy ω = uv ω = w. Consider the run r of A on w tht follows r up to position j, nd from there on forever repets the cycle C r. By the ove definition of i nd j, the run r is lso ccepting. Notice tht since w L(A), it follows tht C r is rejecting for A. As the only difference etween A nd A is the removl of q from the d sets, it follows tht comining C r with ny cycle C tht contins q nd is ccepting for A, yields cycle tht is ccepting for A. Recll tht q hs such n ccepting cycle C, hving tht C r C is ccepting. Since NCW=DCW, there is DCW D equivlent to A. Let n e the numer of sttes in D. Let z e finite word over which A g mkes the cycle C, nd consider the word e = x(y n z n ) ω. We clim tht e L(A) \ L(D), leding to contrdiction. As for the positive prt, e L(A) y the run of A tht reches q nd then follows the C r nd C cycles. Next, we show tht e L(D). For every i N, let e i = x(y n z n ) i y n e suword of e, nd let m i = A g (e i ). Notice tht m i elongs to some stte q i of A nd not necessrily to q. By [6], the fct there exists finite word u such tht q, q i A(u), implies tht L(A q ) = L(A q i ). Thus, since q A(e i ), we hve, y Lemm 2.2, tht L(A m i g ) = L(A q i ) = L(A q ). Since y ω L(A q ) nd L(A m i g ) = L(A q ), it follows tht A g does not ccept x(y n z n ) i y ω. Hence, the run of D on e must visit rejecting stte on every period etween e i nd e i+1, implying tht it is rejecting. Finlly, in α n ll the d sets re empty. Also, α n = α. Thus, A n is relly n NCW with cceptnce condition α, i.e. A. The following exmple shows tht the wek tightness requirement cnnot e omitted, even when the GFG-NSW is ctully GFG-NBW. Exmple 4.2. The utomton A 3 depicted in Figure 8 is GFG-NBW nd recognizes GFG-NCW-relizle lnguge, yet A 3 hs no equivlent NCW on the 15

sme structure. First, it is not hrd to see tht L(A 3 ) = () ω +() + (+) ω {, } ω. Notice tht if we remove the trnsition (q 0,, q 0 ) then A 3 ecomes deterministic utomton for the sme lnguge. In prticulr, A 3 is GFG. Clerly the lnguge of A 3 is GFG-NCW-relizle once the trnsition (q 0,, q 0 ) is removed, we cn mke p 0 the only rejecting stte, nd otin n equivlent DCW. Now ssume towrd contrdiction tht there exists n NCW A 3 equivlent to A 3 over the whole structure of A 3. Let α e its cceptnce condition. Oserve tht q 0 / α s otherwise A 3 rejects the word ω. In tht cse A 3 ccepts the word ω, leding to contrdiction. 5 Büchi Typeness for GFG-NRWs Studying typeness for deterministic utomt, one cn use the dulities etween the Büchi nd co-büchi, s well s the Rin nd Streett conditions, in order to relte the Büchi-typeness of DRWs with the co-büchi typeness of DSWs. In the nondeterministic setting, we cnnot pply dulity considertions, s y dulizing nondeterministic utomton, we otin universl one. As we shll see in this section, our inility to use duliztion considertions is not only technicl. There is n inherent difference etween the co-büchi typeness of GFG-NSWs studied in Section 4, nd the Büchi typeness of GFG-NRWs, which we study here. We first show tht while the proof of Theorem 4.1 only requires wek tightness, Büchi typeness requires full tightness. The following exmple shows tht tightness is necessry lredy for GFG- NCW tht re GFG-NBW-relizle. Exmple 5.1. The utomton A 4 depicted in Figure 7 is wekly tight GFG- NCW tht recognizes GFG-NBW-relizle lnguge, yet A 4 hs no equivlent GFG-NBW on the sme structure. First notice tht the lnguge of A 4 is L 4 = ω + + ( + ) ω {, } ω. Moreover, if we remove the trnsitions (q 0,, q 1 ) nd (q 1,, q 0 ), then A 4 ecomes deterministic utomton for the sme lnguge. In prticulr, A 4 is GFG. Clerly, L 4 is oth DBW- nd DCW-relizle. Now ssume towrd contrdiction tht there exists n NBW A 4 equivlent to A 4 over the (whole) structure of A 4. Let α e its cceptnce condition. Oserve tht the stte q 1 must elong to α, s otherwise A 4 rejects the word ω. But in tht cse, A 4 ccepts the word ω, leding to contrdiction. We now proceed to our min positive result, otining the typeness of GFG- NRWs. Theorem 5.2. Tight GFG-NRWs re Büchi-type: Every tight GFG-NRW tht recognizes GFG-NBW-relizle lnguge hs n equivlent GFG-NBW on the sme structure. 16

p 0 p 1, q 0 q 1, Figure 7: A 4 : A wekly tight GFG- NCW tht is GFG-NBW-relizle yet is not Büchi-type. q 1 q 0 p 0 p 2 p 1 Figure 8: A 3 : A GFG-NBW tht is GFG-NCW-relizle yet is not co- Büchi-type. Consider tight GFG-NRW A tht recognizes GFG-NBW-relizle lnguge. Let g e strtegy tht witnesses A s GFGness nd with respect to which A is tight. By Lemm 2.5, we hve GFG Rin utomton A over the structure of A tht is strongly tight with respect to g. We define n NBW B on top of A s structure, setting its ccepting sttes to e ll the sttes tht pper in good sets of A (nmely in the right components of the Rin ccepting pirs). Clerly, L(A ) L(B), s B s condition only requires the good prt of A s condition, without requiring to visit finitely often in corresponding d set. We should thus prove tht L(B) L(A ) nd tht B is GFG. Once proving the lnguge equivlence, B s GFGness is stright forwrd, s the strtegy g witnesses it. The lnguge equivlence, however, is not t ll strightforwrd. In order to prove tht L(B) L(A ), we nlyze the cycles of A nd of A g, s expressed y the following lemms. Lemm 5.3. Consider GFG-NRW A tht is GFG-NBW-relizle nd strtegy g tht witnesses its GFGness. 1. A g-memory m of stte q of A cnnot elong to oth q-exclusive ccepting cycle nd rejecting cycle. 2. Consider g-memories m nd m of stte q of A, such tht m elongs to q-exclusive ccepting cycle nd m elongs to rejecting cycle. Let P m m nd P m m e the sets of pths from m to m nd from m to m, respectively. Then P m m or P m m stisfies the following property: It is empty or every comintion of its pths is ccepting. Formlly, for P = P m m or P = P m m, we hve tht sttes(p ) = or sttes(s) is ccepting for ll S P. Proof. We strt with the first clim. First, y [8], there is DBW D equivlent to A. Assume, y wy of contrdiction, tht there re finite words p, u nd v, such tht A g (p) = m, A m g (u) = m long q-exclusive ccepting cycle, nd A m g (v) = m long rejecting cycle. 17

Let n e the numer of sttes in D, nd consider the word w = p(u n v n ) ω. For every i 1, the NBW D ccepts p(u n v n ) i u ω. Hence, it is not hrd to prove tht D lso ccepts w. On the other hnd, we clim tht A does not ccept w. Indeed, since v is rejecting cycle tht includes q, it must visit sttes in d set B i for every i such tht q elongs to good set G i. As the cycle u is q-exclusive ccepting, we get tht the cycle u n v n is rejecting. For the second clim, ssume, y wy of contrdiction, tht there re pths π 1,..., π n P m m nd pths π 1,..., π n P m m, such tht oth sets of sttes: n i=1 sttes(π i) nd n i=1 sttes(π i ) re rejecting. Consider the pth π = π 1 π 1 π 2π 2... π nπ n, where w.l.o.g. π n is repeted until reching the lrger index n. Then, since the union of Rin rejecting cycles is rejecting, π is rejecting cycle of m, contrdicting the previous oservtion. Lemm 5.4. Consider strongly tight GFG-NRW A tht is GFG-NBW-relizle. Then, every stte q of A tht ppers in some good set hs single g-memory, nd ll the q-cycles in A g re ccepting, nd t lest one of them is q-exclusive. Proof. Since A is strongly tight nd q ppers in good set, the strong tightening of A, s per the proofs of Lemms 2.4 nd 2.5, gurntees tht q hs g-memory m tht elongs to q-exclusive ccepting cycle. Assume, y wy of contrdiction, tht q hs nother memory m m. Then, due to the removl of redundnt memories in Lemm 2.4, there is rejecting comintion of pths from m to m, s well s from m to m. Hence, m elongs to rejecting cycle, in contrdiction to Lemm 5.3. In ddition, since there is single memory m in q, nd q elongs to good set, we hve, y Lemm 2.4, tht m elongs to q-exclusive ccepting cycle. Hence, y Lemm 5.3, the memory m cnnot lso elong to rejecting cycle. Lemm 5.5. Consider strongly tight GFG-NRW A tht is GFG-NBW-relizle. Then, every stte q of A tht ppers in some good set does not elong to rejecting cycle. Proof. Assume, y wy of contrdiction, tht q elongs to rejecting cycle π = q 0, q 1, q 2,..., q n, q n+1 with q 0 = q n+1 = q. Let S e the set of indices of d sets tht π visits. Tht is, n index j elongs to S if there is stte p in π tht elongs to B j. Notice tht S cnnot e empty, since q ppers in good set. Let h e the mximl index of stte q i in π up to which the strtegy my exhust the cycle sttes, while not dding fresh unrejected good stte. Tht is: There is pth ρ of A g from q to memory m of q h tht visits q i for every 1 i h, nd if stte p ppers in ρ nd in G i \ B i for some cceptnce set i, then i S. (Notice tht the pth my lso visit sttes not in the cycle nd my visit the cycle sttes in different order.) There is no such pth of A g from q to q h+1. 18

Notice tht h 1, since there is trnsition q q 1 tht the strtegy uses, nd h n, since otherwise q elongs to rejecting pth of A g, while such pth does not exist due to Lemm 5.4. Let m e memory of q h tht tkes the trnsition q h q h+1. Notice tht m m, since y the mximlity of h, m does not tke the trnsition q h q h+1. Furthermore, there cnnot e rejecting pth comintions from oth m to m nd from m to m, s merging them would provide rejecting pth ρ from m to m, which is impossile due to the mximlity of h. (Conctenting ρ to ρ provides continution of ρ to q h+1.) Hence, ll pth comintions from either m to m or from m to m re ccepting. However, this leds to contrdiction due to the removl of redundnt memories in Lemm 2.4. We re now in position to finish the proof of Theorem 5.2 y showing tht L(B) L(A ) nd tht B is GFG. Consider word w L(B), nd n ccepting run r of B on it. Let q e n ccepting stte tht ppers infinitely often in r. By Lemm 5.5, ll cycles of A tht include q re ccepting. Hence, r is lso n ccepting run of A on w. As for the GFGness of B, we clim tht the strtegy g lso witnesses B s GFGness. Consider word w L(B). Since L(B) = L(A ) = L(A g), there is n ccepting run r of A g on w. Therefore, there must e some stte q in good set of A tht is visited infinitely often long r. Thus, r is lso n ccepting run of B g on w. This concludes the proof of Theorem 5.2. The following result follows directly from Lemm 2.4, Theorem 5.2, nd the determiniztion procedure for Büchi GFG utomt from [6]. Corollry 5.6. Every GFG-NRW with n sttes tht recognizes DBW-relizle lnguge hs n equivlent DBW with t most O(n 2 ) sttes. Proof. Consider GFG-NRW A with n sttes tht recognizes DBW-relizle lnguge. By Lemm 2.4, A hs n equivlent tight GFG-NRW on sustructure of it, thus with t most n sttes. By Theorem 5.2, A hs n equivlent GFG- NBW on the sme structure, thus with t most n sttes. By [6], GFG-NBWs cn e determinized with qudrtic low-up, nd we re done. 6 Wek Typeness for GFG Automt A Büchi utomton A is wek [13] if for ech strongly connected component C of A, either C α (in which cse we sy tht C is n ccepting component) or C α = (in which cse we sy tht C is rejecting component). Note tht wek utomton cn e viewed s oth Büchi nd co-büchi utomton, s run of A visits α infinitely often iff it gets trpped in n ccepting component iff it visits sttes in Q \ α only finitely often. We use NWW nd DWW to denote nondeterministic nd deterministic wek word utomt, respectively. We show in this section tht ll GFG utomt re type with respect to the wek cceptnce condition. We provide the theorem with respect to GFG-NCWs, 19

from which we cn esily deduce it, y our previous typeness results, lso for the other types. Theorem 6.1. Tight GFG-NCWs re wek-type: every tight GFG-NCW tht recognizes GFG-NWW-relizle lnguge hs n equivlent GFG-NWW on the sme structure. Proof. Consider tight GFG-NCW A tht recognizes lnguge tht is GFG- NWW-relizle. Let S e the set of rejecting sttes of A nd let g e strtegy witnessing A s tight GFGness. Let S e the union of S nd ll the sttes q of A for which no g-memory m hs n ccepting cycle in A g. Let A e the utomton A with the co-büchi condition given y S. Notice tht the strtegy g witnesses tht for every stte q of A we hve L(A q ) L ( (A ) q). The opposite inclusion follows from the fct tht S S. Thus, A is n NCW equivlent to A nd g witnesses its GFGness. We now prove tht A is wek. Assume contrrily tht there exists cycle C in A tht contins oth stte q / S nd stte q S. Since q / S, there is cycle C + in A g tht is ccepting in A g nd contins g-memory m of q. This cycle witnesses tht none of the sttes on C + cn elong to S \ S, therefore the cycle C + is ccepting in A g s well. We construct cycle C of A g tht visits some g-memory m of q nd the g-memory m of q. This cycle is otined y extending the cycle C of A in the following wy. Assume tht (q 0, 0, q 1 ) nd (q 1, 1, q 2 ) re two consecutive trnsitions of C. Since A contins only trnsitions of g, these re ctully trnsitions of A g of the form (m 0, 0, m 0 ) nd (m 1, 1, m 1 ) with g-memories: m 0 of q 0 ; m 0 nd m 1 of q 1 ; nd m 1 of q 2. Notice tht m 0 my possily e different from m 1. However, y the ssumption tht A is tight, there is pth in A g leding from m 0 to m 1. Thus, for ech pir of such consecutive trnsitions we cn dd n pproprite pth to C in such wy to otin cycle C of A g tht extends (s set of sttes) C. Additionlly, we cn dd to C two pths in such wy to visit q exctly in the g-memory m (C visits q, so it is possile s ove). As q S nd q C C, we know tht C is rejecting in A g. Let u + nd u e the finite words over which (A ) m g trverses the cycles C + nd C, respectively. An infinite repetition of u + nd u elongs to L ( (A ) m ) g = L ( (A ) q) = L ( A q) if nd only if it contins only finitely mny copies of u. But this contrdicts the fct tht L(A q ) cn e recognized y DWW. Consider now GFG-NSW A tht is GFG-NWW-relizle. Notice tht it is oviously lso GFG-NBW-relizle. Hence, y Theorem 4.1, there is GFG- NCW on A s structure, nd y Theorem 6.1 lso GFG-NWW. The cses of GFG-NPW nd GFG-NBW oviously follow, since they re specil cses of GFG-NSWs. As for GFG-NRW A tht is GFG-NWW-relizle, notice tht it is oviously lso GFG-NBW-relizle. Hence, y Theorem 5.2, there is GFG-NBW on A s structure, nd y Theorem 6.1 lso GFG-NWW. 20

Corollry 6.2. Tight GFG-NSWs nd GFG-NRWs re wek-type: every tight GFG-NSW nd GFG-NRW tht recognizes GFG-NWW-relizle lnguge hs n equivlent GFG-NWW on the sme structure. Next, we show tht GFG-NWWs re DetByP, generlizing folklore result out sfe nd co-sfe GFG utomt. Theorem 6.3. GFG-NWWs re DetByP. Proof. Consider GFG-NWW A with ccepting set α. By Lemms 2.4 nd 2.5, we my ssume tht A is strongly tight w.r.t. strtegy g. First notice tht y Lemm 5.4, stte q α hs only one g-memory, nd is therefore lredy deterministic. Now we consider the cse of stte q / α such tht there re t lest two g-memories m nd m of q. Let g e the strtegy otined y removing m from g nd redirecting trnsitions to m into m. We now show tht L(A g ) = L(A g ). From tht, y induction it follows tht the numer of memories of ech stte of A cn e reduced to 1. Consider word w A g, nd let r e the run of A g on w. The run r my use the memory m insted of m finitely or infinitely mny times. If r uses it only finitely mny times, then y n rgument similr to the one given in the proof of Lemm 2.4, r is lso ccepting. (The rgument inductively uses Lemm 2.2, ccording to which L(A m g ) = L(A m g ).) We continue with the cse tht the chnge is done infinitely mny times, in positions p 1, p 2,... of r, nd ssume towrd contrdiction tht r is rejecting. Every pth from p i to p i+1 is pth from m to m in A g. Notice tht the suffix of w from position p 1 onwrds is in L(A q ) \ L(A m g ). Since we consider ω-regulr lnguges, we cn ssume without loss of generlity tht this suffix is periodic, in the form of u ω, where A m g (u) = m. Let u e finite word such tht A m g (u ) = m. Consider now word w (u + u ) ω. First ssume tht w contins only finitely mny instnces of u. In tht cse, we hve tht A q ccepts w, ecuse A q hs run tht loops ck to q until reding the lst occurrence of u nd then follows the run witnessing tht u ω L(A q ). We refer to such words s of the first kind. Now ssume tht suffix of w from some point on is equl to (uu ) ω. Since (uu ) ω L(A m g ), we get y Lemm 2.2 tht w / L(A q ). We refer to such words s of the second kind. Now consider the miniml, nmely lst, strongly-connected component of A g tht cn e reched from m y reding words in the lnguge (u + u ). If this component is ccepting, then A m g ccepts word of the second kind. Similrly, if the component is rejecting then A m g rejects word of the first kind. In oth cses we get contrdiction. By comining the ove results, we otin the following corollry. Corollry 6.4. Every GFG-NSW nd GFG-NRW tht recognizes GFG-NWWrelizle lnguge is DetByP. 21