How Deterministic are Good-For-Games Automata?

Transcription

1 How Deterministic re Good-For-Gmes Automt? Udi Boker 1, Orn Kupfermn 2, nd Michł Skrzypczk 3 1 Interdisciplinry Center, Herzliy, Isrel 2 The Herew University, Jeruslem, 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 ACM Suject Clssifiction F.1.1 Models of Computtion Keywords nd phrses Finite utomt on infinite words, determinism, good-for-gmes Digitl Oject Identifier /LIPIcs.FSTTCS 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 A full version of the pper is ville t Typeness_Full_Version.pdf. 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/ ) / ERC grnt greement no Supported y the Polish Ntionl Science Centre (decision UMO-2016/21/D/ST6/00491). Udi Boker, Orn Kupfermn, nd Michł Skrzypczk; licensed under Cretive Commons License CC-BY 37th IARCS Annul Conference on Foundtions of Softwre Technology nd Theoreticl Computer Science (FSTTCS 2017). Editors: Sty Lokm nd R. Rmnujm; Article No. 18; pp. 18:1 18:14 Leiniz Interntionl Proceedings in Informtics Schloss Dgstuhl Leiniz-Zentrum für Informtik, Dgstuhl Pulishing, Germny

2 18:2 How Deterministic re Good-For-Gmes Automt? 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 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 1 GFGness is lso used in [3] in the frmework of cost functions under the nme history-determinism. 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. 4 See Section 2.1 for the full definition of the vrious cceptnce conditions.

3 U. Boker, O. Kupfermn, nd M. Skrzypczk 18:3 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 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 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 F S T T C S

4 18:4 How Deterministic re Good-For-Gmes Automt? 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. Due to lck of spce, some full proofs re missing, nd cn e found in the full version, in the uthors URLs. 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, 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

5 U. Boker, O. Kupfermn, nd M. Skrzypczk 18:5 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. 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 cceptnce-condition 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. Finitestte 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 pndinpers 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. F S T T C S

6 18:6 How Deterministic re Good-For-Gmes Automt? q 2 q 1 q Figure 1 A wekly tight GFG-NPW A 0. The numers elow the sttes descrie their priorities. q 2 m 2 2 q 1 m 1 m 1 q 0 m 0 0 q 2 m 2 2 q 1 m 1 m 1 q 0 m Figure 2 A strtegy witnessing the GFGness of the utomton A 0, depicted in Figure 1. Figure 3 A strtegy witnessing the tightness of su-utomton of A 0. Exmple 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}. Proof sketch. 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. It is esy to check tht if w L 0 (i.e. w contins infinitely mny s) then A 0g ccepts w. The following lemm generlizes known residul properties of GFG utomt (c.f., [6]). Lemm 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 ). 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.

7 U. Boker, O. Kupfermn, nd M. Skrzypczk 18:7 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 3. The GFG-NPW A 0 from Exmple 1 is wekly tight nd is not tight with respect to the strtegy g. Indeed, while ll the trnsitions in A 0 re 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 4. For every GFG utomton A there exists n equivlent tight GFG utomton A. Moreover, A is defined on sustructure of A. Lemm 5. For every tight Rin GFG utomton, there exists n equivlent strongly tight Rin GFG utomton over the sme structure. Exmple 6. In Figure 3 we descrie strtegy g tht witnesses the tightness of GFG- NPW on sustructure of the GFG-NPW A from Exmple 1. The strtegy g is otined from g y following the procedure descried in the proof of Lemm 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). 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 NBWrelizle, 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 lowup 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. F S T T C S

8 18:8 How Deterministic re Good-For-Gmes Automt? q 00 q q 10 q p 0 p 1 p , Figure 4 A 1: An unmiguous NPW tht is DBW-relizle yet is not Büchi-type. q 0 q 1 q 2 q 3 Figure 5 A 2: An unmiguous NBW tht is DCW-relizle yet is not co-büchi-type. 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 DBWrelizle lnguge nd, moreover, when they re unmiguous. Also, we prove the nontypeness results for NPWs, thus they pply to oth Rin nd Streett utomt. Exmple 7. Unmiguous NPWs re not Büchi-type with respect to DBW-relizle lnguges: The utomton A 1 depicted in Figure 4 is unmiguous nd recognizes DBWrelizle lnguge, yet A 1 is not Büchi-type. Moreover, we cnnot prune trnsitions from A 1 nd otin n equivlent Büchi-type NPW. The dul cse of unmiguous NPWs tht re not co-büchi-type with respect to DCWrelizle 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. Exmple 8. [7] Unmiguous NPWs (nd even NBWs) re not co-büchi-type with respect to DCW-relizle lnguges: The NBW A 2 depicted in Figure 5 is unmiguous, nd recognizes DCW-relizle lnguge, yet A 2 is not co-büchi-type. Moreover, we cnnot prune trnsitions from A 2 for otining n equivlent co-büchi-type NPW. 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 9. Unmiguous GFG utomt re DetByP. 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

9 U. Boker, O. Kupfermn, nd M. Skrzypczk 18:9 proof of Theorem 10 only requires the GFG utomt to e wekly tight (rther thn fully tight), implying tht Theorem 10 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 10. 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 sketch. Given GFG-NSW A nd strtegy g tht witnesses its GFGness, we chnge A into n NCW A y defining the co-büchi cceptnce condition α to include the sttes ll of whose g-memories only elong to rejecting cycles in A g. We then prove tht L(A) = L(A ) nd tht A is indeed GFG. (Furthermore, we show tht the originl strtegy g lso witnesses the GFGness of A.) The proof goes y induction, iterting over ll sttes q of A, nd grdully chnging the cceptnce condition until it ecomes co-büchi condition. We show tht t ech step of the induction, the resulting utomton is still GFG-NSW tht recognizes the originl lnguge. We strt the induction with the originl GFG-NSW A, nd dd to its cceptnce condition new empty Streett pir, nmely (, ). Along the induction, hndling stte q, we remove q from ll the originl d sets (nmely the left components) of the cceptnce condition, nd in the cse tht q α, we dd it to the d set of the new cceptnce pir. Oserve tht t the end of the induction, the cceptnce condition consists of single Streett pir, in which α is the d set nd is the good set. Thus, it is in fct the NCW A. The tricky prt of the induction step is when the considered stte q does not elong to α. In this cse, removing it from the d sets might enlrge the lnguge of the utomton. To prove tht the lnguge is not ltered, we tke dvntge of the fct tht there exists deterministic co-büchi utomton D equivlent to A, nd provide pumping scheme tht proceeds long the cycles of A nd D. Anlyzing these cycles, we use the (wek) tightness of A, in order to use Lemm 2 the pumped cycles might go through sttes of A tht were not originlly visited. Yet, due to Lemm 2, we my link the residul lnguges of the originlly visited sttes with tht of the newly visited ones. The following exmple shows tht the wek tightness requirement cnnot e omitted, even when the GFG-NSW is ctully GFG-NBW. Exmple 11. The utomton A 3 depicted in Figure 7 is GFG-NBW nd recognizes GFG-NCW-relizle lnguge, yet A 3 hs no equivlent NCW on the sme structure. 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 Büchitypeness 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 10 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. F S T T C S

10 18:10 How Deterministic re Good-For-Gmes Automt? p 0 p 1, q 0 q 1, Figure 6 A 4: A wekly tight GFG-NCW tht is GFG-NBW-relizle yet is not Büchitype. q 1 q 0 p 0 p 2 Figure 7 A 3: A GFG-NBW tht is GFG- NCW-relizle yet is not co-büchi-type. p 1 Exmple 12. The utomton A 4 depicted in Figure 6 is wekly tight GFG-NCW tht recognizes GFG-NBW-relizle lnguge, yet A 4 hs no equivlent GFG-NBW on the sme structure. We now proceed to our min positive result, otining the typeness of GFG-NRWs. Theorem 13. 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. Proof sketch. 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 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. The proof goes long the following steps: We strt with showing tht memory of stte q cnnot elong to oth q-exclusive ccepting cycle nd rejecting cycle. In order to prove tht, we tke dvntge of the fct tht there is DBW D equivlent to A, nd provide pumping rgument concerning the cycles of A nd D. By the ove nd the strong tightness of A, we show tht every stte q of A tht ppers in some good set hs single memory, ll the cycles of A g tht include q re ccepting, nd t lest one of them is q-exclusive. Oserve tht this implies the lnguge continment L(A g) L(B). Yet, it does not gurntee tht L(A ) L(B), s there might e some rejecting cycle of A tht A g does not use, which ecomes ccepting y our chnges to the cceptnce condition. For precluding the forementioned concern nd concluding our proof, we show tht every stte q of A tht ppers in some good set does not elong to rejecting cycle. We

11 U. Boker, O. Kupfermn, nd M. Skrzypczk 18:11 prove it y pplying mximlity rgument on pths tht the strtegy g exhusts long hypotheticl rejecting cycle of A tht includes q. The following result follows directly from Lemm 4, Theorem 13, nd the determiniztion procedure for Büchi GFG utomt from [6]. Corollry 14. Every GFG-NRW with n sttes tht recognizes DBW-relizle lnguge hs n equivlent DBW with t most O(n 2 ) sttes. 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, from which we cn esily deduce it, y our previous typeness results, lso for the other types. Theorem 15. 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 sketch. Consider tight GFG-NCW A tht recognizes GFG-NWW-relizle lnguge. 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. Anlyzing the cycles of A, we show tht it is GFG-NWW equivlent to A. Consider now GFG-NSW A tht is GFG-NWW-relizle. Notice tht it is oviously lso GFG-NBW-relizle. Hence, y Theorem 10, there is GFG-NCW on A s structure, nd y Theorem 15 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 13, there is GFG-NBW on A s structure, nd y Theorem 15 lso GFG-NWW. Corollry 16. 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 17. GFG-NWWs re DetByP. Proof sketch. Consider GFG-NWW A with ccepting set α, whose GFGness is witnessed y strtegy g. By rguments nlogous to those mde in the proof of Theorem 13, stte q α hs only one g-memory, nd is therefore lredy deterministic. The interesting prt concerns stte q / α tht hs t lest two g-memories m nd m. Let g e the strtegy otined y removing m from g nd redirecting trnsitions to m into m. We clim tht L(A g ) = L(A g ), from which it follows y induction tht the numer of memories of ech stte of A cn e reduced to 1. We prove the clim y induction on the strongly connected components of A nd y inductively pplying Lemm 2. F S T T C S

12 18:12 How Deterministic re Good-For-Gmes Automt? By comining the ove results, we otin the following corollry. Corollry 18. Every GFG-NSW nd GFG-NRW tht recognizes GFG-NWW-relizle lnguge is DetByP. 7 Consequences GFG utomt provide n interesting formlism in etween deterministic nd nondeterministic utomt. Their trnsltion to deterministic utomt is immedite for the wek condition (Theorem 17), polynomil for the Büchi condition [6], nd exponentil for the co-büchi, prity, Rin, nd Streett conditions [6]. They hve the sme typeness ehvior s deterministic utomt, summrized in Tle 1. The positive results of Tle 1 follow from our theorems in Sections 4, 5, nd 6. The negtive results follow from corresponding counterexmples with deterministic utomt [5, 7]. Considering the complementtion of GFG utomt, they lie in etween the deterministic nd nondeterministic settings, s shown in Tle 2. As for the trnsltion of LTL formuls to GFG utomt, it is douly exponentil, like the trnsltion to deterministic utomt (Corollry 21 elow). Complementtion In the deterministic setting, Rin nd Streett utomt re dul: complementing DRW into DSW, nd vice vers, is simply done y switching etween the two cceptnce conditions on top of the sme structure. This is not the cse with GFG utomt. We show elow tht complementing GFG-NSW, nd even GFG-NCW, into GFG-NRW involves n exponentil stte low-up. Essentilly, it follows from the Büchi-typeness of GFG-NRWs (Theorem 13) nd the fct tht while determiniztion of GFG-NBWs involves only qudrtic low-up, determiniztion of GFG-NCWs involves n exponentil one [6]. Corollry 19. The complementtion of GFG-NCW into GFG-NRW involves 2 Ω(n) stte low-up. Using our typeness results, we get n lmost complete picture on complementtion of GFG utomt. Theorem 20. The stte low-up involved in the complementtion of GFG utomt is s summrized in Tle 2. Proof. From wek nd Büchi. A GFG-NBW A with n sttes hs n equivlent DBW D with up to n 2 sttes [6], on which structure there is DCW D for the complement lnguge. Notice tht D is lso GFG-NCW, GFG-NPW, GFG-NRW, nd GFG-NSW. Now, if there is GFG-NBW equivlent to D, then D is DWW-recognizle, nd, y Theorem 15, there is GFG-NWW on sustructure of D. From co-büchi. By Corollry 19, we hve the exponentil stte low-up in the complementtion to GFG-NPW nd GFG-NRW utomt. Since the complement of co-büchi-recognizle lnguge is DBW-recognizle, we get n exponentil stte lowup lso to GFG-NBW. To wek nd co-büchi. Consider GFG-NCW, GFG-NPW, or GFG-NRW A with n sttes tht cn e complemented into GFG-NCW C. Then the lnguge of A is GFG-NBW recognizle. Thus, y Theorem 13, there is GFG-NBW equivlent to A with up to n sttes. Hence, y cse (1), there is GFG-NCW for the complement of A with up to n 2 sttes.

13 U. Boker, O. Kupfermn, nd M. Skrzypczk 18:13 Tle 1 Typeness in trnsltions etween GFG utomt. (Y=Yes; N=No.) Tle 2 The stte low-up involved in the complementtion of GFG utomt. Type To W B C P R S From Comp. To W C B P R S From Wek Wek Büchi Büchi Poly Co-Büchi Yes Co-Büchi Prity Rin No Y N Prity Rin Exp? Streett N Y No Y Streett From Streett to wek. Consider GFG-NSW A tht cn e complemented to GFG- NWW. Then the lnguge of A is DWW-recognizle. Thus, y Theorems 10 nd 15, there is GFG-NWW on sustructure of A, nd we re ck in cse (1). From Streett to co-büchi. Given DRW A tht is NCW relizle, one cn trnslte it to n equivlent NCW y first dulizing A into DSW A for the complement lnguge, nd then complementing A into GFG-NCW C. Since dulizing DRW into DSW is done with no stte lowup nd the trnsltion of DRWs to NCWs might involve n exponentil stte lowup [1], so does the complementtion of GFG-NSW to GFG-NCWs. From Streett to Streett. Anlogous to the ove cse of Streett to co-büchi, due to the exponentil stte lowup in the trnsltion of DRWs to NSWs [1]. Trnslting LTL formuls to GFG Automt Recll tht GFG-NCWs re exponentilly more succinct thn DCWs [6], suggesting they do hve some power of nondeterministic utomt. A nturl question is whether one cn come up with n exponentil trnsltion of LTL formuls to GFG utomt, in prticulr when ttention is restricted to LTL formuls tht re DCW-relizle. We complete this section with negtive nswer, providing nother evidence for the deterministic nture of GFG utomt. This result is sed on the fct tht the lnguge with which the doulyexponentil lower ound of the trnsltion of LTL to DBW in [9] is proven is ounded (tht is, it is oth sfe nd co-sfe). It mens tht y Corollry 18, ny GFG-NSW for it would e DetByP, contrdicting the douly-exponentil lower ound. Corollry 21. The trnsltion of DCW-relizle LTL formuls into GFG-NSW is douly exponentil. References 1 U. Boker. Rin vs. Streett utomt. In Proc. 37th Conf. on Foundtions of Softwre Technology nd Theoreticl Computer Science, pges 17:1 17:15, U. Boker, D. Kupererg, O. Kupfermn, nd M. Skrzypczk. Nondeterminism in the presence of diverse or unknown future. In Proc. 40th Int. Colloq. on Automt, Lnguges, nd Progrmming, volume 7966 of Lecture Notes in Computer Science, pges , T. Colcomet nd C. Löding. Regulr cost functions over finite trees. In Proc. 25th IEEE Symp. on Logic in Computer Science, pges 70 79, F S T T C S

14 18:14 How Deterministic re Good-For-Gmes Automt? 4 T.A. Henzinger nd N. Pitermn. Solving gmes without determiniztion. In Proc. 15th Annul Conf. of the Europen Assocition for Computer Science Logic, volume 4207 of Lecture Notes in Computer Science, pges Springer, S.C. Krishnn, A. Puri, nd R.K. Bryton. Deterministic ω-utomt vis--vis deterministic Büchi utomt. In Algorithms nd Computtions, volume 834 of Lecture Notes in Computer Science, pges Springer, D. Kupererg nd M. Skrzypczk. On deterministion of Good-For-Gmes utomt. In Proc. 42nd Int. Colloq. on Automt, Lnguges, nd Progrmming, pges , O. Kupfermn, G. Morgenstern, nd A. Murno. Typeness for ω-regulr utomt. Interntionl Journl on the Foundtions of Computer Science, 17(4): , O. Kupfermn, S. Sfr, nd M.Y. Vrdi. Relting word nd tree utomt. Ann. Pure Appl. Logic, 138(1-3): , O. Kupfermn nd M.Y. Vrdi. From liner time to rnching time. ACM Trnsctions on Computtionl Logic, 6(2): , O. Kupfermn nd M.Y. Vrdi. Sfrless decision procedures. In Proc. 46th IEEE Symp. on Foundtions of Computer Science, pges , L.H. Lndweer. Decision prolems for ω utomt. Mthemticl Systems Theory, 3: , G. Morgenstern. Expressiveness results t the ottom of the ω-regulr hierrchy. M.Sc. Thesis, The Herew University, D.E. Muller, A. Soudi, nd P.E. Schupp. Wek lternting utomt give simple explntion of why most temporl nd dynmic logics re decidle in exponentil time. In Proc. 3rd IEEE Symp. on Logic in Computer Science, pges , D. Niwiński nd I. Wlukiewicz. Relting hierrchies of word nd tree utomt. In Proc. 15th Symp. on Theoreticl Aspects of Computer Science, volume 1373 of Lecture Notes in Computer Science. Springer, M.O. Rin nd D. Scott. Finite utomt nd their decision prolems. IBM Journl of Reserch nd Development, 3: , S. Sfr. On the complexity of ω-utomt. In Proc. 29th IEEE Symp. on Foundtions of Computer Science, pges , H. Seidl nd D. Niwiński. On distriutive fixed-point expressions. Theoreticl Informtics nd Applictions, 33(4 5): , W. Thoms. On the synthesis of strtegies in infinite gmes. In E.W. Myr nd C. Puech, editors, Proc. 12th Symp. on Theoreticl Aspects of Computer Science, volume 900 of Lecture Notes in Computer Science, pges Springer, M.Y. Vrdi nd P. Wolper. Resoning out infinite computtions. Informtion nd Computtion, 115(1):1 37, 1994.