Typeness for ω-regular Automata

Interntionl Journl of Foundtions of Computer Science c World Scientific Pulishing Compny Typeness for ω-regulr Automt Orn Kupfermn School of Engineering nd Computer Science Herew University, Jeruslem 91904, Isrel orn@cs.huji.c.il Gil Morgenstern School of Engineering nd Computer Science Herew University, Jeruslem 91904, Isrel gil@cs.huji.c.il Aniello Murno Diprtimento di Informtic ed Appliczioni Università degli Studi di Slerno, 84081 Bronissi, Itly murno@unis.it Received (received dte) Revised (revised dte) Communicted y Editor s nme ABSTRACT We introduce nd study three notions of typeness for utomt on infinite words. For n cceptnce-condition clss γ (tht is, γ is wek, Büchi, co-büchi, Rin, or Streett), deterministic γ-typeness sks for the existence of n equivlent γ-utomton on the sme deterministic structure, nondeterministic γ-typeness sks for the existence of n equivlent γ-utomton on the sme structure, nd γ-powerset-typeness sks for the existence of n equivlent γ-utomton on the (deterministic) powerset structure one otined y pplying the suset construction. The notions re helpful in studying the complexity nd compliction of trnsltions etween the vrious clsses of utomt. For exmple, we prove tht deterministic Büchi utomt re co-büchi type; it follows tht trnsltion from deterministic Büchi to deterministic co-büchi utomt, when exists, involves no low up. On the other hnd, we prove tht nondeterministic Büchi utomt re not co-büchi type; it follows tht trnsltion from nondeterministic Büchi to nondeterministic co-büchi utomt, when exists, should e more complicted thn just redefining the cceptnce condition. As third exmple, y proving tht nondeterministic co-büchi utomt re Büchi-powerset type, we show tht trnsltion of nondeterministic co-büchi to deterministic Büchi utomt, when exists, cn e done pplying the suset construction. We give complete picture of typeness for the wek, Büchi, co-büchi, Rin, nd Streett cceptnce conditions, nd discuss its usefulness. Keywords: Automt on Infinite Words, Acceptnce Conditions 1. Introduction A preliminry version of this pper ppers in the Proceedings of the 2nd Interntionl Symposium on Automted Technology for Verifiction nd Anlysis, 2004. This work ws done while the uthor ws visiting the Herew University. 1

Finite utomt on infinite ojects were first introduced in the 60 s. Motivted y decision prolems in mthemtics nd logic, Büchi, McNughton, nd Rin developed frmework for resoning out infinite word nd infinite trees [4, 14, 18]. The frmework hs proved to e very powerful. Automt, nd their tight reltion to second-order mondic logics, were the key to the solution of severl fundmentl decision prolems in mthemtics nd logic [22]. Tody, utomt on infinite ojects re used for specifiction nd verifiction of nonterminting systems. In the utomt-theoretic pproch to verifiction, we reduce questions out systems nd their specifictions to questions out utomt. More specificlly, questions such s stisfiility of specifictions nd correctness of systems with respect to their specifictions re reduced to questions such s nonemptiness nd lnguge continment [23, 10, 24]. The utomt-theoretic pproch seprtes the logicl nd the comintoril spects of resoning out systems. The trnsltion of specifictions to utomt hndles the logic nd shifts ll the comintoril difficulties to utomt-theoretic prolems. Recent industril-strength property-specifiction lnguges such s Sugr [2], ForSpec [1], nd the recent stndrd PSL 1.01 [www.cceller.org] include regulr expressions nd/or utomt, mking the utomt-theoretic pproch even more essentil. Since run of n utomton on n infinite word does not hve finl stte, cceptnce is determined with respect to the set of sttes visited infinitely often during the run. There re mny wys to clssify n utomton on infinite words. One is the clss of its cceptnce condition. For exmple, in Büchi utomt, some of the sttes re designted s ccepting sttes, nd run is ccepting iff it visits sttes from the ccepting set infinitely often [4]. Dully, in co-büchi utomt, run is ccepting iff it visits sttes from the ccepting set only finitely often. More generl re Rin utomt. Here, the cceptnce condition is set α = { G 1,B 1,..., G k,b k } of pirs of sets of sttes, nd run is ccepting if there is pir G i,b i for which the set of sttes visited infinitely often intersects G i nd is disjoint to B i. The condition α cn lso e viewed s Streett condition, in which cse run is ccepting if for ll pirs G i,b i, if the set of sttes visited infinitely often intersects G i, then it lso intersects B i. The numer k of pirs in α is referred to s the index of the utomton. Another wy to clssify n utomton is y the type of its rnching mode. In deterministic utomton, the trnsition function δ mps pir of stte nd letter into single stte. The intuition is tht when the utomton is in stte q nd it reds letter σ, then the utomton moves to stte δ(q,σ), from which it should ccept the suffix of the word. When the rnching mode is nondeterministic, δ mps q nd σ into set of sttes, nd the utomton should ccept the suffix of the word from one of the sttes in the set. The pplictions of utomt theory in resoning out systems hve led to the development of new clsses of utomt. In [17], Muller et l. introduced wek utomt. Wek utomt cn e viewed s specil cse of Büchi or co-büchi utomt in which every strongly connected component in the grph induced y the structure of the utomton is either contined in the ccepting set or is disjoint from it. Since resoning out specifictions is often done y recursively resoning 2

out their su-specifictions, known trnsltions of temporl-logic specifictions to Büchi utomt ctully result in wek utomt [17, 9, 7]. The specil structure of wek utomt is reflected in their ttrctive computtionl properties nd mkes them very ppeling. Essentilly, while the formultion of cceptnce y Büchi or co-büchi utomton involves lterntion etween lest nd gretest fixed-points, no lterntion is required for specifying cceptnce y wek utomton [9]. Deterministic wek utomt hve recently eing used to represent rel numers. A rel numer x in se r is represented y word in the form w i w f where w i is the integer prt of x nd w f is the flot prt of x, nd oth re words over the lphet {0,1,...,r 1}. This wy for instnce, the rel numer 5 1 2 in se r = 10 is represented y 0 5 50 ω or y 0 5 49 ω. In similr wy, vector v = x 1,x 2,...,x n of rel numers is represented y word of the form W i W f where W i is in ({0,1,...,r 1} n ) nd W f is in ({0,1,...,r 1} n ) ω. As rel numers my hve severl representtions, rel vectors my hve severl representtions too. A rel vector utomton is Büchi utomton tht either ccepts ll the representtions of some vector v IR n or none of them. It is proved in [3] tht n RVA is deterministic wek utomton. It turns out tht different clsses of utomt hve different expressive power. For exmple, unlike utomt on finite words, where deterministic nd nondeterministic utomt hve the sme expressive power, deterministic Büchi utomt re strictly less expressive thn nondeterministic Büchi utomt [11]. Tht is, there exists lnguge L over infinite words such tht L cn e recognized y nondeterministic Büchi utomton ut cnnot e recognized y deterministic Büchi utomton. It lso turns out tht some clsses of utomt my e more succinct thn other clsses. For exmple, trnslting nondeterministic co-büchi utomton into deterministic one is possile [16], ut involves n exponentil low up. As nother exmple, trnslting nondeterministic Rin utomton with n sttes nd index k, into n equivlent nondeterministic Büchi utomton my result in n utomton with O(k n) sttes, nd if we strt with Streett utomton, the Büchi utomton my hve n 2 O(k) sttes [21]. Note tht expressiveness nd succinctness depend in oth the rnching type of the utomton s well s the clss of its cceptnce condition. There hs een extensive reserch on expressiveness nd succinctness of utomt on infinite words [22]. In prticulr, since resoning out deterministic utomt is simpler thn resoning out nondeterministic ones, questions like deciding whether nondeterministic utomton hs n equivlent deterministic one, nd the low-up involved in determiniztion re of prticulr interest. These questions get further motivtion with the discovery tht mny nturl specifictions correspond to the deterministic frgments: it is shown in [8] tht n LTL formul ψ hs n equivlent lterntion-free µ-clculus formul iff ψ cn e recognized y deterministic Büchi utomton, nd, s mentioned ove, rel vector utomt re deterministic wek utomt. For deterministic utomt, where Büchi nd co-büchi utomt re less expressive thn Rin nd Streett utomt, reserchers hve come up with the notion 3

of deterministic utomton eing Büchi type, nmely it hs n equivlent Büchi utomton on the sme structure [6]. It is shown in [6] tht Rin utomt re Büchi type. Thus, 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. On the other hnd, Streett utomt re not Büchi type: there is deterministic Streett utomton A tht recognizes lnguge tht cn e recognized y deterministic Büchi utomton, ut ll the possiilities of defining Büchi cceptnce condition on the structure of A result in n utomton recognizing different lnguge. As discussed in [6], Büchi-typeness is very useful notion. In prticulr, Büchi-type deterministic utomton cn e trnslted to n equivlent deterministic Büchi utomton with no low up. In this work, we study typeness in generl: we consider oth nondeterministic nd deterministic utomt, for the five clsses γ of cceptnce conditions descried ove (γ is either Büchi, co-büchi, Rin, Streett, or wek). We define nd exmine three notion of typeness: 1. Deterministic γ-typeness sks for which clsses of deterministic utomt, the existence of some equivlent deterministic γ utomton implies the existence of n equivlent deterministic γ utomton on the sme structure. For exmple, we show tht ll deterministic utomt re wek type. 2. Nondeterministic γ-typeness sks for which clsses of nondeterministic utomt, the existence of some equivlent nondeterministic γ utomton implies the existence of n equivlent nondeterministic γ utomton on the sme structure. For exmple, we show tht nondeterministic Büchi utomt re not co-büchi type. This nswers question on trnslting Büchi to co-büchi utomt tht ws left open in [8]. 3. γ-powerset-typeness sks for which clsses of nondeterministic utomt, the existence of some equivlent deterministic γ utomton implies the existence of n equivlent deterministic γ utomton on the structure otined y pplying the suset construction to the originl utomton. For exmple, while deterministic Rin utomt re Büchi-type, nondeterministic Rin utomt re not Büchi powerset-type. The notion of powerset-typeness is importnt for the study of the low-up involved in the trnsltion of utomt to equivlent deterministic ones. While for some clsses 2 O(n log n) lower ound is known, powerset-typeness implies 2 n upper ound for other clsses. We lso exmine finite-typeness for nondeterministic Büchi utomt cses where the limit lnguge of the utomton when viewed s n utomton on finite words is equivlent to tht of the Büchi utomton, nd we relte finite-typeness with powerset-typeness. Our results, long with previously known results, re descried in Figures 2, 3, nd 5. 4

2. Preliminries Given n lphet Σ, n infinite word over Σ is n infinite sequence w = σ 0 σ 1 σ 2 of letters in Σ. We denote the set of ll infinite words over Σ y Σ ω. A lnguge L is set of words from Σ ω. An utomton over infinite words is tuple A = Σ,Q,δ,Q 0,α, where Σ is the input lphet, Q is finite set of sttes, δ : Q Σ 2 Q is trnsition function, Q 0 Q is set of initil sttes, nd α is n cceptnce condition which is condition tht defines suset of Q ω. We define severl cceptnce conditions elow. Intuitively, δ(q, σ) is the set of sttes tht A my move into when it is in the stte q nd it reds the letter σ. The utomton A my hve severl initil sttes nd the trnsition function my specify mny possile trnsitions for ech stte nd letter, nd hence we sy tht A is nondeterministic. In the cse where Q 0 = 1 nd for every q Q nd σ Σ, we hve tht δ(q,σ) = 1, we sy tht A is deterministic. Given n input infinite word w = σ 0 σ 1 σ 2 Σ ω, run of A on w cn e viewed s function r : IN Q where r(0) Q 0, i.e., the run strts in one of the initil sttes, nd for every i 0, we hve tht r(i + 1) δ(r(i),σ i ), i.e., the run oeys the trnsition function. Note tht while deterministic utomton hs single run on n input word w, nondeterministic utomton my hve severl runs on w or none t ll. Ech run r induces set inf(r) of sttes tht r visits infinitely often. Formlly, inf(r) = {q Q : for infinitely mny i IN, we hve r(i) = q}. As Q is finite, it is gurnteed tht inf(r). The run r is ccepting iff the set inf(r) stisfies the cceptnce condition α. A run tht is not ccepting is rejecting. We consider the following cceptnce conditions. A set S stisfies Büchi cceptnce condition α Q if nd only if S α. A set S stisfies co-büchi cceptnce condition α Q if nd only if S α =. A set S stisfies Rin cceptnce condition α = { G 1,B 1,..., G k,b k } 2 Q 2 Q if nd only if there exists pir G i,b i α for which S G i nd S B i =. A set S stisfies Streett cceptnce condition α = { G 1,B 1,..., G k,b k } 2 Q 2 Q if nd only if for ll pirs G i,b i α we hve tht S G i = or S B i. Note tht the Büchi cceptnce condition is dul to the co-büchi cceptnce condition: set S stisfies Büchi cceptnce condition α iff S does not stisfy α s co-büchi cceptnce condition. Similrly, the Rin cceptnce condition is dul to the Streett cceptnce condition. The numer k ppering in the Rin nd Street conditions is clled the index of the utomton. An utomton A ccepts n input word w iff there exists n ccepting run of A on w. The lnguge of A, denoted L(A), is the set of ll infinite words tht A ccepts. The trnsition function δ induces reltion R δ Q Q, where R δ (q,q ) iff there is σ Σ with δ(q,σ) = q. Accordingly, the utomton A induces grph G A = Q,R δ. For two sttes, q nd q of A, we sy tht q is rechle from q if 5

there is (possily empty) pth in G A from q to q. A strongly connected component (SCC, for short) in G A is set C Q such tht for ll sttes q nd q in C, we hve tht q is rechle from q. The SCC C is non-trivil if the restriction of G A to C contins cycle; tht is, either C hs t lest two sttes, or C hs stte with self loop. A mximl strongly connected component (MSCC, for short ) is n SCC C tht is mximl in the sense tht we cnnot dd to C sttes nd sty with n SCC. Thus, for ll C Q \ C, the set C C is not n SCC. Note tht run of n utomton A eventully get trpped in n MSCC of G A. We sy tht Büchi utomton A is wek if for ech MSCC C of G 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. Indeed, run of A visits α infinitely often iff it gets trpped in n ccepting component, which hppens iff it visits sttes in Q \ α only finitely often. We denote the different types of utomt y three letters cronyms in {D,N} {F, B, C, R, S, W} {W,T}. The first letter stnds for the rnching mode of the utomton (deterministic or nondeterministic); the second letter stnds for the cceptnce-condition type (finite, Büchi, co-büchi, Rin, Streett, or wek). The third letter stnds for the ojects on which the utomt run (words or trees). 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. 2.1. Expressiveness nd Typeness 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. In Figure 1 elow we descrie the known expressiveness hierrchy for utomt on infinite words. As descried in the figure, DRW nd DSW re s expressive s NRW, NSW, nd NBW, which recognize ll ω-regulr lnguge [14]. On the other hnd, DBW re strictly less expressive thn NBW, nd so re DCW. In fct, since y dulizing Büchi utomton we get co-büchi utomton, the two internl ovls complement ech other. The intersection of DBW nd DCW is DWW (note tht while DWW is clerly oth DBW nd DCW, the other direction is not trivil, nd is proven in [3]). Finlly, NCW cn e determinized (when pplied to universl Büchi utomt, the trnsltion in [16], of lternting Büchi utomt into NBW, results in DBW. By dulizing it, one gets trnsltion of NCW to DCW). In ddition to the results descried in the figure, the index of DRW nd DSW lso induces hierrchy, thus DRW[k + 1] re strictly more expressive thn DRW[k], nd similrly for DSW [5]. Consider n utomton A = Σ,Q,δ,Q 0,α. We refer to Q,δ,Q 0 s the structure of the utomton. The powerset structure induced y A is P(A) = 2 Q,δ P, {Q 0 }, where for ll S 2 Q nd σ Σ, we hve tht δ P (S,σ) = s S δ(s,σ). The nottion SCC is sometimes used in the literture to denote mximl SCC. Here, we use oth MSCC (mximl SCC) nd SCC (not necessrily mximl SCC). 6

NBW, NRW, DRW, NSW, DSW NCW DCW NWW DWW DBW Figure 1: The expressiveness hierrchy for ω-regulr utomt. Thus, the powerset structure is otined y the usul suset construction [19]. 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. Tht is, there is n utomton A = Σ,Q,δ,Q 0,α such tht α is n cceptnce condition of clss γ nd L(A ) = L(A). We sy tht A is γ-powerset-type if A hs n equivlent γ utomton with the sme structure s the powerset structure of A. Tht is, there is n utomton A P = Σ,2 Q,δ P, {Q 0 },α P such tht α P is n cceptnce condition of clss γ nd L(A P ) = L(A). Note tht the utomton A P is deterministic. 3. Typeness for Deterministic Automt In this section we consider the following prolem: given two cceptnce-condition types β nd γ, is it true tht every DβW tht is DγW-relizle, is lso γ-type? We then sy tht DβW re γ-type. In other words, DβW re γ-type if every deterministic β-utomton tht hs n equivlent deterministic γ-utomton, lso hs n equivlent deterministic γ-utomton on the sme structure. Our results re descried in Figure 2 elow. Some results re immedite. For exmple, since the Büchi nd the co-büchi cceptnce conditions re specil cses of Rin nd Streett conditions ( Büchi condition α is equivlent to the Rin condition { α, } nd to the Streett condition { Q, α }, nd dully for co-büchi), it is cler tht DBW nd DCW re Rin-type nd Streett-type. Similrly, since wek utomt cn e viewed s Büchi or co-büchi utomt, they cn lso e viewed s specil cse of Rin nd Streett utomt. Thus, DWW re γ-type for ll the types γ we consider. Such cses, where trnsltion of the cceptnce condition exists, nd is independent of the utomton, re indicted in the tle y. Some results re known, or otined esily y dulizing known results, nd the tle contins the pproprite reference. Below we prove the new results. Lemm 1 DβW re wek-type for ll cceptnce-condition types β. Proof. In [3], the uthors introduce the notion of deterministic utomton eing inherently wek (the definition in [3] is for DBW, nd is esily extended to DβW for ll cceptnce-condition types β). A DβW is inherently wek if none of its rechle MSCC contins oth ccepting nd rejecting non-trivil SCCs. It is esy to see tht n inherently wek utomton hs n equivlent DWW on the sme 7

DWW DBW DCW DRW DSW DWW YES YES YES YES Lemm 1 Lemm 1 Lemm 1 Lemm 1 DBW YES YES YES NO Lemm 2 [6] [6] DCW YES YES NO YES Lemm 2 dulizing [6] dulizing [6] DRW YES YES YES NO Lemm 3 DSW YES YES YES NO Lemm 3 DRW[k] re not Rin[k 1]-type, DSW[k] re not Streett[k 1]-type. Lemm 4 Figure 2: Typeness for deterministic utomt. structure. Indeed, y definition, ech of the MSCC of the utomton cn e mde ccepting or rejecting ccording to the clssifiction of ll its non-trivil SCCs. Let A e DWW-relizle DβW. Then, A is oth DBW-relizle nd DCWrelizle. Assume y the wy of contrdiction tht A is not wek type. Then, A is not inherently wek, so there exists rechle MSCC C of A such tht C contins oth n ccepting non-trivil SCC S nd rejecting non-trivil SCC R. Since A is DBW-relizle, then, y [11], every SCC S S is ccepting. In prticulr, C is ccepting. Dully, Since A is DCW-relizle, then every SCC R R is rejecting. In prticulr, C is rejecting. It follows tht C is oth ccepting nd rejecting, nd we rech contrdiction. We note tht [3] prove tht every DBW tht ccepts lnguge in F σ G δ is inherently wek. The proof there, however, does not mke direct use of [11], nd is therefore much more complicted. Lemm 2 DCW re Büchi-type, nd DBW re co-büchi-type. Proof. Since DCW cn e viewed s DRW, nd DRW re Büchi type [6], DCW re Büchi type too. Dully, DBW re co-büchi-type. Note tht if DCW A is DBW-relizle, then it is lso DWW-relizle. Indeed, y [3], DCW DBW = DWW. Hence, y Lemm 1, A hs n equivlent deterministic wek utomton on the sme structure. Thus, Lemm 2 cn e strengthened: DCW tht is DBW-relizle (dully, DBW tht is DCWrelizle) hs n equivlent deterministic wek utomton on the sme structure. Lemm 3 DRW re not Streett-type, nd DSW re not Rin-type. Proof. Since DSW cn recognize ll ω-regulr lnguges, DSW eing Rintype mens tht every DSW hs n equivlent DRW on the sme structure. In [12], Löding shows tht trnsltion of DSW to n equivlent DRW my involve n exponentil low up, thus typeness oviously cnnot hold. The rgument for DRW is dul. 8

In ddition to the results in the tle, we prove tht the expressiveness hierrchy known for the indices of DRW nd DSW induces typeness hierrchy: Lemm 4 For ll k 2, we hve tht DRW[k] re not Rin[k 1]-type, nd DSW[k] re not Streett[k 1]-type. Proof. Let Σ k = {1,2,...,k}. Consider the lnguges L k of exctly ll words contining infinitely mny i s, for ll 1 i k. Consider the DSW[k] A k = Σ k,σ k,δ, {1},α k, with δ(q,i) = i, for ll q,i Σ k, nd α k = { Σ k, {1}, Σ k, {2},..., Σ k, {k} }. Thus, whenever A k reds letter i, it moves to stte i, nd the cceptnce condition requires n ccepting run to visit ll sttes infinitely often. It is esy to see tht A k recognizes L k. Also, since L k cn e viewed s the intersection of k DBWs D i, ech for the lnguge infinitely mny i s, we know tht L k is DBW-recognizle, nd hence lso DSW[k 1]-relizle. On the other hnd, it is impossile to define Streett[k 1] cceptnce condition α k so tht A k with condition α k recognizes L k. To see this, note tht for ech letter i Σ k, the DSW A k ccepts (1 2 k) ω nd rejects (1 2 i 1 i + 1 k) ω. For tht, A k must contin, for ech i Σ k, pir G i,b i such tht G i Σ k nd B i = {i}. Thus, A k must contin t lest k pirs, nd we re done. It follows tht DSW[k] re not Streett[k 1]-type. The rgument for Rin utomt is dul, nd considers the complement of L n. 4. Typeness for Nondeterministic Automt In this section we consider the following prolem: given two cceptnce-condition types β nd γ, is it true tht every NβW tht is NγW-relizle, is lso γ-type? We then sy tht NβW re γ-type. In other words, NβW re γ-type if every nondeterministic β-utomton tht hs n equivlent nondeterministic γ-utomton, lso hs n equivlent nondeterministic γ-utomton on the sme structure. Our results re descried in Figure 3 elow. As in Section 3, some results follow immeditely from trnsltions of the cceptnce condition, nd re indicted in the tle y. The new results re proven in Lemms 5, 6, nd 7. When the results follow from pplying trnsltions to results proven in the Lemms, we indicte it with too. NWW NBW NCW NRW NSW NWW NO NO NO NO Lemm 5 Lemm 6 Lemms 5 nd 6 Lemms 5 nd 6 NBW YES NO NO NO Lemm 6 Lemm 6 Lemm 6 NCW YES NO NO NO Lemm 5 Lemm 5 Lemm 5 NRW YES YES YES NO Lemm 7 NSW YES YES YES NO Lemm 7 Figure 3: Typeness for nondeterministic utomt. 9

Lemm 5 NBW re neither co-büchi- nor wek-type. Proof. Consider the NBW A 1 descried in Figure 4. The NBW recognizes the lnguge ( + ) (t lest one ). This lnguge is in NWW nd NCW, yet it is esy to see tht there is no NCW (nd hence lso no NWW) recognizing L on the sme structure. A 1 : A 2 : q 0 q 1 q 3 q 1 q 2 q 3, q 2 q 0,, Figure 4: NBWs for ( + ). We note tht the utomton in Figure 4 is single-run utomton: every word ccepted y it hs single ccepting run. This is of prticulr interest in the context of specifiction nd verifiction, s the NBW descried in [24] for LTL formuls re single-run utomt. Our exmple shows tht even such utomt re neither co- Büchi- nor wek-type. It is shown in [8] tht n LTL formul ψ hs n equivlent lterntion-free µ-clculus formul ψ iff the lnguge of ψ cn e recognized y DBW A ψ. The construction of the formul ψ in [8] goes vi A ψ, nd therefore it involve douly-exponentil low-up. The construction of ψ my lso go vi n NCW Ãψ, for ψ. While ψ is of length liner in the size of à ψ, the est known trnsltion of LTL to NCW (when exists) ctully constructs DCW nd is douly-exponentil. It is conjectured in [8] tht single-run NBW cn e trnslted to NCW with only liner low up, leding to n exponentil trnsltion of LTL to lterntion-free µ-clculus. In prticulr, the question of otining the NCW y modifying the cceptnce condition of the NBW is left open in [8]. Our result here nswers the question negtively. We lso note tht NCW-typeness nd wek-typeness do not coincide. Figure 4 lso descries different NBW, A 2, for L. This NBW is NCW-type: n NCW with the sme structure ut with the cceptnce condition α = {q 0,q 1 } ccepts L. Yet, it is not wek-type. Lemm 6 NCW re neither Büchi- nor wek-type. Proof. Consider the two-stte DCW A for the lnguge L of ll words with finitely mny s. Since L is not DBW-relizle, nd A is deterministic, A is not Büchi-type. The lnguge L is NWW-relizle. But gin, since A is deterministic nd L is not DWW-relizle, it is not wek-type. Lemm 7 NRW re not Streett-type, nd NSW re not Rin-type. 10

Proof. By Lemm 3, DRW re not Streett-type. Hence, there re DRW tht re DSW-relizle ut do not hve n equivlent DSW on the sme structure. Since DRW re specil cse of NRW, it follows tht NRW re not Streett-type. The proof for NSW not eing Rin-type is similr. By Lemm 4, DRW[k] re not Rin[k 1]-type, nd DSW[k] re not Streett[k 1]-type, for ll k 2. Thus, following the sme considertions s in the proof of Lemm 7, we get tht NRW[k] re not Rin[k 1]-type, nd NSW[k] re not Streett[k 1]-type. 5. Powerset-Typeness for Nondeterministic Automt In this section we consider the following prolem: given two cceptnce-condition types β nd γ, is it true tht every NβW tht is DγW-relizle, is lso γ-powersettype? We then sy tht NβW re γ-powerset-type. In other words, NβW re γ- type if every nondeterministic β-utomton tht hs n equivlent deterministic γ-utomton, lso hs n equivlent deterministic γ-utomton on the powerset structure. Our results re descried in Figure 5 elow. Since A = P(A) for deterministic utomton A, we know tht NβW cnnot e γ-powerset-type if DβW re not γ- type. Thus, the negtive cses in Figure 2 immeditely induce negtive cses here. In prticulr, for ll k 2, we hve tht NRW[k] re not Rin[k 1]-powerset-type, nd NSW[k] re not Streett[k 1]-powerset-type. NWW NBW NCW NRW NSW DWW YES YES YES YES YES [13] [13] [13] [13] [13] DBW YES NO YES NO NO Lemm 8 Lemm 9 Lemm 8 Lemm 9 Lemm 9 DCW NO NO NO NO NO Lemm 10 Lemm 10 Lemm 10 Lemm 10 Lemm 10 DRW NO NO NO NO NO Lemm 10 Lemm 10 Lemm 10 Lemm 10 Lemm 10 DSW NO NO NO NO NO Lemm 10 Lemm 10 Lemm 10 Lemm 10 Lemm 10 Figure 5: Powerset-typeness for nondeterministic utomt. Lemm 8 NWW nd NCW re Büchi-powerset-type. Proof. Consider n NCW A. Recll tht A is DCW-relizle. Therefore, if A is DBW-relizle, then it is lso DWW-relizle. Hence, s NCW re wekpowerset-type, there is DWW, nd thus lso DBW, equivlent to A with structure P(A). Thus, NCW re Büchi-powerset-type. Since NWW re specil cse of NCW, the result for NWW follows. Lemm 9 NBW re not Büchi-powerset-type. Proof. The NBW A in Figure 6 recognizes the lnguge of ll words with infinitely mny occurrences of the suword. The lnguge cn e recognized y 11

DBW, yet no DBW for it cn e defined on top of P(A). A : P(A) :, q 1 q 2 {q 1 } {q 1, q 2 } Figure 6: An NBW for (( + ) ) ω tht is not Büchi-powerset-type. Lemm 10 NWW re neither co-büchi-, Rin-, nor Streett-powerset-type. Proof. The NWW A in Figure 7 recognizes the lnguge of ll words with n ( ) ω til. The lnguge cn e recognized y DCW, nd hence lso y DRW nd DSW. Yet, no DCW, DRW, or DSW for it cn e defined on top of P(A). A :, P(A) : q 1, q 2 q 3 {q1 }, {q 1, q 2 } {q 1, q 2, q 3 } Figure 7: An NBW for ( + ) ( ) ω tht is neither co-büchi-, Rin-, nor Streett-powerset-type. The definition of powerset-typeness requires the deterministic utomton to hve the powerset structure, ut it does not restrict the definition of the set of ccepting sttes. For n utomton A = Σ,Q,δ,Q 0,α, let S(A) = Σ,2 Q,δ P, {Q 0 },α P e the utomton otined from A y pplying to it the suset construction. Thus, the structure of S(A) is the powerset-structure of A, nd stte is in α P if its intersection with α is not empty. We refer to S(A) s the suset utomton of A. Clerly, for n NFW A, we hve tht A nd S(A) re equivlent [19]. We sy tht n NBW A is Büchi-suset-type if A nd S(A) re equivlent. Note tht if A is Büchi-suset-type, then it is lso Büchi-powerset-type. As we shll see elow, the other direction does not necessrily hold. Lemm 11 There is n NBW tht is Büchi-powerset-type ut not Büchi susettype. Proof. The NBW A in Figure 8 recognizes the lnguge of ll words with infinitely mny s ut no two successive s. The DBW otined y ugmenting the powerset structure of A, lso descried in the figure, with the cceptnce condition α P = {{q 1 }} is equivlent to A. Thus, A is powerset type. On the other hnd, S(A) hs α P = {{q 1,q 2 }} nd is not equivlent to A. 5.1. From NBW to NFW 12

A : P(A) : q 1 q 2 {q 1 } {q 1, q 2 } Figure 8: An NBW for ( + ) ω tht is Büchi-powerset-type ut not finite-type. Recll tht DBW re strictly less expressive thn NBW. A lnguge L Σ ω cn e recognized y DBW iff there is regulr lnguge R Σ such tht L = limr; tht is, w L iff w hs infinitely mny prefixes in R [11]. An open prolem is to construct, given n NBW A for L, such tht A is DBW-relizle, n NFW A for the corresponding R. An immedite 2 O(n log n) upper ound follows from the 2 O(n log n) determiniztion construction of [20] for A (since DRW re Büchi type, the DRW constructed in [20] cn e converted to DBW on the sme structure). While the 2 O(n log n) low up in determiniztion is tight [15, 12], no super-liner lower ound is known for the trnsltion of A to A. The chllenges in this prolem re similr to these in the prolem of trnslting n NBW tht is NCW-relizle to n equivlent NCW. While 2 O(n log n) upper ound is immedite, no super-liner lower ound is known. Consider n NBW A = Σ,Q,δ,Q 0,α. We sy tht A is finite-type if there is n NFW A = Σ,Q,δ,Q 0,α such tht L(A) = liml(a ). Thus, A is finite-type if there is n NFW with the sme structure s A (ut possily with different set of ccepting sttes) whose limit lnguge is the lnguge of A. Let A fin e A viewed s n NFW. We sy tht A is strong-finite-type if L(A) = liml(a fin ). Thus, in strong-finite-typeness, we require the NFW to hve oth the structure nd the cceptnce condition of A. Oviously, the trnsition from n NBW A tht is finite-type to n NFW whose limit is A is liner. The notion of suset-typeness turns out to e relted to finite-typeness: Lemm 12 An NBW is Büchi-suset-type iff it is strong-finite-type. Proof. Assume first tht A is not Büchi suset-type. Since S(A) is DBW, then L(S(A)) = liml(s(a) fin ). Since A fin is n NFW, then L(A fin ) = L(S(A) fin ). It follows tht L(S(A)) = liml(a fin ). Since A is not Büchi suset-type, L(A) L(S(A)). It follows tht L(A) liml(a fin ), thus A is not strong-finite-type. Assume now tht A is Büchi suset-type. For every NBW A, we hve tht L(A) liml(a fin ). Indeed, n ccepting run of A on word w points to infinitely mny prefixes of w tht re ccepted y A fin. It is left to prove tht liml(a fin ) L(A). Consider word w liml(a fin ). Thus, w hs infinitely mny prefixes in L(A fin ). Since A fin is n NFW, then L(A fin ) = L(S(A) fin ). It follows tht w hs infinitely mny prefixes in L(S(A) fin ), or equivlently, tht the run of S(A) on w visits the set of ccepting sttes infinitely often, implying tht w L(S(A)). Since A is Büchi-suset-type, w is lso ccepted y A, nd we re done. It is worth noting tht not ll NBW tht re DWW-relizle re strong-finitetype. Indeed, s proved in [13], n NBW tht is DWW-relizle is lso Büchi- 13

powerset-type. On the other hnd, there re DBW tht re DWW-relizle nd re not Büchi-suset-type, nd hence lso not strong-finite-type. To see this, tke the NBW A 1 descried in Figure 4, dd n ccepting stte q tht is rechle from q 2 with trnsition leled, nd lso dd self-loop leled from q to itself. The otined NBW still ccepts L(A 1 ), ut the suset construction results in n utomton tht ccepts Σ ω. Recll tht oth suset-typeness nd strong-finite-typeness restrict the set of ccepting sttes. We could hve then hoped tht the notions of powerset-typeness nd finite-typeness, which oth do not restrict the set of ccepting sttes, would lso e relted. As we now show, this is not the cse. Lemm 13 Büchi-powerset-type NBW re not finite-type. Proof. As discussed in the proof of Lemm 11, the NBW A in Figure 8 is powerset type. On the other hnd, there is no wy to ugment A fin with n cceptnce condition α tht results in n utomton A for which lim(l(a )) = L(A). To see this, note tht either α is empty, in which cse L(A ) is empty, or α is not empty, in which cse L(A ) contins +, thus lim(l(a )) contins ω, which is not in L(A). 6. Discussion We studied three notions of typeness for utomt on infinite words. The notions re helpful in studying the complexity nd compliction of trnsltions etween the vrious clsses of utomt. Of specil interest is the low-up involved in trnsltion of NBW to NCW, when exists. As discussed in Section 4, polynomil trnsltion will enle n exponentil trnsltion of LTL to lterntion-free µ-clculus (for formuls tht cn e expressed in the lterntion-free µ-clculus), improving the douly-exponentil known upper ound. Current trnsltions of NBW to NCW ctully construct DCW with 2 O(n log n) sttes (strting with n NBW with n sttes), wheres even no super-liner lower ound is known. A relted notion hs to do with the trnsltion of n NBW to n NFW whose limit lnguge is equivlent to tht of the NBW. We studied lso this notion, nd chrcterized NBW tht re finite-type, nd for which liner trnsltion exists. We hope to relte finite-typeness with co-büchi typeness, iming t developing more techniques nd understnding for pproching the NBW to NCW prolem. References 1. R. Armoni, L. Fix, A. Flisher, R. Gerth, B. Ginsurg, T. Knz, A. Lndver, S. Mdor-Him, E. Singermn, A. Tiemeyer, M.Y. Vrdi, nd Y. Zr. The For- Spec temporl logic: A new temporl property-specifiction logic. In Proc. 8th Interntionl Conference on Tools nd Algorithms for the Construction nd Anlysis of Systems, volume 2280 of Lecture Notes in Computer Science, pges 296 211, Grenole, Frnce, April 2002. Springer-Verlg. 2. I. Beer, S. Ben-Dvid, C. Eisner, D. Fismn, A. Gringuze, nd Y. Rodeh. The temporl logic sugr. In Proc. 13th Interntionl Conference on Computer Aided 14

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