Topological, Automata-Theoretic and Logical Characterization of Finitary Languages

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1 Topologicl, Automt-Theoretic nd Logicl Chrcteriztion of Finitry Lnguges Krishnendu Chterjee nd Nthnël Fijlkow IST Austri (Institute of Science nd Technology Austri) Am Cmpus 1 A-3400 Klosterneuurg Technicl Report No. IST June 4, 2010

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3 Topologicl, Automt-Theoretic nd Logicl Chrcteriztion of Finitry Lnguges Krishnendu Chtterjee 1 nd Nthnël Fijlkow 1,2 1 IST Austri (Institute of Science nd Technology, Austri) 2 ENS Cchn (École Normle Supérieure de Cchn, Frnce) Astrct The clss of ω-regulr lnguges provide roust specifiction lnguge in verifiction. Every ω-regulr condition cn e decomposed into sfety prt nd liveness prt. The liveness prt ensures tht something good hppens eventully. Two min strengths of the clssicl, infinite-limit formultion of liveness re roustness (independence from the grnulrity of trnsitions) nd simplicity (strction of complicted time ounds). However, the clssicl liveness formultion suffers from the drwck tht the time until something good hppens my e unounded. A stronger formultion of liveness, so-clled finitry liveness, overcomes this drwck, while still retining roustness nd simplicity. Finitry liveness requires tht there exists n unknown, fixed ound such tht something good hppens within trnsitions. In this work we consider the finitry prity nd Streett (firness) conditions. We present the topologicl, utomt-theoretic nd logicl chrcteriztion of finitry lnguges defined y finitry prity nd Streett conditions. We () show tht the finitry prity nd Streett lnguges re Σ 2-complete; () present complete chrcteriztion of the expressive power of vrious clsses of utomt with finitry nd infinitry conditions (in prticulr we show tht non-deterministic finitry prity nd Streett utomt cnnot e determinized to deterministic finitry prity or Streett utomt); nd (c) show tht the lnguges defined y non-deterministic finitry prity utomt exctly chrcterize the str-free frgment of ωb-regulr lnguges. 1 Introduction Clssicl ω-regulr lnguges: strengths nd wekness. The clss of ω-regulr lnguges provide roust lnguge for specifiction for solving control nd verifiction prolems (see, e.g., [PR89,RW87]). Every ω-regulr specifiction cn e decomposed into sfety prt nd liveness prt [AS85]. The sfety prt ensures tht the component will not do nything d (such s violte n invrint) within ny finite numer of trnsitions. The liveness prt ensures tht the component will do something good (such s proceed, or respond, or terminte) in the long-run. Liveness cn e violted only in the limit, y infinite sequences of trnsitions, s no ound is stipulted on when the good thing must hppen. This infinitry, clssicl formultion of liveness hs oth strengths nd weknesses. A min strength is roustness, in prticulr, independence from the chosen grnulrity of trnsitions. Another min strength is simplicity, llowing liveness to serve s n strction for complicted sfety conditions. For exmple, component my lwys respond in numer of trnsitions tht depends, in some complicted mnner, on the exct size of the stimulus. Yet for correctness, we my e interested only tht the component will respond eventully. However, these strengths lso point to wekness of the clssicl definition of liveness: it cn e stisfied y components tht in prctice re quite unstisfctory ecuse no ound cn e put on their response time. Stronger notion of liveness. For the wekness of the infinitry formultion of liveness, lterntive nd stronger formultions of liveness hve een proposed. One of these is finitry liveness [AH94,DJP03]: finitry liveness does not insist on response within known ound (i.e., every stimulus is followed y response within trnsitions), ut on response within some unknown ound (i.e., there exists such tht every stimulus is followed y response within trnsitions). Note tht in the finitry cse, the ound my e ritrrily lrge, ut the response time must not grow forever from one stimulus to the next. In this wy, finitry liveness still mintins the roustness (independence of step grnulrity) nd simplicity (strction of complicted sfety) of trditionl liveness, while removing unstisfctory implementtions.

4 Finitry prity nd Streett conditions. The clssicl infinitry notion of firness is given y the Streett condition: Streett condition consists of set of d pirs of requests nd corresponding responses (grnts) nd the condition requires tht every request tht ppers infinitely often must e responsed infinitely often. The finitry Streett condition requires tht there is ound such tht in the limit every request is responsed within steps. The clssicl infinitry prity condition consists of priority function nd the condition requires tht the minimum priority visited infinitely often is even. The finitry prity condition requires tht there is ound such tht in the limit fter every odd priority lower even priority is visited within steps. Chrcteriztion of infinitry prity nd Streett utomt. There re severl roust lnguge-theoretic chrcteriztion of the lnguges expressile y utomt with infinitry liveness (Büchi), prity nd Streett conditions. Some of the importnt chrcteriztions re s follows: () Topologicl chrcteriztion: it is known tht deterministic utomt with Büchi conditions re Π 2 -complete, wheres non-deterministic Büchi nd oth deterministic nd non-deterministic prity nd Streett utomt lie in the oolen closure of Σ 2 nd Π 2 [MP92]; () Automt theoretic chrcteriztion: non-deterministic utomt with Büchi conditions hve the sme expressive power s deterministic nd non-deterministic prity nd Streett utomt [Cho74,Sf92]; nd (c) Logicl chrcteriztion: the clss of lnguges expressed y deterministic prity (tht is equivlent to non-deterministic Büchi, prity nd Streett utomt) is equivlent to the clss of ω-regulr lnguges nd is lso chrcterized y the mondic second-order logic (MSOL) (see the hndook [Tho97] for detils). Our results. For finitry Büchi, prity nd Streett utomt the topologicl, utomt-theoretic, nd logicl chrcteriztion were ll missing. In this work we present ll the three chrcteriztions. Our min results re s follows. 1. Topologicl chrcteriztion. We show tht the clss of lnguges defined y finitry Büchi, prity nd Streett conditions re Σ 2 -complete, nd thus present precise topologicl chrcteriztion of finitry Büchi, prity nd Streett lnguges. 2. Automt-theoretic chrcteriztion. We show tht lnguges defined y finitry prity nd Streett utomt re incomprle in expressive power s compred to infinitry prity nd Streett utomt. We show tht non-deterministic utomt with finitry prity nd Streett conditions hve the sme expressive power s non-deterministic utomt with finitry Büchi conditions, nd deterministic prity nd Streett utomt hve the sme expressive power nd is strictly more expressive thn deterministic finitry Büchi utomt. However, in contrst to infinitry prity condition, for finitry prity condition, non-deterministic utomt is strictly more expressive thn the deterministic counterprt. We lso present precise chrcteriztion of the closure properties of finitry utomt with respect to union, intersection nd complementtion. 3. Logicl chrcteriztion. Since finitry utomt re incomprle in expressive power s compred to ω-regulr lnguges, the result lso holds for MSOL. We consider the chrcteriztion of finitry utomt through n extension of MSOL nd ω-regulr lnguges defined s MSOLA nd ωb-regulr lnguges y [BC06]. We show tht lnguges defined y non-deterministic finitry prity utomt re exctly the str-free frgment of ωb-regulr lnguges. It follows tht in generl MSOLA nd ωbregulr lnguges re strictly more expressive, nd non-deterministic finitry prity utomt exctly chrcterize the str-free frgment. Hence we otin precise logicl chrcteriztion of the finitry lnguges. 2 Definitions In this section we define lnguges, topology relted to lnguges, then utomt nd lnguges descried y utomt with vrious cceptnce conditions. 2.1 Lnguges, Cntor topology nd Borel hierrchy Lnguges. Let Σ e finite set, we refer to Σ s the lphet, nd its elements s letters. A word w is sequence of letters, which cn e either finite or infinite. A word w will e descried s sequence

5 w 0 w 1... of letters, where w 0, w 1, Σ. Let Σ e the set of ll finite words over Σ nd Σ ω the set of ll infinite words over Σ. A lnguge is set of words, thus L Σ is lnguge over finite words nd L Σ ω is lnguge over infinite words. Cntor topology. The complexity of lnguges cn e studied ccording to the topologicl definition. To present topologicl definition on lnguges we first define open nd closed sets. A lnguge is open if it cn e descried s W Σ ω where W Σ. A closed set is complement of n open set. Then we define the Cntor topology to otin the topology over lnguges. It my e noted tht the ove topology defines the sme topology s the one induced y the following distnce over infinite words: distword(w, w ) = 1 2, i where i is the lrgest nonnegtive integer such tht w j = w j for ll 0 j < i. Borel hierrchy. We now define the Borel hierrchy of lnguges. Let Σ 1 denote the open sets, Π 1 denote the closed sets, nd then inductively we hve the following: Σ i+1 is otined s countle union of Π i sets; nd Π i+1 is otined s countle intersection of Σ i sets. We note tht the closed sets (lnguges in Π 1 ) correspond to sfety properties. For L Σ ω, let pref(l) Σ e the set of finite prefixes of words in L: u Σ elongs to pref(l) iff there exists w L such tht u is finite prefix of w. Then the following property holds. Proposition 1. For ll lnguges L Σ ω, the following sttements re equivlent: () L is closed; () for ll infinite words w, if ll finite prefixes of w re in pref(l), then w L. Topologicl reduction. The clsses Σ 1, Π 1, Σ 2, Π 2,... re the levels of Borel hierrchy. Since they re closed under continuous preimge, we cn define notion of reduction: L reduces to L, denoted y L L, if there exists continuous function f : Σ ω Σ ω such L = f (L ), where f (L ) is the preimge of L y f. This defines the notion of Wdge reduction [Wd84]. A lnguge is hrd with respect to clss if ll lnguges of this clss reduce to it. If it dditionlly elongs to this clss, then it is complete. Clssicl lnguges. We now consider severl clssicl notion of lnguges. For n infinite word w, let Inf(w) Σ denote the set of letters tht pper infinitely often in w. The clss of rechility, sfety, Büchi nd cobüchi lnguges re defined s follows. Let F Σ: Rech(F) = {w i N, w i F }; Sfe(F) = Σ ω \Rech(F) = {w i N, w i / F }; Büchi(F) = {w Inf(w) F }; CoBüchi(F) = Σ ω \Büchi(F) = {w Inf(w) Σ\F }. In other words, the rechility lnguge Rech(F) requires tht letter in F ppers t lest once nd the Büchi lnguge Büchi(F) requires tht some letter in F ppers infinitely often. The Sfe(F) nd CoBüchi(F) re duls of Rech(F) nd Büchi(F), respectively. The clss of prity lnguges is defined s follows. Let p : Σ N e priority function tht mps letters to integer priorities. The prity lnguges re defined s follows: Prity(p) = {w min(p(inf(w))) is even}; i.e., the prity condition ccepts infinite words where the lowest priority infinitely visited is even. The prity conditions re self-dul. The clss of Rin nd Streett lnguges re defined s follows. Let (R, G) = (R i, G i ) 1 i d, where R i, G i Σ re request-grnt pirs. Then we hve Streett(R, G) = {w i, 1 i d, Inf(w) R i Inf(w) G i }; Rin(R, G) = {w i, 1 i d, Inf(w) R i Inf(w) G i = }; i.e., the Streett condition ccepts infinite words w such tht for ll requests R i, if R i ppers infinitely often in w, then the corresponding grnt G i lso ppers infinitely often in w. Rin condition is the dul of Streett condition. Then we hve the following theorem tht presents the topologicl chrcteriztion of the clssicl lnguges. Theorem 1 (Topologicl chrcteriztion of clssicl lnguges [MP92]). The following ssertions hold.

6 For ll F Σ, we hve () Rech(F) is Σ 1 -complete nd Sfe(F) is Π 1 -complete; nd () Büchi(F) is Π 2 -complete nd CoBüchi(F) is Σ 2 -complete. The prity, Streett nd Rin lnguges lie in the oolen closure of Σ 2 nd Π 2. Finitry lnguges. Let p : Σ N priority function. We define: { 0 p(w k ) is even dist k (w, p) = inf{k k k k, p(w k ) is even nd p(w k ) < p(w k )} p(w k ) is odd The finitry prity lnguge FinPrity(p) ws defined s follows in [CHH09]: FinPrity(p) = {w limsup dist k (w, p) < }, k i.e., the FinPrity(p) requires tht the supremum limit of the distnce sequence is ounded. The definition for FinStreett(R, G) lnguges uses similr distnce sequence s follows: { dist j k (w, (R, G)) = 0 w k / R j inf{k k k k, w k G j } w k R j Then we hve dist k (w, (R, G)) = mx{dist j k (w, p) 1 j d}, nd the finitry Streett lnguge FinStreett(R, G) ws defined s follows in [CHH09]: FinStreett(R, G) = {w limsup dist k (w, (R, G)) < }, k i.e., similr to finitry prity lnguges FinStreett(R, G) requires the supremum limit of the distnce sequence to e ounded. 2.2 Automt, ω-regulr nd finitry lnguges In this section we consider utomt with cceptnce conditions nd consider the clss of lnguges defined y utomt with vrious clsses of cceptnce conditions. Definition 1. An utomton is tuple A = (Q, Σ, Q 0, δ,acc), where Q is finite set of sttes, Σ is the finite input lphet, Q 0 Q is the set of initil sttes, δ Q Σ Q is the trnsition reltion nd Acc Q ω is the cceptnce condition. Deterministic nd complete utomt. We consider the specil clss of deterministic nd complete utomt. An utomton A is deterministic if () Q 0 = 1, i.e., there is single initil stte; () for every letter nd every stte there is t most one trnsition, i.e., for ll q Q, for ll σ Σ we hve {q (q, σ, q ) δ} 1. Deterministic utomt will e descried s (Q, Σ, q 0, δ,acc), where δ : Q Σ Q is function. If for ll q Q nd for ll σ Σ, there exists q Q such tht (q, σ, q ) δ, then the corresponding utomton is complete. This is the cse when the trnsition function is totl. Runs. A run ρ = q 0 q 1... is word over Q, where q 0 Q 0. The run ρ is ccepting if it is infinite nd ρ Acc. We will write p q to denote (p,, q) δ. An infinite word w = w 0 w 1... induces possily severl runs of A: word w induces run ρ = q 0 q 1... if we hve q 0 Q 0 nd w q 0 w 0 q 1 w 1 q 2... q n n qn The lnguge ccepted y A, denoted y L(A) Σ ω, is s follows: L(A) = {w there exists run ρ induced y w such tht ρ Acc}.

7 Note tht for deterministic utomton, every word w induces t most one run, wheres in nondeterministic utomton word my induce severl possile runs. Acceptnce conditions. We will consider vrious cceptnce conditions for utomt otined from the lst section y considering Q s the lphet. For F Q, the conditions Rech(F), Sfe(F), Büchi(F), CoBüchi(F), define rechility, sfety, Büchi nd cobüchi cceptnce conditions, respectively. For p : Q N, the conditions Prity(p) nd FinPrity(p) define prity nd finitry prity cceptnce conditions, respectively. For (R, G) = (R i, G i ) 1 i d, where R i, G i Q, the conditionsstreett(r, G), Rin(R, G), nd FinStreett(R, G) define Streett, Rin nd finitry Streett cceptnce conditions, respectively. The set of lnguges recognized y non-deterministic Büchi utomt corresponds to the clss of ω-regulr lnguges [Büc62] nd we will denote the clss of ω-regulr lnguges s L ω. Nottion 1 We use stndrd nottion to denote set of lnguges recognized y some clss of utomt. The first letter is either N or D, where N stnds for non-deterministic nd D stnds for deterministic. The lst lock of letters refers to the cceptnce condition, for exmple, B stnds for Büchi, C stnds for CoBüchi, P stnds for prity nd S stnds for Streett. The cceptnce condition my e prefixed y F for finitry. For exmple, NP denotes non-deterministic prity utomt, nd DFS denotes deterministic finitry Streett utomt. Hence we hve the following comintion: { } { } N F D ε B C P S We now present the following theorem tht summrizes the results of utomt with clssicl lnguges, nd the results of the theorem follows from [Büc62,Sf92,Cho74,GH82]. Theorem 2 (Automt-theoretic results for clssicl lnguges). The following ssertions hold: (1) L ω = NB = NP = NS = DP = DS; (2) DB NB; (3) DC = NC NB. 3 Topologicl Chrcteriztion of Finitry Lnguges In this section we present the topologicl chrcteriztion of finitry Büchi, finitry prity nd finitry Streett lnguges. We first present definition nd then use the definition for chrcteriztion of finitry lnguges. Union of ω-regulr nd closed suset of lnguge. Given lnguge L Σ ω, the lnguge UniCloOmg(L) Σ ω is the union of the lnguges M tht re suset of L, ω-regulr nd closed, i.e., UniCloOmg(L) = {M M Π 1, M L ω, M L}. Proposition 2. The following ssertions hold: () the opertor UniCloOmg is idempotent; i.e., for ll lnguges L we hve UniCloOmg(UniCloOmg(L)) = UniCloOmg(L); () the lnguge UniCloOmg(L) is in Σ 2, i.e., for ll lnguges L we hve UniCloOmg(L) Σ 2. Proof. We prove oth the properties elow. 1. By definition for ll lnguges L we hve UniCloOmg(L ) L. Given lnguge L, let L = UniCloOmg(L). Hence we hve UniCloOmg(L ) L, i.e., UniCloOmg(UniCloOmg(L)) UniCloOmg(L). We now show the other direction. For ny lnguge L nd M L, if M Π 1 nd M L ω, then M UniCloOmg(L ). Consider the lnguge L = UniCloOmg(L). For lnguge M such tht M L, M Π 1 nd M L ω, we hve M UniCloOmg(L), (i.e., M L ) nd hence M UniCloOmg(L ). Hence we hve The result follows. L = {M M Π 1, M L ω, M L} UniCloOmg(L ).

8 2. Since L ω = NB (y Theorem 2), nd the set of finite utomt cn e enumerted in sequence, it follows L ω is countle. It follows tht for ll lnguges L, the set UniCloOmg(L) is descried s countle union of closed sets. Hence UniCloOmg(L) Σ 2. The result follows. We now present pumping lemm for regulr lnguges, nd will use it to present the topologicl chrcteriztion for finitry lnguges. Lemm 1 (A pumping lemm). Let M e ω-regulr lnguge. There exists n 0 such tht for ll words w M, for ll positions j n 0, there exist j i 1 < i 2 j + n 0 such tht for ll l 0 we hve w 0 w 1 w 2... w i1 1 (w i1 w i w i2 1) l w i2 w i M. Proof. Given M is ω-regulr lnguge, let A e complete nd deterministic prity utomt tht recognizes M (such n utomton exists y Theorem 2), nd let n 0 e the numer of sttes of A. Consider word w = w 0 w 1 w 2... such tht w M, nd let ρ = q 0 q 1 q 2... e the unique run induced y w in A. Consider position j in w such tht j n 0. Then there exist j i 1 < i 2 j + n 0 such tht q i1 = q i2, this must hppen s A hs n 0 sttes. For l 0, if we consider the word w l = w 0 w 1 w 2... w i1 1 (w i1 w i w i2 1) l w i2 w i2+1..., then the unique run induced y w l in A is ρ l = q 0 q 1 q 2... q i1 1 (q i1 q i q i2 1) l q i2 q i Since the prity condition is independent of finite prefixes nd the run ρ is ccepted y A, it follows tht ρ l is ccepted y A. Since A recognizes M, it follows w l M, nd the result follows. We now present the min lemm of this section. Lemm 2. For ll (R, G) = (R i, G i ) 1 i d, where R i, G i Σ, we hve UniCloOmg(Streett(R, G)) = FinStreett(R, G); i.e., FinStreett(R, G) is otined y pplying the UniCloOmg opertor to Streett(R, G). Proof. We present the two directions of the proof. 1. We first show tht UniCloOmg(Streett(R, G)) FinStreett(R, G). Let M Streett(R, G) such tht M is closed nd ω-regulr. Let w = w 0 w 1... M, nd ssume towrds contrdiction, tht limsup k dist k (w, (R, G)) =. Hence for ll n 0 N, there exists n N such tht n n 0 nd dist n (w, (R, G)) n 0. Let n 0 N given y the pumping lemm on M, from ove given n 0 we otin j such tht j n 0 nd dist j (w, (R, G)) n 0. By the pumping lemm (Lemm 1), we otin the witness j i 1 < i 2 j + n 0. Let u = w 0 w 1... w i1 1, v = w i1 w i w i2 1 nd w = w i2 w i Since w M, y the pumping lemm for ll l 0 we hve uv l w M. This entils tht ll finite prefixes of the infinite word uv ω re in pref(m). Since M is closed, it follows to uv ω M. Since dist j (w, (R, G)) n 0 it follows tht there is some request i in position j, nd there is no corresponding grnt i for the next n 0 steps. Hence there is position j in v such tht there is request i t j nd no corresponding grnt in v, nd thus it follows tht the word uv ω Streett(R, G). This contrdicts tht M Streett(R, G). Hence it follows tht UniCloOmg(Streett(R, G)) FinStreett(R, G). 2. We now show the converse: UniCloOmg(Streett(R, G)) FinStreett(R, G). We hve FinStreett(R, G) = {w limsup dist k (w, (R, G)) < } = limsup dist k (w, (R, G)) B} k B N{w k = {w k n, dist k (w, (R, G)) B} B N n N The lnguge {w k n, dist k (w, (R, G)) B} is closed, ω-regulr, nd included in Streett(R, G). It follows FinStreett(R, G) UniCloOmg(Streett(R, G)). The result follows.

9 Corollry 1. For ll p : Σ N, we hve UniCloOmg(Prity(p)) = FinPrity(p). Proof. This follows from Lemm 2 nd the fct tht prity condition is specil cse of Streett condition. Corollry 2. For ll F Σ, we hve UniCloOmg(CoBüchi(F)) = CoBüchi(F). Proof. We show tht CoBüchi(F) properties re stle under UniCloOmg opertor. By Lemm 2 we hve UniCloOmg(CoBüchi(F)) nd finitry cobüchi lnguges coincide, nd since finitry cobüchi nd cobüchi lnguges coincide, the result follows. We now present chrcteriztion for finitry Büchi tht will e used in the sequel. For set F Σ, let next k (w, F) = inf{k k k k, w k F }. Corollry 3. For ll F Σ, we hve UniCloOmg(Büchi(F)) = {w limsup k next k (w, F) < }. We now present the results for topologicl chrcteriztion of finitry Büchi, prity nd Streett lnguges. Theorem 3 (Topologicl chrcteriztion of finitry lnguges). The following ssertions hold: 1. For ll p : Σ N, we hve FinPrity(p) Σ For ll (R, G) = (R i, G i ) 1 i d, we hve FinStreett(R, G) Σ For ll F Σ, we hve tht UniCloOmg(Büchi(F)) is Σ 2 -complete. 4. There exists p : Σ N such tht FinPrity(p) is Σ 2 -complete. 5. There exists (R, G) = (R i, G i ) 1 i d such tht FinStreett(R, G) is Σ 2 -complete. Proof. We prove ll the cses elow. 1. It follows from Corollry 1 nd Proposition 2.() tht UniCloOmg(FinPrity(p)) = FinPrity(p), nd then it follows from Proposition 2.() tht FinPrity(p) Σ As ove it follows from Lemm 2 nd Proposition It follows from Proposition 2 tht UniCloOmg(Büchi(F)) Σ 2. We hve tht CoBüchi(Σ \F) is Σ 2 - complete from Theorem 1. We now present topologicl reduction to show tht CoBüchi(Σ \ F) UniCloOmg(Büchi(F)). Recll tht w CoBüchi(Σ \ F) iff Inf(w) F. Let : Σ ω Σ ω e the stuttering function defined s follows: w = w 0 w 1... w n... (w) = w 0 w 1 w }{{} 1... w n w n... w n... }{{} 2 2 n The function is continuous, since distword((w), (w )) distword(w, w ). It remins to show the following: Inf(w) F iff B N, n N, k n, next k ((w), F) B. Left to right direction: ssume tht from the position n of w, letters elong to F. Then from the position 2 n 1, letters of (w) elong to F, then next k ((w), F) = 0 for k 2 n 1. Right to left direction: let B nd n e integers such tht for ll k n we hve next k ((w), F) B. Assume 2 k 1 > B nd k n, then the letter in position 2 k 1 in (w) is repeted 2 k 1 times, thus next k ((w), F) is either 0 or higher thn 2 k 1. The ltter is not possile since it must e less thn B. It follows tht the letter in position k in w elongs to F. 4. This follows from item 3 ove nd the fct tht Büchi condition is specil cse of prity condition. 5. This follows from item 3 ove nd the fct tht Büchi condition is specil cse of Streett condition. The desired result follows.

10 4 Automt-Theoretic Chrcteriztion of Finitry Lnguges In this section we consider the utomt-theoretic chrcteriztion of finitry lnguges. We compre the expressive power of vrious clsses of utomt with finitry cceptnce conditions with respect to utomt with clssicl ω-regulr cceptnce condition. 4.1 Comprison with clssicl lnguges In this section we compre the expressive power of utomt with finitry cceptnce conditions s compred to utomt with clssicl cceptnce conditions. In the exmples we will consider Σ = {, }. Exmple 1 (DFB NB). Consider the finitry Büchi utomton A shown in Fig. 1 nd the stte leled 0 is the ccepting set F. The lnguge of A is L B = {( j0 f(0) ) ( j1 f(1) ) ( j2 f(2) )... f : N N, f ounded, i N, j i N}. Indeed, 0-leled stte is visited while reding the letter, nd the 1- leled stte is visited while reding the letter. An infinite word w is ccepted iff the 0-leled stte is visited infinitely often, nd there must e ound etween ny two consecutive visits of the 0-leled stte. We now show tht L B is not ω-regulr: ssume towrds contrdiction tht L B is ω-regulr. Then y 1 0 Figure 1. A finitry Büchi utomton A Theorem 2 there is deterministic prity utomton A tht recognizes L B, hving N sttes. Without loss of generlity we ssume this utomton to e complete, nd let the strting stte e q 0. Since the word ω elongs to the lnguge, the unique run on this word is ccepting nd cn e decomposed s: q n 0 0 s p 0 0 s p 0 0 s 0... where s 0 is the lowest priority stte visited infinitely often (thus it hs even priority), nd n 0 N, 1 p 0 N. Since the word n0 ω elongs to the lnguge L B, we cn repet the ove construction. By induction, we define s k nd q k s shown in the Figure 2: s k is the lowest priority stte visited infinitely often while n 0 q 0 s 0 p 0 n 0 n 1 q 1 s 1 p 1 p 0 p k q k n k s k n k p k 1 q k+1 Figure 2. Inductive construction showing tht L B / DP. reding n0 n 0 n 1 n 1... n k 1 n k 1 n k ω (thus it hs even priority), nd n k N, 1 p k N,

11 nd similrly for q k, reding n0 n 0 n 1 n 1... n k 1 n k 1 ω. There exists i < j such tht q i = q j, nd hence the infinite word u ( p i 1 v) ( 2p i 1 v)...( kp i 1 v)..., where u = n 0 n 0... n i 1 nd v = ni... n j 1, is ccepted y A, nd hence we hve contrdiction tht A recognizes L B. We now show tht there exist lnguges expressed y deterministic Büchi utomt tht cnnot e expressed y non-deterministic finitry prity utomt. Exmple 2 (DB NFP). Consider the lnguge of infinitely mny s, i.e., L I = {w w hs n infinite numer of }. The lnguge L I is ω-regulr nd there is deterministic Büchi utomton A such tht the lnguge of A is L I. We now show tht there is no non-deterministic finitry prity utomt tht recognizes L I. Assume towrds contrdiction tht A is non-deterministic finitry prity utomt recognizing L I nd let A hve N sttes. Let us consider the infinite word w = n... L I. Since w must e ccepted y A, there must e n ccepting run ρ, nd we represent the ccepting run s follows: nd q 0 p0 q1 p1 2 q 2...p n 1 n q n pn n+1 q n+1... p n 1 qn,1 qn,2... q n,n 1 qn,n = q n... The sequence stisfies tht B N, n N, k n we hve dist k (ρ, p) B. Let c e the lowest priority infinitely visited, nd c must e even. The stte p k 1 is in position k (k+1) 2 in ρ. Let k e n integer such tht () k (k+1) 2 n nd () k (N + 1) B. Let us consider the set of sttes {q k,1,..., q k,k }. Since the distnce function is ounded y B, the priority c ppers t lest once in ech set of consecutively visited sttes of size B. Since k (N + 1) B, it ppers t lest N + 1 times in {q k,1,...,q k,k }. Since there is N sttes in A, t lest one stte hs een reched twice. We cn thus iterte: the infinite word w = k 1 ω, nd the word w is ccepted y A. However, w L I nd hence we hve contrdiction. Remrk 1. From Exmple 2 we deduce the following result: NFB nd NFP re not closed under complementtion. The lnguge {, } ω \L I = {w w hs finite numer of } NFB (see Exmple 3 lter); however, Exmple 2 shows tht the complement is not expressile y non-deterministic finitry prity utomt. We summrize the results in the following theorem. Theorem 4. The following ssertions hold: () DB NFP nd DFP NB; () DB NFB nd DFB NB. 4.2 Deterministic finitry utomt In this susection we consider deterministic utomt with finitry cceptnce conditions. Given deterministic complete utomton A with ccepting condition Acc, we will consider the lnguge otined y the finitry restriction of the cceptnce condition. We first consider function C A s follows: C A : Σ ω Q ω mps n infinite word w to the unique run ρ of A on w (there is unique run since A is deterministic nd complete). Then L(A) = {w C A (w) Acc} = C A (Acc). We will focus on the following property: C A (UniCloOmg(Acc)) = UniCloOmg(C A (Acc)), which follows from the following lemm. Lemm 3. For ll A = (Q, Σ, q 0, δ,acc) deterministic complete utomton, we hve: 1. for ll A Q ω, A is closed C A (A) closed (C A is continuous). 2. for ll L Σ ω, L is closed C A (L) closed (C A is closed).

12 3. for ll A Q ω, A is ω-regulr C A (A) ω-regulr. 4. for ll L Σ ω, L is ω-regulr C A (L) ω-regulr. Proof. We prove ll the cses elow. 1. Let A Q ω such tht A is closed. Let w e such tht for ll n N we hve w 0...w n pref(c A (A)). We define the run ρ = C A (w) nd show tht ρ = q 0 q 1... A. Since A is closed, we will show for ll n N we hve q 0... q n pref(a). From the hypothesis we hve w 0...w n 1 pref(c A (A)), nd then there exists n infinite word u such tht C A (w 0...w n 1 u) A. Let C A (w 0...w n 1 u) = q 0 q 1...q n..., then we hve q w 0 0 q w 1 1 q 2 w n 1 q n. Since A is deterministic, we get q i = q i, nd hence q 0... q n pref(a). 2. Let L Σ ω such tht L is closed. Let ρ = q 0 q 1... such tht for ll n N we hve q 0... q n w pref(c A (L)). Then for ll n N, there exists word w 0 w 1...w n 1 such tht q 0 w 0 q 1 1 w n 1 q 2... qn, nd w 0 w 1... w n 1 pref(l). We define y induction on n n infinite nested sequence of finite words w 0 w 1... w n pref(l). We denote y w the limit of this nested sequence of finite words. We hve tht ρ = C A (w). Since L is closed, w L. 3. Let A Q ω such tht A recognized y Büchi utomton B = (Q B, Q, P 0, τ, F). We define the Büchi utomton C = (Q Q B, Σ, {q 0 } P 0, γ, Q B F), where (q 1, p 1 ) σ σ (q 2, p 2 ) iff q 1 q2 in A nd q 1 p 1 p2 in B. We now show the correctness of our construction. Let w = w 0 w 1... ccepted y C, then there exists n ccepting run ρ, s follows: (q 0, p 0 ) w0 (q 1, p 1 ) w1 (q 2, p 2 )...(q n, p n ) wn (q n+1, p n+1 )... where the second component visits F infinitely often. Hence: w q 0 w 0 q 1 w 1 q 2... q n q 0 q 1 q n n qn+1... in A nd p 0 p1 p2...p n pn+1... in B ( ) Hence from ( ), we hve C A (w) = q 0 q 1 L(B) = A, nd it follows tht w C A (A). Conversely, let w C A (A), then we hve ρ = C A(w) = q 0 q 1 A = L(B). Then the ove sttement ( ) holds, which entils tht w is ccepted y C. It follows tht C recognizes C A (A). 4. Let L Σ ω such tht L is recognized y Büchi utomton B = (Q B, Σ, P 0, τ, F). We define the Büchi utomton C = (Q Q B, Q, {q 0 } P 0, γ, Q F), where (q, p 1 ) q (q, p 2 ) iff there exists σ Σ, such tht q σ q σ in A nd p 1 p2 in B. A proof similr to ove show tht C recognizes C A (L). The desired result follows. Theorem 5. For ll deterministic complete utomt A = (Q, Σ, q 0, δ,acc) recognizing lnguge L, the finitry restriction of this utomton UniCloOmg(A) = (Q, Σ, q 0, δ, UniCloOmg(Acc)) recognizes UniCloOmg(L). Proof. A word w is ccepted y UniCloOmg(A) iff w C A (UniCloOmg(Acc)) = UniCloOmg(C A (Acc)) = UniCloOmg(L). Theorem 5 llows to extend ll known results on deterministic clsses to finitry deterministic clsses, nd we hve the following corollry. Corollry 4. We hve: () DFP = DFS; () DFB DFP; (c) DC DFP. We now show tht non-deterministic finitry prity utomt is more expressive thn deterministic finitry prity utomt. Exmple 3 (DFP NFP). Consider the following lnguge L F of finitely mny s, L F = {, } ω \L I = {w w hs finite numer of } NFP.

13 , 1 0 Figure 3. A NFB for L F. The lnguge L F is recognized y the non-deterministic finitry Büchi utomt shown in Fig 3. To show tht deterministic finitry prity utomt re strictly less expressive thn non-deterministic finitry prity utomt, i.e., DFP NFP we show L F DFP. Assume towrds contrdiction tht there is deterministic finitry prity utomton A with N sttes tht recognizes L F. Without loss of generlity we ssume this utomton to e complete, nd let the strting stte e q 0. Since the word ω elongs to the lnguge, the unique run on this word is ccepting nd cn e decomposed s: q n 0 0 s p 0 0 s p 0 0 s 0... where s 0 is the lowest priority stte visited infinitely often (thus it hs even priority), nd n 0, p 0 N. Let s 0 r0. Since the word n0 ω elongs to the lnguge L F, we cn repet the ove construction. By induction, we define s k nd q k s shown in the Figure 4: s k is the lowest priority stte visited infinitely n 0 q 0 s 0 p 0 p 1 n 1 q 1 s 1 p k q k n k s k q k+1 Figure 4. Inductive construction showing tht L F / DFP. often while reding n0 n1... n k 1 n k ω (thus it hs even priority), nd n k, p k N. There exists i < j such tht q i = q j, nd hence the infinite word u ( ni+pi v) ( ni+2pi v)...( ni+kpi v)... where u = n0 n1... ni 1 nd v = ni... nj 1, is ccepted y A. u q 0 n i qi s p i v i s i n i qi s 2p i i... Indeed, iterting on s i s loop ensures tht there is no ound etween two consecutive visits of stte, for those which re not in this loop. In s i s loop, s i hs the lowest priority, nd it is even. There is ound etween two consecutive visits of s i : the loop hs less thn N sttes, nd the wy from s i y v to q i nd ck to s i hs constnt size v + n i. Hence we hve contrdiction tht A recognizes L F. Theorem 6. We hve DFP NFP. Remrk 2. Oserve tht Theorem 5 does not hold for non-deterministic utomt, since we hve DP = NP ut DFP NFP.

14 4.3 Non-deterministic finitry utomt We now show tht non-deterministic finitry Streett utomt cn e reduced to non-deterministic finitry Büchi utomt, nd this would complete the picture of utomt-theoretic chrcteriztion. We first show tht non-deterministic finitry Büchi utomt re closed under conjunction, nd use it to show Theorem 7. Lemm 4. NFB is closed under conjunction. Proof. Let A 1 = (Q 1, Σ, δ 1, Q 1 0, F 1) nd A 2 = (Q 2, Σ, δ 2, Q 2 0, F 2) e two non-deterministic finitry Büchi utomt. Without loss of generlity we ssume oth A 1 nd A 2 to e complete. We will define construction similr to the synchronous product construction, where switch etween copies will hppen while visiting F 1 or F 2. The finitry Büchi utomton is A = (Q 1 Q 2 {1, 2}, Σ, δ, Q 1 0 Q2 0 {1}, F 1 Q 2 {2} Q 1 F 2 {1}). We define the trnsition reltion δ elow: δ = {((q 1, q 2, k), σ, (q 1, q 2, k)) q 1 / F 1, q 2 / F 2, (q 1, σ, q 1) δ 1, (q 2, σ, q 2) δ 2, k {1, 2}} {((q 1, q 2, 1), σ, (q 1, q 2, 2)) q 1 F 1, (q 1, σ, q 1 ) δ 1, (q 2, σ, q 2 ) δ 2} {((q 1, q 2, 2), σ, (q 1, q 2, 1)) q 2 F 2, (q 1, σ, q 1 ) δ 1, (q 2, σ, q 2 ) δ 2} Intuitively, the trnsition function δ is s follows: the first component mimics the trnsition for utomt A 1, the second component mimics the trnsition for A 2, nd there is switch for the third component from 1 to 2 visiting stte in F 1, nd from 2 to 1 visiting stte in F 2. We now prove the correctness of the construction. Consider word w tht is ccepted y A 1, nd then there exists ound B 1 nd run ρ 1 in A 1 such tht eventully, the numer of steps etween two visits to F 1 in ρ 1 is t most B 1 ; nd similrly, there exists ound B 2 nd run ρ 2 in A 2 such tht eventully the numer of steps etween two visits to F 2 in ρ 2 is t most B 2. It follows tht in our construction there is run ρ (tht mimics the runs ρ 1 nd ρ 2 ) in A such tht eventully within mx{b 1, B 2 } steps stte in F 1 Q 2 {2} Q 1 F 2 {1} is visited in ρ. Hence w is ccepted y A. Conversely, consider word w tht is ccepted y A, nd let ρ e run nd B e the ound such tht eventully etween two visits to the ccepting sttes in ρ is seprted y t most B steps. Let ρ 1 nd ρ 2 e the decomposition of the run ρ in A 1 nd A 2, respectively. It follows tht oth in A 1 nd A 2 the respective finl sttes re eventully visited within t most 2 B steps in ρ 1 nd ρ 2, respectively. It follows tht w is ccepted y oth A 1 nd A 2. Hence we hve L(A) = L(A 1 ) L(A 2 ). Theorem 7. We hve NFS NFP NFB. Proof. We will present reduction of NFS to NFB nd the result will follow. Since the Streett condition is finite conjunction of conditions Inf(w) R i Inf(w) G i, y Lemm 4 it suffices to hndle the specil cse when d = 1. Hence we consider non-deterministic Streett utomton A = (Q, Σ, δ, Q 0, (R, G)) with (R, G) = (R 1, G 1 ). Without loss of generlity we ssume A to e complete. We construct non-deterministic Büchi utomton A = (Q {1, 2, 3}, Σ, δ, Q 0 {1}, Q {2}), where the trnsition reltion δ is given s follows: δ = {(q, 1), σ, (q, j) (q, σ, q ) δ, j {1, 2}} {(q, 2), σ, (q, 2) q / R 1, (q, σ, q ) δ} {(q, 2), σ, (q, 3) q R 1, (q, σ, q ) δ} {(q, 3), σ, (q, 3) q / G 1, (q, σ, q ) δ} {(q, 3), σ, (q, 2) q G 1, (q, σ, q ) δ} In other words, the stte component mimics the trnsition of A, nd in the second component: () the utomton cn choose to sty in component 1, or switch to 2; () there is switch from 2 to 3 upon visiting stte in R 1 ; nd () there is switch from 3 to 2 upon visiting stte in G 1. Consider word w ccepted y A nd n ccepting run ρ in A, nd let B e the ound on the distnce sequence. We show tht w is ccepted y A y constructing n ccepting run ρ in A. We consider the following cses:

15 1. If infinitely mny requests R 1 re visited in ρ, then in A immeditely switch to component 2, nd then mimic the run ρ s run ρ in A. It follows tht from some point j on every request is grnted within B steps, nd it follows tht fter position j, whenever the second component is 3, it ecomes 2 within B steps. Hence w is ccepted y A. 2. If finitely mny requests R 1 re visited in ρ, then fter some point j, there re no more requests. The utomton A mimics the run ρ y stying in the second component s 1 for j steps, nd then switches to component 2. Then fter j steps we lwys hve the second component s 2, nd hence the word is ccepted. Conversely, consider word w ccepted y A nd consider the ccepting run ρ. We mimic the run in A. To ccept the word w, the run ρ must switch to the second component s 2, sy fter j steps. Then, from some point on whenever stte with second component 3 is visited, within some ound B steps stte with second component 2 is visited. Hence the run ρ is ccepting in A. Thus the lnguges of A nd A coincide, nd the desired result follows. Corollry 5. We hve DFB DFP NFB = NFP = NFS. Our results estlishing the precise utomt-theoretic chrcteriztion of lnguges defined y utomt with finitry cceptnce condition is shown in Fig 5. In generl NFP cnnot e determinized to DFP, however, for every lnguge L L ω there is A DP such tht A recognizes L, nd hence the deterministic finitry prity utomt UniCloOmg(A) recognizes UniCloOmg(L). Corollry 6. For every lnguge L L ω there is deterministic finitry prity utomt A such tht A recognizes UniCloOmg(L). L ω DB DC DFB DFP = DFS NFB = NFP = NFS Figure 5. Automt-theoretic chrcteriztion

16 5 Logicl Chrcteriztion of Finitry Lnguges In this section we consider the logicl chrcteriztion of finitry lnguges. Closure properties. For logicl chrcteriztion of lnguges defined y utomt with finitry cceptnce conditions, we first study the closure properties of deterministic nd non-deterministic utomt with finitry cceptnce conditions. We will consider DFP nd NFP. Theorem 8 (Closure properties). The following closure properties hold: 1. DFP is closed under intersection. 2. DFP nd NFP re not closed under complementtion. 3. DFP is not closed under union. 4. NFP is closed under union nd intersection. Proof. We prove ll the cses elow. 1. Intersection closure for DFP follows from Theorem 5 nd from the oservtion tht for ll L, L Σ ω we hve UniCloOmg(L L ) = UniCloOmg(L) UniCloOmg(L ). The oservtion is proved s follows. Let M Π 1 L ω nd M L L, then M UniCloOmg(L) UniCloOmg(L ), nd hence UniCloOmg(L L ) UniCloOmg(L) UniCloOmg(L ). Conversely, let M 1 UniCloOmg(L) nd M 2 UniCloOmg(L ), then M 1 M 2 Π 1 L ω nd M 1 M 2 L L. Hence M 1 M 2 UniCloOmg(L L ), thus UniCloOmg(L) UniCloOmg(L ) UniCloOmg(L L ). 2. It follows from Exmple 2 nd Exmple 3 tht there is non-deterministic finitry prity utomt tht recognizes the lnguge L F, nd the complementry lnguge {, } ω \L F = L I is not recognized y non-deterministic finitry prity utomton. It follows tht NFP is not closed under complementtion. The result for DFP is similr. 3. As for Exmple 1 we consider the lnguges L 1 = {( j0 f(0) ) ( j1 f(1) ) ( j2 f(2) )... f : N N, f ounded, i N, j i N} nd L 2 = {( f(0) j0 ) ( f(1) j1 ) ( f(2) j2 )... f : N N, f ounded, i N, j i N}, lso descried y the ωb-regulr expressions ( B )ω nd ( B )ω, respectively. It follows from Exmple 1 tht oth L 1 nd L 2 elong to DFP, nd we show tht L 1 L 2 / DFP. The proof is the very similr to Exmple 1. Assume towrds contrdiction tht L 1 L 2 DFP, nd let A e deterministic complete finitry prity utomton tht recognizes L 1 L 2. Let A hs N sttes, nd let q 0 e the strting stte. Since ω elongs to this lnguge, its unique run on A is ccepting: q n 0 0 s p 0 0 s p where n 0 N, 1 p 0 N nd s 0 is the lowest priority visited infinitely often while reding ω. Then, n0 ω elongs to this lnguge, its unique run on A is ccepting: q n 0 0 s n 0 0 q p 0 1 q p where n 0 N, 1 p 0 N nd q 1 is the lowest priority visited infinitely often while reding n0 ω. Repeting this construction nd y induction we hve: q n 0 0 s n 0 q n 1 1 s n 1... q k k 0 1 n k s k n k q k+1 with s k p k s k nd q p k+1 q k+1, where n k, n k N nd 1 p k, p k N; nd s k is the lowest priority visited infinitely often while reding n0 n 0... n k ω ; nd q k+1 is the lowest priority visited infinitely often while reding n0 n 0... n k n k ω. There must e i < j, such tht q i = q j. Let u = n0 n 0... n i 1 nd v = n i... n j 1. The word w = u ( p i n i+p i v) ( 2p i n i+2p i v)...( kp i n i+kp i v)... is ccepted y A, ut does not elong to L 1 L 2. Hence we hve contrdiction, nd the result follows. 4. Union closure for NFP is ovious, intersection closure for NFP follows from Lemm 4, since NFP = NFB (Corollry 5). The result follows.

17 Comprison with ωb-regulr expressions. We now study the expressive power of NFP s compred to ωb-regulr expressions. The clss of ωb-regulr expressions ws introduced in the work of [BC06] s n extension of ω-regulr expressions. Regulr expressions defines exctly regulr lnguges over finite words, nd hs the following grmmr: L := ε σ L L L L + L; σ Σ In the ove grmmr, stnds for conctention, for Kleene str nd + for union. Then ω-regulr lnguges re finite union of L L ω, where L nd L re regulr lnguges of finite words. The clss of ωb-regulr lnguges, s defined in [BC06], is exctly descried y finite union of L M ω, where L is regulr lnguge over finite words nd M is B-regulr lnguge over infinite sequences of finite words. The grmmr for B-regulr lnguges is s follows: M := ε σ M M M M B M + M; σ Σ The semntics of regulr lnguges over infinite sequences of finite words will ssign to B-regulr expression M, lnguge in (Σ ) ω. The infinite sequence u 0, u 1,... will e denoted y u. The semntics is defined y structurl induction s follows. is the empty lnguge, ε is the lnguge contining the single sequence (ε, ε,...), is the lnguge contining the single sequence (,,...), M 1 M 2 is the lnguge { u 0 v 0, u 1 v 1,... u M 1, v M 2 }, M is the lnguge { u 1... u f(1) 1, u f(1)... u f(2) 1,... u M, f : N N}, M B is defined like M ut we dditionlly require the vlues f(i + 1) f(i) to e ounded, M 1 + M 2 is {w u M 1, v M 2, i, w i {u i, v i }}. Finlly, the ω opertor on sequences with nonempty words on infinitely mny coordintes: u 0, u 1,... ω = u 0 u This opertion is nturlly extended to lnguges of sequences y tking the ω power of every sequence in the lnguge (ignoring those with nonempty words on finitely mny coordintes). The clss of ωb-regulr lnguges is more expressive thn NFP, nd this is due to the -opertor. We will consider the following frgment of ωb-regulr lnguges where we do not consider the -opertor for B-regulr expressions (however, the -opertor is llowed for L, regulr lnguges over finite words). We cll this frgment the str-free frgment of ωb-regulr lnguges. In the following two lemms we show tht strfree ωb-regulr expressions express exctly NFP = NFB. Lemm 5. All lnguges in NFP cn e descried y str-free ωb-regulr expression. Proof. Let A = (Q, Σ, δ, Q 0, p) e non-deterministic finitry prity utomton. Without loss of generlity we ssume Q = {1,..., n}. Let C = p(q) e the set of priorities, Q even = {q Q p(q) even} the set of sttes with even priority nd Q c = {q Q p(q) c} the set of sttes with priority t lest c. Let L q,q = {u Σ q u q } nd Mq c = {u ( u i ) i is ounded nd i, q ui q where ll intermedite visited sttes hve priority greter thn c}. Then L(A) = L q0,q (Mq p(q) ) ω. q 0 Q 0,q Q even For ll q, q Q we hve L q,q Σ is regulr. We now show tht for ll q Q nd c C the lnguge Mq c is B-regulr. We fix c C, nd then for simplicity of nottion revite Mq c to M q. For ll 0 k n nd q, q Q, let Mq,q k = {u ( u u i ) i is ounded nd i, q i q where ll intermedite visited sttes re from {1,...,k} nd hve priority greter thn c}. We show y induction on 0 k n tht for ll q, q Q the lnguge Mq,q k is B-regulr. The se cse k = 0 follows from oservtion: l if q q nd (q,, q ) δ i {1,...,l}, = i Mq,q 0 = ε l if q = q nd (q,, q ) δ i {1,...,l}, = i otherwise

18 The inductive cse for k > 0 follows from oservtion: M k q,q = Mk 1 q,k (Mk 1 k,k )B M k 1 k,q + Mk 1 q,q Since M n q,q = M q, we conclude tht L(A) is descried y str-free ωb-regulr expression. Lemm 6. All lnguges descried y str-free ωb-regulr expression is recognized y nondeterministic finitry Büchi utomton. Proof. To prove this result, we will descrie utomt reding infinite sequences of finite words, nd corresponding cceptnce conditions. Let A = (Q, Σ, δ, Q 0, F) finitry Büchi utomton. While reding n infinite sequence u of finite words, A will ccept if the following conditions re stisfied: (1) q 0 u Q 0, q 1, q 2,... F, i N, we hve q i i qi+1 nd (2) ( u n ) n is ounded. We show tht for ll M str-free B-regulr expression, there exists non-deterministic finitry Büchi utomton ccepting M B, lnguge of infinite sequence of finite words, s descried ove. We proceed y induction on M. The cses, ε nd Σ re esy. From M to M B, the sme utomton for M works for M B s well, since B is idempotent. From M 1, M 2 to M 1 + M 2 : this involves non-determinism. The utomton guesses for ech finite word which word is used. Let A 1 = (Q 1, Σ, δ 1, Q 0 1, F 1) nd A 2 = (Q 2, Σ, δ 2, Q 0 2, F 2) two nondeterministic finitry Büchi utomt ccepting M B 1 nd M B 2, respectively. For k {1, 2} nd T Q k, we define Finl(T) = {q F k q T, u Σ, q u Ak q } to e the stte of finl sttes rechle from stte in T. We denote y Finl k the k-th itertion of Finl, e.g., Finl 3 (T) = Finl(Finl(Finl(T))). We define finitry Büchi utomton: where A = ((Q 1 2 Q1 ) (Q 2 2 Q1 ) 2 }{{}} Q1 {{ 2 Q2 }, Σ, δ, (Q 0 1, Q 0 2), F) computtion sttes guess sttes δ = {((Q, Q ), ε, (q, Finl(Q ))) q Q} (guess is 1) {((Q, Q ), ε, (q, Finl(Q))) q Q } (guess is 2) {((q, T), σ, (q, T)) (q, σ, q ) δ 1 δ 2 } {((q 1, T), ε, ({q 1 }, T)) q 1 F 1 } {((q 2, T), ε, (T, {q 2 })) q 2 F 2 } There re two kinds of sttes. Computtion sttes re (q, T) where q Q 1 nd T Q 2 (or symmetriclly q Q 2 nd T Q 1 ), where q is the current stte of the utomton tht hs een decided to use for the current finite word, nd T is the set of finl sttes of the other utomton tht would hve een rechle if one hd chosen this utomton. Guess sttes re (Q, Q ), where Q is the set of sttes from A 1 one cn strt reding the next word, nd similrly for Q. We now prove the correctness of our construction. Consider n infinite sequence w ccepted y A, nd consider n ccepting run ρ. There re three cses: 1. either ll guesses re 1; 2. or ll guesses re 2; 3. else, oth guesses hppen. The first two cses re symmetric. In the first, we cn esily see tht w is ccepted y A 1, nd similrly in the second w is ccepted y A 2. We now consider the third cse. There re two symmetric sucses: either the first guess is 1, then ρ = (Q 0 1, Q0 2 ) (q0 1, Finl(Q0 2 ))..., with q1 0 Q0 1 ; or symmetriclly the first guess is 2, then ρ = (Q 0 1, Q0 2 ) (q0 2, Finl(Q0 1 ))...,

19 with q2 0 Q0 2. We consider only the first sucse. Then ρ = (Q 0 1, Q0 2 ) (q0 1, Finl(Q0 2 ))...(q1 1, Finl(Q0 2 )) ({q1 1 }, Finl(Q0 2 ))..., u 0 where u 0 is finite prefix of w ω such tht q1 0 q 1 1 in A 1 nd q1 1 F 1. We denote y ρ 0 the finite prefix of ρ up to (q1 1, Finl(Q0 2 )). Let k e the first time when guess is 2: then ρ = ρ 0 ρ 1 ρ k 1 ({q k 1 }, Finlk (Q 0 2 )) (q0 2, Finl({q k}))..., where q 0 2 Finl k (Q 0 2) nd for 1 i k 1, we hve ρ i = ({q i 1}, Finl i (Q 0 2)) (q i 1, Finl i+1 (Q 0 2))...(q i+1 1, Finl i+1 (Q 0 2)), u i nd u i is finite word such tht q1 i q i+1 1 in A 1, q1 i+1 F 1 nd u 0 u 1... u k 1 finite prefix of w ω. Since q2 0 Finlk (Q 0 2 ), there exists v 0, v 1,..., v k 1 finite words nd q2 1,..., qk 2 F 2 such tht: q2 0 v 0 q 1 v 1 v k q k 2. Then we cn repet this y induction, constructing u M1 B nd v MB 2, such tht for ll i N, we hve w i {u i, v i }. Conversely, let u M1 B nd v M2 B, nd w such tht i N, w i {u i, v i }. Using A 1 when w i = u i nd A 2 otherwise, one cn construct n ccepting run for w nd A. Hence A recognizes (M 1 +M 2 ) B. From M 1, M 2 to M 1 M 2 : the utomton keeps trcks of pending sttes while reding the other word. Let A 1 = (Q 1, Σ, δ 1, Q 0 1, F 1) nd A 2 = (Q 2, Σ, δ 2, Q 0 2, F 2) two non-deterministic finitry Büchi utomt ccepting M1 B nd MB 2, respectively. Let A = ((Q 1 F 2 ) (Q 2 F 1 ), Σ, δ, Q 0 1 Q0 2, F 1 F 2 ), where δ = {((q, f), σ, (q, f)) (q, σ, q ) δ 1, f F 2 } {((q, f), σ, (q, f)) (q, σ, q ) δ 2, f F 1 } {((q 1, f), ε, (f, q 1 )) q 1 F 1 } {((q 2, f), ε, (f, q 2 )) q 2 F 2 } Intuitively, the trnsition reltion is s follows: either one is reding using A 1 or A 2. In oth cses, the utomton rememers the lst finl stte visited while reding in the other utomton in order to restore this stte for the next word. Let w ccepted y A, n ccepting run is s follows: (q1, 0 q2) 0 w0 (q1, 1 q2) 1 w1... (q1, i q2) i wi (q1 i+1, q2 i+1 )... where (q1, 0 q2) 0 Q 0 1 Q 0 2, for ll i 1, we hve (q1, i q2) i F 1 F 2 nd ( w n ) n ounded. From the construction, for ll i N, we hve w i = u 0 i v0 i u1 i v1 i... uki i v ki i, where q1 i = q1(0) i u0 i q1(1) i u1 i q1(2)... i uk i i q1(k i i + 1) = q1 i+1 in A 1 q2 i = q2(0) i v0 i q2(1) i v1 i q2(2)... i vk i i q2(k i i + 1) = q2 i+1 in A 2 the sttes (q1 i(k), qi 2 (k)) elong to F 1 F 2. We define u i = u 0 i u1 i... uki i nd v i = v 0 i v1 i... vki i. From the ove follows tht u nd v re ccepted y A 1 nd A 2, respectively. Then w (M 1 M 2 ) B. Conversely, sequence in (M 1 M 2 ) B is clerly ccepted y A. Hence A recognizes (M 1 M 2 ) B. We now prove tht ll str-free ωb-regulr expressions re recognized y non-deterministic finitry Büchi utomton. Since NFB re closed under finite union (Theorem 8), we only need to consider expressions L M ω, where L Σ is regulr lnguge of finite words nd M str-free B-regulr expression. The constructions ove ensure tht there exists A M = (Q M, Σ, δ M, Q 0 M, F M), non-deterministic finitry Büchi utomton tht recognizes the lnguge M B of infinite sequences. Let A L = (Q L, Σ, δ L, Q 0 L, F L) e finite utomton over finite words tht recognizes L. We construct non-deterministic finitry Büchi utomton s follows: A = (Q L Q M, Σ, δ, Q 0 L, F M) where δ = δ L δ M {(q, ε, q ) q F L, q Q 0 M }. In other words, first A simultes A L, nd when finite prefix is recognized y A L, then A turns to A M nd simultes it.