A Complexity Mesure on Büchi Automt Dn Fismn University of Pennsylvni fismn@ses.upenn.edu Abstrct. We define complexity mesure on non-deterministic Büchi utomt, bsed on the notion of the width of the skeleton tree introduced by Kähler nd Wilke. We show tht the induced hierrchy tightly correltes to the Wgner Hierrchy, corner stone in the theory of regulr ω-lnguges tht is derived from complexity mesure on deterministic Muller utomt. The reltion between the hierrchies entils, for instnce, tht nondeterministic Büchi utomton of width k cn be trnslted to deterministic prity utomton of degree t most 2k + 1. Keywords: utomt nd logic, utomt for system nlysis nd progrm verifiction, clssifiction of regulr ω-lnguges. 1 Introduction There re vrious wy to define cceptnce on infinite words, deriving different types of ω-utomt. Büchi utomt hve the simplest cceptnce criterion: run is declred ccepting if the set of sttes visited infinitely often intersect designted set of ccepting sttes F. Their dul, co-büchi utomt declre run ccepting if the set of sttes visited infinitely often does not intersect designted set of rejecting sttes F. Muller utomt hve the most generl cceptnce criterion: run is ccepting if the set of sttes visited infinitely often is exctly one of set of designted subsets of sttes F 1, F 2,..., F k. Other types of ω-utomt include Rbin, Streett nd prity. The different ω-utomt hve vrying levels of complexity nd expressivity. Using the convention tht DT (NT) for T {B, C, R, S, M, P} denotes the clss of lnguges ccepted by deterministic (nondeterministic) utomt of type Büchi, co-büchi, Rbin, Street, Muller or prity, resp., the following reltions re known. The clss DB is less expressive thn NB which is s expressive s DM, DP, DR, DS, which recognize ll regulr ω-lnguges. The clsses DB nd DC hve incomprble expressive power, in the sense tht there exists lnguges in DB \ DC nd in DC \ DB. Wgner [1] hs suggested complexity mesure on Muller utomt, nd showed tht this complexity mesure is lnguge-specific nd is invrint over ll utomt ccepting the sme lnguge. This result, referred to s the Wgner Hierrchy, is corner stone in the theory of regulr ω-lnguges. The clsses DB This reserch ws supported by US NSF grnt CCF-118996.
nd DC coincide with clsses of the lower levels of this hierrchy. For lnguge L the miniml number of colors required by recognizing deterministic prity utomton, nd the miniml number of pirs required by deterministic Rbin or Streett utomton tightly correltes to the miniml clss in the hierrchy in which the recognized lnguge L resides. In this pper we define complexity mesure on nondeterministic Büchi utomt (nb). The mesure is bsed on the notion of the width of the skeletontree, structure for summrizing runs of nbs, introduced by Kähler nd Wilke [7]. A lnguge L is sid to be of degree k, or belong to NB k, if there exists n nb N recognizing L such tht the width of the skeleton-tree of ll words with respect to N is t most k. We show tht this mesure induces strict hierrchy of clsses of lnguges, nd tht this hierrchy tightly correltes to the Wgner hierrchy. The reltion between the hierrchies entils, for instnce, tht n nb of width k cn be trnslted to prity utomton using t most 2k + 1 colors, nd lnguge L NB k+1 \ NB k cnnot be trnslted to prity utomton using less thn 2k colors. We provide definitions for ω-utomt, summry of the Wgner hierrchy, nd definition of the skeleton-tree in Section 2. The contribution of the pper strts in Section where we define the complexity mesure on Büchi utomt, nd stte the min theorem of the pper, tht the induced hierrchy tightly correltes to the Wgner hierrchy. Section 4 proves the direction from the proposed hierrchy to the Wgner hierrchy nd Section 5 the other direction. In sense, the direction from the Wgner hierrchy to the proposed hierrchy provides n insight on the need for non-determinism in the Büchi model, nd quntifiction of the mount of non-determinism needed reltive to the complexity of the Muller utomton. 2 Preliminries Automt on Infinite Words An utomton is tuple A = Σ, Q, q 0, δ, α consisting of finite lphbet Σ of symbols, finite set Q of sttes, n initil stte q 0, trnsition function δ : Q Σ 2 Q, nd n cceptnce condition α. A run of n utomton on n infinite word v = 1 2... is n infinite sequence of sttes p 0, p 1, p 2..., such tht p 0 = q 0 nd p i+1 δ(p i, i ) for every i N. The trnsition function cn be extended to function from Q Σ (to Q) by defining δ(q, ɛ) = q nd δ(q, v) = δ(δ(q, ), v) for q Q, Σ nd v Σ, where ɛ denotes the empty word. We sy tht A is deterministic if δ(q, ) 1 for every q Q nd Σ. An utomton ccepts word if t lest one of the runs on tht word is ccepting. We use [A] to denote the set of words ccepted by A. We use L to denote the lnguge Σ ω \ L. For finite words the cceptnce condition is set F Q nd run on v is ccepting if it ends in n ccepting stte, i.e., if δ(q 0, v) F. For infinite words, there re mny cceptnce conditions in the literture; here we mention four: Büchi, co-büchi, prity nd Muller. In Büchi nd co-büchi utomt the cceptnce condition refers to subset F of the 2
sttes set. In prity utomt the cceptnce condition refers to mpping κ : Q [1..k] from the set of sttes to the set of colors [1..k]. 1 In Muller utomt the cceptnce condition is function τ : 2 Q {+, } ssigning positive/negtive polrity to subsets of sttes. Let ρ be n infinite sequence of sttes s 0, s 1, s 2.... We use κ(ρ) to denote the respective sequence of colors κ(s 0 ), κ(s 1 ), κ(s 2 ).... We use Inf(ρ) to denote the set of sttes tht pper infinitely often in ρ. An infinite pth ρ stisfies the Büchi condition w.r.t F iff Inf(ρ) F co-büchi condition w.r.t F iff Inf(ρ) F = prity condition w.r.t κ iff min(inf(κ(ρ))) is odd. Muller condition w.r.t τ iff τ(inf(ρ)) = +. We use db, dc, dp, nd dm, to denote deterministic Büchi, co-büchi, prity nd Muller utomt nd nb, nc, np, nd nm to denote the respective non-deterministic utomt. Similrly, we use DB, DC, DP nd DM to denote the clss of lnguges recognized by db, dc, dp nd dm, respectively, nd NB, NC, NP, NM to denote the clss of lnguges ccepted by nb, nc, np, nd nm, respectively. For prity utomt we use DP k to denote the set of lnguges recognized by dp with k colors, nd refer to it s the DP k hierrchy. The Chin Mesure Let D = Σ, Q, q 0, δ, τ be complete dm (i.e. dm where δ(q, ) = 1 for every q Q nd Σ). A set of sttes S Q is sid to be dmissible if S is rechble strongly connected component (scc) of D. An dmissible set S is sid to be positive or ccepting (resp. negtive or rejecting) iff τ(s) = + (resp. τ(s) = ). A chin of dmissible sets S 0 S 1 S m 1 is D-chin iff the sets re lterntely positive nd negtive. The length of such chin is m. The polrity of D-chin is determined ccording to the polrity of its bottom set. Tht is, the bove D-chin is sid to be positive (resp. negtive) iff S 0 is positive (resp. negtive). We use k(d), k + (D) nd k (D) to denote the mximl length of D-chin, positive D-chin nd negtive D-chin, resp. In the sequel we will focus on k + (D) to which we refer s the positive-chin mesure. Note tht 0 k(d) Q. Also k + (D) k (D) 1 simply by omitting the bottom set. Definition 1 (The Clsses DM + k nd DM k [1]) Let k N. The clss of lnguges DM + k nd DM k re defined s follows. DM + k = { L dm D : L = [D ], k+ (D) k} DM k = { L dm D : L = [D ], k (D) k} The Wgner hierrchy consists of n dditionl mesure, the length of the longest sequence of chins tht re rechble from ech other nd hve lternting polrities, nd tkes into ccount lso the polrity of the chins. We omit the detils for lck of spce nd since this mesure is of less importnce to this pper. 1 For j, k N s.t. j k, we use [j..k] to denote the set {j, j+1, j+2,..., k}.
Wgner [1] hs shown tht the hierrchy is strict nd tht these mesures re invrint overll dms ccepting the sme lnguge. He further showed tht the chin hierrchy tightly correltes to the miniml numbers of chins in Rbin utomton. A similr result regrding prity utomt (see f.g. [2, ]) sttes tht L DM + k iff L DP k. The reltion between dbs nd the clss DM + 1 ws lredy shown by Lndweber [10]. The Wgner hierrchy hs been rediscovered severl times [8, 1]. Theorem 1 ([1, 10, 2, ]). 1. L DM + k L DM k 2. DM k DM k+1 nd DM+ k DM+ k+1. L DM + 1 L DB nd L DM 1 L DC. 4. L DM + k L DP k The Skeleton Tree The complexity mesure we propose is bsed on the notion of the width of the skeleton tree introduced by Kähler nd Wilke [7]. Kähler nd Wilke, iming to provide constructions unifying Büchi determiniztion, complementtion nd dismbigution introduced the notions of the split tree, the reduced tree nd the skeleton tree, where the ltter hs been sid to be identified from the work of Muller nd Schupp [11]. All three re mechnisms to summrize runs of nbs. For lck of spce we suffice here with n informl description; the forml definition of these trees, nd n illustrting exmple, cn be found in the ppendix. The split-, reduced- nd skeleton-trees re defined per given word w nd w.r.t. given nb N. A key invrint tht is mintined is tht if there exists n ccepting run of N on w then there is n ccepting infinite pth in ll of these trees. Roughly speking, the split tree refines the subset construction by seprting ccepting nd non-ccepting sttes. From ech node of the tree the left son holds its ccepting successors nd the right son its non-ccepting successor. Thus, n ccepting pth hs infinitely mny left turns, nd is lso referred to s left recurring. The width of lyer of the split-tree is generlly unbounded. The reduced tree bounds the number of nodes on lyer of the tree to n, the number of sttes of the given Büchi utomton N, by eliminting from node of the tree ll sttes tht ppered in node to its left. The skeleton-tree is the smllest sub-tree of the reduced-tree tht contins ll its infinite pths. The width of the skeleton tree thus equls the number of infinite pths in the reduced tree. We use width(n, w) to denote the width of the skeleton tree for w w.r.t to N. A Complexity Mesure on NBAs Given n nb N we sy tht the width of N is the mximl width of skeleton tree on ny given word. Formlly width(n ) = mx {width(n, w) w Σ ω } 4
N 1 : 1 N 2 : 1 2 Fig. 1. Two nbs N 1 nd N 2 such tht [[N 1]] = [N 2]] yet width(n 1) width(n 2). It is not hrd to see tht two nbs N 1 nd N 2 my hve different widths even if [N 1 ] = [N 2 ] since we cn dd nondeterministic trnsitions to ccepting nd non-ccepting sttes without chnging the lnguge. For instnce, consider the nbs N 1 nd N 2 over lphbet Σ = {} depicted in Fig. 1. We hve [N 1 ] = [N 2 ] = { ω }, yet width(n 1, w ) = 1 nd width(n 2, w ) = 2. Moreover, since ω is the only word in Σ ω we hve width(n 1 ) = 1 nd width(n 2 ) = 2. Definition 2 (The Clsses NB k ) Let k 1 be nturl number. The clss of lnguges NB k is defined s follows. NB k = { L nb N : L = [N ], width(n ) k} The min contribution of this pper cn be summrized by the following two theorems. Theorem 2 sttes tht the hierrchy is strict. For every level k of the hierrchy there exists lnguge L such tht L cnnot be recognized by ny nb of width k or smller. Theorem sttes tht this hierrchy is tightly correlted to the Wgner hierrchy. Theorem 2. NB k NB k+1 Proof. This is corollry of Theorem 1 nd the forthcoming Theorem. Theorem. Let L be n ω-lnguge, nd let k 0 be nturl number. Then L NB k = L DM + 2k+1 L DM + k = L NB k 2 +1 Proof. The first nd second items re respectively given by the forthcoming Corollry 7 nd Proposition 12. Deciding the Width In view of these reltions, n importnt question is to find the width of given nb N. Proposition 4 The problem of finding the width of n nb is solvble in time n O(n) where n is the number of sttes in the given nb. Proof. Fismn nd Lustig [4, Proposition 2] provide construction for db B k tht ccepts word w iff width(n, w) < k when N is given nb. A db cn be seen s dp with 2 colors (ccepting sttes re colored 1 nd nonccepting sttes re colored 2). Given dp P k one cn construct dp P k for 5
its complement by ssigning κ(q) = κ(q) + 1 (where κ is the coloring function of P k nd κ is the coloring function of P k ). The constructed dp P k ccepts word w iff width(n, w) k. Thus, width(n ) < k iff P k is empty. Emptiness of dp cn be solved in time polynomil in the number of sttes nd colors. Applying binry serch would dd fctor of log n where n is the number of sttes in N. When n is the number of sttes in N, the db B k my hve upto n O(n) sttes. Since the dp P k hs the sme number of sttes s B k (nd the time spent to build B k is polynomil in the size of B k ), we cn decide the width in time n O(n). Krishnn, Puri nd Bryton [9] show tht determining the Rbin index (Streett index) of given dr (ds) is np-complete, yet it is decidble in polynomil time whether dr (ds) with f pirs hs Rbin index (Streett index) f or ny constnt c. 2 Wilke nd Yoo [14] show tht the chin mesure of given dm over n lphbet of size l with n sttes, m strongly connected components, nd f ccepting sets cn be decided in time O(f 2 nl + m log m). Note tht m nd f my be exponentil in n. Since the width is not n invrint mong ll nbs ccepting given lnguge, the question of finding the miniml k for which [N ] NB k where the input is N is not nswered by Proposition 4. It cn be nswered e.g. by trnslting it to dr or dm nd inferring the result using the bove mentioned results nd the reltions of the hierrchies, but lower bound remins open. 4 From NB k to DM k The proof of the first prt of Theorem goes vi dps, using the reltions between their rnk nd the chin mesure s stted in Theorem 1. Proposition 5 L NB k = L DP 2k+1. Proof. The known constructions from nb to dp, given n nb N with n sttes produce dp of rnk 2n [12, 4]. The construction of Fismn nd Lustig [4] is bsed on the notion of the width of the skeleton tree. Given n nb N, they first show how to construct dp P k tht provides correct nswer only for words of width exctly k w.r.t. N, s stted in Lemm 6. Lemm 6 ([4, Proposition 4]) Let N be n nb with n sttes, nd let k [1..n]. There exists dp P k using colors {0, 1, 2} such tht for ny word w if width(n, w) = k then P k ccepts w iff N does, if width(n, w) < k then P k rejects w, if P k ccepts w then w is ccepted by N, nd P k visits 0 infinitely often iff width(n, w) < k. 2 The Rbin index (Streett index) is the lest possible number of ccepting pirs used in dr (ds) recognizing the lnguge. 6
M 2 : {{0, 1}} 0 1 b b, c, c B 2 : 1, c b 2 0, b, c b, c Fig. 2. A dm M 2 in DM + 1 nd miniml equivlent db B2. They then show tht dp recognizing the sme lnguge s N cn be constructed by running the dps for width 1 to n in prllel, where n is the size of N. The colors re distributed so tht P k uses colors 2k, 2k + 1 nd 2n + 2, nd the color of the compound stte (s 1, s 2,..., s n ) where s i is the stte of P i is min{c 1, c 2,..., c n } where c i is the color of s i. The obtined dp hs rnk 2n + 2. Clerly the sme resoning shows tht given the width of N is t most k, the dp obtined by running in prllel the utomt P 1, P 2,..., P k provides correct result, nd uses 2k + 1 colors. The following is direct corollry of Proposition 5 nd Theorem 1. Corollry 7 L NB k = L DM + 2k+1. 5 From DM + k to NB k We show here the second item of Theorem. We strt with the simple cses DM + 1 nd DM 1, nd then generlize the ides of these constructions to obtin the construction for DM + k for rbitrry k. By Theorem 1, L DM + 1 L DB. Any db hs width 1 when regrded s n nb. It follows tht L DM + 1 implies L NB 1. We note, however, tht given dm M such tht M DM + 1 it is not lwys the cse tht we cn define db on the sme structure s M. Consider, for instnce, the dm M 2 of Figure 2 defined over lphbet Σ = {, b, c}. It ccepts the lnguge (Σ Σ b) ω. No rejecting set subsumes n ccepting set, thus the mximl length of positive chin is 1 nd [M 2 ] DM + 1. While M 2 hs two sttes there is no db with two sttes tht ccepts the sme lnguge. A miniml db for [M 2 ] requires t lest sttes. The db B 2 of the sme figure is miniml db for [M 2 ]. Proposition 8 below follows from the fct tht L DM + 1 L DB. We provide direct construction bsed on the lr (ltest ppernce record) dt structure due to Gurevich nd Hrrington [6], for completeness nd to introduce lr which will be used in the proof of the reltions of higher levels of the hierrchy s well. Proposition 8 L DM + 1 implies L NB 1 7
Proof. Let M be dm in DM + 1. From the definition of the hierrchy clss DM + 1 it follows tht no superset of n ccepting scc in M cn be rejecting. On the other hnd, subset of n ccepting set my be rejecting. To be ble to use Büchi condition we need to define set of sttes F such tht visit to F gurntees tht ll sttes of some ccepting set were visited. The solution uses the lr (ltest ppernce record) dt structure due to Gurevich nd Hrrington [6]. The ide is to construct deterministic utomton whose sttes re permuttions of the sttes of M ugmented by hit position, denoted. If the current stte is p 1 p 2... p i p i+1... p n nd δ(p n, ) = q nd q = p j, then δ B ( p 1 p 2... p i p i+1... p n, ) = p 1 p 2... p j 1 p j+1... p n p j Tht is, the trnsition reltion moves the stte p j currently visited by M to the rightmost position in the list, nd moves the symbol to the position on the list where p j resided previously. We formlize this using the following definitions, tht will lso be used in lter proofs. Formlly, let Q = {q 0, q 1,..., q n 1 }, δ : Q Σ Q, τ : 2 Q {+, }, symbol not in Q, nd A Q subset of crdinlity l with sttes p 1, p 2,..., p l so tht if p i = q j nd p i+1 = q k then j < k. Let lrset(a) = {w (A { }) q A { }, w q = 1} lrinit(a, p i ) = p 1 p 2... p i 1 p i+1 p i+2... p l p i lrtrns(a) = {( p 1 p 2... p i p i+1... p l,, p 1 p 2... p j 1 p j+1... p l p j ) δ(p l, ) = p j } lrcc(a, τ) = {u v set(uv) = A, τ(set(v)) = +} where set(v) denotes the set of sttes in the word v. Let M = Σ, Q, q 0, δ, τ. Let B be the db Σ, lrset(q), lrinit(q, q 0 ), lrtrns(q), lrcc(q, τ). Lemm 9 ([5, Lemm 1.21]) Let ρ be run of M on given word w nd let ρ B = s 0 s 1 s 2... be the run of B on w where s i = u i v i. Then Inf(ρ) = S iff the following conditions hold for some i 0 N for ll i > i 0 we hve set(v i ) S nd for infinitely mny i s we hve set(v i ) = S Since M DM + 1 gurntees tht no superset of n ccepting set my be rejecting, it is enough to require tht we infinitely often visit stte u v where ll sttes of v form n ccepting set of M, this is exctly the cceptnce condition of B. Therefore, [B ] = [M]. Since B is deterministic, the width of B is one. Hence, L NB 1. For L DM 1 there is no gurntee tht n equivlent db exists. We re gurnteed, though, tht n equivlent dc exists. In Proposition 11 we show tht this entils tht L NB 2. We use w to denote the number of occurrences of the letter in w. 8
First we need some terms nd lemm. Let ρ = q 0 q 1 q 2... be run of given utomton A on given word w. We sy tht the run ρ gets trpped in n scc S if strting from some z N for every z > z the stte q z of this run belongs to S. In n utomton with finitely mny sttes, every run on n infinite word should eventully get trpped in some scc. We let trp(ρ) denote the miniml scc tht ρ gets trpped in. If the utomton is non-deterministic, there my be severl runs on given word nd ech run my get trpped in different scc. For skeleton pth ϱ = Q 0 Q 1 Q 2... we sy tht it gets trpped in {S 1,..., S k } if strting from some z N for every z > z we hve Q z i [1..k] S i. We let trp(ϱ) = {S 1,..., S k } denote the miniml set of miniml sccs tht ϱ gets trpped in. Lemm 10 Let N = Σ, Q, q 0, δ, F be n nb with n scc S stisfying the following two conditions: For every letter Σ nd every stte q S, δ(q, ) = 1. For every pir of sttes q, p S we hve [N q0,q ] [N q0,p ] =. 4 Let SK A w be the skeleton-tree of w w.r.t N. Let ϱ 1 nd ϱ 2 be two skeleton pths in SK A w. Then S trp(ϱ 1 ) implies S / trp(ϱ 2 ). Proof. Consider word w = 1 2..., nd let ρ 1 nd ρ 2 be two different runs of N on w. Assume now both ρ 1 nd ρ 2 get trpped in S. We clim tht there exists point in which both runs rech the sme stte. Tht is, if ρ 1 = q 0 q 1 q 2... nd ρ 2 = p 0 p 1 p 2... then there exists z N such tht for every z > z we hve q z = p z. Assume ρ 1, ρ 2 enter the scc S t time points z 1, z 2, resp. nd w.l.o.g. z 1 z 2. Then ρ 1 (resp. ρ 2 ) entered S fter reding the prefix 1 2... z1 of w (resp. 1 2... z2 ). We clim tht p z2 = q z2. If not, then exists two sttes p, q S such tht p = p z2 q z2 = q nd 1 2... z2 [N q0,p ] [N q0,q ] contrdicting the second premise of the lemm. Now tht we estblished p z2 = q z2, since by the first premise ll trnsitions within S re deterministic, it follows tht p z = q z for every z > z 2. Since ll runs tht get trpped in S eventully rech the sme stte, it follows tht they must conjoin to the sme pth ϱ of SK A w. Proposition 11 L DM 1 implies L NB 2 Proof. Let M be dm in DM 1. From the definition of the hierrchy clss DM 1 it follows tht no subset of n ccepting scc in M cn be rejecting. We show tht we cn build n equivlent nb B of width 2 using the following ide. The nb B will consists of ll of M s sttes nd trnsitions. In ddition, for ech mximl ccepting scc A of M, B will hve the bility to non-deterministiclly trnsit to copy A of A. The primed copy of A will hve ll the inner trnsitions of A, but no trnsitions out of A. This wy B cn t ny point during the run choose to move to primed copy A of one of the ccepting sccs A. Once this 4 The nottion N q,q is used to denote the nf Σ, Q, q, δ, {q } obtined from N by mking q the initil stte, q the finl stte, nd regrding it s nondeterminsitc utomton on finite words. 9
choice ws mde, B cn only remin in A or fll off the utomton. The set of ccepting sttes of B consists of ll sttes in the primed copies. If B visits such stte in sy the ccepting scc A i infinitely often then the run of M gets trpped in A i, nd if the run of M gets trpped in A i, then there exists run of B tht will get trpped in A i. Therefore, both recognize the sme lnguge. Formlly, let M = Σ, Q, λ, δ, τ. Let A 1, A 2,..., A k be the mximl ccepting sccs of M. 5 The nb B = Σ, Q B, q 0B, δ B, F B is defined s follows. The sttes of Q B re Q A 1 A 2... A k where A i = {q q A i }. The initil stte q 0B is q 0. The ccepting sttes F B re i [1..k] A i. The trnsitions δ B re defined s follows. If (q,, p) δ then (q,, p) δ B. In ddition, if q Q nd p A i then (q,, p ) δ B. If q A i nd p A i then (q,, p ) δ B. Suppose run q 0 q 1 q 2... of M on given word w gets trpped in n ccepting scc A i. Since no subset of A i is rejecting, this run is ccepting. Let z 0 be such tht q z A i for every z > z 0. Then B cn mimic M up to point z 0 1, t z 0 it cn mke nondeterministic trnsition to A i nd then sty there forever long. Tht is, the run q 0 q 1 q 2... q z0 1q z 0 q z 0+1q z 0+2... where q z 0+i is the primed version of q z0+i is run of B, nd this run is ccepting. For the other direction, if there exists n ccepting run of B on given word w, it mens tht strting from some point the run moved to one of the A i s nd styed there forever long. This entils tht when M reds w, it will get trpped in the scc A i. (If this ws not the cse, tht run of B on w would encounter letter for which no trnsitions is vilble.) Thus [M] = [B ]. Next we show tht the width of B is 2. Let ρ w be the run of M on given word w nd let trp(ρ w ) = T. If T is rejecting then B cn only get trpped in T, since on ny choice to move to some A i it will end up flling of A i since M DM 1 implies T A i. (If this ws not the cse then, since for ny trnsition in A i there is trnsition from A i to A i, this would entil tht M would hve got trpped in A i which is not the cse.) If T is ccepting then it equls some A j for j [1..k]. Then run of B on w cn get trpped either in A j or in A j, but it cnnot get trpped in ny other A i or A i. Since, from the sme rguments s in the rejecting cse, if t some point B chooses to trnsit to A i it will eventully fll off of it. We hve shown tht ech run of B my get trpped in t most 2 msccs. Since ll msccs of B (the originls of M nd the new msccs A i ) stisfy the premises of Lemm 10, nd since for ech skeleton pth ϱ we hve trp(ϱ), the mximum width of ny run of B is 2. Thus [M] NB 2. We cn now generlize these two ides to obtin construction for DM + k rbitrry k. for Proposition 12 L DM + k = L NB k 2 +1 Proof. Let M be dm in DM + k. From the definition of the hierrchy clsses it follows tht the mximum length of positive chin is k. We build DM + k 5 An scc A of M is sid to be mximl ccepting if no ccepting scc A subsumes it. Note tht A is not required to be n mscc. E.g. if R = A {q} is n mscc, nd R is rejecting nd A is ccepting, then A is mximl ccepting scc, but not n mscc. 10
n equivlent nb B of width t most k 2 + 1 using the following ide. As in the proof of Proposition 11, B will consists of ll sttes of the given dm M, nd will hve nondeterministic trnsitions to copies of ccepting sccs. Unlike in tht proof, we will need to consider not just the mximl ccepting sccs but ll ccepting sccs. As in the proof of Proposition 8, since n ccepting scc A my contin rejecting sccs, we will use lr construction to mke sure tht ll sttes of n ccepting scc re visited infinitely often. Formlly, let M = Σ, Q, λ, δ, τ. Assume A 1, A 2,..., A m re the ccepting sccs of M. We define the nb B = Σ, Q B, q 0B, δ B, F B s follows. q 0B = q 0 Q B = Q i [1..m] lrset(a i ) δ B = δ i [1..m] lrtrns(a i ) {(q,, lrinit(a i, p)) (q,, p) δ, q Q \ A i, p A i } F B = i [1..m] lrcc(a i, τ i ) where τ i (S) = + iff S = A i. We clim tht B recognizes the sme lnguge s M. Let w be word nd let ρ be the run of M on w. Let Inf(ρ) = S. If τ(s) = + then S = A i for some i [1..m]. Assume ρ gets trpped in S fter time point z. Thus B, t some time point fter z, cn choose trnsition of the form (q,, lrinit(a i, p)) nd move to lrset(a i ). Since M gets trpped in A i, B will not fll off lrset(a i ), nd by Lemm 9, infinitely often B will visit stte of the form i v such tht set(v) = A i. Thus B will ccept. Suppose now τ(s) =. If B did not move to ny of the lrset(a i ), clerly it will not ccept. Suppose it did move to lrset(a i ) for some i. If A i does not subsume S then B will fll off lrset(a i ). Assume thus A i S. Then by Lemm 9, eventully only sttes of the form u i v where set(v) S will be visited. Since A i S, by Lemm 9, it will not be the cse tht infinitely often sttes of the form i v with set(v) = A i will be visited, thus B will reject. Hence, [B ] = [M]. We turn to reson bout B s width. Let ρ w be the run of M on w nd ssume trp(ρ w ) = T. Let ρ be run of B, then from the sme rguments s in the proof of Proposition 11 we hve tht either trp(ρ) = T or trp(ρ) = lrset(a i ) for some A i T. Note tht by definition of B, for every i [1..m] the set lrset(a i ) is n mscc, nd ll the msccs of B stisfy the premises of Lemm 10. Thus if ϱ 1 nd ϱ 2 re two skeleton pths of SK B w then trp(ϱ 1 ) trp(ϱ 2 ) =. We clim further tht if lrset(a 1 ) trp(ϱ 1 ) nd lrset(a 2 ) trp(ϱ 2 ) then either A 1 A 2 or A 2 A 1. Assume this is not the cse. Note tht ϱ SK B w such tht T trp(ϱ). Since there is single run leding to T, nd ll of its sttes re non-ccepting, ϱ must be the rightmost skeleton-pth in SK B w. Since Q is the only non sink mscc of B, ll skeleton pths split from ϱ t some point. Assume ϱ 1 nd ϱ 2 split from ϱ t time points z 1 nd z 2, resp. The nodes t the splits must be ccepting (since their sibling in ϱ is non-ccepting). Thus t z 1 it must be tht ϱ 1 recently visited ll sttes of A 1 nd t z 2 it must be tht ϱ 2 recently visited ll sttes of A 2. Assume w.l.o.g. z 1 < z 2. If t some time point between z 1 nd z 2, the originl pth of M, ρ w visited node in A 2 \ A 1 then ϱ 1 will not hve ny descendnts (since lrset(a 1 ) hs no corresponding trnsitions). Assume thus ll nodes of A 2 \ A 1 hve recently been visited before z 1. Then 11
t z 1 not only ll A 1 re visited but lso ll of A 2, thus there exists split corresponding to lrset(a 2 ) before or t z 2, which contrdicts the split t z 2 since lrset(a 2 ) cn only be trpped in one skeleton-pth. Therefore if the set of skeleton pths of SK B w is {ϱ} {ϱ 1,..., ϱ l } where ϱ i split from ϱ before ϱ i+1 then trp(ϱ 1 ),..., trp(ϱ l ) will correspond to ccepting sets A 1,..., A l long one inclusion chin of M. Since the length of positive chin is bounded by k, the number of ccepting sets long such chin is bounded by k 2. Adding ϱ, the rightmost brnch, we obtin tht the width of B is k 2 + 1. This completes the proof of Theorem. Acknowledgments I would like to thnk Jvier Esprz for sking me if I cn chrcterize words/lnguges of certin skeleton-width, when I presented [4] t CONCUR 15. This question initited this study. I would like to thnk Yod Lustig for mny interesting discussions on Büchi determiniztion, Dn Angluin for mny interesting discussions on the Wgner Hierrchy, nd Orn Kupfermn for importnt comments on n erly drft. References 1. Bru, R.: The Husdorff-Kurtowski hierrchy of omeg-regulr lnguges nd hierrchy of Muller utomt. Theoreticl Computer Science 96(2), 45 60 (1992) 2. Crton, O., Perrin, D.: The Wdge-Wgner hierrchy of omeg-rtionl sets. In: ICALP. pp. 17 5 (1997). D. Perrin, J.E.P.: Infinite Words: Automt, Semigroups, Logic nd Gmes. Springer (2004) 4. Fismn, D., Lustig, Y.: A modulr pproch for Büchi determiniztion. In: 26th Int. Conf. on Concurrency Theory. pp. 68 82 (2015) 5. Grädel, E., Thoms, W., Wilke, T.: Automt, Logics, nd Infinite Gmes: A Guide to Current Reserch, Lecture Notes in Computer Science, vol. 2500. Springer (2002) 6. Gurevich, Y., Hrrington, L.: Trees, utomt, nd gmes. In: Proc. 14th ACM Symp. on Theory of Computing. pp. 60 65. ACM Press (1982) 7. Kähler, D., Wilke, T.: Complementtion, dismbigution, nd determiniztion of Büchi utomt unified. In: Proc. 5th Int. Colloq. on Automt, Lnguges, nd Progrmming. Lecture Notes in Computer Science, vol. 5125, pp. 724 75. Springer (2008) 8. Kminski, M.: A clssifiction of ω-regulr lnguges. Theoreticl Computer Science 6, 217 229 (1985) 9. Krishnn, S., Puri, A., Bryton, R.: Structurl complexity of ω-utomt. In: Proc. 12th Symp. on Theoreticl Aspects of Computer Science. Lecture Notes in Computer Science, vol. 900. Springer (1995) 10. Lndweber, L.: Decision problems for ω utomt. Mthemticl Systems Theory, 76 84 (1969) 11. Muller, D., Schupp, P.: Simulting lternting tree utomt by nondeterministic utomt: New results nd new proofs of theorems of Rbin, McNughton nd Sfr. Theoreticl Computer Science 141, 69 107 (1995) 12. Pitermn, N.: From nondeterministic Büchi nd Streett utomt to deterministic prity utomt. In: Proc. 21st IEEE Symp. on Logic in Computer Science. IEEE Computer Society Press (2006) 12
1. Wgner, K.W.: A hierrchy of regulr sequence sets. In: MFCS. pp. 445 449 (1975) 14. Wilke, T., Yoo, H.: Computing the Wdge degree, the Lifschitz degree, nd the Rbin index of regulr lnguge of infinite words in polynomil time. In: Proc. 6th Joint Conf. Theory nd Prctice of Softwre Development. pp. 288 02 (1995) A Forml definition of the skeleton tree We provide here the definitions of the split, reduced nd skeleton trees of Kähler nd Wilke. In the following we ssume fixed non-deterministic Büchi utomton A = (Σ, Q, q 0, δ, F ) with Q = n nd n infinite word w = σ 1 σ 2 σ... over Σ. We use F to denote the set of non-ccepting sttes, i.e. Q \ F. Annotted Binry Trees We first provide some elementry definitions of nnotted binry trees. The nodes of such trees re strings in {0, 1}. The root node is the empty string ɛ. The left successor of node t is t0 nd the right successor is t1. An nnotted binry tree is function from {0, 1} to some domin D. A node mpped to is regrded s bsent from the tree. The depth of node t, denoted t, is its distnce from the root, nd it equls the length of the string t. So the depth of the root is 0, nd the depth of its successors re 1. For t 1, t 2 {0, 1} we sy tht t 1 < lex t 2 if the string t 1 is lexicogrphiclly smller thn t 2. We sy tht t 1 < lft t 2 if t 1 = t 2 nd t 1 < lex t 2. The i-th level of n nnotted tree T consists of ll nodes in depth i. If t 1 < lex t 2 < lex < lex t n re ll the nodes of the i-th level of T then the i-th slice of T is the sequence T (t 1 ), T (t 2 ),..., T (t n ). The width of the i-th level is the number of nodes in tht level. The Split Tree The split tree cn be thought of s refinement of the subset construction by differentiting ccepting nd rejecting sttes. It is binry tree, where node holds subset of A s sttes. If S is subset in level i of the split-tree, then its left successor holds ll the ccepting successors of sttes in S nd its right successor holds ll the non-ccepting successors of sttes in S. Formlly, the split tree of A on w is function from {0, 1} to 2 Q stisfying SP A w(ɛ) = I, nd for t {0, 1} such tht t = i nd SP A w(t) = S, SP A w(t0) = δ(s, σ i+1 ) F nd SP A w(t1) = δ(s, σ i+1 ) F. We sy tht ρ {0, 1} ω is n infinite pth of SP A w(t) if SP A w(t) for every prefix t of ρ. We sy tht n infinite pth ρ of SP A w(t) is left-recurring or ccepting if ρ hs infinitely mny zeros (we use the term left recurring to emphsize the pth hs infinitely mny left turns, nd the term ccepting to emphsize it hs infinitely mny ccepting nodes). Clim ([7]). A ccepts w iff SP A w hs left recurring pth. The width of SP A w s lyers is unbounded, in generl. 1
W : 0 T ω : 0 T (b) ω : 0 1 b 2, b, b 012 012 012 2 1 02 1 2 1 02 2 1 02 1 2 1 02 2 Fig.. A Büchi utomton W nd its reduced nd skeleton trees for words w 1 = ω nd w 2 = (b) ω. Solid nodes nd edges belong to both the reduced nd skeleton trees, nd dotted nodes nd edges belong to the reduced tree but not to the skeleton tree. A node lbeled with multiple digits stnds for the subset contining the corresponding sttes (e.g. 012 stnds for {0, 1, 2}). The Reduced Split Tree The width of the split-tree is unbounded, in generl, which mkes it hrd to hndle. The reduced split tree (or simply the reduced tree) reduces the split tree by removing from node ll sttes tht pper to the left of tht node, in the sme level. This reduction elimintes mny of the pths of the split-tree, but it keeps the left-most ccepting pth. And if there re ccepting runs of A on w, then t lest one is embedded in the left-most ccepting pth. Formlly, the reduced split tree of A on w, denoted RS A w, is function from {0, 1} to 2 Q stisfying RS A w(ɛ) = I, nd for t {0, 1} such tht t = i, RS A w(t) = S nd S = {S t < lft t, δ(rs A w(t ), σ i+1 ) = S }, Clim ([7]). RS A w(t0) = δ(s, σ i+1 ) F \ S nd RS A w(t1) = δ(s, σ i+1 ) F \ S. A ccepts w iff RS A w hs left recurring pth. The width of RS A w s lyers is bounded by n. The Skeleton Tree The reduced tree cn be effectively constructed nd hndled, thnks to its bounded width. It my, however, hve nodes with only finitely mny descendnts. Conceptully, it is esier to think bout tht tree without these nodes. The skeleton tree of A on w, denoted SK A w, is the mximl subtree of RS A w stisfying tht every node hs infinitely mny descendnts. Clim ([7]). A ccepts w iff SK A w hs left recurring pth. The width of SK A w s lyers is monotoniclly non-decresing nd bounded by n. 14
The second item sys tht there exists nturl number z 0 such tht forll z z 0, the width of SK A w t level z is k. This k is thus the number of infinite pths of the skeleton (or reduced) tree. We use width(a, w) to denote the width of the skeleton tree on w. Exmple 1. A Büchi utomton W nd its reduced nd skeleton trees for words w 1 = ω nd w 2 = (b) ω re depicted in Fig.. Both words re of width 2, i.e. width(w, w 1 ) = 2 = width(w, w 2 ), but w 1 / [W ] wheres w 2 [W ]. Indeed, the skeleton tree for w 1 hs no left-recurring pths, wheres the skeleton tree for w 2 does hve left-recurring pth. 15