Regular Expressions for Muller Context-Free Languages

Transcription

1 ct Cyernetic 23 (2017) Regulr Expressions for Muller Context-Free Lnguges Kitti Gelle nd zolcs ván strct Muller context-free lnguges (MCFLs) re lnguges of countle words, tht is, leled countle liner orders, generted y Muller context-free grmmrs. Equivlently, they re the frontier lnguges of (nondeterministic Muller-)regulr lnguges of infinite trees. n this rticle we survey the known results regrding MCFLs, nd show tht lnguge is n MCFL if nd only if it cn e generted y so-clled µη-regulr expression. Keywords: Muller context-free lnguges, well-ordered induction, regulr expressions 1 ntroduction word, lso clled rrngement in [9], is the isomorphism type of leled liner order. Thus, this notion is generliztion of finite nd ω-words, permitting e.g. lelings of the integers of the rtionls. Finite utomt on ω-words hve y now vst literture, see [21] for comprehensive tretment. Finite utomt cting on well-ordered words longer thn ω hve een investigted in [1, 6, 7, 24, 25], to mention few references. Recently, the theory of utomt on well-ordered words hs een extended to utomt on ll countle words, including scttered nd dense words. n [2, 5, 4], oth opertionl nd logicl chrcteriztions of the clss of lnguges of countle words recognized y finite utomt were otined. Context-free grmmrs generting ω-words were introduced in [8] nd susequently studied in [3, 19]. Context-free grmmrs generting ritrry countle words were defined in [10, 11]. ctully, two types of grmmrs were defined, context-free grmmrs with üchi cceptnce condition (CFG), nd context-free grmmrs with Muller cceptnce condition (MCFG). These grmmrs generte the üchi nd the Muller context-free lnguges of countle words, revited s CFLs nd MCFLs. Every CFL is clerly n MCFL, ut there exists n This work ws supported y NKF grnt no University of zeged, Hungry, E-mil: {kgelle,szivn}@inf.u-szeged.hu DO: /ctcy

2 350 Kitti Gelle nd zolcs ván MCFL of well-ordered words tht is not CFL, for exmple the set of ll countle well-ordered words over some lphet. n fct, it ws shown in [10] tht for every CFL L of well-ordered words there is n integer n such tht the order type of the underlying liner order of every word in L is ounded y ω n. n this pper we survey the results on Muller context-free lnguges, nd we give n opertionl ( regulr-expression-like ) chrcteriztion of this clss. 2 Nottion 2.1 Liner orderings (strict) liner ordering is pir (, <) where < is strict totl ordering on, tht is, n irreflexive, trnsitive nd trichotomous reltion. et-theoreticl properties of the domin set, such s finiteness, memership nd crdinlity re lifted to the ordering (, <) thus we cn sy e.g. tht liner ordering is countle, finite etc. n order to ese nottion, we will omit the ordering < from the pir nd identify the ordering (, <) with its domin, if the ordering reltion is not importnt or is cler from the context. n this pper we will (unless stted otherwise) only del with countle orderings. good reference for liner orderings is [23]. n emedding of liner ordering (, <) into liner ordering (J, ) is (necessrily) injective mpping h : J preserving the ordering, i.e. x < y implying h(x) h(y). We write (, <) (J, ) if cn e emedded into J. Clerly, this reltion is preorder ( reflexive nd trnsitive reltion) etween orderings. surjective emedding is clled n isomorphism etween the orderings nd J; if there exists n isomorphism etween nd J, then they re clled isomorphic. somorphism of liner orderings is n equivlence reltion, the clsses of which re clled order types. Well-known isomorphism types re the order type ω of the nturl numers N, the type ω of the negtive integers, the type ζ of integers nd the type η of rtionls. ince isomorphism is comptile with emeddility, we cn sy tht n order type α cn e emedded into n order type β, written s α β. Note tht this reltion is not ntisymmetric in generl s the orderings (0, 1) nd [0, 1] cn e emedded into ech other ut they re not isomorphic s the ltter one hs lest element while the former one does not. The ordering (, <) is su-ordering of (J, ) if J nd < is the restriction of onto. n ordering is well-ordering if it hs no su-ordering of type ω, is scttered if it hs no su-ordering of type η, qusi-dense if it is not scttered nd dense if it hs t lest two elements nd for ny x < y there exists some z with x < z < y. ll dense countle orderings hve the type η, possily enriched with either lest or gretest element (or oth). Order types of well-orderings re clled ordinls. mongst the clss of ordinls, the emeddility reltion itself is well-ordering, moreover, ech ordinl α is either successor ordinl α = β + 1 for some ordinl β, in which cse there is no other ordinl etween α nd β, or is limit ordinl with α = β, i.e. is the lest upper ound of the set of ordinls strictly β<α smller thn α with respect to the emeddility reltion <. Note tht lthough

3 Regulr Expressions for Muller Context-Free Lnguges 351 the ordinls themselves form proper clss, whenever α is n ordinl, then the clss of ordinls smller thn β is set nd whenever X is set of ordinls, then their lest upper ound X exists nd is n ordinl. Moreover, the clss of countle ordinls is the lest clss which contins the order types 0 nd 1 nd is closed under tking ω-sums, tht is, tking suprem of ω-chins. When (, ) is some linerly ordered index set nd for ech i, (J i, < i ) is liner order, then their ordered sum (J, <) = i (J i, < i ) hs the disjoint union J = {(i, j) : i, j J i } s domin with (i, j) < (i, j ) if either i i or i = i nd j < i j. n prticulr, (J 1, < 1 ) + (J 2, < 2 ) denotes the sum (J i, < i ). t is cler i {1,2} tht well-ordered sum of well-ordered orderings is well-ordered, nd scttered sum of scttered orderings is scttered. The sum opertion is lso comptile with the isomorphism, thus it cn e extended to order types. For exmple, ω + ω = ζ nd η + η = η η = η where 1 is the order type of the singleton orderings; in generl, n is the order type of the n-element orderings for n N 0 = {0, 1,...}. f α is the order type of nd β is the order type of ech J i, i, then β α stnds for the order type of the sum i J i. Hence, product of ordinls is n ordinl. Ordinls re lso equipped with n exponenttion opertor ut we will only use finite powers of the form α n, with n eing n integer, tht is, α 0 = 1 nd α n+1 = α n α. Husdorff clssified the scttered order types into n infinite hierrchy. We mke use of the following vrint [17] of this hierrchy: let V D 0 e the clss of ll finite order types, nd when α is some ordinl, then let V D α consist of the clss of ll order types tht cn e written s i where ech i is memer of V D βi i ζ for some ordinl β i < α. Husdorff s theorem sttes tht (countle) order type is scttered if nd only if it is contined in V D α for some (countle) ordinl α. The Husdorff-rnk rnk(o) of scttered order type o is the lest ordinl α with o V D α. 2.2 Words, tree domins, trees n lphet is finite nonempty set of symols, or letters. lphets re usully denoted Σ, Γ in this pper, nd letters re denoted y,, c,.... Σ-leled liner ordering is tuple w = (dom(w), l w ) where dom(w) = (, <) is some liner ordering nd l w : Σ is the leling function of w. We usully identify w with l w nd write w(i) for l w (i), i dom(w). Two words u nd v re isomorphic if there is n isomorphism h : dom(u) dom(v) which preserves lso the lels, tht is, u(i) = v(h(i)) for ech i dom(u). word over Σ is n isomorphism clss of countle Σ-leled liner orderings. For convenience, when u is word, we tke representnt of u nd use the nottion dom(u), l u referring the domin nd leling of the representnt. Order theoretic properties re lifted to words. The set of ll countle, ω- nd finite words over Σ re respectively denoted Σ #, Σ ω nd Σ. n prticulr, ε denotes the empty word (hving the empty set s domin). Note tht s for ny Σ, there is unique η-word u such tht for ny x < y dom(u) nd

4 352 Kitti Gelle nd zolcs ván letter Σ there is some z dom(u) with x < z < y nd u(z) =, this u eing clled the perfect shuffle of Σ, nd every countle Σ-word v is suword of thus u (tht is, dom(v) is su-ordering of dom(u) nd l v is the restriction of l u to dom(v)), thus Σ # is indeed set. lnguge over Σ is n ritrry suset of Σ #. Lnguges will usully e denoted y K, L,.... When (, <) is liner ordering nd for ech i, w i is Σ-word, then their product or (con)ctention is the word w = w i with domin dom(w i ) nd i i the leling of l w eing the source tupling of the lelings l wi, tht is, for (i, j), i, j dom(w i ) let w(i, j) e w i (j). Product is extended to lnguges in the expected wy: when (, <) is the indexing ordering nd to ech i, L i Σ # is lnguge, then L i consists of those words w i where w i L i for ech i. i inry products re simply written s u v or K L, or simply uv nd KL. When α is some order type, then L α is the lnguge L, nd L stnds for the union i α of the lnguges L n, n eing nturl numer. lso, L + stnds for L n. tree domin is prefix closed nonempty (ut possily infinite) suset T of N. Tht is, whenever x i is in T for x N nd i N, then so is x, in which cse x is the prent of x i nd x i is child of x. Memers of T re lso clled nodes of T. f x y T for x, y N, then x is clled n ncestor of x y nd x y is descendnt of x, denoted x x y. f y is nonempty, then we tlk out proper ncestor / descendnt, denoted x x y. Nodes of T hving no child re clled leves of T, the other nodes re clled inner nodes of T. Clerly, ε is memer of every tree domin. usets of tree domin T which re tree domins themselves re clled prefixes of T. pth π of T is prefix in which every node hs t most one child. Clerly, to every pth π there exists unique word u π in N ω = N N ω such tht π = {x N : x u π }. We will use π nd u π interchrgly. When T is tree domin nd x T, then the su-tree domin of T is T x = {y N : xy T }. t is cler tht T x is lso tree domin, nd is pth if so is T. (Σ-)tree over some lphet Σ is leled tree domin t, tht is, pir (dom(t), l t ) where dom(t) is tree domin nd l t : dom(t) Σ is leling function. imilrly to the cse of words, we often identify t with l t nd write t(x) in plce of l t (x) for x dom(t). Notions of tree domins (nodes, pths, su-tree domins etc.) re lifted to trees; the sutree of t rooted t some node x dom(t) hs the domin dom(t) x nd leling y t(xy). When X dom(t) is set of nodes of t, then lels(t, X) = {t(x) : x X} is the set of lels occurring on the nodes elonging to X, nd inflels(t, X) is the set of lels occurring infinitely mny times. n prticulr, if π is pth of t, then letter Σ elongs to inflels(t, π) if nd only if for ech x π there exists some descendnt y π of x with t(y) =. When t is cler from the context, we just write lels(x) nd inflels(x), respectively. For ny infinite pth π, there exists -miniml node x of π with inflels(π x ) = lels(π x ), this node x is denoted hed(π). Clerly, i n>0

5 Regulr Expressions for Muller Context-Free Lnguges 353 inflels(π) = inflels(π x ) lels(π x ) lels(π) for every pth π nd node x π. 2.3 Muller context-free grmmrs nd lnguges Muller context-free grmmr or MCFG for short, is tuple G = (N, Σ, P,, F) where N nd Σ re the disjoint lphets of nonterminls (or vriles) nd terminls, respectively, N is the strt symol, P is the finite set of productions (or rules) of the form α with N nd α (N Σ) +, nd F P (N) is the Muller cceptnce condition. Here P (N) = {N N} is the power set of N. Oserve tht we explicitly disllow rules of the form ε here; this mkes the tretment of lef lels more uniform, nd s it turns out, such rules cn e mimicked y introducing fresh nonterminl, rules nd nd dding {} to the cceptnce condition. Nevertheless, in our exmples we will mke use of rules ε in order to help redility. n (N Σ)-tree t is loclly consistent (with G) if it stisfies the following condition: ech inner node x of t is leled y some nonterminl nd the set of children of x is {x 1,..., x n} for some integer n > 0 with t(x 1)t(x 2)... t(x n) eing production in P. loclly consistent tree is complete if its leves re leled in Σ. The leves of ny tree domin re linerly ordered y the lexicogrphic ordering < l, tht is, u < l v if nd only if u = u 1 i u 2 nd v = u 1 j u 3 for some words u 1, u 2, u 3 N nd integers i < j. The frontier word of tree t is the word fr(t) hving the set of leves s domin, ordered lexicogrphiclly, nd leling inherited from t. Tht is, dom(fr(t)) = {x dom(t) : x is lef of t} nd fr(t)(x) = t(x) for ech lef x. loclly consistent tree t is derivtion tree of G if for ech infinite pth π of t the set inflels(π) elongs to the cceptnce condition F. Given symol X N Σ, we let (G, X) denote the set of ll complete derivtion trees t of G whose root symol t(ε) is X. We write G α for symol N Σ nd word α (N Σ) # if α is the frontier word of some derivtion tree of G hving root symol. The lnguge generted y G is L(G) = {w Σ # : G w}. lnguge L Σ # of Σ-words is Muller context-free lnguge, or MCFL for short, if L = L(G) for some MCFG G. n fct, Muller context-free lnguges re precisely the frontier lnguges of (nondeterministic Muller-)regulr lnguges of infinite trees. Exmple 1. f G = ({, }, Σ, P,, {{}}), with P = { : Σ} { ε,, }, then L(G) consists of ll the well-ordered words over Σ. ndeed, ssume t 1, t 2,... re derivtion trees. Then so is the tree t depicted in Figure 1 with frontier word fr(t 1 )fr(t 2 ).... Thus, L(G) contins the empty word (y ε), the words of length 1 (y nd ), nd is closed under tking ω-products. ince the lest clss of order types which contins 0, 1 nd which

6 354 Kitti Gelle nd zolcs ván t 1 t 2 t () Tree for Exmple 1 () Tree for Exmple 2 Figure 1: Derivtion trees corresponding to Exmples 1 nd 2 is closed under ω-sums is the clss of ll countle ordinls (see e.g. [23]), L(G) contins ll the well-ordered words over {, }. For the other direction, ssume t is derivtion tree hving frontier word contining n infinite descending chin... < l u 2 < l u 1. Then let us define the pth v 0, v 1,... in t: v 0 = ε nd v i+1 is v i 1 if this node is n ncestor of infinitely mny u j nd v i 2 otherwise (which hppens if v i corresponds to the production nd the node v i 1 (which is leled ) hs no descendnt of the form u j t ll). Note tht for ech u j there exists unique v ij such tht v ij is n ncestor of u j nd v ij+1 is not, since the length of the words v i grows without ound. Now these nodes v ij correspond to the production nd v ij+1 = v ij 1, so tht the successor of v ij long the pth is leled y. Hence v 0, v 1,..., is pth π in t such tht inflels(π) contins, which is contrdiction since the only ccepting set is {}. Exmple 2. f G = ({, }, Σ, P,, {{}}), with P = { : Σ} { ε,, }, then L(G) consists of ll the scttered words over Σ. ndeed, the derivtion tree depicted on Figure 1 shows tht L(G) is closed under ω + ( ω)-products of words. ince ε L(G), the lnguge is thus closed under ω-products nd ω-products s well, thus closed under ζ-products. ince the one-letter words elong to L(G), we get y Husdorff s Theorem tht L(G) consists of ll the scttered Σ-words. We note tht if insted of the Muller condition we define üchi-type cceptnce condition, then we get weker device: the resulting clss of üchi contextfree lnguges is strictly contined within the clss of MCFLs, see [11, 10]. 3 MO-definle properties re decidle The logic usully rising when deling with regulr structures is tht of mondic second-order logic, or MO. n [13] the following generl decidility theorem ws

7 Regulr Expressions for Muller Context-Free Lnguges 355 proved: Theorem 1. The following prolem is decidle: given n MCFG G generting Σ-words nd n MO formul ϕ evluted Σ-words, does it hold tht w = ϕ for every w L(G)? Thus in prticulr, it is decidle whether G genertes scttered, or wellordered, or dense words only. We sketch the outline of the proof. First, to ech MCFG G = (N, Σ, P,, F) we ssocite the grmmr G = (N P, Σ, P, ) where P contins the following set of productions: For ech nonterminl N, there is production ( α 1 )( α 2 )... ( α n ) in P where α 1,..., α n re the productions in P hving on their left-hnd side, in some fixed order. For ech production X 1... X n, there is production ( X 1... X n ) X 1... X n in P. Tht is, we cn rewrite nonterminl to the sequence of its lterntives, nd rewrite production to its right-hnd side. Thus, ech nonterminl of G hs exctly one lterntive (it is ssumed tht for ech nonterminl there is t lest one production hving left-hnd side ), thus there is exctly one loclly consistent tree of G hving root symol. We cll this unique tree t G the grmmr tree of G. Exmple 3. For the MCFG of Exmple 1, this tree t G is depicted in Figure 2. ε ε ε ε Figure 2: Grmmr tree of the MCFG of Exmple 1

8 356 Kitti Gelle nd zolcs ván The cruicl fct is tht the grmmr tree is regulr tree, hving finitely mny (more precisely, t most Σ + N + P ) sutrees. y [22], the MO theory of regulr tree is decidle, tht is, given regulr tree t (which is t G in our cse) nd n MO formul ψ, it is decidle whether t ψ holds. s t G cn e effectively constructed from G, it remins to construct formul ψ from G nd ϕ such tht t G ψ holds if nd only if for ech w L(G) we hve w ϕ. nformlly, the formul ψ constructed in [13] hs the semntics whenever T is derivtion tree of G, its frontier word stisfies ϕ. Now derivtion trees re encoded s susets of dom(t G ) s follows: suset X dom(t G ) encodes derivtion tree of G if the following conditions ll hold: X contins the root node. f some x X is leled y some nonterminl N, then exctly one child of x elongs to X. f some x X is leled y some production α, then ll the children of x elong to X. On ech infinite pth π X, the set of symols from N occurring infinitely mny times elongs to F. These properties cn e expressed in MO nd such set encodes derivtion tree in the ovious wy. Exmple 4. Figure 3 shows (prt of ) derivtion tree t of the grmmr of Exmple 1 nd the corresponding suset T of dom(t G ) (s nodes in oldfce). Then, given set X of nodes of t G, we cn define the suset Y X of the leves in X nd the lexicogrphic ordering over this Y is lso MO-definle. Hence the formul whenever X is suset of dom(t G ) encoding derivtion tree, nd Y is the set of leves within X, then the word corresponding to Y stisfies ϕ is expressile in MO (moreover, is effectively computle from G nd ϕ), thus proving Theorem 1. 4 norml form The generl decidility theorem of the previous section does not give us n exct complexity result s model checking MO is decidle, ut nonelementry in generl. n this section we give complexity results for severl decision prolems (nd severl undecidility results s well) regrding MCFLs, surveying the results of [11]. The decidle properties surveyed here re MO-definle, thus their decidility is immedite from Theorem 1. For exmple, L(G) is empty if nd only if every memer w of L(G) stisfies the flse formul. (On the other hnd, universlity is not definle this wy nd indeed, universlity of MFCGs is lredy undecidle for singleton lphets.) ( slightly modified vrint of) the norml form of MCFGs introduced in [11] is the following:

9 Regulr Expressions for Muller Context-Free Lnguges 357 ε ε ε ε Figure 3: prt of derivtion tree t of Exmple 1 nd the corresponding suset T of t G Definition 1. n MCFG G = (N, Σ, P,, F) is in norml form if either P is empty, or P consists of the single production ε, or for ech N there exists nonempty word w with G w nd words u, v with G uv. Moreover, to ech F F there exists pth π of some derivtion tree t with inflels(π) = F. n the terminology of [11], ech nonterminl hs to e +-productive nd rechle, nd ech ccepting set hs to e vile. uch norml form of ny MCFG is computle: Theorem 2 ([11]). For ny MFCG G, n equivlent MCFG G in norml form cn e computed in PPCE. The resulting grmmr G hs size polynomilly ounded y the size of G. We sketch the lgorithm here. The strightforwrd modifiction of the corresponding lgorithms for ordinry context-free grmmrs works: first we check for ech symol X whether X is productive (is there complete derivtion tree with root symol X t ll). However, the complexity of this prolem is PPCEcomplete due to the fct tht the emptiness prolem of Muller regulr tree lnguges, given y nondeterministic Muller tree utomton, is PPCE-complete. ince there is polynomil-time trnsformtion from n MCFG to corresponding Muller tree utomton nd vice vers, we gin PPCE-completeness for deciding productiveness of individul nonterminls, nd emptiness of MCFLs s well: Proposition 1. [11] Deciding whether L(G) = is PPCE-complete. Then, we cn throw wy ll the non-productive nonterminls nd get rid of ll the productions tht hve non-productive nonterminl on either of its sides.

10 358 Kitti Gelle nd zolcs ván fter tht, this set is further reduced to the set of rechle nonterminls, which cn e done y the usul fixed-point construction for CFGs, s for ech rechle nonterminl there is finite (not necessrily complete) derivtion tree with root symol nd occurring s lef lel. Then we cn throw wy ll the nonrechle nonterminls. This cn e done in polynomil time. n the next step, nonterminl genertes nonempty word if nd only if there is finite (not necessrily complete) derivtion tree rooted tht hs some letter Σ occurring s lef lel. This cn lso e decided in polynomil time y solving rechility prolem. Then, we cn simply erse ll the symols from the right-hnd sides from which only the empty word cn e generted (nd if the right-hnd side of rule ecomes empty, then erse the rule itself s well), rriving to grmmr in the required norml form, prt from viility of ech F F. (This wy we might lose the word ε from L(G) if we need the nonempty word, we cn llow the production ε to e present, ut in tht cse should not pper on the right-hnd side of ny rule, s in the clssicl cse.) The norml form cn e generted in PPCE. Then, L(G) contins nonempty word if nd only if there re still productions in G. Proposition 2 ([11]). t cn e decided in PPCE whether L(G) contins nonempty word. Now for retining only the vile ccepting sets, the following ssocited grph Γ G is hndy: Definition 2. Given n MCFG G = (N, Σ, P,, F), we define the following edgeleled multigrph Γ G : the vertices of Γ G re the nonterminls, nd there is n edge from to leled (α, β) for α, β (N Σ) if αβ is production of G. Now if G lredy contins only productive nd rechle nonterminls, then set F F is vile if nd only if the sugrph of Γ G induced y F is strongly connected, which is efficiently decidle, finishing the construction of the norml form. However, lnguge universlity (nd thus lnguge inclusion nd equivlence) is undecidle lredy for singleton lphets (contrry to the cse of context-free grmmrs, where undecidility holds only for lphets of size t lest two): Proposition 3 ([10]). t is undecidle whether L(G) = Σ # for n MCFG G, even when Σ is singleton lphet. n fct, the prolem is undecidle lredy for üchi context-free grmmrs. The key for proving this is reduction from the universlity prolem of context-free lnguges of finite words over the inry lphet {, }: first we encode y ω nd y ω. Then, the lnguge L of those words in # not elonging to { ω, ω } is MCFG nd thus, s MCFLs re effectively closed under homomorphisms nd finite unions, we get tht L(G) = {, } for the CFG G if nd only if L(G ) = # for some MCFG G effectively constructed from G.

11 Regulr Expressions for Muller Context-Free Lnguges Lnguges of finite words Given n MCFG G = (N, Σ, P,, F) in norml form, one cn decide in (low-degree) polynomil time whether L(G) consists of finite words only. (Note tht decidility is cler s the property of eing scttered is MO-definle.) The key oservtion here is tht L(G) contins n infinite word if nd only if there is some F F nd n edge α,β in Γ G with αβ ε nd, F which cn e verified efficiently. lso, it is quite strightforwrd to see tht lnguge L Σ is context-free if nd only if it is n MCFL: for one direction we only hve to set F =. For the other direction somewht more cre is needed since the frontier word of n infinite tree cn e empty. However, it cn e decided in PPCE whether G ε holds for nonterminl : we only hve to remove the productions from G hving some terminl symol occurring on the right-hnd side (tht is, we retin the productions of the form X α with α N ), nd pply n emptyness check for the generted lnguge. Then, for ech generting ε we cn include the production ε nd the resulting (clssicl) context-free grmmr will generte L(G) Σ if G is in norml form. Thus, Proposition 4 ([11]). lnguge L Σ is context-free if nd only if it is Muller context-free. lso, since emptiness of CFGs is efficiently decidle, we hve: Proposition 5 ([11]). t is decidle in PPCE whether n MCFG genertes t lest one finite word. 4.2 Lnguges of well-ordered words Given n MCFG G = (N, Σ, P,, F) in norml form, one cn decide in (lowdegree) polynomil time whether L(G) consists of well-ordered words only. gin, decidility itself is lredy cler, since the property is MO-definle. Proposition 6 ([11]). For n MFCG G in norml form, L(G) contins word which is not well-ordered if there is some set F F, nonterminls, F nd n edge α,β in Γ G with β ε. To see this, suppose there is derivtion tree t of G with frontier word not hving well-ordered domin. Then there exists n infinite descending chin... < x 3 < x 2 < x 1 < x 0 of leves of t. trting from the root, one cn then uild up n infinite pth π = y 0, y 1,... such tht for ech node u of π, n infinite numer of these leves x i re descendnts of u. (This property holds for the root, nd t ech step we set y i+1 to e the first child of y i which is n ncestor of some x j.) Then, s ech such lef x i hs some finite depth, there exists n y j for ech x i such tht y j is n ncestor of x i ut y j+1 is not; it is esy to see tht in this cse x i is on the right side of π.

12 360 Kitti Gelle nd zolcs ván Hence, for this π it holds tht F = inflels(π) is contined within nontrivil strongly connected component of Γ G, moreover, there is n edge α,β with, F nd β ε, the other direction eing lso strightforwrd, simply y following closed pth in F visiting ech edge t lest once, iterting ω times nd complete the resulting derivtion tree (which lredy hs n infinite descending chin mong its leves due to β ε). Now if L(G) contins well-ordered words only, one cn compute n intervl of ordinls contining ll the order types of L(G): Proposition 7 ([16]). f the MCFG G genertes well-ordered words only, then oth the minimum nd the supremum of the order types of the memers of L(G) re effectively computle ordinls. For the minimum, first we note tht to ech nonterminl, if there is finite word w with G w, then the length n of the shortest such word is computle. Then we cn replce ech occurrence of these nonterminls y n : the minimum order types for the nonterminls in the resulting grmmr will coincide with those of G. Let us fix for ech symol X complete derivtion tree t X with root symol X, minimizing the order type of fr(t X ). t turns out tht the memers of N cn e prtilly ordered y some properties of these trees t X nd tht these trees t X cn e chosen in wy tht ech sutree of t X is some t Y for Y N Σ. The key construction to see this is the following. When t is complete derivtion tree of such n MCFG G, then we cll t simple if it is either finite or hs some infinite pth π with inflels(π) = lels(π) nd moreover, ech production corresponding to the nodes of π occur infinitely mny times on π. s uv P for some, F F implies v = ε, this pth π hs to e the rightmost pth of t. Then, the order type of fr(t) is the ω-sum of the order types of the frontiers of the sutrees of t eing djcent to π (tht is, rooted t some node x not on π whose prent is on π). s ech left-hnd side occurs infinitely mny times, we get tht ech nonterminl djcent to π hs strictly smller minimum. Thus, if t X is simple, then we cn define t X fter eing defined ech t Y where the minimum order type of Y is strictly smller thn tht of X, nd the order type of its frontier is computle. Now if t X is not simple, then the order type of its frontier is the sum of the order types corresponding to its direct children. Now ll these order types ut the lst hve to e smller then the order type of fr(t X ) nd the lst child hs to hve one level closer for eing simple (nd this level is finite), estlishing the inductive cse. Thus, stndrd itertive lgorithm recomputing the minim from the current estimtions eventully termintes nd produces the minimum ordinls (in Cntor norml form, sy). For the supremum, the cse nlysis is slightly more involved. First, one seeks for reproductive nonterminls: is clled reproductive if G α for some α in which occurs infinite times. t turns out tht is reproductive if nd only if there is production X 1... X n nd n ccepting set F F with, F nd X i G uv for some i [n] nd u, v Σ, which is decidle.

13 Regulr Expressions for Muller Context-Free Lnguges 361 Then, if is reproductive, then it is esy to see tht ritrrily lrge countle ordinls cn e generted from, thus in tht cse, the supremum in question is ω 1, the smllest uncountle ordinl. lso, if G uv for some reproductive nonterminl, then the sme holds for s well. Otherwse, to ech production of the ove form, the nonterminls X i elong to strongly connected component strictly elow the component of, gin setting strightforwrd induction rgument: the reder is referred to [16] for the technicl detils. s ω-lnguges re lso well-ordered, we note tht nd lnguge of ω-words is context-free in the sense of Cohen nd Gold [8] if nd only if it is n MCFL [11], nd moreover, it is decidle whether n MCFG genertes only well-ordered words hving order type t most ω. 4.3 Lnguges of scttered words nlogously for the cse of well-ordered words, it is decidle whether n MCFG G genertes scttered words only, s this property is lso MO-definle. Proposition 8 ([11]). t is decidle in polynomil time whether n MCFG G in norml form genertes scttered words only. The key oservtion is the following: L(G) contins qusi-dense word if nd only if there exists finite derivtion tree t, two leves x nd y of t nd vile ccepting set F F such tht lels(π x ) = lels(π y ) = F nd t(x) = t(y) = t(ε) where π x nd π y respectively denote the pths from the root to x nd y. s this property is further equivlent to the existence of some production αβcγ with,, C F for vile F F, we hve n efficient decision procedure. For lnguges of scttered words, the min ingredient of mny proofs is tht of the Husdorff-rnk. siclly, given derivtion tree t, we cn tg ech node x of t y the rnk of fr(t x ). Then, consider the suset D of the nodes tgged y the sme ordinl s the root. s n infinite sum of scttered orderings of the sme rnk α hs rnk strictly greter thn α, this suset D cnnot contin n infinite ntichin, yielding tht D is finite union of pths. Then, one cn prtilly order the set of derivtion trees primrily y the rnk of their frontier, secondry y the numer of pths covering their respective sets D, nd third, y the depth of the first node of D hving t lest two children in D. The defined ordering ecomes then well-ordering of the derivtion trees, llowing us to pply well-founded induction. The first such ppliction is the following gp theorem of MCFLs of scttered words: Proposition 9 ([11]). The supremum of the Husdorff-rnk of the memers of L(G) is computle when G genertes scttered words only. Moreover, this supremum is either ω 1 or some nturl numer. The key oservtion for this result is gin tht reproductive nonterminls, nd only those, cn produce words of ritrrily lrge (countle) rnk, nd for the others, simple induction works over the strongly connected components of Γ G.

14 362 Kitti Gelle nd zolcs ván nterestingly, it is lso known [14] tht n MCFL consisting only of sctterred words is CFL if nd only if the second cse pplies, i.e. if it hs finite upper ound n on the Husdorff-rnk of its memers. n order to define the regulr expression-like expressions cpturing these MCFLs, we will lso consider pirs of words over n lphet Σ, equipped with finite conctention nd n ω-product opertion. For pirs (u, v), (u, v ) in Σ # Σ #, we define the product (u, v) (u, v ) to e the pir (uu, v v), nd when for ech i ω, (u i, v i ) is in Σ # Σ #, then we let (u i, v i ) e the word ( )( ) u i v i. Let i ω i ω i ω P (Σ # Σ # ) denote the set of ll susets of Σ # Σ #. Then P (Σ # Σ # ) is equipped with the opertions of set union, conctention L L = {(u, v) (u, v ) : (u, v) L, (u, v ) L } nd Kleene str L = {ɛ} L L 2. We lso define n ω-power opertion P (Σ # Σ # ) P (Σ # ) y L ω consisting of the words of the form (u i, v i ) with (u i, v i ) L for ech i ω. i ω The motivtion ehind this notion is the following. f t is derivtion tree with distinguished lef x leled y some nonterminl, nd frontier word fr(t) = uv, then this frontier word is represented y the pir (u, v). Now if we sustitute tree s with root symol in plce of the distinguished lef, hving frontier word fr(s) = u v, yielding the tree t, then we hve fr(t ) = uu v v which is (u, v) (u, v ) ccording to the product opertion we defined on pirs. imilrly, if the root of t is lso leled, then we cn iterte sustituting t in plce of the distinguished lef: if we iterte finite numer of times, then the Kleene str contins the pir representnt of the resulting tree; if we iterte ω times, then the frontier is u ω v ω = (u, v) ω. Then, let the set of µωt s -expressions over the lphet Σ e defined y the following grmmr (with T eing the initil nonterminl): T ::= ε x T + T T T µx.t P ω P ::= T T P + P P P P Here, Σ nd x X for n infinite countle set of vriles. n occurrence of vrile is free if it is not in the scope of µ-opertion, nd ound, if it is not free. closed expression does not hve free vrile occurrences. The semntics of these expressions re defined s expected using the monotone functions over P (Σ # ) nd P (Σ # Σ # ) introduced erlier. The chrcteriztion theorem of [12] sttes tht these notions correspond to ech other: Theorem 3 ([12]). lnguge L is n MCFL of scttered words if nd only if it cn e denoted y some closed µωt s -expression. The direction tht such expressions lwys denote MCFLs is done y strightforwrd construction, while the converse direction is gin done vi the wellordering of the derivtion trees we introduced erlier.

15 Regulr Expressions for Muller Context-Free Lnguges Opertionl chrcteriztion of generl MCFLs n this section we give n opertionl chrcteriztion of MCFLs in generl, using the opertionl chrcteriztion of Muller regulr lnguges of infinite trees given in [18, 20] which we reproved using slightly different methods in [15]. There, we introduced for ech rnked lphet Σ (tht is, ech symol Σ hs some rity n 0; the set of n-ry symols of Σ is denoted Σ n ) the set of µηregulr tree expressions (tree expressions for short) over Σ s the lest set stisfying ll the following conditions: f Σ n, n 0 nd E 1,..., E n re tree expressions over Σ, then (E 1,..., E n ) is tree expression over Σ. When n = 0, we write in plce of (). f E nd F re tree expressions over Σ, then so is (E + F ). f E is tree expression over Σ {x} for the nullry symol x, nd F is tree expression over, then (E x F ) is tree expression over Σ. f E is tree expression over Σ {x} for the nullry symol x, then (µx.e) nd (ηx.e) re tree expressions over Σ. n order to define the semntics of these expressions, we hve to define the opertions of x-product, µx nd ηx on tree lnguges, for which we use cuts nd decompositions of trees. o let Σ e n lphet nd let Σ stnd for the disjoint copy {â : Σ} of Σ. The htted symols re of rity zero. Given tree t, nd suset X dom(t) of its nodes, the X-cut of t is the following Σ Σ-tree t/x: node x elongs to dom(t/x) if nd only if x is node of t which is not proper descendnt of ny non-root memer of X, i.e. dom(t/x) = dom(t) {y u X {ε} dom(t) : u y}. The leling of t/x is defined s follows: for node x dom(t/x) let (t/x)(x) e t(x) if x / X {ε} nd t(x) if x X {ε}. Tht is, we cut the tree t the nodes of X nd dd ht to the symols occurring t the cut-points. Exmple 5. Figure 4 shows tree t with frontier ω ω. Choosing the set X to contin ll the inner nodes of t, ll the trees of the form (t x )/(X x ) re the sme (shown on the right hnd side of the Figure). Figure 5 shows (finite) tree t which gets decomposed into set of six trees. Now when K is lnguge of Σ Σ-trees nd L is lnguge of -trees for some lphets Σ nd, then K[ Σ/L] is the following lnguge of Σ -trees: tree t elongs to K[ Σ/L] if there exists some suset X dom(t) such tht t/x elongs to K nd for ech x X, t x elongs to L. (Usully is either Σ or Σ Σ). Tht is, if we cn cut t such tht the retined prt of the tree elongs to K nd ll the sutrees tht re cut down elong to L. Oserve tht if there exists such n X, then the suset X of -miniml elements of X is lso fine (s t/x = t/x then, nd the condition for the sutrees is lso vlid), thus in tht cse we cn ssume tht t is cut y some ntichin of its nodes.

16 364 Kitti Gelle nd zolcs ván  Figure 4: tree nd t one of its possile decompositions. Given tree lnguge L over Σ Σ, the function K L[ Σ/K] is monotone function (with respect to lnguge inclusion), which mps the poset of Σ Σ-tree lnguges to itself, thus it hs lest fixed point tht cn e reched y the Kleene itertion L 0 =, L α = L[ Σ/L β ] for successor ordinls α = β + 1 nd L α = L β β<α for limit ordinls α, tht is, there exists some (lest) ordinl α such tht L α is the lest fixed point of this function. This lest fixed point is denoted L µ nd is Σ-tree lnguge. t is reltively esy to show tht Σ-tree t elongs to L µ if nd only if there is some suset X t of its nodes such tht there is no infinite chin x 1 x 2... in X nd for ech x X, the tree (t x )/(X x ) elongs to L. The lst opertion on trees is the η-product. Given Σ Σ-tree lnguge L, the lnguge L η is lnguge of Σ-trees: tree t elongs to L η if nd only if there is some X dom(t) of its nodes such tht for ech x X, the tree (t x )/(X x ) elongs to L. Tht is, now n ritrry set of cut-points in the domin. Now if L is lnguge of Σ {x}-trees for nullry symol x, then µx.l nd ηx.l re the tree lnguges L µ nd L η where L is the following lnguge of Σ Σtrees: tree t elongs to L if nd only if its projection defined y, â x elongs to L. Tht is, the Σ {x}-tree t we get from t y releling ech htted symol to x. imilrly, if K is lnguge of Σ {x}-trees nd L is some -tree lnguge, then let K x L e the lnguge K[ Σ/L]. Exmple 6. When K consists of the tree single tree (x, Â), then Kη contins single tree with root symol nd frontier x ω. Then, K η x {Â} contins single tree with root symol nd frontier Âω. Finlly, the frontier lnguge of (K η x {Â} {(), (Â, Â)})µ contins ll the nonempty well-ordered words over the singleton lphet {}. fter these definitions we re redy to define the semntics of tree expressions in the expected wy. tree expression E denotes tree lnguge E, set of Σ-trees defined s follows:

17 Regulr Expressions for Muller Context-Free Lnguges 365 Â Â Figure 5: decomposition of finite tree. (E 1,..., E n ) consists of those Σ-trees t whose root symol is leled, the root hve exctly the children 1,..., n nd for ech i [n], t i E i. (E+F ) = E F, E xf = E x F, µx.e = µx. E nd ηx.e = ηx. E. Now using the terms of [15], tree lnguge is Muller-regulr if nd only if it cn e denoted y some µη-regulr tree expression. Hence it is esy to derive µη-regulr word expressions for MCFLs s MCFLs re exctly the frontier lnguges of Mullerregulr tree lnguges. For this, we hve to define opertions corresponding to the x, µx nd ηx-opertions ove. nformlly, for x we cn define the sustitution opertion: K[x/L] contins those words we get from memers of K in which we replce ech occurrence of x y some word in L; then, µx.l is the lest fixed point of the monotone function X L[x/X]; nd, memers of ηx.l re the Σ-words which we get y strting from the word x, then replcing ech occurrence of x y some memer of L, nd repet this process the words occurring s limits of this (possily infinite) replcement sequence re memers of ηx.l. To tret the cse of ηx.l formlly, we introduce the clss of generlized Σ-trees s follows. generlized tree domin is modified tree domin where we do not restrict the set of children of ny node to e finite linerly ordered set, ut llow

18 366 Kitti Gelle nd zolcs ván x x x Figure 6: n L-x-sustitution tree. ritrry countle liner orders. Nevertheless, ech node hs to hve finite depth. More formlly, given prtilly ordered set P = (P, <), P -tree domin is suset D of P stisfying the following conditions: D is nonempty nd prefix-closed. For ech node d D, the set {p P : d p D} of the children of d is linerly ordered suset of P. When n lphet Σ is lso given, then P -Σ-tree is mpping t : dom(t) Σ from P -tree domin to Σ, tht is, Σ-leled P -tree domin nd the frontier word of t is the Σ-word fr(t) whose domin is the set of the leves of t (those nodes hving no children) equipped with the lexicogrphic ordering: p 1... p n < l p 1... p m if nd only if for some i m, n we hve p i < p i nd for ech j < i, p j = p j. Oserve tht this ordering is totl on the leves since for two different leves u = p 1... p n nd v = p 1... p m neither of them cn e prefix of the other, hence there exists unique lest index i m, n with p i p i ; nd s the set of the children of the node p 1... p i 1 is linerly ordered, it hs to e either the cse p i < p i or p i < p i. Oserve lso tht if ech node hs countle children set, the fr(t) is countle word. Given lnguge L ( Σ {x} ) #, we define the lnguges µx.l nd ηx.l over Σ s follows. Let P = dom(u) e the disjoint union of the domins of u L ll the words elonging to L. Then, n L-x-sustitition tree is P - ( Σ {x} ) -tree stisfying the following conditions: the root is not lef node, ech inner node is leled y x, ech lef node is leled in Σ nd for ech inner node u, the word formed y the lels of the children of u elongs to L. Exmple 7. Figure 6 depicts n L-x-sustitution tree where L is the lnguge { ω, (x) ω }. Then, let ηx.l contin the frontier words of the L-x-sustitution trees, nd let µx.l contin the frontier words of those L-x-sustitution trees hving no infinite pths. Exmple 8. Figure 7 depicts n L-x-sustitution tree for L = {x}. This tree shows tht ω ω is memer of ηx.l (ut does not, in fct, elong to µx.l s

19 Regulr Expressions for Muller Context-Free Lnguges 367 x x x x Figure 7: The word ω ω elongs to ηx.{x}. it hs n infinite pth. For this lnguge, µx.l =. n contrst, µx.{x, c} is { n c n : n 0} nd ηx.{x, c} is µx.{x, c} { ω ω }.) These opertions µ nd η on lnguges over words correspond to the opertions µ nd η on tree lnguges in the sense fr(µx.l) = µx.fr(l) nd fr(ηx.l) = ηx.fr(l) for ech tree lnguge L. We lso mke use of the x product opertion: when K (Σ {x}) # nd L # for the lphets Σ nd, then K x L (Σ ) # contins those words one cn get from word u in K y replcing ech occurrence of x in u y some memer of L. Or more techniclly, the frontier words of those (K L)-x-sustitution trees of depth t most two in which the word formed y the children of the root symol elongs to K nd ech word formed y the children of the depth-one inner nodes elongs to L. n prticulr, the tree depicted in Figure 6 shows its frontier ( ω ) ω elongs to (x) ω x ω. Then lso, fr(k x L) = fr(k) x fr(l) for ritrry tree lnguges K nd L. y the chrcteriztion of Muller regulr tree lnguges we get the following chrcteriztion: Theorem 4. lnguge L Σ # is n MCFL if nd only if it cn e generted from the singleton lnguges of one-letter words y finite numer of conctention, inry union, x-product, µx nd ηx-opertions. References [1] edon, Nicols. Finite utomt nd ordinls. Theor. Comput. ci., 156(1 2): , [2] edon, Nicols, ès, lexis, Crton, Olivier, nd Rispl, Chloé. Logic nd Rtionl Lnguges of Words ndexed y Liner Orderings, pges pringer erlin Heidelerg, erlin, Heidelerg, 2008.

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