The Cucl Hierrchy of Infinite Grphs in Terms of Logic nd Higher-order Pushdown Automt Arnud Cryol 1 nd Stefn Wöhrle 2 1 IRISA Rennes, Frnce rnud.cryol@iris.fr 2 Lehrstuhl für Informtik 7 RWTH Achen, Germny woehrle@informtik.rwth-chen.de Astrct. In this pper we give two equivlent chrcteriztions of the Cucl hierrchy, hierrchy of infinite grphs with decidle mondic second-order (MSO) theory. It is otined y iterting the grph trnsformtions of unfolding nd inverse rtionl mpping. The first chrcteriztion sticks to this hierrchicl pproch, replcing the lngugetheoretic opertion of rtionl mpping y n MSO-trnsduction nd the unfolding y the treegrph opertion. The second chrcteriztion is non-itertive. We show tht the fmily of grphs of the Cucl hierrchy coincides with the fmily of grphs otined s the ε-closure of configurtion grphs of higher-order pushdown utomt. While the different chrcteriztions of the grph fmily show their roustness nd thus lso their importnce, the chrcteriztion in terms of higher-order pushdown utomt dditionlly yields tht the grph hierrchy is indeed strict. 1 Introduction Clsses of finitely generted infinite grphs enjoying decidle theory re strong suject of current reserch. Interest rises from pplictions in model checking of infinite structures (e.g. trnsition systems, unfoldings of Kripke structures) s well s from theoreticl point of view since the order to undecidility is very close nd even for very regulr structures mny properties ecome undecidle. We re interested in hierrchy of infinite grphs with decidle mondic second-order (MSO) theory which ws introduced y D. Cucl in [7]. Strting from the clss of finite grphs two opertions preserving the decidility of the MSO-theory re pplied, the unfolding [9] nd inverse rtionl mppings [6]. Iterting these opertions we otin the hierrchy (Grph(n)) n N where Grph(n) is the clss of ll grphs which cn e otined from some finite grph y n n-fold itertion of unfolding followed y n inverse rtionl mpping. This hierrchy of infinite grphs contins severl interesting fmilies of grphs (see [7, 17]) nd hs lredy een suject to further studies [2].
The first level contins exctly the prefix-recognizle grphs [6], which cn in turn e chrcterized s the grphs definle in 2 (the infinite inry tree) y n MSO-trnsduction [1], or s the ε-closure of configurtion grphs of pushdown utomt [16] (see [1] for n overview). We extend these chrcteriztions to higher levels. In Sect. 3 we show tht the itertion of the treegrph opertion, vrint of the tree-itertion [18], nd MSO-trnsductions genertes exctly the Cucl hierrchy. These two opertions re to our knowledge the strongest grph trnsformtions which preserve the decidility of the MSO-theory. In [17] the hierrchy ws defined strting from the infinite inry tree y iterting MSOinterprettions (prticulr MSO-trnsductions) nd unfolding. Since the unfolding is definle inside the grph otined y the treegrph opertion, it follows from our result these definitions re indeed equivlent. Pushdown utomt cn lso e seen s the first level of hierrchy of higherorder pushdown utomt, whose stck entries re not only single letters (s for level 1), ut words (level 2), words of words (level 3).... Similr hierrchies hve lredy een considered in [14, 11, 15]. In Sect. 4 we show tht grph is on level n of the Cucl hierrchy iff it is the ε-closure of configurtion grph of higher-order pushdown utomton of level n. This result is incorrectly ttriuted to [2, 7] in [17]. In [2], in the context of gme simultion, only the esier direction from higher-order pushdown grphs to grphs in the hierrchy is shown. All the proofs in Sections 3 nd 4 re effective. In Sect. 5 we use the chrcteriztion of the hierrchy in terms of higher-order pushdown utomt to show tht it is strict. Moreover we exhiit genertor for every level, i.e. every grph on this level cn e otined from the genertor y pplying rtionl mrking nd n inverse rtionl mpping, or n MSOinterprettion. Finlly we give n exmple of grph with decidle MSO-theory which is not in the hierrchy. 2 Preliminries 2.1 Opertions on Grphs nd the Cucl Hierrchy We fix countle set A, lso clled lphet. Let Σ A e finite suset of edge lels. A Σ-leled grph G is tuple (V G, (E G ) Σ ) where V G is set of vertices nd for Σ we denote y E G V G V G the set of -leled edges of G. We ssume tht V G is t most countle, nd tht there re no isolted vertices in G, i.e. for every v V G there exists n w V G such tht (v, w) E G or (w, v) E G for some Σ. If the grph G nd the set of edge lels Σ is cler from the context we drop the superscript G nd spek just of leled grph. A grph is clled deterministic if (v, w) E nd (v, w ) E implies w = w for ll v, w, w V nd Σ. A pth from vertex u to vertex v leled y w = 1... n 1 is sequence v 1 1... n 1 v n V (ΣV ) such tht v 1 = u, v n = v nd (v i, v i+1 ) E i for every i {1,..., n 1}. In this cse we will lso write u w v. A tree T is
grph contining vertex r clled the root such tht for ny vertex v V T there exists unique pth from r to v. The unfolding Unf(G, r) of grph G = (V G, (E G ) Σ ) from node r V G is the tree T = (V T, (E T ) Σ ) where V T is the set of ll pths strting from r in G nd for ll Σ, (w, w ) E T iff w = w v for some v V G. The treegrph Treegrph(G, ) of grph G = (V, (E ) Σ ) y symol Σ is the grph G = (V +, (E ) Σ { } ) where V + designtes the set of ll finite non-empty sequences of elements of V, for ll Σ nd ll w V, (wu, wv) E iff (u, v) E, nd E = {(wu, wuu) w V, u V }. The tree-itertion s defined in [18] lso contins son-reltion given y {(w, wu) w V nd u V }. If G is connected then the son-reltion cn e defined in the treegrph. Let Σ e set of symols disjoint from ut in ijection with Σ. We extend every Σ-leled grph G to (Σ Σ)-leled grph Ḡ y dding reverse edges Eā := {(u, v) (v, u) E }. Let Γ A e set of edge lels. A rtionl mpping is mpping h : Γ P(Σ Σ) which ssocites to every symol from Γ regulr suset of (Σ Σ). If h() is finite for every Γ we lso spek of finite mpping. We pply rtionl mpping h to Σ-leled grph G y the inverse to otin Γ -leled grph h 1 (G) with (u, v) E iff there is pth from u to v in Ḡ leled y word in h(). The set of vertices of h 1 (G) is given implicitly y the edge reltions. We lso spek of h 1 (G) s the grph otined from G y the inverse rtionl mpping h 1. The mrking M (G, X) of grph G = (V, (E ) Σ ) on set of vertices X V y symol Σ is the grph G = (V, (E ) Σ { } ) where V = {(x, 0) x V } {(x, 1) x X}, E = {((x, 0), (y, 0)) (x, y) E } for Σ, nd E = {((x, 0), (x, 1)) x X}. A rtionl mrking of grph G = (V, (E ) Σ ) y symol Σ with rtionl suset R over Σ Σ from ( w vertex r V is the grph M G, {x V r Ḡ x, w R} ). An MSOmrking ( of grph G y symol with n MSO-formul ϕ(x) is the grph M G, {v V G G = ϕ(v)} ). Following [7], we define Grph(0) to e the clss contining for every finite suset Σ A ll finite Σ-leled grphs, nd for ll n 0 Tree(n + 1) := { Unf(G, r) G Grph(n), r V G}, Grph(n + 1) := { h 1 (T ) T Tree(n + 1), h 1 n inverse rtionl mpping }, where we do not distinguish etween isomorphic grphs. 2.2 Mondic Second-order Logic nd Trnsductions We define the mondic second-order logic over Σ-leled grphs s usul, (see e.g. [13]), i.e. we view grph s reltionl structure over the signture consisting of the inry reltion symols (E ) Σ. A formul ϕ(x 1,..., X k ) contining t most the free vriles X 1,..., X k is evluted in (G, V) where G = (V, (E ) Σ ) is Σ-leled grph nd V : V P({1,..., k}) is function which ssigns to every vertex v of G set V(v) such tht v X i iff i V(v). We write (G, V) = ϕ(x 1,..., X k ), or equivlently
G = ϕ[v 1,..., V k ] where V i := {v V i V(v)}, if ϕ holds in G under the given vlution V. An MSO-interprettion of Γ in Σ is fmily I = (ϕ (x, y)) Γ of MSOformuls over Σ. Applying n MSO-interprettion I = (ϕ (x, y)) Γ of Γ in Σ to Σ-leled grph G we otin Γ -leled grph I(G) where the edge reltion E I(G) is given y the pirs of vertices for which ϕ (x, y) is stisfied in G, nd V I(G) is given implicitly s the set of ll vertices occurring in the reltions E I(G). Note tht the ddition of n MSO-formul δ(x) to I defining the vertex set explicitly does not increse the power of n interprettion if we require tht there re no isolted vertices in the resulting grph. Interprettions cnnot increse the size of structure. To overcome this wekness the notion of trnsduction ws introduced, cf. [8]. Let G = (V, (E ) Σ ) e Σ-leled grph nd K e finite suset of A disjoint from Σ. A K-copying opertion for Σ ssocites to G (Σ K)-leled grph G = (V, (E ) Σ K ) where V = V (V K), E := E for Σ, nd E := {(v, (v, )) v V } for K. An MSO-trnsduction T = (K, I) from Σ to Γ is K-copying opertion for Σ followed y n MSO-interprettion I of Γ in Σ K. Note tht n inverse rtionl mpping is specil cse of n MSO-interprettion nd n MSO-mrking is specil cse of n MSO-trnsduction. 2.3 Higher-order Pushdown Automt We follow the definition of [15]. Let Γ e finite set of stck symols. A level 1 pushdown stck over Γ is word w Γ in reversed order, i.e. if w = 1... m the corresponding stck is denoted y [ m,..., 1 ]. For n 2 level n pushdown stck over Γ is inductively defined s sequence [s r,..., s 1 ] of level n 1 pushdown stcks s i for 1 i r. [ε] denotes the empty level 1 stck, the empty level n stck, denoted y [ε] n, is stck which contins for 1 i < n only single empty level i stck. The following instructions cn e executed on level 1 stck [ m,..., 1 ]: push 1([ m,..., 1 ]) := [, m,..., 1 ] for every Γ pop 1 ([ m, m 1..., 1 ]) := [ m 1,..., 1 ] Furthermore we define the following function which does not chnge the content of stck: top([ε]) := ε nd top([ m,..., 1 ]) := m for m 1. For stck [s r,..., s 1 ] of level n 2 we define the following instructions push 1([s r,..., s 1 ]) := [push 1(s r ), s r 1,... s 1 ] for every Γ push n ([s r,..., s 1 ]) := [s r, s r,..., s 1 ] push k ([s r,..., s 1 ]) := [push k (s r ), s r 1,..., s 1 ] for 2 k < n pop n ([s r,..., s 1 ]) := [s r 1,..., s 1 ] pop k ([s r,..., s 1 ]) := [pop k (s r ), s r 1,..., s 1 ] for 1 k < n
nd extend top to level n stck y setting top([s r,..., s 1 ]) := top(s r ). We denote y Instr n the set of instructions tht cn e pplied to level n stck (without the top function). For the ske of esiness we lso dd n identity function denoted y which does not chnge the stck t ll. The instruction push 1 dds the symol to the topmost level 1 stck, while push k duplictes the topmost level k 1 stck completely. Similrly pop 1 removes the top symol of the topmost level 1 stck, while pop k for 1 < k n removes the corresponding level k 1 stck completely. Note tht the instruction pop k for 2 k n cn only e pplied if the resulting stck is gin level n stck, i.e. it does not remove ottom level k 1 stck. A higher-order pushdown utomton of level n is tuple A = (Q, Σ, Γ, q 0, ) where Q is finite set of sttes, Σ is n input lphet, Γ is stck lphet, q 0 Q is n initil stte, nd Q (Σ {ε}) (Γ {ε}) Q Instr n is trnsition reltion. A configurtion of A is pir (q, [s r,..., s 1 ]) where q is stte of A nd [s r,..., s 1 ] is stck of level n. The initil configurtion (q 0, [ε] n ) consists of the initil stte q 0 nd the empty level n stck [ε] n. A cn rech configurtion (q, [s r,..., s 1]) from (q, [s r,..., s 1 ]) y reding Σ {ε} if there is trnsition (q,, top([s r,..., s 1 ]), q, i) such tht i([s r,..., s 1 ]) = [s r,..., s 1]. The utomton A ccepts word w Σ if A reches from the initil configurtion the empty level n stck fter reding w. We denote y HOPDA(n) the clss of ll higher-order pushdown utomt of level n. 3 Closure Properties In this prt, we prove tht the hierrchy is closed under MSO-trnsductions nd the treegrph opertion. We first consider the cse of deterministic trees. 3.1 The Deterministic Cse We consider su-hierrchy otined y unfolding only deterministic grphs. Grph d (0) is equl to Grph(0). Tree d (n + 1) contins the unfoldings of every deterministic grph G Grph d (n) from vertex in V G. Grph d (n) is defined in the sme wy s Grph(n). Note tht Grph d (n) lso contins non-deterministic grphs. Closure under MSO-trnsductions Using results from [3], we prove tht for ll n N, Grph d (n) is closed under MSO-trnsductions. This result ws otined for the first level y A. Blumensth in [1]. Oviously, Tree d (n) is not closed under MSO-trnsductions ut if we consider only MSO-mrkings, we otin lso closure property for Tree d (n). Proposition 1. For ll n 0, ll tree T Grph d (n), we hve tht: Tree d (n) nd ll grph G 1. M(T ) lso elongs to Tree d (n), for ny MSO-mrking M,
2. T (G) lso elongs to Grph d (n), for ny MSO-trnsduction T. Proof (sketch): These results re proved y induction on the level using prtil commuttion results of MSO-trnsductions nd unfolding otined in [3]. 1. For every deterministic grph G nd every MSO-mrking M, there exists n MSO-trnsduction T nd vertex r such tht M(Unf(G, r)) Unf(T (G), r ). 2. For every deterministic grph G nd every MSO-trnsduction T, there exists n MSO-trnsduction T, rtionl mpping h nd vertex r such tht T (Unf(G, r)) h 1 (Unf(T (G), r )). Note tht in oth cses T preserves determinism. Closure Under the Treegrph Opertion The unfolding is prticulr cse of the treegrph opertion in the sense tht for ny grph G the unfolding from definle vertex r, Unf(G, r), cn e otined y n MSO-interprettion from Treegrph(G, ) (see [9]). In the cse of deterministic trees, we show converse result: how to otin treegrph using MSO-interprettions nd unfolding. This construction is due to T. Colcomet. Lemm 1. For ny finite set of lels Σ, there exist two finite mppings h 1,h 2 nd rtionl mrking M such tht for ny deterministic tree T with root r: Treegrph(T, ) h 1 ( ( ( 2 M Unf h 1 1 (T ), r))). Proof (sketch): The finite mpping h 1 dds ckwrd edges leled y elements of Σ nd loop leled y to every vertex. Thus, for ll Σ, h 1 is defined y h 1 () = {}, h 1 (ã) = {ā} nd h 1 ( ) = {ε}. Let H e the deterministic tree equl to Unf ( h 1 1 (T ), r), every node x of H is uniquely chrcterized y word in (Σ Σ { }). The rtionl mrking M $ mrks ll the vertices corresponding to word which does not contin x x or xx for x Σ. Finlly, h 2 is used to erse unmrked vertices nd to reverse the remining edges with lels in Σ. h 2 is given y h 2 ( ) = { } nd h 2 () = { $ $$ $, $ $ ã$ $ } for Σ. Figure 1 illustrtes the construction ove on the semi-infinite line. The filled dots represent the vertices mrked y M $. The closure of the deterministic hierrchy under the treegrph opertion is otined from Lem. 1 nd Prop. 1, using the fct tht for ll trees T nd ll rtionl mppings h which do not contin, Treegrph(h 1 (T ), ) = h 1 (Treegrph(T, )) where h designtes the rtionl mpping tht extends h with h ( ) = { }. Proposition 2. For ll n 0, if G Grph d (n) then Treegrph(G, ) Grph d (n + 1).
ã ã ã ã ã ã ã ã ã ã ã Fig. 1. The semi-infinite line fter pplying h 1 nd its unfolding 3.2 Deterministic Trees re Enough We now prove tht for ll n, Grph(n) is equl to Grph d (n). From the technicl point of view, this mens tht even if the hierrchy contins non-deterministic grphs nd even grphs of infinite out-degree, we cn lwys work with n underlying deterministic tree. Lemm 2. For ll n > 0, if G Grph(n), then there exists deterministic tree T Tree d (n) nd rtionl mpping h such tht G = h 1 (T ). Proof (sketch): The proof proceeds y induction on the level n. Let T Tree(n+ 1), we wnt to prove tht T elongs to Grph d (n + 1). By definition of Tree(n + 1) nd y induction hypothesis, we hve T Unf(h 1 (T d ), s) for some deterministic tree T d Tree d (n) nd some rtionl mpping h. Using the fct tht the unfolding cn e defined in the treegrph opertion (see [9]), we hve T T (Treegrph(h 1 (T d ), )) for some MSO-trnsduction T. If h denotes the rtionl mpping otined y extending h with h ( ) = { }, we hve T = T (h 1 (Treegrph(T d, )). Applying Lem. 1, we hve T = T (Unf(h 1 1 (T d), r)) T. It is esy to check tht Unf ( h 1 1 (T d), r ) elongs where T = M h 1 2 h 1 to Tree d (n + 1). Using Prop. 1, we prove tht T elongs to Grph d (n + 1). The cse of G Grph(n + 1) is esily derived from this. We cn now prove tht every grph of the hierrchy hs decidle MSOtheory. Note tht this does not follow directly from the definition ecuse unfolding from n ritrry (i.e. not necessrily MSO-definle) vertex does not preserve the decidility of MSO-logic. However, using Lem. 2 we cn lwys
come ck to the cse where we unfold from MSO-definle vertex (see [4] for more detils). Theorem 1. Ech grph of the hierrchy hs decidle MSO-theory nd this remins true if we dd to MSO-logic the predictes X <, X = k mod p for ll k nd p N which re interpreted s X is finite respectively X hs size equl to k modulo p for k, p N. Comining Prop. 1, Prop. 2 nd Lem. 2, we now hve two equivlent chrcteriztions of the hierrchy: one miniml in terms of unfolding nd inverse rtionl mppings nd one mximl in terms of the treegrph opertion nd MSO-trnsductions. The mximl chrcteriztion shows the roustness of the hierrchy nd its interest ecuse it comines the two, to our knowledge, most powerful MSO-preserving opertions. On the other side, the miniml chrcteriztion llows us to mke the link etween the hierrchy nd the grphs of higher-order pushdown utomt. Theorem 2. The Cucl hierrchy is equl to the hierrchy otined y iterting the treegrph opertion nd MSO-trnsductions. 4 Higher-order Pushdown Grphs vs. Cucl Grphs In this section we give n utomt-theoretic chrcteriztion of the clsses of the Cucl hierrchy. This chrcteriztion provides us with flt model for descriing grph of ny level, i.e. we do not hve to refer to sequence of opertions. Furthermore it extends the chrcteriztion of the first level of the hierrchy s the ε-closure of configurtion grphs of pushdown utomt given in [16] to ny level. We recll some definitions. The configurtion grph C(A) of A HOPDA(n) is the grph of ll configurtions of A rechle from the initil configurtion, with n edge leled y Σ {ε} from (q, s) to (q, s ) iff there is trnsition (q,, top(s), q, i) such tht i(s) = s. Let C(A) e the configurtion grph of A HOPDA(n). We will ssume for the reminder of the pper tht for every pir (q, α) of stte q nd top stck symol α only ε-trnsitions or only non-ε-trnsitions re possile. The ε-closure of C(A) is the grph G otined from C(A) y removing ll vertices with only outgoing ε-trnsitions nd dding n -leled edge etween v nd w if there is n -leled pth from v to w in C(A). A higher-order pushdown grph G of level n is the ε-closure of the configurtion grph of some A HOPDA(n). We cll G the higher-order pushdown grph generted y A nd denote y HOPDG(n) the clss of ll higher-order pushdown grphs of level n (up to isomorphism). This notion of ε-closure ws used in [16] to show tht the clss HOPDG(1) coincides with the clss of prefix recognizle grphs, i.e. with the the grphs on the first level of the hierrchy. We extend this result to every level of the hierrchy.
The esier prt of the equivlence is to show tht every HOPDG of level n is grph on the sme level of the hierrchy. The min ide is to find grph in Grph(n) such tht every node of this grph cn e identified with configurtion of higher-order pushdown utomton, nd to construct n inverse rtionl mpping which genertes the edges of the configurtion grph of the utomton. Such construction is lredy contined in [2] in slightly different setting. We propose here to use the fmily n m of grphs otined y n (n 1)-fold ppliction of the treegrph opertion to the infinite m-ry tree m. This hs the dvntge tht there is lmost one-to-one correspondence etween configurtions of the higher-order pushdown utomton nd the vertices of the grph. Using the fct tht n m Grph(n) we otin: Lemm 3. If G HOPDG(n) then G Grph(n). We now turn to the converse direction: every grph G on level n of the hierrchy is indeed the ε-closure of configurtion grph of higher-order pushdown utomton of level n. We show this using the following two Lemms. Lemm 4. If G HOPDG(n) nd r V G, then Unf(G, r) HOPDG(n + 1). Lemm 5. If G HOPDG(n), r V G h 1 (Unf(G, r)) HOPDG(n + 1). nd h is rtionl mpping, then While the proof of Lem. 4 consists of strightforwrd modifiction of the HOPDA for G, the proof of Lem. 5 requires some technicl preprtion. We need to show tht for n utomton s constructed in the proof of Lem. 4 there exists higher-order pushdown utomton which genertes exctly the grph Unf(G, r) extended y reverse edges, i.e. for ll v, w Unf(G, r), v w in Unf(G, r) iff w ā v in the extended grph. To show tht such n utomton exists we introduce the notion of wek popping higher-order pushdown utomton. A wek popping utomton is only llowed to execute pop instruction of level j 2 if the two top level j stcks coincide. We skip forml definition of wek popping higher-order pushdown utomton nd just mention tht even though this utomton model is equipped with uilt-in test on the equlity of two stcks of the sme level, it is equivlent to the usul model. All proofs re given in the full version of this rticle [4]. Theorem 3. For every n N, G HOPDG(n) iff G Grph(n). 5 More Properties of the Cucl Hierrchy We give genertor for ech level of the hierrchy. Then we use the trces of the grphs of the hierrchy to prove its strictness nd to exhiit grph hving decidle MSO-theory which is not in the hierrchy.
5.1 Genertors For the first level of the hierrchy, the infinite inry tree 2 is genertor for rtionl mrkings (without ckwrd edges) from the root nd inverse rtionl mppings. As hinted y the proof of Lem. 3, similr result cn e otined t ny level. Recll tht n 2 is the grph otined from 2 y n (n 1)-fold ppliction of the treegrph opertion. Proposition 3. Every grph G Grph(n) cn e otined from n 2 y pplying rtionl mrking (with ckwrd edges) from its source nd n inverse rtionl mpping. 5.2 On Trces The Strictness of the Hierrchy A direct consequence of Theo. 3 is tht the trces of the grphs of level n re recognized y higher-order pushdown utomton of level n. These fmilies of lnguges hve een studied y W. Dmm nd form the OI-hierrchy [11]. The equivlence etween the OI-hierrchy nd the trces of higher-order pushdown utomt is proved in [12]. In [10, 14], this hierrchy is proved to e strict t ech level. Theorem 4. For ll n 1, () for ll T Tree(n) the rnch lnguge of T (i.e. the set of ll words leling pth from the root to lef) is recognized y HOPDA of level n 1. () for ll G Grph(n) nd u, v V G, L(u, v, G) = {w Γ u w v} is recognized y HOPDA of level n. According to Theo. 4, the strictness level-y-level of the OI-hierrchy implies the strictness of the Cucl hierrchy. An ovious exmple of grph which is t level n ut not t level n 1 is the genertor n 2. To otin more nturl grphs tht seprte the hierrchy, we consider the trees ssocited to monotoniclly incresing mppings f : N N. The tree T f ssocited to f is defined y the following set of edges: E = {((i, 0), (i + 1, 0)) i N} nd E = {((i, j), (i, j + 1)) i N nd j + 1 f(i)}. The rnch lnguge of T f is { n f(n) n 0}. Using property of rtionl indexes of k-oi lnguges (see [10]), we otin the following proposition. Proposition 4. If { n f(n) n N} is recognized y higher-order pushdown utomton of level k then f O(2 k 1 (p(n))) for some polynomil p where 2 0 (n) = n nd 2 k+1 (n) = 2 2 k (n). Let us consider the mpping exp k (n) = 2 k (n). It hs een proved in [7] tht T expk elongs to Grph(k + 1). Note tht using Theo. 3 the construction given in [7] cn e voided y providing deterministic higher-order pushdown utomton of level k + 1 tht recognizes { n expk(n) n N}. It is nturl to consider the digonl mpping exp ω (n) = exp n (1). Figure 2 shows n initil segment of the tree ssocited to exp ω. By Prop. 4, the ssocited tree T expω is not in the hierrchy. However, using techniques from [5], we cn prove tht T expω hs decidle MSO-theory.
n exp ω (n) Fig. 2. The grph T expω of the function exp ω Proposition 5. There exists grph with decidle MSO-theory which is not in the Cucl hierrchy. 6 Conclusion We hve given two chrcteriztions of the Cucl hierrchy. We hve shown tht it coincides with the hierrchy otined y lternting the treegrph opertion nd MSO-trnsductions, nd thus hve prtly nswered question posed in [17]. It remins open whether one cn extend this result to structures other thn grphs, i.e. with symols of higher rity. We hve lso chrcterized the Cucl hierrchy s the ε-closure of configurtion grphs of higher-order pushdown utomt nd hve used this result to otin tht the hierrchy is indeed strict, ut does not contin ll grphs with decidle MSO-theory. Despite these chrcteriztion results we know surprisingly few out the grphs otined on level n 2. This deserves further study. Also thorough comprison with other methods to generte infinite grphs with decidle theory misses (see [17] for more precise ccount on this). Futhermore we like to mention tht neither the constructions used to uild the hierrchy nor Proposition 5 contrdicts Seese s conjecture tht every infinite grph (or every set of finite grphs) hving decidle MSO-theory is the imge of tree (or set of trees) under n MSO-trnsduction. Finlly, mny of the questions posed in [7] on the corresponding hierrchy of trees remined unsolved so fr. Acknowledgment We thnk D. Cucl nd W. Thoms for stimulting discussions while working on this pper nd for finncil support enling our visits in Achen respectively Rennes. We lso thnk T. Colcomet for pointing to solution for Lem. 1, T.
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