Automata on Lempel-Ziv Compressed Strings

Transcription

1 Automt on Lempel-Ziv Compressed Strings Hns Leiß 1 nd Michel de Rougemont 2 1 Universität München, CIS, D München, Germny, leiss@cis.uni-muenchen.de 2 Université Pris-II & LRI Bâtiment 490, F Orsy Cedex, Frnce, mdr@lri.fr Astrct. Using the Lempel-Ziv-78 compression lgorithm to compress string yields dictionry of sustrings, i.e. n edge-lelled tree with n order-comptile enumertion, here clled n LZ-trie. Queries out strings trnslte to queries out LZ-tries nd hence cn in principle e nswered without decompression. We compre notions of utomt ccepting LZ-tries nd consider the reltion etween cceptle nd MSOdefinle clsses of LZ-tries. It turns out tht regulr properties of strings cn e checked efficiently on compressed strings y LZ-trie utomt. 1 Introduction We re interested in the compressed model checking prolem: which properties of strings cn e checked given the compressed strings? The chllenge is to et the decompress-nd-then-check method. We restrict ourselves to the clssicl Ziv-Lempel[8] string compression lgorithm LZ-78. It compresses string w Σ + to generlly much shorter sequence LZ (w) (N Σ) +, where the numers point to previous elements of the sequence. As usul, we view string w s colored finite liner order S w = (D, <, U ) Σ, where D = {0,..., w 1} is the set of positions, ordered y < s usul, nd U (i) mens tht occurs t position i of w. Properties of strings re expressed in first-order (FO) or second-order (SO) logic of colored liner orders. In similr wy, we give two representtions of LZ-78-compressed strings α y reltionl structures, one y node-lelled LZ-grphs G α nd nother one y LZ-tries T α, which re kind of edge-lelled trees. A nturl pproch to the compressed model checking prolem is to trnslte properties of strings to properties of compressed strings.

2 In fct, mondic second-order (MSO) formuls ϕ in the lnguge of strings cn e trnslted to dydic second-order (DSO) formuls ϕ LZ in the lnguge of LZ-grphs G α. One cn therefore nswer queries ϕ out S w y evluting ϕ LZ on the smller structure G LZ (w). However, since the trnsltion doules the rity of reltion vriles, this is not gurnteed to provide n efficient solution. By Büchi s well-known theorem (cf. [4], Theorem 5.2.3), MSO for strings is eqully expressive s regulr expressions, or finite utomt, re. This rises the question whether MSO for LZ-grphs leds to notion of LZ-utomton tht provides n efficient method for checking resonly rich clss of properties of compressed strings. We study this question in the slightly more suitle formt of LZ-tries. These come with n enumertion of their nodes nd cn e viewed s simple cyclic directed grphs. We introduce notions of LZtrie utomt y modifying corresponding notions of tree-utomt. We show tht deterministic ottom-up LZ-trie-utomt re less powerful thn non-deterministic ones, nd tht the ltter cpture -MSO for LZ-tries, where -MSO is the set of MSO-formuls of the form Xψ where X re set vriles nd ψ is n FO-formul. Finlly, we show tht MSO-properties of strings cn e checked efficiently on LZ-compressed strings y deterministic top-down LZutomt. Thus, regulr expression serch in strings cn e done on the LZ-compressed strings, without decompression. 2 Lempel-Ziv-78 Compression We fix finite lphet Σ nd, to void structures with empty universe, only consider non-empty strings w Σ +. The clssicl Lempel-Ziv-78 compression lgorithm hs mny vritions (cf. [2], [3]). It decomposes string w Σ + into sequence of sustrings or locks B i Σ +, so tht w = B 0 B m 1. The first lock B 0 consists of the first letter of w. Suppose for some n > 0, we hve constructed locks B 0,..., B n 1 such tht w = B 0 B n 1 v for some v Σ +. Then B n is the shortest non-empty prefix of v tht is not mong {B 0,..., B n 1 }, if this exists 1, otherwise B n is v. The LZ-compression LZ (w) of w is the sequence p 0 p m 1 of pirs 1 nd then B n extends one of B 0,..., B n 1 y exctly one letter. 2

3 p n = (k, ) such tht B n = B k 1 (where B 1 := ɛ) nd w = B 0 B m 1. The decompression is given y B k = decode(p k ) where decode((0, )) = nd decode((n + 1, )) = decode(p n ). Exmple 1. The locks of w = re..., nd its compression is LZ (w) = (0, )(0, )(2, )(1, )(3, )(3, ). 2.1 LZ-Grphs Definition 1. A compressed string α = p 0 p m 1 s finite lelled ordered grph is represented G α := (D m, <, U, E) Σ, where m = α is the numer of locks, D m = {0,..., m 1}, < the nturl order on D m, U (i) is true iff the lst letter of lock B i is. The inry reltion E descries the reference to previous pirs: if p i = (k, ) for some Σ nd k > 0, (i.e. the k-th lock B k 1 is the longest strict prefix of B i ), then there is n edge E(i, k 1) from node i to k 1. Exmple 1 (Cont.). For w =... = B 0 B 1 B 2 B 3 B 4 B 5, the grph G LZ (w) is Oserve tht S w cn e interpreted in G LZ (w) s inry reltion: position i in w is mpped to the pir h(i) = (k, j) in LZ (w) iff i lies in lock B k nd B j is the nonempty prefix of B k ending in i. Note tht in G LZ (w), node j cn e reched from k y pth of E-edges. Exmple 2. If w =..., the positions 5,6,7 occurring in lock B 3 = re represented y (3, 0), (3, 2), (3, 3), ecuse the nonempty prefixes of B 3 re = B 0, = B 2, nd = B 3. Theorem 1 ([1]). For every MSO-formul ϕ(x 1,..., x n, X (1) ) out strings there is DSO formul ϕ LZ (x 1, y 1,..., x n, y n, X (2) ) out LZ-grphs, such tht for ech S w nd ll i 1,..., i n S w nd S S w, S w = ϕ[i, S] G LZ (w) = ϕ LZ [h(i), h(s)]. 3

4 Proof. (Sketch) In DSO, we cn define E, the reflexive trnsitive closure of E, nd then define ϕ LZ inductively using (U (x i )) LZ := U (y i ), (x i x j ) LZ := x i < x j (x i = x j y i y j ), ( x n+1 ϕ) LZ := x n+1 y n+1 (E (x n+1, y n+1 ) ϕ LZ ), ( X 1 ϕ) LZ := X 2 ( x y(x 2 (x, y) E (x, y)) ϕ LZ ). (1) For the tomic cses, note tht in ϕ LZ vrile x i stnds for lock of w nd y i for reltive position in this lock. A property L of non-empty strings is definle on strings (resp. on compressed strings or LZ-grphs), if for some formul ϕ of the pproprite lnguge, L = {w S w = ϕ} (resp. L = {w G LZ (w) = ϕ}). Remrk 1. There re properties of strings tht re F O-definle on strings, ut not on LZ-grphs, like x(u (x) U (x + 1)). There re properties of strings tht re FO-definle on LZ-grphs, ut not even MSO-definle on strings (cf. [1]). 2.2 LZ-Tries While compressing w = B 0 B n 1 v, the locks B 0,..., B n 1 found re mintined s dictionry of suwords of w nd stored s tree y shring common prefixes. The liner order of the locks in LZ (w) mounts to n enumertion of the nodes of the tree. Definition 2. A (finite) Σ-tree (T,,, 0) Σ is (finite) tree (T,, 0) with root 0, where { T T Σ} re pirwise disjoint miniml reltions such tht is the reflexive trnsitive closure of their union. A Σ-tree is Σ-trie if to ech node n T nd ech Σ there is t most one n T such tht n n. A (finite) enumerted Σ-trie T = (T,,, 0, Succ) Σ, or n LZ-trie for short, is Σ-trie (T,,, 0) Σ with successor reltion 2 Succ on T tht is comptile with the prtil order. We ssume tht T = {0, 1, 2,..., m} nd Succ(i, j) iff i + 1 = j in N. 2 i.e. miniml inry reltion Succ whose trnsitive reflexive closure Succ is totl ordering of T 4

5 Exmple 1 (Cont.). Enumerting the pirs of LZ (w) y 1,2, etc. in third component, we otin sequence (0,, 1)(0,, 2)(2,, 3) (1,, 4)(3,, 5)(3,, 6) of triples. These represent tree in which lock B k lels the pth from the root 0 to node k + 1: A tuple p n = (k, ) of LZ (w) is drwn s n edge k n + 1. We write T α for the enumerted trie representing the compressed word α. We lwys ssume tht our strings w hve distinguished end symol; then the finl lock of LZ (w) is different from the previous ones nd the tree of locks indeed is trie. Remrk 2. Wht differs in choosing G α or T α is the logicl lnguge used to tlk out LZ-compessed strings α. Bsiclly, we hve G α = E(i, j) U (i) T α = (j + 1) (i + 1). Modulo the dditionl root node in the trie, in the trie mounts to E in the grph, nd in the grph to Succ in the trie. Since E resp. Succ is - nd -MSO-definle in the lnguge of LZ - grphs resp. LZ -tries, -MSO-properties of LZ-grphs trnslte to -MSO-properties of LZ-tries nd vice vers. Using (1), quntifiers Qx nd QX out strings trnslte to ounded quntifiers Q(x, y) E nd QX E. So when trnslting to the lnguge of LZ-tries we only quntify over tuples (nd reltions of such) whose components lie on pth of T LZ (w). 3 MSO-Equivlence for LZ-Grphs For reltionl structures A, B of the sme signture, A MSO r B sys tht A nd B stisfy the sme MSO-sentences of quntifier rnk r. Two fcts out MSO for strings imply the existence of finite utomt tht cn check MSO-properties of strings (cf. [4]): 5

6 ) for ech r, there re only finitely mny MSO r -equivlence clsses for word structures S w, nd ) for compound strings w, the MSO r -clss of S w depends only on the MSO r -clsses of S w nd S. (The nlogous situttion holds for trees over Σ.) LZ-compressed words α = p 0 p m 1 re words over the infinite lphet N Σ. Cn we check MSO-properties of compressed strings y kind of finite utomton for LZ-grphs? Since we del with finite reltionl lnguge, we still hve ): Proposition 1. For ech r, the equivlence reltion MSO r LZ-grphs hs finite index. etween But wht out )? By extending winning strtegies for duplictor in the Ehrenfeucht-Frisse-gme G r (G α, G β ), we cn show: Lemm 1. (i) For LZ-compressed words α(0, ), β(0, ) over Σ, G α MSO r G β = = G α(0,) MSO r G β(0, ). (ii) For LZ-compressed words α(k + 1, ) nd β(k + 1, ) over Σ, (G α, k) MSO r (G β, k ) = = G α(k+1,) MSO r G β(k +1, ). However, in (ii) one hs to ssume tht the elements k, k pointed to from the new mximl elements shre the sme properties. Insted, one would need the stronger clim G α G β G α k G β k = = G α(k+1,) G β(k +1, ), where G α k is the restriction of G α with k s its mximl element. The equivlence clss of G α k would e the utomton stte ssigned to k. (Notice tht G α k is LZ-grph, ut (k + 1, ) is not LZcompressed word.) The prolem is tht duplictor s winning strtegies for G r (G α, G β ) nd G r (G α k, G β k ) my pick different elements to nswer spoilers plying of some element of, sy, G α k. Thus, unlike in the cse of strings or trees, for compound compressed words α(k, ) we hve component LZ-grphs G α nd G α k tht re not disjoint, nd winning strtegies in gmes for these do not comine to winning strtegies for composed LZ-grphs. From this we conclude tht we cnnot use Büchi-Myhill-Nerode construction to otin from the MSO r -clsses finite sequentil utomton for LZ-grphs, nd likewise for LZ-tries. 6

7 4 LZ-Trie-Automt If we view LZ-tries s trees with n dditionl edge Succ etween nodes, we otin directed cyclic grphs of specil kind: the successor child my e equl to some decendnt with respect to the -child-reltions. Since these re still very close to trees, it is nturl to use vrition of tree-utomt s n pproximtive notion of LZ-utomton for checking properties of LZ-tries. Definition 3. Let n G e node in grph (G, E). For m N, the sphere of rdius m round n, s m (n), is the set of nodes k G such tht there is E-pth of length m from n to k or vice vers. The hemisphere of rdius m round n, hs m (n), is the set of nodes k such tht there is n E-pth of length m from n to k. Definition 4. Let n T e node in the LZ -trie T. The ottomup LZ-hemisphere of rdius m round n, u-hs T m (n), is the restriction of T to the m-hemisphere round n in the grph (T, E), where E := { Σ} {Succ}. The top-down LZ-hemisphere of rdius m round n, td-hs T m (n), is the restriction of T to the m-hemisphere round n in the grph (T, Ĕ), where Ĕ is the converse of E. An LZ-hemisphere is n LZhemisphere of some rdius round some node in some LZ-trie T. Exmple 1 (Cont.). An LZ-trie nd the ottom-up resp. top-down 2-hemispheres of node 3 (with dshed edges for Succ resp. Pred): Definition 5. A finite ottom-up (resp. top-down) m-lz-utomton A = (Q, Σ, δ, q in, F ) consists of finite set Q of sttes, sets I, F Q of initil nd finl sttes, finite lphet Σ, finite 7

8 trnsition reltion δ consisting of pirs (P, q), written P q, where q Q nd P is ottom-up (resp. top-down) LZ-hemisphere of rdius m whose nodes except the root re lelled y elements of Q. A run of A on n LZ-trie T is function r : T Q where r(mx) I (resp. r(0) I) nd for ech n T there is some (P, q) δ such tht u-hs T m (n) (resp. td-hst m (n)), expnded y the lelling of nodes given y r, is isomorphic to P with lel q t its root. A ccepts T if there is run r of A on T such tht r(0) F (resp. r(mx) F ). Let L(A) := {T A ccepts T } e the clss of LZ-tries ccepted y A. A is deterministic if I = 1 nd q = q when (P, q), (P, q ) δ. While n m-lz-utomton A sequentilly follows the enumertion of trie T α, it cn ccess some sttes reched t suffixes (resp. prefixes) of α. Strictly speking, it does not hve finite memory. 4.1 Bottom-up LZ-Trie-Automt A 1-LZ-utomton working ottom-up the LZ-trie towrds the root hs trnsitions tht determine the stte t node from the sttes t the node s Σ-children nd successor. But it lso hs to distinguish which of the Σ-children is the successor of the node, if ny. Exmple 2. Consider the clss K of LZ-tries over Σ = {, } which hve node whose successor nd -child gree, i.e. which stisfy the sentence ϕ := x y [y = x + 1 (x y)]. We give ottom-up 1-LZ-utomton A ccepting K. We write trnsition in the form (q, q, q succ, i) p, where q, q, q succ re the sttes of the -, - nd Succ-child or, when there is no such child, nd i {1, 2, 3, } sys which of the children is equl to the successor node, if ny. Thus, (p, q, p, 1) q corresponds to the trnsition P q which cn e shown s q (n) p(n+1) q(m) 8

9 A hs finl stte q 1, which is ssigned to ll ncestors of the root of sutrie stisfying ϕ, nd n initil stte q 0, which is ssigned to ll other nodes of the input trie. Letting q, p, q rnge over {q 0, q 1 }, the trnsition tle is ) (,,, ) q 0 e) (q, p, q, 3) q ) (q,, q, 1) q 1 f) (q,, q, 3) q c) (q, p, q, 1) q 1 g) (, q, q, 3) q d) (q, p, q, 2) q h) (,, q, 3) q. Rule ) mens tht if there is no successor-node, A is in stte q 0. Rules ) nd c) sy tht if the successor-node is the -child, then A goes to q 1 s we just sw the pttern ϕ. Rules d) f) sy tht if the successor node differs from the -child, A remins in the stte of the successor node. Similr for g) nd h), when there is no -child. Theorem 2. There is property of LZ-tries tht is recognized y non-deterministic ottom-up 1-LZ-utomton ut not y ny deterministic ottom-up m-lz-utomton. Proof. (Sketch) Let Σ = {,, c} nd consider the following property ϕ of enumerted Σ-tries: There re two susequent nodes i 1 nd i such tht some node j is oth the -predecessor of i nd -ncestor of i 1. An LZ-trie stisfies ϕ iff it contins nodes linked s follows (in the trie nd in the LZ-grph, respectively), where j < i 1: i j j+1 i 2 i 1 j When finding nodes connected like i 1 nd i, ottom-up 1-LZutomton cn (non-deterministiclly) guess tht -ncestor of i 1 will e the -prent of i nd check this while proceeding. But no i 1 i (2) 9

10 deterministic ottom-up m-lz-utomton cn check the property ϕ ecuse, intuitively speking, the m-hemisphere of node j is not ig enough to see if the predecessor of its -child is one of its - descendnts. The proof detils re not quite ovious ut hve to e omitted for lck of spce. 4.2 Top-down LZ-Trie-Automt G.Nvrro [5] hs shown how to do regulr expression serch on LZ-78-compressed texts y simulting n utomton reding the originl text, eting the decompression-nd-serch pproch y fctor of 2. Actully, this simultion is deterministic top-down 1- LZ-utomton on the LZ-compressed text: Theorem 3. The LZ-compression {T LZ (w) w R} of ny regulr set R Σ + is ccepted y deterministic top-down 1-LZ-utomton. Proof. Let A = (Q, Σ, q 0, δ, F ) e deterministic finite utomton ccepting R. Define deterministic top-down 1-LZ-utomton A = (Q, Σ, δ, q in, F ) y Q := Q (Q Q), q 0 := (q 0, λq.q), F := {(q, f) q F } nd δ ccording to the following trnsitions: 0 (q 0, λq.q) (p, f) (p, f ) i i+1 (δ(f (p), ), λq.δ(f (q), )) for ech Σ (including the cse i = k). Suppose r is run of A on T LZ (w) for compressed word LZ (w) = p 0 p m 1 where p k represents lock B k of w = B 0 B m 1. By induction we see tht for ech k < m, r (k) = (δ(q 0, B 0 B k 1 ), λp.δ(p, B k 1 )). (3) For k > 0, suppose r (i) = (p, f) = (δ(q 0, B 0 B i 1 ), f) nd p i = (k, ). Then B i = B k 1, nd with r (k) = (p, f ) = (p, λp.δ(p, B k 1 )) r (i + 1) = (δ(f (p), ), λq.δ(f (q), )) = (δ(p, B k 1 ), λq.δ(q, B k 1 )) = (δ(δ(q 0, B 0 B i 1 ), B i ), λq.δ(q, B i )) = (δ(q 0, B 0 B i ), λq.δ(q, B i )). 10 k

11 Hence A ccepts {T LZ (w) w R}, ecuse w R δ(q 0, B 0 B m 1 ) F r (m) F. 4.3 Grph Acceptors nd -MSO for LZ-Tries If we consider the sttes q of n utomton A s mondic predictes on nodes of tries, the condition there is n ccepting A-run cn e expressed y n -MSO-sentence qψ in the lnguge of LZ-tries: Proposition 2. For every m-lz-utomton A there is n -MSOsentence ϕ A defining the clss of LZ-tries ccepted y A. For sentences, this improves on the trnsltion of Theorem 1: Corollry 1. Every property of strings which is definle in MSO on strings is definle in -MSO on LZ-tries. Proof. By Büchi s theorem, Theorem 3 nd Proposition 2. The converse of Corollry 1 is wrong: {... n. n. n N} is not regulr lnguge, ut hs n FO-definle clss of LZ-tries. To show the converse of Proposition 2, we use the grph cceptors for directed cyclic grphs presented y W. Thoms [7]. On grphs G = (V, E), tile δ over set Q is n m-neighourhood of node with lelling in Q, i.e. finite node-lelled grph. A grph cceptor is triple A = (Q,, Occ) where is finite set of tiles over the finite set Q nd Occ is oolen comintion of conditions χ p,δ := there re p occurences of tile δ. A run of A on G is function r : V Q such tht ech m-neighourhood of G ecomes tile δ. Then A ccepts grph G if there exists run of A on G whose tiles stisfy Occ. Note tht the existence of n ccepting run is non-constructive. The min result of [7] sys tht clss of grphs of ounded degree is definle in -MSO iff it is ccepted y grph cceptor. Theorem 4. A clss K of LZ-tries is -MSO-definle iff for some m, K is ccepted y n m-lz-utomton. Proof. Note tht ech LZ-trie T stisfies the following conditions: 11

12 (i) ech node of T hs degree Σ + 1, (ii) T is n cyclic (directed) grph with designted out-edge (the successor) for ech node, (iii) T hs node tht is rechle from ny node y pth. Using (i), y Theorem 3 of [7], K is -MSO-definle iff K is recognizle y grph cceptor. Moreover, from (i)-(iii) nd Proposition 6 of [7], it follows tht K is recognizle y grph cceptor iff it is recognizle y grph cceptor without occurrence constrints. : Suppose K is ccepted y some m-lz-utomton A. We my ssume tht finl sttes of A do not occur in the m-hemispheres P of trnsitions (P, q) of A. Consider the trnsitions of A s tiling system. Then n ccepting run of A is tiling tht uses t lest one of the tiles (P, q) where q F. Thus, K hs grph cceptor. : By the ove remrks, K is recognizle y grph cceptor without occurrence constrints. On the LZ-tries, the tiles hve root node, so we cn view ech tile s trnsition rule sying tht the utomton enters stte t the root depending on the m-sphere of the root nd the sttes t the descendnts in the sphere. Let ech stte e finl; then the utomton ccepts iff there is tiling. A grph G = (V, E) is k-colorle if there is prtitioning of the set V of nodes into t most k clsses (colors) such tht ny two djcent nodes elong to different clsses. Clerly, k-colorility cn e expressed y n -MSO sentence, nd hence hs grph cceptor. While 3-colorility on ritrry grphs is n NP-complete prolem, it is trivil on LZ-tries, ecuse the mximl node m is connected to t most two nodes: Proposition 3. Every LZ-trie is 3-colorle. Potthoff e.. [6] hve shown tht on the clss of directed cyclic grphs whose edges re uniquely lelled with ounded numer of lels, 2-colorility is not recognizle y deterministic grph cceptor, i.e. one with t most one ccepting run on ech grph. For the suclss of LZ -tries, however, it is (ctully y 4-stte deterministic ottom-up-lz-utomton): Proposition 4. On the clss of LZ-tries, 2-colorility is recognizle y deterministic grph cceptor. 12

13 4.4 Strictness of the Automt Hierrchy The ottom-up resp. top-down hierrchies of m-lz-utomt re strict, nd no level exhusts the MSO-properties of tries: Theorem 5. For every m, there is n MSO-sentence defining clss of LZ-tries tht is not ccepted y ny m-lz-utomton. Proof. For m = 1, let L = { 1 2 n 1 2 n 1 n N} {, } +. Ech w L hs LZ-lock decomposition s indicted y nd compression w n = n n 1 LZ (w n ) = (0, )(1, ) (n 1, )(0, )(1, ) (n 1, ). The corresponding tries T LZ (wn) look like n (4) n+2 n 1 2n n where the successor reltion is given y the node numers. The clss of enumerted tries in LZ (L) cn e defined y the MSO-sentence ϕ sying tht the set B of nodes tht re the root 0 or -child, hs the following properties (of which (i) is neither -MSO nor -MSO): (i) B {0} is the smllest set eing closed under -children, (ii) node is not in B iff it is n -child of node in B, (iii) for ll nodes x, y, we hve y = x + 1 iff one of the following holds: () y is the -child of x, () y is the -child of the -child y of some x whose -child is x, (c) y is the -child of 0 nd x the memer of B tht hs no -child. Clim. LZ (L) is not ccepted y 1-ottom-up-LZ-utomton. 13

14 Suppose tht A is 1-ottom-up-LZ-utomton tht ccepts the clss defined y ϕ. Let m > Q nd w = w 2m. An ccepting run of A on T LZ (w) ssigns the sme stte, sy q, to t lest two different -childs, sy nodes n+k+2 nd n+2. The stte of their predecessors is determined y rule (,, q, 3) p. Let T e like T LZ (w), except tht nodes n + 1 nd n + k + 1 re switched in the ordering. Then A will lso ccept the modified structure, s indicted y the sttes ssigned to nodes s follows: m m p(n+1) m+1 p(n+k+1) m+1 q(n+2) m+k q(n+2) m+k p(n+k+1) m+k+1 p(n+1) m+k+1 q(n+k+2) q(n+k+2) But the enumertion of -children in T does not conform to ϕ, so T T (A) \ LZ (L). For the cse m > 1, we modify the exmple s follows: long the B-prt, etween two nodes tht hve oth n -child nd -child, we dd m 1 nodes tht hve no -child, i.e. the sugrphs k k+1 n n +1 re replced y k n k+1 k+m 1 k+m n +1 The LZ-tries rising this wy re the compressions of the words w n,k = nm+k.. m. 2m.. nm. 14

15 Then {T LZ (wn,k ) n, k N, k m} is ccepted y (m + 1)-ottomup-LZ-utomton, hence MSO-definle. But it is not ccepted y ny m-ottom-up-lz-utomton: intuitively, the m-hemisphere of node k does not tell whether the pth from k following nd successor nd the pth following m end t the sme node. A similr rgument shows tht the clss defined y ϕ is lso not ccepted y ny m-top-down-lz-utomton with the sme m. 5 Conclusion We hve shown some reltions etween properties of strings, properties of their Lempel-Ziv-78-compressions, nd utomt ccepting compressed strings, ut the picture is not complete yet. For exmple: Conjecture 1. On the clss of ll LZ-tries, not every MSO-formul is equivlent to n -MSO-formul. It seems likely tht the clss of cceptle LZ-tries is not closed under complement (cf. Theorem 2). We lso elieve tht deterministic ottom-up-lz-utomt re much weker thn deterministic top-down-lz-utomt (cf. Theorem 3): Conjecture 2. There is no deterministic ottom-up m-lz-utomton tht cn check on LZ (w) whether w {, } + hs suword. References 1. Foto Afrti, Hns Leiß, nd Michel de Rougemont. Definility nd compression. In 15th Annul IEEE Symposium on Logic In Computer Science, LICS Snt Brr, CA, June 22-26, 2000, pges Computer Society Press, Timothy C. Bell, John G. Clery, nd In H. Witten. Text Compression. Prentice Hll, Englewood Cliffs, NJ, T. Cover nd J. Thoms. Elements of Informtion Theory. John Wiley, H.-D. Einghus nd J. Flum. Finite Model Theory. Springer-Verlg, Gonzlo Nvrro. Regulr expression serching on compressed text. Journl of Discrete Algorithms, 2003 (to pper). 6. Andres Potthoff, Sestin Seiert, nd Wolfgng Thoms. Nondeterminism versus determinism of finite utomt over directed cyclic grphs. Bull. Belg. Mth. Soc., 1: , Wolfgng Thoms. Automt theory on trees nd prtil orders. In eds. M. Bidoit, M. Duchet, editor, TAPSOFT 97, LNCS 1214, pges Springer Verlg, J. Ziv nd A. Lempel. Compression of individul sequences vi vrile-rte coding. IEEE Trnsctions on Informtion Theory, pges ,