MSO Querying over Trees via Datalog Evaluation

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1 MSO Querying over Trees via Datalog Evaluation Eugénie Foustoucos and Labrini Kalantzi M.Π.Λ.A., Department of Mathematics National and Capodistrian University of Athens Panepistimiopolis, Athens, Greece Abstract. The MSO evaluation problem on trees is the problem of computing, for any MSO formula φ and any finite tree T,thesetofsatisfying assignments of φ(x 1,...,X k )overt. In [3] we considered the MSO evaluation problem on full binary trees as a Datalog query evaluation problem, and gave a Datalog-theoretic solution to it for any MSO formula φ; we thus subsumed previous results obtained for a restricted class of MSO, namely for unary MSO queries. In the present paper we extend the results of [3] both to r-ary trees and to unranked trees, thus providing a general Datalog-theoretic solution to the problem for the class of finite trees. 1 Introduction Monadic second order logic (MSO) is an extension of first order logic (FO) with set variables over which quantification is allowed (see also [4]). The MSO evaluation problem or MSO querying problem on trees is the problem of evaluating k-ary (k 0) MSO formulas, i.e. it is the problem of computing, for any MSO formula φ(x 1,...,X k ) and any finite tree T, the set of satisfying assignments sat assign(φ, T )={(B 1,...,B k ) T = φ(b 1,...,B k )} of φ(x 1,...,X k )overt. In this paper we first deal with finite colored trees that are k-ranked and then with finite colored trees that are unranked. MSO over trees is a well studied logic (Doner [2], Thatcher & Wright [10]) presenting great interest due to its expressive power which makes it suitable for many computer science applications. Such an example is a very important database querying problem in the context of XML, namely the problem of selecting nodes in trees (representing semi-structured documents). Since node selecting queries Q can be expressed by MSO formulas φ with only one free variable which moreover is a first-order one (called unary MSO queries), answering Q on an XML document D means evaluating φ onthetreerepresentationofd. Itis due to the aforementioned connection with XML that the unary MSO querying problem has received so much attention recently and many representation formalisms have been proposed to address it: for instance Monadic Datalog [7], query automata [9], selection tree automata [6]. More precisely, in was proved in [7] that MSO and Monadic Datalog are equivalent in their power to define

2 MSO Querying over Trees via Datalog Evaluation 69 unary queries (over both ranked and unranked trees). The proof was based on the simulation of query automata (proved in [9] to capture the expressiveness of MSO with respect to unary queries) in Monadic Datalog. Moreover selection automata and Monadic Datalog have been related in [8] with respect to evaluation of unary queries over trees. In [3] we considered the whole class of MSO formulas, i.e. the class of formulas having k (k 0) free (either first- or second-order) variables, over full binary trees and we introduced a new automaton called assignment automaton, which evaluates MSO k-ary queries over trees. We then used assignment automaton, translating both its construction procedure (program Π φ ) and its run evaluation procedure (program Π eval φ ), to connect MSO to Datalog and further to solve the MSO querying problem via Datalog query evaluation. More precisely, we showed that for every MSO formula φ, we can define a Datalog program Π eval φ, such that the evaluation of φ over a finite tree T (viewed as database D T )is obtained by evaluating Π eval φ on input databases D φ (computed by program Π φ )andd T ; in the present paper we essentially follow the same lines to solve the MSO evaluation problem first on ranked trees and then on unranked trees. 2 Basic Notions of Trees, Automata and MSO-Logic Ranked Colored Trees, MSO over Ranked Trees. Let r be a natural number; a tree t of rank r (or r-ranked tree) is a finite tree (i.e. its set of nodes dom(t ) is finite) which is ordered (i.e. for each node n, the set of children of n is ordered) and where each inner node has degree (i.e. number of children) at most r. LetΓ be a finite set of symbols, called alphabet of colors. Anr-ranked Γ -colored tree T is a pair (t, c),wherec:dom(t) Γ is the coloring function and c(n) isthecolor of node n. Anr-ranked Γ -tree T is usually represented as a relational τ (Γ,r) -structure, τ (Γ,r) = {S 1,...,S r, (P γ ) γ Γ }, having domain dom(t ); S i (x, y), i =1,...,r,meansthaty is the i th child of x and P γ (x) means that node x has color γ. It is convenient for our Datalog approach, to view T as a slightly richer relational structure D T called the database of T ; D T has domain dom(t ) and signature (leaf, root, (degree i ) i=1,...,r, (child i ) i=1,...,r, (color γ ) γ Γ ), where predicates leaf, root, degree i, color γ are of arity 1, predicates child i are of arity 2 and their meaning is the obvious one. Without loss of generality, we consider MSO formulas having no FO variables at all, assuming that a formula has been translated to an equivalent formalism, called MSO [τ (Γ,r) ] (very similar to MSO 0 of [11] when r = 2) where the free variables of atomic formulas are interpreted as singleton sets. In particular, the atomic formulas of MSO [τ (Γ,r) ]arey (X), Si (X, Y ), P γ (X), with respective meaning: 1) the unique element of set X is included into set Y, 2) the unique element of set Y is the i th child of the unique element of set X, 3) the unique element of set X has color γ. Distinguishing between MSO and MSO is a technical detail (see also [3]) and we refer to it by simply saying MSO.

3 70 Eugénie Foustoucos and Labrini Kalantzi Unranked Colored Trees and MSO over Unranked Trees. An unranked Γ -colored tree T u, is a finite ordered tree having no a priori bound on the number of children of its nodes. T u is usually represented as a relational τ Γ -structure, τ Γ = {S first,next, Last, (P γ ) γ Γ }, having domain dom(t u ); the binary predicate symbol S first corresponds to the first (leftmost) child relation, Next (also binary) to the next-sibling relation and Last (unary) to the has-no-next-sibling relation. We view T u, as a slightly richer relational structure D T u, called the database of T u and defined analogously to the database D T of a ranked tree T. MSO formulas that are interpreted over unranked trees are formulas over the signature τ Γ (same remarks hold as before). MSO/Automata Connection: the Assignment Automaton. Definition 1 A non deterministic bottom up r-ranked colored tree automaton A is a tuple (Γ, Q,, F), whereγ is an alphabet of colors, Q is a finite set of states, F Q is the set of final states and is a family of transition relations ( i ) i=0,...,r,where 0 Q Γ and i Q Γ Q i.arun ρ of A on an r-ranked Γ -tree T is a mapping, assigning states to nodes s.t. i) if n is a leaf with color a, thenρ(n) =q if there exists a transition (q, a) 0, and ii) if n is a node of color a, having children n 1,...,n m, then the value ρ(n) is such that (ρ(n),a,ρ(n 1 ),...,ρ(n m )) m.arunissuccessful if it maps the root to a final state. A Γ -tree T is recognized by a Γ -automaton A if there exists a successful run of A on T.AclassofΓ -trees T is recognizable if there exists an automaton that recognizes it. Definition 2 (introduced in [3]) A k-assignment ranked automaton A is a non deterministic bottom up tree automaton (Γ, Q,, F) where the set of states Q has the special form Q = Q 0 {0, 1} k for some finite set Q 0.Thek-assignment automaton A computes an assignment B =(B 1,..., B k ) over T,ifthereisa successful run ρ of A on T computing it i.e. such that for i =1,...,k, B i = {n dom(t ) ρ i (n) =1}. Theorem 3 Given a k-ary MSO formula φ, ak-assignment automaton Assign φ can be constructed that computes the satisfying assignments of φ. Assign φ is called the assignment automaton of φ. Proof (Sketch). Let T =(t, c) bearankedγ-tree and let B =(B 1,...,B k )be a tuple of k subsets of dom(t ). We define a new Γ k -tree (Γ k = Γ {0, 1} k ), denoted (T,B), by extending the coloring function c of T with a new function c B : dom(t ) {0, 1} k. More precisely, c B (n) = (c B1 (n),...,c Bk (n)) where c Bi (n) =1iffn B i, i =1,...,k;(T,B) =(t, c )withc (n) =(c(n),c B (n)). Let φ be a k-ary MSO formula; the class of Γ k -trees (T,B) such that T = φ(b), is recognizable (Doner [2], Thatcher & Wright [10]). Let A φ be a Γ k -tree automaton recognizing this class and let A φ be the k-assignment automaton satisfying the following correspondence: (T,B) is recognized by A φ iff there is a successful run of A φ over T computing B. Clearly Assign φ = A φ.

4 MSO Querying over Trees via Datalog Evaluation 71 Example 4 The satisfying assignments of formula Pw (X) over a 3-ranked {w, b}- colored tree (w stands for white and b stands for black) with four nodes (n 1 is root of color w having 1st, 2nd and 3rd child n 2, n 3, n 4 of color w, b, b respectively) are computed by Assign P w via its successful runs n 1 (q a, 0) n 1 (q a, 1) n 2 (q a, 1) n 3 (q 0, 0) n 4 (q 0, 0) n 2 (q 0, 0) n 3 (q 0, 0) n 4 (q 0, 0) corresponding to assignments {n 2 }, {n 1 } respectively. 3 From MSO to Datalog : the Ranked Case 3.1 MSO/Datalog Connection over Ranked Trees Based on our assignment automaton, we translate the MSO/automata connection into an MSO/Datalog connection. We assume some familiarity with Datalog; the reader is also referred to [1]. Definition 5 (Program Π φ ) Given an MSO[τ (Γ,r) ] formula φ (with maximal subformulas φ 1 and φ 2 if φ = φ 1 φ 2 ; with maximal subformula φ 1 if either φ = φ 1 or φ = φ 1 ), program Π φ is a ground Datalog program consisting of 1) programs Π ψ where ψ maximal subformula of φ and 2) of a last stratum defining φ-predicates i.e. IDB predicate symbols either of arity 0, of the form φ f q,φ s q or of the same arity as φ and of the form φ [q] γ 0 or φ [q, q 1,...,q i ] γ i, 1 i r, γ Γ.Symbolsq, q 1,...,q i appearing in the φ-predicates form set Q φ called the set of state symbols of program Π φ. More precisely, Π φ is constructed by induction on the structure of φ. Wegive the 3 cases where φ is atomic. If φ = Y (X), thenπ φ consists of facts written below in compact form ( denotes any value of the set of valid values for the corresponding argument, i.e. in the following {0, 1}): - φ [q 0 ] γ 0 (0, ), φ [q f ] γ 0 (1, 0), φ [q a ] γ 0 (1, 1). - φ [q 0, q 0 i ] γ i (0, ), φ [q f, q 0 i ] γ i (1, 0), φ [q a, q 0 i ] γ i (1, 1), 1 i r, where q 0 i denotes a sequence of length i of q 0 s, i.e. of the form q 0,...,q 0. - φ [q a, q 0 n, q a, q 0 m ] γ i (0, ), φ [q f, q 0 n, q a, q 0 m ] γ i (1, ), 1 i r and n, m 0, n + m +1=i. - φ [q f, s] γ i (, ), i 2, fors sequence of length i over Q φ, containing at least two occurrences of state q a or at least one occurrence of state q f. - φ s q a, φ s q 0, φ s q f, φ f q a. If φ = Sj (X, Y ), thenπ φ consists of facts written below in condensed form: - φ [q f, s] γ i (1, 1), fors sequence of length i over Q φ, 0 i r.

5 72 Eugénie Foustoucos and Labrini Kalantzi - φ [q 0 ] γ 0 (0, 0), φ [q 1 ] γ 0 (0, 1), φ [q f ] γ 0 (1, 0). - φ [q 0, q 0 i ] γ i (0, 0), φ [q 1, q 0 i ] γ i (0, 1), φ [q f, q 0 i ] γ i (1, 0), 1 i r. Again q 0 i denotes a sequence of the form q 0,...,q 0 of length i. - φ [q a, s] γ i (1, 0) and φ [q f, s] γ i (0, ), j i r, wheres=s 1,...,s i with s j =q 1,ands l =q 0 when l j. - φ [q a, q 0 n, q a, q 0 m ] γ i (0, 0) and φ [q f, q 0 n, q a, q 0 m ] γ i (e 1,e 2 ), 1 i r; n, m 0, n + m +1=i and (e 1,e 2 ) {(0, 1), (1, 0)}. - φ [q f, s] γ i (, ),1 i r, wheres is any sequence of length i over Q φ not considered in the previous cases. - φ s q a, φ s q f, φ s q 0, φ s q 1, φ f q a. For φ = P α (X), α Γ, Π φ consists of facts written below in condensed form, where γ Γ, α Γ \{α}: - φ [q 0 ] γ 0 (0), φ [q a ] α 0 (1), φ [q f ] α 0 (1). - φ [q 0, q 0 i ] γ i (0), φ [q a, q 0 i ] α i (1), φ [q f, q 0 i ] α i (1), 1 i r. - φ [q a, q 0 n, q a, q 0 m ] γ i (0) and φ [q f, q 0 n, q a, q 0 m ] γ i (1), 1 i r and n, m 0, n + m +1=i. - φ [q f, s] γ i ( ),1 i r, fors sequence of length i over Q φ, either having more than one occurrence of q a or having at least one occurrence of q f. - φ s q a, φ s q 0, φ s q f, φ f q a. When φ is not atomic, Π φ is constructed (by induction on the structure of φ) along the lines of [3] with slight modifications that take into account the degree of nodes. Definition 6 (Database D φ ) We call database D φ of the MSO formula φ(x 1,...,X k ),thesetdφ 1 D 2 φ ; Dφ 1 is the set of ground atoms over φ-predicates in program Π φ where argument vector (e 1,...,e k ) of atoms φ [q] γ i (e 1,...,e k ) has been replaced by word e 1 e k and Dφ 2 is the union of sets of facts {φ s [q]γ 0 φ [q] γ 0 (e) Dφ 1} and {φ s [q, q 1,...,q i ] γ i φ [q, q 1,...,q i ] γ i (e) Dφ 1}. It can be proved that D φ exactly describes Assign φ =(Γ, Q,, F); in particular φ [q, q 1,...,q i ] γ i (e) D φ iff there is a transition ((q, e),γ,(q 1, e 1 ),...,(q i, e i )) i, φ s q D φ iff (q, e) Q and φ f q D φ iff (q, e) F (for φ = P w (X) recall Example 4). 3.2 MSO-Evaluation via Datalog Evaluation We are now ready to present our Datalog solution to the MSO evaluation problem. Proposition 7 For every MSO formula φ(x) and for every integer m, adatalog program Π eval φ with goal predicate φ assign (of arity 2 m) can be constructed, such that for every ranked tree T of size m, T = φ(b) iff φ assign(n 1,

6 MSO Querying over Trees via Datalog Evaluation 73 ɛ 1,...,n m, ɛ m ) is computed by program Π eval φ on input database {D φ, D T }, where n 1 ɛ 1,...,n m ɛ m is a mapping codifying B = (B 1,...,B k ) i.e. satisfying ɛ j i =1iff n i B j,fori =1,...,m and j =1,...,k. Proof. Program Π eval φ consists of three strata, namely P, S, A, corresponding to the three steps of linear-time algorithm given in [5]. Definition of stratum P is the following: for each atom φ s [q] γ 0, φ s [q, q 1,...,q i ] γ i in D φ,createrules pot q(x) leaf(x), color γ (x),φ s [q] γ 0, pot q(x) degree i (x), child 1 (x, x 1 ),...,child i (x, x i ), color γ (x), pot q 1 (x 1 ),...,pot q i (x i ),φ s [q, q 1,...,q i ] γ i. Stratum S is defined: for φ f q, φ s [q, q 1,...,q i ] γ i in D φ,create suc q(x) pot q(x), root(x),φ f q, suc q j (x j ) suc q(x), pot q 1 (x 1 ),...,pot q i (x i ), child 1 (x, x 1 ),...,child i (x, x i ), degree i (x), color γ (x),φ s [q, q 1,...,q i ] γ i, j =1,...,i. Last, stratum A is defined: for each atom φ [q] γ 0 (e 1 e k )ind φ,create assign q 1 (x, e) suc q(x), leaf(x), color γ (x),φ [q] γ 0 (e), and for each atom φ [q, q 1,...,q i ] γ i (e 1 e k )ind φ,createrules assign q m1+ +mi+1 (x, e, x e 1,...,x e i ) suc q(x), degree i (x), color γ (x), child 1 (x, x 1 ),...,child i (x, x i ), m assign q 1 1 (x e 1 ),...,assign q mi i (x e i ),φ [q, q 1,...,q i ] γ i (e), m j 1, 1 + i j=1 m j dom(t ) and with x e j we denote the following sequence (of length 2 m j ): x 1 j, e1 j,...,x mj j, e mj j, j =1,...,i. φ assign(x 1, e 1,...,x m, e m ) assign q m (x 1, e 1,...,x m, e m ). Example 8 Consider MSO [τ ({w,b},2) ] formula ψ(x, Y )=Pw(X) S2(X, Y ). Then database Dψ 1 (recall Definition 6) of formula ψ(x, Y ) consists of the following facts (where γ {w, b}, e {0, 1} and U = {00, 01, 10, 11}) - ψ [q 0 ] γ 0 (00), ψ [q 1 ] γ 0 (01), ψ [q f ] γ 0 (1e). - ψ [q 0, q 0 ] γ 1 (00), ψ [q 1, q 0 ] γ 1 (01), ψ [q f, q 0 ] γ 1 (1e), ψ [q a, q a ] γ 1 (00), ψ [q f, q a ] γ 1 (g) for g U \{00}, ψ [q f, q 1 ] γ 1 (u), ψ [q f, q f ] γ 1 (u), u U. - ψ [q 0, q 0, q 0 ] γ 2 (00), ψ [q 1, q 0, q 0 ] γ 2 (01), ψ [q f, q 0, q 0 ] γ 2 (1e), ψ [q a, q 0, q a ] γ 2 (00), ψ [q a, q a, q 0 ] γ 2 (00), ψ [q f, q 0, q a ] γ 2 (g), ψ [q f, q a, q 0 ] γ 2 (g) for g U \{00}, ψ [q a, q 0, q 1 ] w 2 (10), ψ [q f, q 0, q 1 ] b 2 (10), ψ [q f, q 0, q 1 ] γ 2 (d) for d U \{10}, ψ [q f, q 1, q] γ 2 (u), ψ [q f, q a, q a ] γ 2 (u), ψ [q f, q a, q 1 ] γ 2 (u),

7 74 Eugénie Foustoucos and Labrini Kalantzi ψ [q f, q f, q] γ 2 (u), ψ [q f, q, q f ] γ 2 (u), u U, q {q 0, q 1, q a, q f }. - ψ s q 0, ψ s q 1, ψ s q f, ψ s q a, ψ f q a. Consider the 2-ranked {w, b}-tree T given below. We give the arguments of goal predicate ψ assign of Π eval ψ computed on input database {D ψ, D T }. v 1 v 2 v 3 v 4 v 5 v 6 v 7 ψ assign : (v 1, 00,v 2, 10,v 3, 00,v 5, 00,v 4, 01,v 6, 00,v 7, 00), (v 1, 00,v 2, 00,v 3, 00,v 5, 00,v 4, 10,v 6, 00,v 7, 01). sat assign(ψ, T) :({v 2 }, {v 4 }), ({v 4 }, {v 7 }). Proposition 9 For every unary MSO formula φ(x) we can construct a monadic Datalog program Π eval φ with only two strata and with goal predicate φ assign such that for every r-ranked tree T, T = φ(n) iff φ assign(n) is computed by program Π eval φ on input database {D φ, D T }. Proposition 10 Let D pre T be an enriched, with the pre-order node ordering information, tree database. For every MSO formula φ(x 1,...,X k ) we can construct a monadic Datalog program Π mon eval φ with goal predicates φ assign i ρ (ρ successful runs of Assign φ ) such that for every ranked tree T, T = φ(b 1,...,B k ) iff B i = {n Π mon eval φ {Dφ, D pre T } computes φ assign i ρ(n)}, i =1,...,k. Corollary of the above result is that for formulas with just first-order free variables, a Datalog program Π can be constructed such that T = φ(n 1,...,n k )iff φ assign(n 1,...,n k ) is computed by Π on input databases {D pre φ, D T }. 4 Unranked Trees In order to handle MSO queries over unranked trees (recall Sect. 2), we map unranked trees into binary trees of a special kind and readapt Definition 5. Definition 11 (Special Binary Tree T bin, Program Πφ bin ) A special bin- ary colored tree, denoted T bin,isa2-ranked colored tree where an inner node is allowed to have a second child although not having a first one. The corresponding tree database D T bin, is a database of signature (leaf, root, (child i ) i=1,2, (child i ) i=1,2, (color γ ) γ Γ );predicateschild 1, child 2 correspond to unique firstchild and unique second-child relations respectively. Program Πφ bin (producing database Dφ bin ) is derived by a slight modification of program Πφ 2 (Π2 φ denotes program Π φ of Definition 5 for 2-ranked colored trees). More precisely, φ-predicates of Πφ bin have form φ [q] γ 0, φ [q, q 1, q 2 ] γ 2 (as in Πφ 2)andalsoφ [q, q ] γ i, i =1, 2 (instead of φ [q, q 1 ] γ 1 in the Πφ 2 case). Furthermore facts corresponding to predicate symbols of the form φ [q] γ 0 and φ [q, q 1, q 2 ] γ 2 are the same for both Π bin φ and Π 2 φ.

8 MSO Querying over Trees via Datalog Evaluation 75 In particular, when φ is an atomic formula, for facts of Πφ bin corresponding to predicate symbols of the form φ [q, q i ] γ i, i =1, 2, the following hold: φ [q, q ] γ i (e) Πφ bin, i =1, 2, whenever φ [q, q ] γ 1 (e) Πφ 2, with the only exception when φ = Sj (X, Y ), j = 1, 2, q =q 1 and e = (1, 0). Recall that for this case we had φ [q a, q 1 ] γ 1 (1, 0) ΠS 2 and φ [q 1 f, q 1 ] γ 1 (1, 0) ΠS 2 ; 2 here we have φ [q a, q 1 ] γ 1 (1, 0),φ [q f, q 1 ] γ 2 (1, 0) ΠS bin and φ [q f, q 1 1 ] γ 1 (1, 0), φ [q a, q 1 ] γ 2 (1, 0) ΠS bin. 2 Lemma 12 Given an unranked colored tree database D T u, there exists a Datalog program Π u bin computing in linear time a special binary colored tree database D enc(t u ) such that the special binary tree enc(t u ) codifies the unranked tree T u. The basic idea of the above encoding is the interpretation of first-child, nextsibling relations as first- and second-child relations respectively. Example 13 Consider the unranked {w, b}-tree T u with seven nodes having database D T u={root(n 1 )} {leaf(n i ) i = 4, 5, 6, 7} {first child(n 1,n 2 ), first child(n 2,n 5 ), first child(n 3,n 7 )} {next sibling(n i,n i+1 ) i =2, 3, 5} {last sibling(n i ) i =4, 6, 7} Color T u,wherecolor T u={color w (n i ) i =1, 2, 3, 5} {color b (n i ) i =4, 6, 7}.ThenD enc(t u )={root(n 1 )} {leaf(n i ) i =4, 6, 7} {child1 (n 1,n 2 ), child 1 (n 2,n 5 ), child 1 (n 3,n 7 )} {child 2 (n i,n i+1 ) i =2, 3, 5} {child 1 (n 1,n 2 )} {child 2 (n 5,n 6 ) Color T u. T u n 1 enc(t u ) n 1 n 2 n 3 n 4 n 5 n 6 n 7 n 2 n 5 n 3 n 6 n 7 n 4 Proposition 14 For every MSO[τ Γ ] formula φ(x) and for every unranked colored tree T u of size m, a Datalog program Πeval u φ can be constructed, such that T u = φ(b) iff φ assign(n 1, ɛ 1,...,n m, ɛ m ) is computed by Πeval u φ on input database {Dφ, D T u} for an appropriate database D φ ; n 1 ɛ 1,...,n m ɛ m is the mapping codifying B. Proof. We translate MSO[τ Γ ]formulaφ(x)intoanmso[τ Γ,2 ]formulaencφ(x) such that T u = φ(b) iffenc(t u ) = encφ(b). Obviously such a formula is obtained from φ by replacing each occurrence of the form Sfirst (X, Y )bys 1 (X, Y ), Next (X, Y )bys2 (X, Y ), Last (X) bys 2 (X) = YS 2 (X, Y ). Let D φ = Denc bin φ and Πu eval φ =Π u bin Π eval enc φ,whereπeval enc φ is analogous to Datalog program of Proposition 7 for 2-ranked trees; then the requirements of the proposition hold.

9 76 Eugénie Foustoucos and Labrini Kalantzi Example 15 Consider the MSO[τ {w,b} ] formula φ(x, Y )=Pw (X) Next (X, Y ); enc φ(x, Y )=Pw(X) S2 (X, Y ). Formulaenc φ(x, Y ) isthesameasformula ψ(x, Y ) considered in Example 8; database Dψ bin 1 is derived from Dψ 1 by replacing 1) facts ψ [q f, q 1 ] γ 1 (10), γ {w, b}, byψ [q a, q 1 ] w 2 (10), ψ [q f, q 1 ] b 2 (10) and 2) all other facts of the form ψ [q, q ] γ 1 (e), byψ [q, q ] γ i (e), i =1, 2. Consider also the unranked {w, b}-tree T u of Example 13. Evaluating φ(x, Y ) over T u reduces to evaluating ψ(x, Y ) over enc(t u ).Wegivebelowthearguments of goal predicate ψ assign facts computed by Πeval ψ on input database {D enc(t u ), D bin } and the satisfying assignments they codify ψ ψ assign sat assign(ψ, enc(t u )) (n 1, 00,n 2, 00,n 5, 10,n 6, 01,n 3, 00,n 7, 00,n 4, 00) ({n 5 }, {n 6 }) (n 1, 00,n 2, 00,n 5, 00,n 6, 00,n 3, 10,n 7, 00,n 4, 01) ({n 3 }, {n 4 }) (n 1, 00,n 2, 10,n 5, 00,n 6, 00,n 3, 01,n 7, 00,n 4, 00) ({n 2 }, {n 3 }) Note the similarity in the structure of the 2-ranked tree T of Example 8 and the special binary tree enc(t u ). Identifying node sequence v 1,v 2,v 3,v 5,v 4,v 6,v 7 of T by node sequence n 1,n 2,n 5,n 6,n 3,n 7,n 4 of enc(t u ), observe that instead of child 1 (v 3,v 5 ) (first-child) we now have child 2 (n 5,n 6 ) (unique second-child) and thus one more satisfying assignment. Propositions analogous to Proposition 9 and Proposition 10 also hold for unranked trees. References 1. Abiteboul, S., Hull, R., Vianu V.: Foundations of Databases. Addison-Wesley, Doner, J.E.: Tree Acceptors and some of their Applications. Journal of Computer and System Sciences 4 (1970) Foustoucos, E., Kalantzi, L.: The MSO-evaluation problem on colored binary trees: a database-theoretic approach (2005, submitted) 4. Ebbinhaus, H.D., Flum, J.: Finite Model Theory. Springer-Verlag, 2nd edition, Flum, J., Frick, M., Grohe, M.: Query Evaluation via tree decompositions. Journal of the ACM 49 (2002), no. 6, Frick, M., Grohe, M., Koch, C.: Query Evaluation on Compressed Trees. Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science LICS (2003, to appear) 7. Gottlob, G., Koch, C.: Monadic Datalog and the Expressive Power of Web Information Extraction Languages. Proceedings of PODS (2002), also Journal of the ACM 51 (2004), no. 1, Grohe, M., Schweikardt, N.: Comparing the succinctness of monadic query languages over finite trees. CLS (2003)

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