Weighted Monadic Datalog Tree Transducers

Transcription

1 Weighted Monadic Datalog Tree Transducers Daniel Geisler, Torsten Stüber Faculty of Computer Science, Technische Universität Dresden D Dresden, Germany December 16, 2010 We introduce a weighted tree transducer model that combines aspects of weighted top down tree transducers and monadic datalog; this allows us to specify information transport for non-adjacent nodes in a single rule. We will investigate four different semantics for our weighted tree transducer model and prove that they coincide under certain conditions. We present and prove the correctness of an algorithm which determines an approximate solution to the best translation problem of our tree transducer model and which determines a precise solution to the n-best hyperpaths problem of a given hypergraph. Finally, we show that our transducer model is at least as expressive as weighted top-down tree transducers. 1 Introduction In the database community there is an increasing interest in monadic query languages over trees [17]. These are formalisms for selecting a subset of nodes of a given tree. Since data on the web are usually exchanged by means of tree-like structures, monadic query languages are well-suited to extract data satisfying a given property. Commonly, the tree-like structure is represented by XML. An XML document can be considered as a labeled ordered tree, where the number of subtrees of a node is arbitrary; this is called an unranked tree [22, 1]. It is common to encode unranked using ranked trees [33, 35]; for this reason we will focus on ranked trees. The work of the authors was partially supported by Deutsche Forschungsgemeinschaft, project DFG VO 1011/4-1 Corresponding author 1

2 A common approach for processing tree-structured data amounts to using logicbased query languages, like monadic second order logic (for short: MSOL) and monadic datalog [18]. Both of these formalisms have the same expressive power [18]. The result of a query allows us to determine whether a certain node meets a given condition. Unfortunately, Boolean valued logics are weak in the sense that they do not allow for extracting quantitative information from trees. A more advanced quantitative approach has been introduced by Bistarelli et al. [4], where weights taken from a semiring are allowed to occur in Horn clauses. This idea was adapted in [37] in order to introduce a quantitative extension of monadic datalog, called weighted monadic datalog. They showed that the expressiveness of the extension is the same as of weighted automata on unranked trees. For the purpose of exchanging and processing data it is essential to determine parts of the input tree that satisfy certain structural properties and to transform their structure. Hence, querying is strongly related to the transformation of tree-like structured data. In practice these transformations are specified by transformation languages, like XSLT and XQuery. Such transformation processes can formally be specified by tree transducers, which is a formalism for transforming a given tree into a tree language. In [6] the concept of monadic datalog tree transducers has been introduced, which form a tree transducer model combining aspects of attributed tree transducers and monadic datalog. In their paper the authors showed that their approach is strictly more powerful than attributed tree transducers. In this paper we introduce a quantitative tree transducer model called weighted monadic datalog tree transducers which we obtain by joining the concepts of weighted monadic datalog and monadic datalog tree transducers. Let us sketch the underlying idea. A set of user-defined predicates is given by the programmer. By using these predicates, which are either nullary or unary, the programmer specifies a finite set of rules, where each rule is associated with a quantity of a semiring. Consider the following example of such a rule: p(x) 0.8 σ(p(x 1 ), p(x 2 )); {child 1 (x, x 1 ), child 2 (x, x 2 )}. This rule is weighted with the real value 0.8. The symbols x, x 1 and x 2 are variables representing nodes in the input tree. Intuitively, this rule means that the output tree of the predicate p at node x is be obtained by putting a σ on top of the output trees of p at nodes x 1 and x 2 ; where the weight of the resulting tree is the product of the weights of the trees of p at nodes x 1 and x 2 and the semiring value 0.8. However, these nodes are not chosen arbitrarily, because there are constraints given by the set on the right-hand side of the rule. It is a subset of atoms of structural predicates, which refer to the structure of the input tree. In this example the nodes x 1 and x 2 have to be the first and the second child of x, respectively. The semantics of a weighted monadic datalog tree transducer (for short: wmdtt) is a weighted tree transformation, i.e., a mapping from T Σ T to S, where S is a semiring, and T Σ and T are the sets of input and output trees, respectively. To define the weighted tree transformation of wmdtt we pursue two basic approaches. The first one uses the concept of the least fixpoint. Depending on the circularity of the wmdtt 2

3 and the properties of the underlying semiring, we distinguish between (1a) the fixpoint semantics for non-circular wmdtt and (1b) for wmdtt based on ω-continuous semirings. The second approach uses a generalization of finite directed graphs called hypergraphs. Again we distinguish between (2a) the hypergraph semantic for non-circular wmdtt and (2b) for wmdtt based on ω-complete semirings. In this paper we prove three main results. In the first main result we identify conditions that guarantee that semantics (1a), (1b), (2a), and (2b) coincide (see Theorem 4.1). We will show that these four versions of semantics are equal; the only exception is the relationship between semantics (1b) and (2b). These two semantics are shown to be equivalent whenever the employed ω-continuous semiring and the ω-complete semiring are related in some way or when the considered wmdtt is non-circular. The second main result is concerned with the best translation problem: given an order on the semiring, a wmdtt M, and an input tree, what is the highest-valued output tree of the weighted tree transformation that is induced by M? We will discuss the relationship between the best translation problem and the n-best derivations problem [21, 5]. The latter problem is concerned with determining the n highest-valued derivations in a weighted hypergraph. We will present an algorithm that solves the n- best derivations problem and that gives an approximate solution to the best translation problem; moreover, we will prove the correctness of this algorithm (see Theorem 5.11). We will denote our algorithm in a functional programming style. Other algorithms that solve the n-best derivations problem have been proposed in [21, 5]; however, to our knowledge the present paper is the first exposition of a rigorous correctness proof of such an algorithm. The third main result states that every weighted tree transformation that is definable by a polynomial top-down weighted tree transducer [26, 9, 13, 15] (for short: td-wtt) is also definable by a wmdtt (see Theorem 6.5). This implies that wmdtt are at least as expressive as td-wtt. The latter claim is of particular interest in the area of natural language processing, in particular, statistical machine translation [24, 23, 40, 19]. This paper is organized as follows. In Section 2 we recall basic concepts that we need throughout later sections. In Section 3 we will introduce the syntax and all versions of the semantics of wmdtt. In Section 4 we will establish a connection between the fixpoint and the hypergraph semantics. In Section 5 we will give an approximate solution of the best translation problem of wmdtt. Finally in Section 6 we will study the class of weighted tree transformations induced by top down weighted tree transducer and show that it is contained in the class of weighted tree transformations that are definable by wmdtt. 2 Preliminaries Let A, B, C, and D be sets. By P(A) we denote the power set of A, and by P fin (A) we denote the set of all finite subsets of A. The cardinality of A is denoted by A. Moreover, we denote the empty set by. We will denote the set of non-negative integers by N and the set of positive integers by N +. For every n N we abbreviate 3

4 the set {m N + m n} by [n]; observe that [0] =. Let (A, ) be a poset. A mapping c : N A is called an ω-chain (wrt ) if for every n, n N with n n we have c(n) c(n ). In this context we omit the parentheses and simply write c n instead of c(n) for every n N. The poset (A, ) is called ω- complete (for short: ω-cpo) if A has a least element and every ω-chain wrt has a supremum. If B A is non-empty and every pair of elements in B has an upper bound in B, then B is called a directed set (wrt ). Observe that, whenever (A, ) is an ω-cpo, then every countable directed subset of A has a supremum [31]. Let (A, ) be an ω-cpo and k N. We will denote the supremum of a set A A by A. An operation f : A k A is said to be ω-continuous if for every a 1,..., a k A, i [k], and ω-chain c: N A we have f ( a 1,..., a i 1, {c n n N}, a i+1,...,a k ) = {f(a 1,..., a i 1, c n a i+1,..., a k ) n N}. Strings and Trees. Let A be a set. By A we denote the set of strings over A. We denote the empty string by ε and for every string w A we denote by w the length of w. A ranked alphabet is an alphabet Σ together with a mapping rk: Σ N. For every k N, we denote the set {σ Σ rk(σ) = k} by Σ (k), and a symbol σ Σ (k) is sometimes denoted by σ (k). We say that Σ is monadic if Σ = Σ (0) Σ (1). For the remainder of this paper Σ and will denote arbitrary ranked alphabets if not stated otherwise. Let Σ be a ranked alphabet and H be a set. The set of trees (over Σ indexed by H), denoted by T Σ (H), is the smallest set T such that (i) H T and (ii) for every k N, σ Σ (k), and t 1,..., t k T also σ(t 1,...,t k ) T. We abbreviate T Σ ( ) by T Σ. If no confusion arises, we will denote the tree α() by α, for every α Σ (0). We define the mapping pos: T Σ (H) P(N ) recursively as follows: (i) for every h H we let pos(h) = {ε}, and (ii) for every k N, σ Σ (k), and t 1,..., t k T Σ (H) we let pos(σ(t 1,..., t k )) = {ε} {iw i [k], w pos(t i )}. Let t T Σ (H). We call pos(t) the set of positions in t. The size of t and the height of t are defined by size(t) = pos(t) and height(t) = max{ w w pos(t)}, respectively. For every w pos(t) we define the label t(w) Σ H of t at position w and the subtree t w T Σ (H) of t at position w as usual. The index-yield of t, denoted by iyield(t) H, is recursively defined as follows: (i) if t H, then we let iyield(t) = t, and (ii) if t = σ(t 1,...,t k ) for some k N, σ Σ (k), and t 1,...,t k T Σ (H), then iyield(t) = iyield(t 1 )... iyield(t k ). For every k N, σ Σ (k), and T 1,..., T k T Σ (H) the σ-top concatenation of T 1,...,T k, denoted by σ(t 1,...,T k ) T Σ (H), is defined as follows: σ(t 1,..., T k ) ={σ(t 1,..., t k ) t 1 T 1,..., t k T k }. 4

5 Semirings. A semiring (S, +,, 0, 1) consists of a commutative monoid (S, +, 0) and a monoid (S,, 1) such that, for every a, b, c S, a (b+c) = a b+a c, (a+b) c = a c+b c, and a 0 = 0 = 0 a. As usual, we identify the semiring (S, +,, 0, 1) with the set S. For the remainder of this paper (S, +,, 0, 1) will denote an arbitrary semiring if not stated otherwise. An ω-continuous semiring is a tuple (S, ) such that (i) (S, ) is an ω-cpo with least element 0 and (ii) + and are ω-continuous operations wrt. The following observation is easy to prove. Observation 2.1 Let (S, ) be an ω-continuous semiring. Moreover, let I be a finite index set, k N, and (c i i I) and (d i i [k]) be families of ω-chains wrt. Then {c i n n N} = { i I i I ci n n N }, {d 1 n n N}... {d k n n N} = ( {d 1 n... d k n n N}). An ω-infinitary sum operation for S associates with every countable index set I and family (a i i I) over S an element i I a i such that the following properties are satisfied for every family (b i i {j, k}) over S, all countable sets I, J, every partition (I j j J) of I, every family (a i i I) over S, and every c S: i b i = 0, i I a i = j I c i I a i = i I c a i, i {j} b i = b j, i I j a i, i {j,k} b i = b j + b k, ( i I a ) i c = i I a i c. A tuple (S, ) is called an ω-complete semiring if is an ω-infinitary sum operation for S. Weighted tree languages, weighted tree transformations Let D be a set. A (formal) series (over S and D) is a mapping ψ : D S. Let d D. The support of ψ, denoted by supp(ψ), is defined by supp(ψ) = {d D ψ(d) 0}. Moreover, ψ is called a polynomial series if the set supp(ψ) is finite. The set of all series and the set of all polynomial series over S and D are denoted by S D and S D, respectively. The series ϕ S D with supp(ϕ) = is the empty series and is denoted by 0. For two series ϕ, ψ S D we define the sum ϕ + ψ S D pointwise, i.e., for every d D, (ϕ + ψ)(d) = ϕ(d) + ψ(d). Moreover, for every s S and every ψ S D we define s ψ S D by letting (s ψ)(d) = s ψ(d) for every d D. Let I be an index set. We call a family (ϕ i i I) over S D locally finite if for every d D there are only finitely many i I such that d supp(ϕ i ). We lift the sum of series to locally finite families as follows. If the family (ϕ i i I) over S D is locally finite, then we denote by i I ϕ i S D the series such that for every d D: ( ) (d) = i I ϕ i i I d supp(ϕ i) ϕ i (d). 5

6 Let s S and d D. The monomial series (over s and d) is the series s.d S D such that (s.d)(d) = s and (s.d)(d ) = 0 for every d D different from d. We will refer to the elements of S T and S T Σ T as weighted tree languages (over ) and weighted tree transformations (over Σ and ), respectively. Let k N, δ (k), and ψ 1,...,ψ k S T. The δ-top concatenation of weighted tree languages ψ 1,..., ψ k, denoted by top δ (ψ 1,...,ψ k ) S T, is defined as follows: top δ (ψ 1,..., ψ k ) = t 1,...,t k T (ψ 1 (t 1 )... ψ k (t k )).δ(t 1,..., t k ). The sum is well-defined because ( ) (ψ 1 (t 1 )... ψ k (t k )).δ(t 1,..., t k ) t 1,..., t k T is a locally finite family. Observe that top α () = 1.α for every α (0). Let H be a set and s T (H). Let s T such that iyield(s) = s. The tree substitution of s into s is the tree s s in T defined by recursion as follows: if s H, then iyield(s) = 1 (i.e., s T ) and we let s s = s, if s = δ(s 1,...,s k ) for some k N, δ (k), and s 1,...,s k T (H), then there are unique s 1,..., s k T with s 1 s k = s and iyield(s i ) = s i for every i [k]; we let s s = δ(s 1 s 1,..., s k s k ). Next we define the OI substitution [10, 11] of tree series into a tree. Let I : H S T. The OI substitution of I into s, denoted by s OI I S T, is defined recursively as follows: (i) if s H, then s OI I = I(s), and (ii) if s is of the form δ(s 1,...,s j ), then s I OI OI OI = top δ (s 1 I,..., s j I). We lift the OI substitution to S T (H) such that ϕ I OI = OI s T (H) ϕ(s) (s I) for every ϕ S T (H). As an example of tree substitution and OI substitution consider the ranked alphabet = {α (0), γ (1), σ (2) }, the set H = {x, y}, and the trees s = σ(σ(α, y), x), s 1 = α, and s 2 = γ(α). Then s s 1 s 2 = σ(σ(α, α), γ(α)). Moreover, let S = (N, +,, 0, 1) be the semiring of natural numbers and I : H S T be defined by I(x) = 2.s 1 +3.s 2 and I(y) = 5.s s 2. Then s I OI = top σ (top σ (top α, I(y)), I(x)) = top σ (top σ (1.α, 5.α + 7.γ(α)), 2.α + 3.γ(α)) = top σ (5.σ(α, α) + 7.σ(α, γ(α)), 2.α + 3.γ(α)) = 10.σ(σ(α, α), α) + 15.σ(σ(α, α), γ(α)) + 14.σ(σ(α, γ(α)), α) + 21.σ(σ(α, γ(α)), γ(α)). Observation 2.2 Let I : A S T and ϕ S T (A). Then ϕ I OI S T. 3 Weighted monadic datalog tree transducers In this section we introduce the main device of this paper. First we will define the syntax of weighted monadic datalog tree transducers (for short: wmdtt). The syntactic structure of wmdtt is similar to the syntax of monadic datalog tree transducers [6], 6

7 which are based on weighted monadic datalog [37]. Weighted monadic datalog is a combination of the concepts of monadic datalog [17, 18] and semiring-based constraint logic programming [4]. We will define two different types of semantics, which we call fixpoint semantics and hypergraph semantics. The fixpoint semantics is reminiscent of the initial algebra semantics of bottom-up weighted tree automata [3, 16], whereas the hypergraph semantics is related to the run semantics of weighted tree automata (or similar concepts such as m-weighted tree automata [29, 38]). The fixpoint semantics is inspired by the definition of the semantics of monadic datalog tree transducers [6], weighted monadic datalog [37], and monadic datalog [17, 18]. The concept of the hypergraph semantics is novel. Roughly speaking, each type of semantics of wmdtt takes three inputs: an wmdtt, an input tree, and an output tree. The semantics are defined in such a way that they evaluate the input tree according to the wmdtt by applying operations from the semiring, and afterwards select the component of the resulting tree series that belongs to the output tree. Thus, when keeping the wmdtt fixed, the semantics are mappings from pairs of input and ouput trees to semiring elements. It turns out that there are wmdtt whose semantics cannot be evaluated for arbitrary semirings. Such wmdtt exhibit circular behavior when computing their semantics; as a consequence they require an ω-continuous semiring (for the fixpoint semantics) or an ω-complete semiring (for the hypergraph semantics) for the evalution of the semantics; moreover, we need to develop alternate variants of the fixpoint and the hypergraph semantics that can employ the strength of ω-continuous and ω-complete semirings, respectively. Hence, we will define, study, and compare four different variants of semantics in this section. Throughout this work we fix an infinite set V, the elements of which being called variables. In the sequel let maxrk(σ) denote the maximal rank of symbols in Σ. We define the ranked alphabet sp Σ as follows: sp Σ = {root (1), leaf (1) } {label (1) σ σ Σ} {child (2) i i [maxrk(σ)]}. We refer to the elements of sp Σ as structural predicates over Σ. Before we define the syntax of our main device, let us introduce an auxiliary notion. For every ranked alphabet Γ and set H let Γ(H) = {γ(h 1,..., h k ) k N, γ Γ (k), h 1,...,h k H}. Definition 3.1 A triple (P, R, p) is called a weighted monadic datalog tree transducer (for short: wmdtt) over Σ,, and S if P is a monadic ranked alphabet (of user-defined predicates), R P(V) T (P(V)) P fin (sp Σ (V)) S is a finite set (of weighted rules), p P (1) (called query predicate). A rule r = (h, b, G, w ) R is denoted by h w b ; G and h, b, G, and w are called head, body, guard, and weight of r. For every r R we refer to the head, body, guard, and weight of r by r h, r b, r G, and r w, respectively. The elements in P(V) sp Σ (V) are called atoms. 7

8 Example 3.2 Let Σ = {α (0), β (0), σ (2) } be a ranked alphabet and S = (R [0,1], max,, 0, 1) be the Viterbi-semiring [8] over the real numbers from 0 to 1. Consider the wmdtt M nc = ({p (1) }, {r 1, r 2, r 3, r 4 }, p) over Σ, Σ, and S, where r 1 = p(x) 1 α; {label α (x)}, r 2 = p(x) 1 β; {label β (x)}, r 3 = p(x) 0.8 σ(p(x 1 ), p(x 2 )); {child 1 (x, x 1 ), child 2 (x, x 2 )}, r 4 = p(x) 0.2 σ(p(x 2 ), p(x 1 )); {child 1 (x, x 1 ), child 2 (x, x 2 )}. In addition, we define the wmdtt M c over Σ, Σ, and R [0,1], that is obtained from M nc by adding the following rule: r c = p(x) 0.1 p(x 1 ); {label β (x), root(x 1 )}. Intuitively speaking, this rule allows jumping to the root of the input tree if a β-labeled position is reached. In the sequel let t T Σ and M = (P, R, p) be a wmdtt over Σ,, and S. We refer to the elements in P(pos(t)) sp Σ (pos(t)) as atom instances (over t). The tree t constitutes the set B t sp Σ (pos(t)) of atom instances compatible with t defined by B t = {root(ε)} {leaf(w) w pos(t), t(w) Σ (0) } {child i (w, wi) w pos(t), i [rk(t(w))]} {label σ (w) w pos(t), t(w) = σ}. Let r R and let var(r) denote the set of variables occurring in r h, r b, and r G. An r, t-variable assignment is a mapping ρ : var(r) pos(t). We lift the mapping ρ to T ( P(var(r)) ) and P ( spσ (var(r)) ) in an obvious manner; for example, ρ(v 1 ) = 1 and ρ(v 2 ) = ε yield ρ ( δ(p(v 1 ), q(v 2 )) ) = δ(p(1), q(ε)) and ρ ( {root(v 1 ), child 2 (v 2, v 1 )} ) = {root(1), child 2 (ε, 1)}. We call ρ valid if ρ(r G ) B t. By Φ M,t we denote the set of all pairs (r, ρ) such that r R and ρ is a valid r, t-variable assignment. For every a P(pos(t)) we denote by Φ M,t,a the set {(r, ρ) Φ M,t ρ(r b ) = a}. If M is understood, we omit the reference to M and just write Φ t and Φ t,a. Example 3.3 (Cont. of Example 3.2) Let t = σ(α, σ(α, β)). Then set of atom instances compatible with t is given as follows: B t = {root(ε)} {leaf(1), leaf(21), leaf(22)} {child 1 (ε, 1), child 2 (ε, 2), child 1 (2, 21), child 2 (2, 22)} {label α (1), label α (21), label β (22), label σ (ε), label σ (2)}. Let ρ 1, ρ 21, ρ 22 : {x} pos(t), and ρ ε, ρ 2 : {x, x 1, x 2 } pos(t) such that ρ ε (x) = ε ρ 1 (x) = ρ ε (x 1 ) = 1 ρ 2 (x) = ρ ε (x 2 ) = 2 Then the following facts hold: ρ 21 (x) = ρ 2 (x 1 ) = 21 ρ 22 (x) = ρ 2 (x 2 ) = 22 8

9 Φ Mnc,t,p(ε) = {(r 3, ρ ε ), (r 4, ρ ε )}, since label α (ε) / B t and label β (ε) / B t. Φ Mnc,t,p(1) = {(r 1, ρ 1 )}, since label β (1) / B t and α has no children. Φ Mnc,t,p(2) = {(r 3, ρ 2 ), (r 4, ρ 2 )}, since label α (2) / B t and label β (2) / B t. Φ Mnc,t,p(21) = {(r 1, ρ 21 )}, since label β (21) / B t and α has no children. Φ Mnc,t,p(22) = {(r 2, ρ 22 )}, since label α (22) / B t and β has no children. Obviously, for every w pos(t) different from 22 one can see that the sets Φ Mnc,t,p(w) and Φ Mc,t,p(w) coincide. Let ρ c : {x, x 1 } pos(t) such that ρ c (x) = 22 and ρ c (x 1 ) = ε. Observe that Φ Mc,t,p(22) = {(r c, ρ c )} Φ Mnc,t,p(22). The dependency graph D M,t of M and t is the directed graph (P(pos(t)), E) such that for every a, b P(pos(t)) we have (a, b) E iff there is an (r, ρ) Φ t,a such that b occurs in iyield(ρ(r b )). We say that M is non-circular iff there is no s T Σ such that the directed graph D M,s is cyclic; otherwise M is called circular. Example 3.4 (Cont. of Example 3.3) Let us determine the dependency graph of M nc and t. To this end we will refer to the set of edges of D Mnc,t as E nc. First, observe that iyieldρ ε ((r 3 ) b ) = p(1) p(2), iyieldρ ε ((r 4 ) b ) = p(2) p(1), iyieldρ 2 ((r 3 ) b ) = p(21) p(22), iyieldρ 2 ((r 4 ) b ) = p(22) p(21), iyieldρ 1 ((r 1 ) b ) = iyieldρ 21 ((r 1 ) b ) = iyieldρ 22 ((r 2 ) b ) = ε. Thus, E nc contains precisely the edges (p(ε), p(1)), (p(ε), p(2)), (p(2), p(21)), and (p(2), p(22)). Note that there are no edges starting from p(1), p(21), or p(22), since there are no variables in the bodies of r 1 and r 2. Now let us determine the dependency graph D Mc,t = (P(pos(t)), E c ) of M c and t. It is easy to see that E c = E nc { (p(22), p(ε)) }. The dependency graphs D Mnc,t and D Mc,t are shown in Figures 1 and 2, respectively. Clearly, D Mc,t is cyclic, and thus, M c is circular. It is easy to show that, for every s T Σ, the graph D Mc,s does not contain cycles. Thus, M nc is non-circular. 3.1 Fixpoint semantics of wmdtt The fixpoint semantics, which is based on the application of an immediate consequence operator, is inspired by the definition of the semantics for the Horn calculus [28, 20] and monadic datalog [17, 18]. The idea of the fixpoint semantics is as follows. Given a wmdtt and an input tree, we associate every atom instance with a tree series; such an association is called an interpretation. Among all interpretations is one designated interpretation, which is computed by the fixpoint semantics in a stepwise manner. Finally, the semantics 9

10 Figure 1: Dependency graph of M nc and t Figure 2: Dependency graph of M c and t returns the element of the semiring that is associated with the output tree and the atom instance p(ε) by the designated interpretation, i.e., the element that the designated interpretation associates with the query predicate at the root of the input tree. The computation of the designated interpretation is based on the immediate consequence operator; this is a mapping from the set of interpretations into itself; it takes an interpretation I and returns an interpretation, called the consequence interpretation of I. The computation of the consequence interpretation is guided by the rule instances. Let t T Σ and M = (P, R, p) be a wmdtt over Σ,, and S. An interpretation over M and t is a mapping I : P(pos(t)) S T. The empty interpretation, denoted by I 0, is defined by I 0 (a) = 0 for every a P(pos(t)). The set of all interpretations over M and t is denoted by I M,t. The immediate consequence operator T M,t : I M,t I M,t over M and t is defined for every I I M,t and every a P(pos(t)) as follows: T M,t (I)(a) = r w (ρ(r b ) I). OI (r,ρ) Φ t,a If M and t are clear from the context, we simply write I and T instead of I M,t and T M,t, respectively. For every n N we denote by T n the n-fold composition of T The non-circular case Now let us define the fixpoint semantics formally. For arbitrary semirings the fixpoint semantics is only defined if the considered wmdtt is non-circular. Let t T Σ and M = (P, R, p) be a non-circular wmdtt over Σ,, and S. Lemma 3.5 Let I, J I and n = P(pos(t)). Then T n (I) = T n (J). Proof. Let D = ( P(pos(t)), E ) be the dependency graph of M and t. Since M is non-circular, D is acyclic. Moreover, for every a P(pos(t)) let n a be the length of the longest path in D starting in a; this number is well-defined because D is acyclic. For every a P(pos(t)) and I, J I, we claim that T m (I)(a) = T m (J)(a) for every m N with m > n a. We prove this claim by induction on n a, i.e., we let a P(pos(t)) and assume that we have already proved the claim for every atom instance b P(pos(t)) with n b < n a. Thus, for every (r, ρ) Φ t,a and b occurs in iyield(ρ(r b )) we have 10

11 n b < n a and, hence, T m 1 (I)(b) = T m 1 (J)(b). This yields that ρ(r b ) OI T m 1 (I) is equal to ρ(r b ) OI T m 1 (J). Therefore T (T m 1 (I))(a) = T (T m 1 (J))(a). Hence, T m (I)(a) = T m (J)(a). Lemma 3.6 For every I I, the interpretation T P(pos(t)) (I) is the unique fixpoint of T. Proof. Let n = P(pos(t)). By Lemma 3.5, T (T n (I)) = T n (T (I)) = T n (I). Let J be a fixpoint of T. Then J = T n (J) = T n (I) by Lemma 3.5. Definition 3.7 The weighted tree transformation fix-defined by M, denoted by [M ] fix S T Σ T, is defined for every (t, u) T Σ T by [M ] fix (t, u) = T P(pos(t)) M,t (I 0 )(p(ε))(u). We say that a weighted tree transformation τ S T Σ T is definable by non-circular wmdtt if there is a non-circular wmdtt M over Σ,, and S such that [M ] fix = τ. By nc WMDTT(Σ,, S) we denote the class of weighted tree transformations τ S T Σ T that are definable by non-circular wmdtt. Example 3.8 (Cont. of Example 3.4) Now we consider the weighted tree transformation fix-defined by M nc. As an example we will determine the value [M nc ] fix (t, u), where t = σ(α, σ(α, β)) and u = σ(α, σ(β, α))). To this end we abbreviate T n (I 0 ) by I n, for every n N. Since pos(t) = 5, we need to determine the value of I 5 (p(ε))(u). Since Φ p(ε) = {(r 3, ρ ε ), (r 4, ρ ε )}, we obtain I 5 (p(ε)) = r w (ρ(r b ) I OI 4 ) (r,ρ) {(r 3,ρ ε),(r 4,ρ ε)} = 0.8 (σ(p(1), ) ( ) p(2)) I OI σ(p(2), p(1)) OI I 4 = 0.8 (top σ (I 4 (p(1)), I 4 (p(2))) ) (top σ (I 4 (p(2)), I 4 (p(1))) ) = 0.8 top σ ( 1.α, 0.8.σ(α, β) σ(β, α) ) + (by Subtasks A and B) 0.2 top σ ( 0.8.σ(α, β) σ(β, α), 1.α ) = 0.8 (0.8.σ(α, σ(α, β)) σ(α, σ(β, α))) (0.8.σ(σ(α, β), α) σ(σ(β, α), α)) = 0.64.σ(α, σ(α, β)) σ(α, σ(β, α))) σ(σ(α, β), α) σ(σ(β, α), α)). Subtask A. We evaluate I 4 (p(1)). Since Φ p(1) = {(r 1, ρ 1 )}, we obtain I 4 (p(1)) = r w (ρ(r b ) I OI 3 ) = 1 (α I OI 3 ) = top (r,ρ) {(r α () = 1.α. 1,ρ 1)} Subtask B. We evaluate I 4 (p(2)). Since Φ p(2) = {(r 3, ρ 2 ), (r 4, ρ 2 )}, we obtain I 2 (p(2)) = 0.8 (σ(p(21), ) ( ) p(22)) I OI σ(p(22), p(21)) OI I 3 = 0.8 (top σ (I 3 (p(21)), I 3 (p(22))) ) (top σ (I 3 (p(22)), I 3 (p(21))) ) = ( 0.8 top σ (1.α, 1.β) ) + ( 0.2 top σ (1.β, 1.α) ) (by Subtasks C and D) = 0.8.σ(α, β) σ(β, α). 11

12 Subtask C. We evaluate I 3 (p(21)). Since Φ p(21) = {(r 1, ρ 21 )}, we obtain I 3 (p(21)) = r w (ρ(r b ) I OI 2 ) = 1 (α I OI 0 ) = top α () = 1.α. (r,ρ) {(r 1,ρ 21)} Subtask D. We evaluate I 3 (p(22)). Since Φ p(22) = {(r 2, ρ 22 )}, we obtain I 3 (p(22)) = r w (ρ(r b ) I OI 2 ) = 1 (β I OI 0 ) = top β () = 1.β. (r,ρ) {(r 2,ρ 22)} Thus, [M nc ] fix (t, u) = I 5 (p(ε))(u) = Let us briefly give an intuitive explanation of the weighted tree transformation [M nc ] fix. For all trees t, u T Σ, we say that u is a permuted tree of t if u is obtained by flipping the children of none, some, or all σ- labeled nodes in t. For example, the tree permutations of σ(α, β) and σ(α, σ(α, β)) are the elements in the sets T 1 = {σ(α, β), σ(β, α)} and T 2 = {σ(α, t ) t T 1 } {σ(t, α) t T 1 }, respectively. Additionally, we assume that each flipping occurs with the probability 0.2. Then for every t, u T Σ, [M nc ] fix (t, u ) is the probability that t is permuted to u The ω-continuous case If the considered wmdtt is not non-circular, then the fixpoint semantics is defined if the considered semiring is ω-continuous. Now we will define this ω-continuous version of the fixpoint semantics formally. Let (S, ) be an ω-continuous semiring, t T Σ, and M = (P, R, p) be a wmdtt over Σ,, and S. We lift to I pointwise; more precisely, for every I, J I we let I J iff I(a)(u) J(a)(u) for every a P(pos(t)), and u T. Lemma 3.9 The poset (I, ) is an ω-cpo. Moreover, every ω-chain c : N I has a supremum I I such that for every a P(pos(t)) and u T we have I(a)(u) = {c n (a)(u) n N}. Proof. Since S is an ω-continuous semiring, 0 is the least element of S. This implies that I 0 is the least element in I. Observe that every ω-chain c : N I has a supremum I with I(a)(u) = {c n (a)(u) n N} for every a P(pos(t)) and u T. Hence, (I, ) is an ω-cpo. Lemma 3.10 Let s T (P(pos(t))) and c : N I be an ω-cpo. Then for every u T we have (s OI {c n n N})(u) = {(s c OI n )(u) n N}. Proof. We give a proof by induction on s. Assume that s P(pos(t)) and let u T. Then (s OI {c n n N})(u) = ( {cn n N} ) (s)(u) = {c n (s)(u) n N} = {(s c OI n )(u) n N}. (by Lemma 3.9) Now assume that s is of the form δ(s 1,..., s k ). As induction hypothesis assume for every i [k] and u OI T that (s i {c n n N})(u ) = OI {(s i c n )(u ) n N}. 12

13 Moreover, assume that u is of the form γ(u 1,..., u l ). If γ δ, then the definition of top δ implies (s OI {c n n N})(u) = 0 = {(s OI c n )(u) n N}. Now assume that δ = γ (and, hence, k = l). Then (s OI {c n n N})(u) ( OI = top δ (s1 OI {c n n N}),...,(s k {c n n N}) ) (u) = ( OI s 1 {c n n N} ) (u 1 )... (s OI k {c n n N} ) (u k ) = OI {(s 1 c n )(u 1 ) n N}... {(s OI k c n )(u k ) n N} (by ind. hyp.) = OI OI {(s 1 c n )(u 1 )... (s k c n )(u k ) n N} (by Observation 2.1) = {(s c OI n )(u) n N}. (by definition of top δ ) Lemma 3.11 The immediate consequence operator T is ω-continuous. Proof. Let a P(pos(t)), u T, and c : N I be an ω-chain. It is sufficient to show that T ( {c n n N})(a)(u) = {T (c n ) n N}(a)(u). We obtain T ( {c n n N})(a)(u) = ( r w (ρ(r b ) OI {c n n N} )) (u) (r,ρ) Φ t,a = r w (ρ(r b ) OI {c n n N} ) (u) (r,ρ) Φ t,a = r w {(ρ(r b ) c OI n )(u) n N} (by Lemma 3.10) (r,ρ) Φ t,a = { r w (ρ(r b ) c OI n )(u) n N } (r,ρ) Φ t,a (by Observation 2.1 and since is ω-continuous) = {( ) } r w ρ(r b ) c OI n (u) n N (r,ρ) Φ t,a = {T (c n )(a)(u) n N} = ( {T (cn ) n N} ) (a)(u). (by Lemma 3.9) The following theorem is well-known (see, e.g., [39, Theorem 1]). Theorem 3.12 (Fixpoint Theorem of Knaster and Tarski) Let (A, ) be an ω- cpo and let f : A A be ω-continuous. Then {f n ( ) n N} is the least fixpoint of f, where is least element of A. The following lemma is an immediate consequence from Lemmas 3.11 and 3.9 and Theorem Lemma 3.13 The least fixpoint of T is {T n (I 0 ) n N}. We denote the least fixpoint of T M,t by T ω M,t (or simply T ω if M and t are clear from the context). 13

14 Definition 3.14 The weighted tree transformation fix-defined by M and, denoted by [M ] fix S T Σ T, is defined for every (t, u) T Σ T by [M ] fix (t, u) = TM,t ω (p(ε))(u). 3.2 Hypergraph semantics of wmdtt The idea of the hypergraph semantics is as follows. For a given wmdtt and an input tree we consider the set of derivations ending in p(ε). For every such derivation η we compute a tree tr(η) and a weight wt(η), which is an element of the semring. The weight that the hypergraph semantics assigns to the input tree and a given output tree u is the semiring element that results from summing over the weights of all derivations η with tr(η) = u. Now let us explain how the tree tr(η) and the weight wt(η) of a derivation η is defined. Every derivation is a tree whose labels are rule instances. The weight of η is obtained by multiplying over the weights of all these rule instances. Such a tree over rule instances can be transformed into a tree tr(η) over by, roughly speaking, replacing every label (r, ρ) by the tree ρ(r b ). Now let us define the notions of hypergraphs and derivations. Definition 3.15 A hypergraph is a triple (V, E, λ) such that V is a finite set (of vertices), E is ranked alphabet (of hyperedges), λ: E V V such that for every k N and e E (k) : λ(e) V k V, Let e E, k N, v 1,...,v k, v V such that λ(e) = (v 1 v k, v). We will denote v by out(e) and for every i [k] we denote v i by in i (e). Definition 3.16 Let G = (V, E, λ) be a hypergraph. The set of all derivations of G, denoted by H G, is defined by H G = {η T E w pos(η): i [rk(η(w))]: in i (η(w)) = out(η(wi))}. Let v V and n N. The set of derivations of H ending in v, denoted by HG v, is defined by HG v = {η H G out(η(ε)) = v}. The set of derivations of G ending in v bounded by height n, denoted by HG,n v, is defined by Hv G,n = {η Hv G height(η) < n}. If G is clear from the context, then we omit the reference to G and just write H v and H v n instead of Hv G and Hv G,n. Observation 3.17 Let G = (V, E, λ) be a hypergraph, v V, and n N +. Then H v = k N e E {e(η (k) 1,..., η k ) η 1 H in1(e),..., η k H ink(e) } and out(e)=v Hn v = {e(η 1,..., η k ) η 1 H in1(e) k N n 1,..., η k H in k(e) n 1 }. e E (k) out(e)=v 14

15 Let t T Σ and M = (P, R, p) be a wmdtt over Σ,, and S. Definition 3.18 The hypergraph induced by M and t, denoted by G M,t, is the hypergraph (V, E, λ) that is defined by V = P(pos(t)), E = Φ M,t, where rk((r, ρ)) = iyield(ρ(r b )) for every (r, ρ) E, for every (r, ρ) E, λ((r, ρ)) = (iyield(ρ(r b )), ρ(r h )). In the sequel we will write λ(r, ρ) instead of λ((r, ρ)), for every (r, ρ) E. We define two mappings tr : H GM,t T and wt : H GM,t S by structural recursion as follows. Let k N, (r, ρ) E (k), and η 1,..., η k H GM,t. Then we let tr((r, ρ)(η 1,..., η k )) = r b tr(η 1 ) tr(η k ) and wt((r, ρ)(η 1,..., η k )) = r w wt(η 1 )... wt(η k ). Let u T. If M and t are clear from the context, then we denote by H(u) the set of derivations η H p(ε) such that tr(η) = u. Example 3.19 (Cont. of Example 3.4) The hypergraph G nc = (V nc, E nc, λ nc ) induced by M nc and t, is given as follows: V nc = P(pos(t)), E nc = {(r 3, ρ ε ), (r 4, ρ ε ), (r 3, ρ 2 ), (r 4, ρ 2 ), (r 1, ρ 1 ), (r 1, ρ 21 ), (r 2, ρ 22 )}, λ nc (r 3, ρ ε ) = (p(1)p(2), p(ε)), λ nc (r 4, ρ ε ) = (p(2)p(1), p(ε)), λ nc (r 3, ρ 2 ) = (p(21)p(22), p(2)), λ nc (r 4, ρ 2 ) = (p(22)p(21), p(2)), λ nc (r 1, ρ 1 ) = (ε, p(1)), λ nc (r 1, ρ 21 ) = (ε, p(21)), and λ nc (r 2, ρ 22 ) = (ε, p(22)). The hypergraph G c = (V c, E c, λ c ) induced by M c and t, is given as follows: V c = P(pos(t)), E c = E nc {(r c, ρ c )}, λ c (r c, ρ c ) = (p(ε), p(22)) and λ c (r, ρ) = λ nc (r, ρ) for every (r, ρ) E nc. The hypergraphs G Mnc,t and G Mc,t are depicted by Figures 3 and 4, respectively. For reasons of brevity, we will denote (r i, ρ w ) E by e i,w, for every i [4] and w pos(t). Then H p(ε) G nc = {e 3,ε (e 1,1, e 3,2 (e 1,21, e 2,22 )), e 3,ε (e 1,1, e 4,2 (e 2,22, e 1,21 ))} {e 4,ε (e 3,2 (e 1,21, e 2,22 ), e 1,1 ), e 4,ε (e 4,2 (e 2,22, e 1,21 ), e 1,1 )}, H p(ε) G c = H p(ε) G nc {e 3,ε (e 1,1, e 3,2 (e 1,21, q)) q H p(ε) G c } {e 3,ε (e 1,1, e 4,2 (q, e 1,21 )) q H p(ε) G c } {e 4,ε (e 3,2 (e 1,21, q), e 1,1 ) q H p(ε) G c } 15

16 {e 4,ε (e 4,2 (q, e 1,21 ), e 1,1 ) q H p(ε) G c }. Now let η 1 = e 3,ε (e 1,1, e 4,2 (e 2,22, e 1,21 )) and let us determine tr(η 1 ) and wt(η 1 ): tr(e 3,ε (e 1,1, e 4,2 (e 2,22, e 1,21 ))) = σ(p(x 1 ), p(x 2 )) tr(e 1,1 )tr(e 4,2 (e 2,22, e 1,21 )) = σ ( tr(e 1,1 ), tr(e 4,2 (e 2,22, e 1,21 )) ) = σ ( α ε, σ(p(x 2 ), p(x 1 )) tr(e 2,22 )tr(e 1,21 ) ) = σ ( α, σ(tr(e 2,22 ), tr(e 1,21 )) ) = σ ( α, σ(β ε, α ε) ) = σ(α, σ(β, α)), wt(e 3,ε (e 1,1, e 4,2 (e 2,22, e 1,21 ))) = 0.8 wt(e 1,1 ) wt(e 4,2 (e 2,22, e 1,21 )) = wt(e 2,22 ) wt(e 1,21 ) = = Moreover, let η 2 = e 3,ε (e 1,1, e 3,2 (e 1,21, e 2,22 )), η 3 = e 4,ε (e 3,2 (e 1,21, e 2,22 ), e 1,1 ), and η 4 = e 4,ε (e 4,2 (e 2,22, e 1,21 ), e 1,1 ). Then it is easy to check that tr(η 2 ) = σ(α, σ(α, β)), tr(η 3 ) = σ(σ(α, β), α), tr(η 4 ) = σ(σ(β, α), α), wt(η 2 ) = 0.64, wt(η 3 ) = 0.16, wt(η 4 ) = Figure 3: Hypergraph induced by M nc and t Figure 4: Hypergraph induced by M c and t Lemma 3.20 Let M be non-circular and let G = (V, E, λ) be the hypergraph induced by M and t. Then for every a P(pos(t)) the set HG a is finite. 16

17 Proof. Let D = (P(pos(t)), E ) be the the dependency graph of M and t. We prove this lemma by wellfounded induction on D. More precisely, let a P(pos(t)) and assume that we have already proved the claim for every atom instance b P(pos(t)) with (a, b) E. By Observation 3.17 it is sufficient to show for every k N, i [k], and e E (k) with out(e) = a that the set H ini(e) is finite. Let b = in i (e). By Definition 3.18 we have that b occurs in iyield(ρ(r b )), where (r, ρ) = e. Thus, by definition of D there is an edge (a, b) E. Hence, the induction hypothesis yields that H b is finite. Now let us define the hypergraph semantics formally. For arbitrary semirings the hypergraph semantics is only defined if the considered wmdtt is non-circular. Definition 3.21 Let M be a non-circular. The weighted tree transformation hypdefined by M, which is denoted by [M ] hyp S T Σ T, is defined for every (t, u) T Σ T as follows: [M ] hyp (t, u) = η H(u) wt(η). Note that the weighted tree transformation [M ] hyp is well-defined because the set H(u) H p(ε) G and H p(ε) G is finite due to Lemma Example 3.22 (Cont. of Example 3.19) Let us determine [M nc ] hyp (t, u), where u = σ(α, σ(β, α)). In Example 3.19 we have established that H p(ε) G nc contains four derivations η 1, η 2, η 3, η 4. Clearly, tr(η 1 ) = u and for every other derivation η {η 2, η 3, η 4 } we obtain tr(η) u. Hence, H(u) = {η 1 } and, thus, [M nc ] hyp (t, u) = η H(u) wt(η) = wt(η 1 ) = Recall that in Example 3.8 we showed that [M nc ] fix (t, u) = It is easy to check that for other combination of trees t and u we have [M nc ] hyp (t, u ) = [M nc ] fix (t, u ). Hence, [M nc ] hyp = [M nc ] fix. In Section 4 we will show that the fixpoint and hypergraph semantics coincides for every non-circular wmdtt (see Lemma 4.4). If the considered wmdtt is not non-circular, then the considered semiring must be ω-complete. Definition 3.23 Let (S, ) be an ω-complete semiring. The weighted tree transformation hyp-defined by M and, denoted by [M ] hyp P S TΣ T, is defined for every (t, u) T Σ T as follows: [M ] hyp P (t, u) = η H(u) wt(η). Observe that the definitions of [M ] hyp and [M ] hyp P look the same. However, in the former case the symbol denotes the extension of the additive operation of S to finite families, whereas in the latter case is the ω-infinitary sum operation. Lemma 4.2 (see the following section) states that both definitions are equivalent whenever both of them are applicable. 17

18 4 Relations of the four semantics In this section we will investigate the relationships between the four semantics that we defined in the previous section. In fact, we will show that all applicable semantics coincide; the only exception is the relationship between the fixpoint semantics defined on an ω-continuous semiring (S, ) and the hypergraph semantics defined on an ω- complete semiring (S, ). These two semantics are only guaranteed to be equivalent whenever the considered wmdtt is non-circular or when (S, ) and (S, ) are related in the following sense. We say that (S, ) and (S, ) are related if for every countable set I and family (a i i I) over S: i I a i = { j J a j J P fin (I) }. Observe that { j J a j J P fin (I)} is directed and, thus, has a supremum. In this section we will prove the following theorem. Theorem 4.1 Let Σ and be ranked alphabets, and S be a semiring. Moreover, let M be a wmdtt over Σ,, and S. Assume that M is non-circular. Then [M ] hyp = [M ] fix. If (S, ) is an ω- complete semiring, then [M ] hyp = [M ] hyp P. If (S, ) is an ω-continuous semiring, then [M ] fix = [M ] fix. Let (S, ) be an ω-complete semiring, and (S, ) be an ω-continuous semiring such that (S, ) and (S, ) are related. Then [M ] hyp P = [M ] fix. Theorem 4.1 follows from Lemmas 4.2, 4.4 and 4.5. For the remainder of this section let M = (P, R, p) be a wmdtt over Σ,, and S. Lemma 4.2 Let M be non-circular. 1. Let (S, ) be an ω-continuous semiring. Then [M ] fix = [M ] fix. 2. Let (S, ) be an ω-complete semiring. Then [M ] hyp = [M ] hyp P. Proof. 1. It suffices to show that for every t T Σ we have T P(pos(t)) M,t (I 0 ) = T M,t ω. This follows from Lemma 3.6 and the fact that TM,t ω is a fixpoint of T M,t. 2. This statement follows immediately from the fact that any ω-infinitary sum operation for S agrees with the additive operation of S for finite families. For the following discussion let t T Σ and G = (V, E, λ) be the hypergraph induced by M and t. In order to simplify notation in the following proof let us introduce an auxiliary notion. Let a P(pos(t)), n N, and u T. Then let us denote the set of derivations η HG,n a such that tr(η) = u by H(a, n, u). We will now prove a technical lemma that relates the computation of the fixpoint semantics to the computation of the hypergraph semantics. 18

19 Lemma 4.3 For every a P(pos(t)), n N, and u T we have η H(a,n,u) wt(η) = TM,t n (I 0 )(a)(u). Proof. We give a proof by induction on n. The base case of n = 0 is trivial because HG,0 a = and T M,t 0 (I 0 ) = I 0. Now we prove the induction step. To this end let n > 0, a P(pos(t)), and u T. Moreover, let I = T n 1 M,t (I 0 ) and for every k N and (r, ρ) E (k) let P (r,ρ) = H in1((r,ρ)) n 1 H in k((r,ρ)) n 1. We obtain wt(η) = η H a wt(η) η H(a,n,u) G,n tr(η)=u = k N (r,ρ) E (k) (η 1,...,η k ) P (r,ρ) wt((r, ρ)(η 1,..., η k )) out((r,ρ))=a tr((r,ρ)(η 1,...,η k ))=u (by Observation 3.17) = k N (r,ρ) E (k) (η 1,...,η k ) P (r,ρ) r w wt(η 1 )... wt(η k ) out((r,ρ))=a r b tr(η 1)...tr(η k )=u = k N (r,ρ) E r (k) w ( (η 1,...,η k ) P (r,ρ) wt(η 1 )... wt(η k ) ) out((r,ρ))=a r b tr(η 1)...tr(η k )=u (by distributivity) = r w ((ρ(r b ) I)(u) ) OI ( ) k N (r,ρ) E (k) out((r,ρ))=a = r w ((ρ(r b ) I)(u) ) OI (by definition of Φ t,a and E) (r,ρ) Φ t,a = ( r w (ρ(r b ) I) ) OI (u) (r,ρ) Φ t,a = T M,t (I)(a)(u) = TM,t n (I 0 )(a)(u). It remains to prove Equation ( ). Let k N and (r, ρ) E (k) such that out((r, ρ)) = a. Moreover, for every i [k] let a i = in i ((r, ρ)). We will prove the following equation: (η 1,...,η k ) P (r,ρ) r b tr(η 1)...tr(η k )=u wt(η 1 )... wt(η k ) = (ρ(r b ) I)(u) OI. (1) First assume that there are no u 1,...,u k T such that r b u 1 u k = u. Then it is easy to check that both sides of Equation (1) are equal to 0. For the remainder of this proof assume that there are u 1,...,u k T with r b u 1 u k = u. Note that u 1,...,u k are uniquely determined. Equation (1) follows from the following derivation: (η 1,...,η k ) P (r,ρ) r b tr(η 1)...tr(η k )=u wt(η 1 )... wt(η k ) = (η 1,...,η k ) P (r,ρ) wt(η 1 )... wt(η k ) tr(η 1)=u 1,...,tr(η k )=u k (since u 1,..., u k are uniquely determined) = wt(η 1)... wt(η k ) η 1 H(a 1,n 1,u 1),...,η k H(a k,n 1,u k ) (by definition of P (r,ρ) and H(a i, n 1, u i )) 19

20 = ( η 1 H(a 1,n 1,u 1) wt(η 1) )... ( η 1 H(a k,n 1,u k ) wt(η k) ) = ( I(a 1 )(u 1 ) )... (I(a k )(u k ) ) (by the induction hypothesis) = (ρ(r b ) I)(u) OI. It remains to prove Equation ( ). We show by structural induction on s that for every s T (P(pos(t))), a 1,...,a k P(pos(t)), and u 1,..., u k T with k = iyield(s) and a 1 a k = iyield(s) we have ( I(a 1 )(u 1 ) )... (I(a k )(u k ) ) = (s I)(s OI u 1 u k ). First consider the case that s P(pos(t)). Then k = 1, a 1 = s, and s u 1 = u 1. Therefore, I(a 1 )(u 1 ) = I(s)(s u 1 ) = (s I)(s OI u 1 ). Now assume that s = δ(s 1,...,s j ) for some j N, δ (j), and s 1,..., s j T (P(pos(t))). For every i [j] there are k i N, a i 1,...,a i k i P(pos(t)), and u i 1,...,ui k i T such that k i = iyield(s i ), a 1 1 a1 k 1 a j 1 aj k j = a 1 a k, and u 1 1 u1 k 1 u j 1 uj k j = u 1 u k. Observe that a i 1 ai k i = iyield(s i ). Then we obtain ( I(a1 )(u 1 ) )... (I(a k )(u k ) ) = ( I(a 1 1)(u 1 1) )... (I(a 1 k 1 )(u 1 k 1 ) )... (I(a j 1 )(uj 1 ))... (I(a j k j )(u j k j ) ) ( ) = ( OI (s 1 I)(s 1 u 1 1 u 1 k 1 ) )... ((s OI j I)(s j u j 1 uj k j ) ) (by the induction hypothesis (of the proof of Equation ( )) OI OI = top δ (s 1 I,...,s j I) ( δ(s 1 u 1 1 u1 k 1,...,s j u j 1 uj k j ) ) = (s I) ( ) OI s u 1 u k. Using the previous lemma, we are now prepared to prove the relationships between the fixpoint and the hypergraph semantics (cf. Lemmas 4.4 and 4.5). Lemma 4.4 If M is non-circular, then [M ] hyp = [M ] fix. Proof. Let t T Σ, u T, a = p(ε), and G = (V, E, λ) be the hypergraph induced by M and t. Since the set HG a is finite by Lemma 3.20, there is an n N such that HG,n a is equal to Ha G. Let m = max(n, P(pos(t)) ). This implies H(u) = H(a, m, u). Then [M ] hyp (t, u) = η H(u) wt(η) = η H(a,m,u) wt(η) = TM,t(I 0 m P(pos(t)) )(a)(u) = T M,t (I 0 )(a)(u) = [M ]fix (t, u). (by Lemmas 3.5 and 4.3) The proof of the following lemma uses the concept of cofinal subsets. Let (A, ) be an ω-cpo and C B A. We call C a cofinal subset of B if for every b B there is a c C such that b c. Observe that for every cofinal subset C of B we have C = B, if B or C has a supremum. Lemma 4.5 Let (S, ) be ω-complete and (S, ) be ω-continuous such that (S, ) and (S, ) are related. Then [M ] hyp P = [M ] fix. 20

21 Proof. Let t T Σ, u T, a = p(ε), and G = (V, E, λ) be the hypergraph induced by M and t. We prove that [M ] fix (t, u) = [M ]hyp P (t, u). We obtain that [M ] fix (t, u) = T ω M,t (a)(u) = {TM,t n (I 0 ) n N}(a)(u) (by Lemma 3.13) = {TM,t n (I 0 )(a)(u) n N} (by Lemma 3.9) = { η H(a,n,u) wt(η) n N} (by Lemma 4.3) = { η J wt(η) J P fin(h(u)) } ( ) = wt(η) = [M ]hyp P (t, u). (since (S, ) and (S, ) are related) η H(u) It remains to prove Equation ( ). It suffices to show that C = { η H(a,n,u) wt(η) n N } is a cofinal subset of B = { η J wt(η) J P fin(h(u)) }. Clearly, C is a subset of B, because H(a, n, u) P fin (H(u)) for every n N. Let J P fin (H(u)). Since J is finite, there is an n N such that J H(a, n, u). Then the properties of (S, ) imply that η J wt(η) η H(a,n,u) wt(η). This finishes the proof of Theorem The n-best problem In this section we will study the problem to determine the best translation of an input tree under a wmdtt. More precisely, let be an order on the semiring S and M be an wmdtt over Σ,, and S. Given an input tree t, the best translation problem of M and t is to determine the output tree u that maximizes τ(t, u) wrt, where τ is the weighted tree transformation defined by M (in the previous section we have shown that we do not need to distinguish between the fixpoint and the hypergraph semantics of M); more precisely, it is the problem to compute u = argmax u T τ(t, u). The best translation problem has important applications, e.g., in the statistical syntaxbased translation of natural languages [21, 32, 34]. It turns out that the best translation problem is NP-hard (this has been shown in [36] and [7] for similar problems). Therefore, we will consider the popular approach to approximate the best translation [32, 27] as follows: u = argmax u T τ(t, u) = argmax u T wt(η) (hypergraph semantics) η H(u) = argmax u T η H wt(η) p(ε) argmax u T tr(η)=u η H tr(η)=u wt(η), (2) 21

22 where, for some n N, H contains the n derivations in H p(ε) with the highest weight. If H is known, then the term in Equation (2) is easy to compute. Hence, it remains to study the problem to determine H. This is called the n-best derivations problem: given a hypergraph G = (V, E, λ), an order on the set of derivations of G, and a vertex v V, what are the n-best derivations ending in v? In the sequel, we will assume that the order on the set of derivations is given explicitely (instead of an order that is given in terms of derivation weights and an order on the semiring). More precisely, we fix a linear pre-order on H G (for the definition of pre-orders we refer to [2]) such that the following properties are satisfied for every η H G of the form e(η 1,..., η k ): (subtree property) η i η for every i [k], and (compatibility) for every η H G of the form e(η 1,...,η k ) we have that η i η i, for every i [k], implies η η. Since is a pre-order, the sequence of n best derivations is not necessarily unique. Instead we have to deal with the the set of sequences of n best derivations. In the following definition we introduce a function that returns the set of sequences of n best derivations of a given set of derivations. Throughout this section, we fix a hypergraph G = (V, E, λ). Definition 5.1 For every n N the mapping min n : P(H) P(H ) is defined recursively as follows: if n = 0 or P =, then min n (P) = {ε}, otherwise min n (P) = {η l η P, l min n 1 (P \ {η}), η P : η η }. Note that for every v V, n N, and P H v we have that min n (P) (H v ). Now we can define the n-best derivation problem formally. n-best derivations problem Given a hypergraph G, a vertex v, a number n N Determine some sequence in the set min n (H v ). In this section we will present an algorithm for solving the n-best derivations problem. Similar algorithms for the n-best derivations problem have been studied in [21, 5]. In fact, our algorithm is based on the algorithm presented in [5]. However, to our knowledge the present paper is the first exposition of a rigorous correctness proof of such an algorithm. For the sake of simplicity, our algorithm is not given in an imperative programming style. Instead, in Definition 5.7 we give a recursive definition of a sequence B v n (H v ) and in Theorem 5.11 we show that B v n min n(h v ). The recursive definition of B v n can be understood as a functional program that solves the n-best derivations problem. Let us begin with introducing some auxiliary concepts. Definition 5.2 For every l H with l = h t for some h H and t H the head of l and the tail of l are defined by head(l) = h and tail(l) = t, respectively. Observation 5.3 For every n N +, non-empty P H, and l min n (P), we have that head(l) is minimal in P and tail(l) min n 1 (P \ {head(l)}). For every n N + and P H it holds that min n (P) = {ε} iff ε min n (P) iff P is empty. 22