Typechecking Top-Down Uniform Unranked Tree Transducers

Transcription

1 Typeheking Top-Down Uniform Unranked Tree Transduers Wim Martens and Frank Neven University of Limburg Abstrat. We investigate the typeheking problem for XML queries: statially verifying that every answer to a query onforms to a given output shema, for inputs satisfying a given input shema. As typeheking quikly turns undeidable for query languages apable of testing equality of data values, we return to the limited framework where we abstrat XML douments as labeled ordered trees. We fous on simple top-down reursive transformations motivated by XSLT and strutural reursion on trees. We parameterize the problem by several restritions on the transformations (deleting, non-deleting, bounded width) and onsider both tree automata and DTDs as output shemas. The omplexity of the typeheking problems in this senario range from ptime to exptime. 1 Introdution The emergene of XML as the likely standard for representing and exhange of data on the Web onfirmed the entral role of semistrutered data. However, it has also marked the return of the shema. In the ontext of the Web, shemas an be used to validate data exhange. In a typial senario, a user ommunity agrees on a ommon shema and on produing only data onforming to that shema. This raises the issue of typeheking: verifying at ompile time that every XML doument whih is the result of a speified query applied to a valid input doument, satisfies the output shema [23, 24]. Obviously, typeheking depends on the transformation language and the shema language at hand. As shown by Alon et al. [1, 2], when transformation languages have the ability to ompare data values, the typeheking problem quikly turns undeidable. Milo, Suiu, and Vianu argued that the apability of most XML transformation languages an be enompassed by k-pebble transduers when data values are ignored and XML douments an be abstrated by labeled ordered trees [15]. Further, the authors showed that the typeheking problem in this ontext is deidable. More preisely, given two types τ 1 and τ 2, represented by tree automata, and a k-pebble transduer T, it is deidable whether T (t) τ 2 for all t τ 1. Here, T (t) is the tree obtained by running T on input t. The omplexity, however, is very bad: non-elementary. In an attempt to lower the omplexity, we onsider muh simpler tree transformations: those defined by deterministi top-down uniform tree transduers

2 Table 1. The presented results: the top row of the table shows the representation of the input and output shemas and the left olumn shows the type of tree transduer. NTA DTA DTD(NFA) DTD(DFA) DTD(SL) general exptime exptime exptime exptime exptime non-deleting exptime exptime pspae pspae onp bounded width exptime in exptime/pspae-hard pspae ptime onp on unranked trees. Suh transformations orrespond to strutural reursion on trees [5] and to simple top-down XSLT transformations [3, 6]. The transduers are alled uniform as they annot distinguish between the order of siblings. In brief, a transformation onsists of a top-down traversal of the input tree where every node is replaed by a new tree (possibly the empty tree). We show that the ability of transduers to delete interior nodes (i.e., replaing them by the empty tree) already renders the typeheking problem exptimehard for very simple DTDs (e.g. DTDs with deterministi regular expressions on their right hand side). To obtain a lower omplexity, for the remainder of the paper, we fous on non-deleting transformations. Our inquiries reveal that the omplexity of the typeheking problem of non-deleting transduers is determined by two features: (1) non-determinism in the shema languages; and, (2) unbounded opying of subtrees by the transduer. Only when we disallow both features, we get a ptime-omplexity for the typeheking problem. An overview of our results is given in Table 1. Unless speified otherwise, all omplexities are both upper and lower bounds. The top row of the table shows the representation of the input and output shemas and the left olumn shows the type of tree transduer. NTA and DTA stand for non-deterministi and deterministi tree automata, respetively. Suh automata abstrat the expressiveness of XML Shema [7]. DTD(X) stands for DTDs whose right-hand sides onsist of regular languages in X. The exat definitions are given in setion 2. Related Work. A problem related to typeheking is type inferene [14, 19]. This problem onsists in onstruting a tight output shema, given an input shema and a transformation. Of ourse, solving the type inferene problem implies a solution for the typeheking problem: hek ontainment of the inferred shema into the given one. However, haraterizing output languages of transformations is quite hard [19]. The transduers onsidered in the present paper are restrited versions of the ones studied by Maneth and Neven [13]. They already obtained a non-elementary upper bound on the omplexity of typeheking (due to the use of monadi seond-order logi in the definition of the transduers). Although the struture of XML douments an be faithfully represented by unranked trees (these are trees without a bound on the number of hildren of nodes), Milo, Suiu, and Vianu hose to study k-pebble transduers over binary trees as there is an immediate enoding of unranked trees into binary ones. The

3 top-down variants of k-pebble transduers are well-studied on binary trees [11]. However, these results do not aid in the quest to haraterize preisely the omplexity of typeheking transformations on unranked trees. Indeed, the lass of unranked tree transdutions an not be aptured by ordinary transduers working on the binary enodings. Maro tree transduers an simulate our transduers on the binary enodings [13, 9], but as very little is known about their omplexity this observation is not of muh help. For these reasons, we hose to work diretly with unranked tree transduers. Tozawa onsidered typeheking w.r.t. tree automata for a fragment of topdown XSLT similar to ours [25]. He adapts the bakward type inferene tehnique of [15] and obtains a double exponential time algorithm. Due to spae limitations, we only provide skethes of proofs. 2 Definitions In this setion we provide the neessary bakground on trees, automata, and uniform tree transduers. 2.1 Trees and Hedges We fix a finite alphabet Σ. The set of unranked Σ-trees, denoted by T Σ,isthe smallest set of strings over Σ and the parenthesis symbols ) and ( suh that for σ Σ and w TΣ, σ(w) isint Σ. We write σ rather than σ(). Note that there is no a priori bound on the number of hildren of a node in a Σ-tree; suh trees are therefore unranked. In the following, whenever we say tree, we always mean Σ-tree. A hedge is a finite sequene of trees. The set of hedges, denoted by H Σ, is defined as TΣ. For every hedge h H Σ,theset of nodes of h, denoted by Dom(h), is the subset of N defined as follows: if h = ε, then Dom(h) = ; if h = t 1 t n where eah t i T Σ, then Dom(h) = n i=1 {iu u Dom(t i)}; and, if h = σ(w), then Dom(h) ={ε} Dom(w). In the sequel we adopt the following onvention: we use t, t 1,t 2,... to denote trees and h, h 1,h 2,... to denote hedges. Hene, when we write h = t 1 t n we taitly assume that all t i s are trees. For every u Dom(h), we denote by lab h (u) the label of u in h. Atree language is a set of trees. 2.2 DTDs and Tree Automata We use extended ontext-free grammars and tree automata to abstrat from DTDs and the various proposals for XML shemas. Further, we parameterize the definition of DTDs by a lass of representations M of regular string languages like, e.g., the lass of DFAs or NFAs. For M M, we denote by L(M) the set of strings aepted by M.

4 Definition 1. Let M be a lass of representations of regular string languages over Σ. A DTD is a tuple (d, s d ) where d is a funtion that maps Σ-symbols to elements of M and s d Σ is the start symbol. For simpliity, we usually denote (d, s d )byd. Atreet satisfies d if lab t (ε) =s d and for every u Dom(t) with n hildren lab t (u1) lab t (un) L(d(lab t (u))). By L(d) we denote the tree language aepted by d. We parameterize DTDs by the formalism used to represent the regular language M. Therefore, we denote by DTD(M) the lass of DTDs where the regular string languages are represented by elements of M. Thesize of a DTD is the sum of the sizes of the elements of M used to represent the funtion d. To define unordered languages we make use of the speifiation language SL inspired by [17] and also used in [1, 2]. The syntax of the language is as follows. Definition 2. For every σ Σ and natural number i, σ =i and σ i are atomi SL-formulas; true is also an atomi SL-formula. Every atomi SL-formula is an SL-formula and the negation, onjuntion, and disjuntion of SL-formulas are also SL-formulas. A string w over Σ satisfies an atomi formula σ =i if it has exatly i ourrenes of σ; w satisfies σ i if it has at least i ourrenes of σ. Further, true is satisfied by every string. 1 Satisfation of Boolean ombinations of atomi formulas is defined in the obvious way. As an example, onsider the SL formula o-produer 1 produer 1. This expresses the onstraint that a o-produer an only our when a produer ours. The size of an SL-formula is the number of symbols that our in it (every i in σ =i or σ i is written in binary notation). We reall the definition of non-deterministi tree automata from [4]. We refer the unfamiliar reader to [16] for a gentle introdution. Definition 3. A nondeterministi tree automaton (NTA) is a tuple B =(Q, Σ, δ, F ), where Q is a finite set of states, F Q is the set of final states, and δ is a funtion Q Σ 2 Q suh that δ(q, a) is a regular string language over Q for every a Σ and q Q. A run of B on a tree t is a labeling λ : Dom(t) Q suh that for every v Dom(t) with n hildren, λ(v1) λ(vn) δ(λ(v), lab t (v)). Note that when v has no hildren, then the riterion redues to ε δ(λ(v), lab t (v)). A run is aepting iff the root is labeled with an aepting state, that is, λ(ε) F.Atree is aepted if there is an aepting run. The set of all aepted trees is denoted by L(B). We extend the definition of δ to trees and denote this by δ (t): if t onsists of only one node labeled with a then δ (t) ={q ε δ(q, a)}; ift is of the form a(t 1 t n ), then δ (t) ={q q 1 δ (t 1 ),..., q n δ (t n )and q 1 q n δ(q, a)}. So,t is aepted if δ (t) F. A tree automaton is bottom-up deterministi if for all q, q Q with q q and a Σ, δ(q, a) δ(q,a)=. We denote the set of bottom-up deterministi 1 The empty string is obtained by σ Σ σ=0 and the empty set by true.

5 NTAs by DTA. A tree automaton is top-down deterministi if for all q, q Q with q q, a Σ, andn 0, δ(q, a) ontains at most one string of length n. Like for DTDs, we parameterize NTAs by the formalism used to represent the regular languages in the transition funtions δ(q, a). So, for M a lass of representations of regular languages, we denote by NTA(M) the lass of NTAs where all transition funtions are represented by elements of M. Thesize of an automaton B is then Q + Σ + q,a δ(q, a) + F. Here, by δ(q, a) we denote the size of the automaton aepting δ(q, a). Unless expliitly speified otherwise, δ(q, a) is always represented by an NFA. Let 2AFA be the lass of two-way alternating finite automata [12]. We give without proof the following theorem whih will be a useful tool for obtaining upper bounds. Theorem Emptiness of NTA(NFA) is in ptime; 2. Emptiness of NTA(2AFA) is in pspae. 2.3 Transduers We next define the tree transduers used in this paper. To simplify notation, we restrit to one alphabet. That is, we onsider transdutions mapping Σ-trees to Σ-trees. It is, however, possible to define transdutions where the input alphabet differs from the output alphabet. For a set Q, denote by H Σ (Q) (resp.t Σ (Q)) the set of Σ-hedges (resp. trees) where leaf nodes an be labeled with elements from Q. Definition 4. A uniform tree transduer is a tuple (Q, Σ, q 0,R), where Q is a finite set of states, Σ is the input and output alphabet, q 0 Q is the initial state, and R is a finite set of rules of the form (q, a) h, where a Σ, q Q, and h H Σ (Q). When q = q 0, h is restrited to T Σ (Q) \ Q. The restrition on rules with the initial state ensures that the output is always a tree rather than a hedge. For the remainder of this paper, when we say tree transduer, we always mean uniform tree transduer. Example 1. Let T =(Q, Σ, p, R) where Q = {p, q} and R ontains the rules (p, a) d(e) (p, b) (q p). (q, a) q (q, b) d(q) Our definition of tree transduers orresponds to strutural reursion [5] and a fragment of top-down XSLT. For instane, the XSLT program equivalent to the above transduer is given in Figure 1 (we assume the program is started in mode p). The translation defined by T =(Q, Σ, q 0,R)onatreet in state q, denoted by T q (t), is indutively defined as follows: if t = ε then T q (t) :=ε; ift = a(t 1 t n ) and there is a rule (q, a) h R then T q (t) is obtained from h by replaing

6 <xsl:template math="a" mode ="p"> <d> <e/> </d> </xsl:template> <xsl:template math="b" mode ="p"> <> <xsl:apply-templates mode="q"/> <xsl:apply-templates mode="p"/> </> </xsl:template> <xsl:template math="a" mode ="q"> </> <xsl:apply-templates mode="q"/> </xsl:template> <xsl:template math="b" mode ="q"> <d> <xsl:apply-templates mode="q"/> </d> </xsl:template> Fig. 1. The XSLT program equivalent to the transduer of Example 1. every node u in h labeled with p by the hedge T p (t 1 ) T p (t n ). Note that suh nodes u an only our at leaves. So, h is only extended downwards. If there is no rule (q, a) h R then T q (t) :=ε. Finally, define the transformation of t by T, denoted by T (t), as T q0 (t). For a Σ, q Q and (q, a) h R, we denote h by rhs(q, a). If q and a are not important, we say that h is a rhs. The size of T is Q + Σ + (q,a) rhs(q, a). Example 2. In Figure 2 we give the translation of the tree t defined as b b a b by the transduer of Example 1. a a a We disuss two features whih are of importane in the remainder of the paper: opying and deleting. The rule (p, b) (qp) in the above example opies the hildren of the urrent node in the input tree two times: one opy is proessed in state q and the other in state p. Thesymbol is the parent node of the two opies. So the urrent node in the input tree orresponds to the latter node. The rule (q, a) q also opies the hildren of the urrent node two times. However, in this ase, one opy is replaed by the single symbol tree, the other opy is obtained by proessing the hildren in state q. No parent node is given for this opy. So, there is no orresponding node for the urrent node in the input tree. We, therefore, say it is deleted. For instane, T q (a(b)) = dwhere d orresponds to b and not to a. 2.4 The Typeheking Problem We define the problem entral to this paper.

7 Definition 5. A tree transduer T typeheks w.r.t. to an input tree language S in and an output tree language S out,ift (t) S out for every t S in. We parameterize the typeheking problem by the kind of tree transduers and tree languages we allow. Let T be a lass of transduers and S be a lass of tree languages. Then TC[T, S] denotes the typeheking problem where T T and S in,s out S. The size of the input of the typeheking problem is the sum of the sizes of the input and output shema and the tree transduer. A transduer T has width k if there are at most k ourrenes of states in every rhs of T.ByBW k we denote the lass of transduers of width k. A transduer is non-deleting if no states our at the top-level of a rhs. We denote by T g the lass of all transduers and by T nd the lass of non-deleting transduers. For a lass of representations of regular string languages M, we write TC[T,M] rather than TC[T,DTD(M)]. 3 The General Case When we do not restrit our transduers in any way, the typeheking problem is in exptime and is exptime-hard for even the simplest DTDs: those where the right-hand sides are speified with SL-formulas or with DFAs. The main reason is that the deleting states allow the transduer to simulate deterministi top-down tree automata in suh a way that the transduer produes no output besides aeptane information. In this way, even a very simple DTD an hek whether the output was rejeted or not. By opying the input several times, we an exeute several deterministi tree automata in parallel. These are all the ingredients we need for a redution from non-emptiness of the intersetion of an arbitrary number of deterministi tree automata whih is known to be exptime-hard. Theorem TC[T g,nta] is in exptime; 2. TC[T g,sl]isexptime-hard; 3. TC[T g,dfa] is exptime-hard. Proof. (Sketh) (1) Let T =(Q T,Σ,q 0 T,R T ) be a transduer and let A in and A out =(Q A,Σ,δ A,F A ) be two NTAs representing the input and output shema, respetively. We next desribe a non-deleting transduer S andanntab out whih an be onstruted in logspae, suh that T typeheks w.r.t. A in and A out iff S typeheks w.r.t. A in and B out. From Theorem 3(1) it then follows that TC[T g,nta] is in exptime. Intuitively, S puts a # whenever T would proess a deleting state. For instane, the rule (q, a) q is replaed by (q, a) #(q). We introdue some notation to haraterize the behavior of B out. Define the #-eliminating funtion γ as follows: γ(σ(h)) is γ(h) when σ =# and σ(γ(h)) otherwise; further, γ(t 1 t n ):=γ(t 1 ) γ(t n ). Then, learly, for all t T Σ, T (t) =γ(s(t)). B out then aepts a tree t iff γ(t) L(A out ).

8 T p (t) T q (b) T q (a(aa)) T q (b(a)) T p (b) T p (a(aa)) T p (b(a)) d T q (a) T q (a) d d T q (ε) T q (a) T q (ε) T p (ε) e T q (a) T p (a) d T q (ε) T q (ε) d d T q (ε) e T q (ε) d e d d d e d e Fig. 2. The translation of t = b(ba(aa)b(a)) by the transduer T of Example 1.

9 (2) We use a redution from the intersetion problem of deterministi binary top-down tree automata A i (i =1...n), whih is known to be hard for exptime [21]. The problem is stated as follows, given deterministi binary top-down automata A 1,...,A n,is n i=1 L(A i)=? We define a transduer T and two DTDs d in and d out suh that n i=1 L(A i) iff T does not typehek w.r.t. d in and d out. In the onstrution, we exploit the opying power of transduers to make n opies of the input tree: one for eah A i. By using deleting states, we an exeute eah A i on its opy of the input tree without produing output. When an A i does not aept, we output an error symbol under the root of the output tree. The output DTD should then only hek that an error symbol always appears. The proof of (3) is similar to the one for (2). 4 Non-deleting Transformations In an attempt to lower the omplexity, we onsider, in the present setion, nondeleting transformations w.r.t. various shema formalisms. We observe that when shemas are represented by tree automata, the omplexity remains exptimehard. When tree languages are represented by DTDs, the omplexity of the typeheking problem drops to pspae and is hard for pspae even when righthand sides of rules are represented by DFAs. The main reason for this is that the tree transduers an still make an unbounded number of opies of the input tree. This allows to simulate in parallel an unbounded number of DFAs and makes it possible to redue the intersetion emptiness problem of DFAs to the typeheking problem. In the next setion, we therefore onstrain this opying power. In summary, we prove the following results: Theorem TC[T nd,nta] is exptime-omplete; 2. TC[T nd,dta] is exptime-omplete; 3. TC[T nd,nfa] is pspae-omplete; 4. TC[T nd,dfa] is pspae-omplete; 5. TC[T nd,sl]isonp-omplete. 4.1 Tree Automata Consider the ase where the input and output shemas are represented by NTA(NFA)s. One way to obtain an appropriate typeheking algorithm, would be to build a omposite automaton A o that on input t, runs A out on T (t) without atually onstruting T (t). The given instane typeheks iff L(A in ) L(A o ). However, onstruting A o would lead to an exponential blowup beause the state set of A o would be 2 QT 2 Qout. Sine A o is nondeterministi beause A out is nondeterministi, solving the inlusion problem on this instane would lead to a double exponential time algorithm. Thus, to show that TC[T nd,nta] an be solved in exptime, we need a slightly more sophistiated approah. Theorem 3(1). TC[T nd,nta] is exptime-omplete.

10 Proof. (Sketh) Hardness is immediate as ontainment of NTAs is already hard for exptime [20]. We, therefore, only prove membership in exptime. The proof is similar in spirit to a proof in [18], whih shows that ontainment of Query Automata is in exptime. LetT =(Q T,Σ,q 0 T,R T ) be a non-deleting tree transduer and let A in =(Q in,σ,δ in,f in )anda out =(Q out,σ,δ out,f out )bethe NTAs representing the input and output shema, respetively. For ease of exposition, we restrit hedges in the rhs of T to be trees. In brief, our algorithm omputes the set P = {(S, f) S Q in,f : Q T 2 Qout, t suh that S = δin(t) and q Q T,f(q) =δout(t q (t))}. Intuitively, in the definition of P, t an be seen as a witness of (S, f). That is, S is the set of states reahable by A in at the root of t, while for eah state q of the transduer, f(q) is the set of states reahable by A out at the root of T q (t) (reall that this is the translation of t started in state q). So, the given instane does not typehek iff there exists an (S, f) P suh that F in S and F out f(qt 0 )=. In Figure 3, an algorithm for omputing P is depited. By rhs(q, a)[p f 1 (p) f n (p) p Q T ], we denote the tree obtained from rhs(q, a) by replaing every ourrene of a state p by the sequene f 1 (p) f n (p). For {in, out}, δ : T Σ (2 Q ) 2 Q is the transition funtion extended to trees in T Σ (2 Q ). To be preise, for a Σ, δ (a) :={q ε δ (q, a)}; forp Q, δ (P ) := P ; and, δ (a(t 1 t n )) := {q q i δ (t i ) : q 1 q n δ (q, a)}. The orretness of the algorithm follows from the following lemma whih an be easily proved by indution. Lemma 1. Apair(S, f) has a witness tree of depth i iff (S, f) P i. It remains to show that the algorithm is in exptime. The set P 1 an be omputed in time polynomial in the sizes of A in, A out,andt.asp i P i+1 for all i, the loop an only make an exponential number of iterations. It, hene, suffies to show that eah iteration an be done in exptime. Atually, we argue that it an be heked in pspae whether a tuple (S, f) P i. Indeed, the question whether there are (S 1,f 1 ) (S n,f n ) Pi 1 an be redued to the emptiness test of a 2AFA A whih works on strings over the alphabet P i 1. On input (S 1,f 1 ) (S n,f n )the2afaa operates as follows: for every p S it heks whether there are r i S i suh that r 1...r n δ(p, a). This an be done by S traversals through the input string. Next, A heks for every q Q in \S whether for all r i S i, r 1...r n δ(q, a). This an be done by Q in \S traversals through the input string while using alternation. In a similar way f(q) is heked. The automaton A is exponential in the input. However, we an onstrut A on the fly when exeuting the pspae algorithm for non-emptiness. The latter algorithm is an adaptation of the tehnique used by Vardi [26]. As there are only exponentially many tuples (S, f), the overall algorithm is in exptime. In the remainder of this setion, we examine what happens when tree automata are restrited to be deterministi. From the above result, it is immediate

11 P 0 := ; i := 1; P 1 := {(S, f) a : r S : ε δ in(r, a), q Q T : f(q) =δ out(t q (a))}; while P i P i 1 do P i := { (S, f) (S 1,f 1) (S n,f n) P i 1, a Σ : S = {p r k S k,k =1...n,r 1 r n δ in(p, a)} q Q T : f(q) =δ out( rhs(q, a)[p f1(p) f n(p) p Q T ] )} ; i := i +1; end while P := P i; Fig. 3. The algorithm of Theorem 3(1) omputing P. that TC[T nd,dta] is in exptime. To show that it is hard, we an use a redution from the intersetion problem of deterministi binary top-down tree automata like in the proof of Theorem 2(2). The redution is almost idential to the one in Theorem 2(2): A in defines the same set of trees as d in does with the exeption that A in enfores an ordering of the hildren. The transduer in the proof of Theorem 2(2) starts the in parallel simulation of the n automata, but then, using deleting states, delays the output until it has reahed the leaves of the input tree. In the present setting, we an not use deleting states. Instead, we opy the input tree and attah error-symbols to the leaves when an automaton rejets. The output automaton then heks whether at least one error ourred. We obtain the following theorem. Theorem 3(2) TC[T nd,dta] is exptime-omplete. 4.2 DTDs When we onsider DTDs as input shemas the omplexity drops to pspae and onp. Theorem 3(3) TC[T nd,nfa] is pspae-omplete. Proof. (Sketh) The hardness result is immediate as ontainment of regular expressions is known to be pspae-hard [22]. For the other diretion, let T be a non-deleting tree transduer. Let d in and d out be the input and output DTDs, respetively. We onstrut an NTA(2AFA) B suh that L(B) ={t L(d in ) T (t) d out }. Moreover, the size of B is polynomial in the size of T, d in,and d out.thus,l(b) = iff T typeheks w.r.t. d in and d out. By Theorem 1(2), the latter is in pspae. To explain the operation of the automaton we introdue the following notion: let q be a state of T and a Σ then define q(a) :=z where z is the onatenation of the labels of the top most nodes of rhs(q, a). For a string w := a 1 a n, we define p(w) :=p(a 1 ) p(a n ). Intuitively, the automaton B now works bottom-up as follows: (1) B heks that t L(d in ); (2) at the same time, B guesses a node v labeled σ with n hildren and piks a state q in whih

12 v is proessed: B then aepts if h does not satisfy d out, where h is obtained from rhs(σ, q) by replaing every state p by p(lab t (u1) lab t (un)). As d out is speified by NFAs and we have to hek that d out is not satisfied, we need alternation to speify the transition funtion of B. Additionally, as T an opy its input, we need two-way automata. Formally, let T =(Q T,Σ,q T 0,δ T ). Define B =(Q B,Σ,F B,δ B ) as follows. The set of states Q B is the union of the following sets: Σ, {(σ, q) q Q T,σ Σ}, and{(σ, q, hek) q Q T,σ Σ}. If there is an aepting run on a tree t, thenanodev labeled with a state of the form σ, (σ, q), (σ, q, hek) has the following meaning: σ: the urrent node is labeled with σ and the subtree rooted at this node satisfies d in. (σ, q): same as in previous ase with the following two additions: (1) v is proessed by T in state q; and, (2) a desendant of v will produe a tree that will not satisfy d out. (σ, q, hek): same as the previous ase only now v itself will produe a tree that does not satisfy d out. The set of final states is defined as follows: F B := {(σ, q0 T ) σ Σ}. The transition funtion is defined as follows: 1. δ B (a, b) =δ B ((a, q),b)=δ B ((a, q, hek),b)= for all a b; 2. δ B (a, a) =d in (a) and δ B ((a, q),a)={σ (b, p)σ +Σ (b, p, hek)σ p ours in rhs(q, a),b Σ}, for all a Σ and q Q T. 3. Finally, δ B ((a, q, hek),a)={a 1 a n h L(d out )anda 1 a n d in (a)}. Here, h is obtained from rhs(q, a) by replaing every q by q(a 1 a n ). We are left with the proof that δ B ((a, q, hek),a) an be omputed by a 2AFA A with only a polynomial blowup. Before we define A, we define some other automata. First, for every b Σ, leta b be the NFA aepting d out (b). For every v in rhs(q, a), let w be the largest string in (Σ Q T ) suh that lab h (v)(w) is a subtree rooted at v in h. Define the 2NFA B v as follows: suppose w is of the form z 0 p 1 z 1 p l z l, then a 1 a n L(B v ) if and only if z 0 p 1 (a 1 a n )z 1 p l (a 1 a n )z l L(A lab (v)). As w is fixed, B h v an reognize this language by reading a 1 a n l times while simulating A lab (v). Intuitively, h the automaton simulates A lab h (v) on z i 1 p i (a 1 a n )ontheith pass. It remains to desribe the onstrution of A. To this end, let A a in be the NFA suh that d in (a) = L(A a in ). On input a 1 a n, A first heks whether a 1 a n L(A a in ) by simulating Aa in. After this, A goes bak to the beginning of the input string, guesses an internal node u in rhs(q, a) and simulates the negation of B u.asb u is a 2NFA, A is a 2AFA. The intersetion problem of deterministi finite automata is known to be pspae-hard [10] and is easily redued to TC[T nd,dfa]. This implies the following result:

13 Theorem 3(4) TC[T nd,dfa] is pspae-omplete. Using SL-expressions to define right-hand sides of DTDs redues the omplexity of typeheking to onp. Theorem 3(5) TC[T nd,sl]isonp-omplete. Proof. (Sketh) First, we prove the hardness result. Let φ be a sat-formula and let v 1,...,v n be the variables ourring in φ. We define the typeheking instane as follows. Σ = {σ 1,...,σ n }. We only define d in and d out for σ 1, sine this is all we require. d in (σ 1 )=φ, where φ is the formula φ with every ourrene for i = 1...n. The transduer T is the identity, and d out (σ 1 )=. Hene, this instane typeheks iff φ is not satisfiable. To prove the upper bound, let T =(Q T,Σ,qT 0,R T )andlet(d in,s in )and (d out,s out ) be the input and output DTD respetively. We desribe an np algorithm that aepts iff the given instane does not typehek. We introdue some notation. For a DTD (d, s d )andσ Σ, we denote by d σ the DTD d with start symbol σ, that is, (d, σ). Let k be the largest number ourring in an SL-formula in d in. Set r =(k +1) Σ. The algorithm onsists of three main parts: of v i replaed by σ =1 i 1. First, we sequentially guess a subset L of the derivable symbols {b Σ L(d b in ) }. 2. Next, we guess a path of a tree in d in. In partiular, we guess a sequene of pairs (a i,q i ) L Q T, i =0,...,m, with m Σ Q T, suh that (a) a 0 = s in and q 0 = q 0 T ; (b) t, u Dom(t) suh that a 0 a m is the onatenation of the labels of the nodes on the path from the root to u; and, () i =1,...,m: T visits a i in state q i. 3. Finally, we guess a string w L of length at most r suh that T qm (a m (w)) L(d σ out) with σ the root symbol of T qm (a m (w)). All guesses an be done at one and an be heked by a polynomial verifier. This ompletes the desription of the algorithm. 5 Transduers of Bounded Width When we put a bound on the width (or opying power, reall the disussion at the end of setions 2.3 and 2.4) of transduers we get a ptime algorithm for typeheking when the right-hand sides of DTDs are represented by DFAs. All other results have the same omplexity as in the ase of unrestrited opying. Theorem TC[BW k,nta] is exptime-omplete; 2. TC[BW k,re] is pspae-omplete; 3. TC[BW k,dfa] is ptime-omplete; 4. TC[BW k,sl]isonp-omplete.

14 The lower bounds of (1), (2), and (4) follow immediately from the onstrution in the proofs of Theorem 3(1), (3), and (5). Theorem 4(3) TC[BW k,dfa] is ptime-omplete. Proof. (Sketh) In the proof of Theorem 3(3), TC[T nd,nfa] is redued to the emptiness of NTA(2AFA)s. Alternation was needed to express negation of NFAs; two-wayness was needed beause T ould make arbitrary opies of the input tree. However, when transduers an make only a bounded number of opies and DFAs are used, TC[BW k,dfa] an be logspae-redued to emptiness of NTA(NFA)s. From Theorem 1(1), it then follows that TC[BW k,dfa] is in ptime. Aptime lower bound is obtained by a redution from path systems [8]. 6 Conlusion Motivated by strutural reursion and XSLT, we studied typeheking for topdown XML transformers in the presene of both DTDs and tree automata. In this setting the omplexity of the typeheking problem ranges from ptime to exptime. In partiular, the ptime algorithm is obtained by restriting to nondeleting tree transduers of bounded width and DTD(DFA)s. The main open question for future researh is how these restritions an be relaxed while still having a ptime algorithm. Another question we left open is the exat omplexity of TC[BW k,dta] whih is in exptime and pspae-hard. Referenes 1. N. Alon, T. Milo, F. Neven, D. Suiu, and V. Vianu. Typeheking XML views of relational databases. In Pro. 16th IEEE Symposium on Logi in Computer Siene (LICS 2001), pages , N. Alon, T. Milo, F. Neven, D. Suiu, and V. Vianu. XML with data values: Typeheking revisited. In Pro. 20th Symposium on Priniples of Database Systems (PODS 2001), pages , G. J. Bex, S. Maneth, and F. Neven. A formal model for an expressive fragment of XSLT. Information Systems, 27(1):21 39, A. Brüggemann-Klein, M. Murata, and D. Wood. Regular tree and regular hedge languages over unranked alphabets: Version 1, april 3, Tehnial Report HKUST-TCSC , The Hongkong University of Siene and Tehnology, P. Buneman, M. Fernandez, and D. Suiu. UnQl: a query language and algebra for semistrutured data based on strutural reursion. The VLDB Journal, 9(1):76 110, James Clark. XSL transformations version august World Wide Web Consortium. XML Shema S.A. Cook. An observation on time-storage trade-off. Journal of Computer and System Sienes, 9(3): , J. Engelfriet and H. Vogler. Maro tree transduers. Journal of Computer and System Sienes, 1985.

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