October 23, concept of bottom-up tree series transducer and top-down tree series transducer, respectively,

Transcription

1 Bottom-up and Top-down Tree Series Transformations oltan Fulop Department of Computer Science University of Szeged Arpad ter., H-670 Szeged, Hungary Heiko Vogler Department of Computer Science Technical University of Dresden D-0106 Dresden, Germany October 3, 000 Abstract: We generalize bottom-up tree transducers and top-down tree transducers to the concept of bottom-up tree series transducer and top-down tree series transducer, respectively, by allowing formal tree series as output rather than trees, where a formal tree series is a mapping from output trees to some semiring. We associate two semantics with a tree series transducer: a mapping which transforms trees into tree series (for short: tree to tree series transformation or t-ts transformation), and a mapping which transforms tree series into tree series (for short: tree series transformation or ts-ts transformation). We show that the standard case of tree transducers is reobtained by choosing the boolean semiring under the t-ts semantics. Also, for each of the two types of tree series transducers and for both types of semantics, we prove a characterization which generalizes in a straightforward way the corresponding characterization result for the underlying tree transducer class. More precisely, we prove that polynomial bottom-up tree series transducers can be characterized by the composition of nite state relabeling tree series transducers and homomorphism tree series transducers. Moreover, we prove that the total deterministic top-down tree series transducers can be characterized by the composition of homomorphism tree series transducers and linear total deterministic top-down tree series transducers. 1 Introduction The investigation of this paper was inspired by [Kui99] in which a rst attempt was made to integrate formal power series over trees into a restricted form of top-down tree transducer. The resulting tree series transducers transform an input tree into a formal tree series which is a mapping from the set of output trees into some semiring. By this means it is possible to measure the computation of output trees. Here we introduce the concepts of bottom-up tree series transducer and of (unrestricted) top-down tree series transducer. Before we discuss our investigation, we will now briey review the origins of tree series transducers which are a) tree transducers and b) automata with multiplicity (or cost functions). Tree transducers have been introduced in [Rou68, Rou70, Tha70]. They can be viewed as generalizations of generalized sequential machines [Ber79] to trees where the trees are either read and processed from their leaves towards their root (bottom-up tree transducer) or from their roots towards their leaves (top-down tree transducer). A rule (or: transition) of a bottom-up tree transducer looks like (q 1 (x 1 ); : : : ; q k (x k ))! q(t) and a rule of a top-down tree transducer has the form q((x 1 ; : : : ; x k ))! where q; q 1 ; : : :; q k are states of the tree transducers, is an input symbol of rank k, x 1 ; : : : ; x k are variables ranging over trees (more precisely, over output trees in the bottom-up case, and over input trees Research of this author was supported by the Hungarian Scientic Foundation (OTKA) under Grant T

2 in the top-down case), t is an output tree which may contain variables from the set X k fx 1 ; : : : ; x k g, and is an output tree which may contain constructs of the form p(x i ) at its leaves where p is a state and 1 i k. In the usual way, the rules of a tree transducer constitute a binary term rewriting relation (or: derivation relation) ) M by means of which the tree transformation M T T computed by M can be dened (where T and T are the sets of input trees and output trees, respectively). Since the seventies, tree transducers have been studied intensively. The rst substantial papers dealt with (de-)composition and hierarchy results [Bak79, Eng75, Eng77, Eng8]. In [FV9] a method of deciding the equivalence of the compositions of classes of tree transformations is overviewed. Survey articles and books are [GS84, GS97, CDG + 97, FV98]. Recently, a characterization of tree transformation classes in terms of monadic second order logic has been proved [BE00, EM99]. Now let us briey review the second origin of tree series transducers: automata with multiplicities. Let M (Q; ; ; q 0 ; Q d ) be a usual nite state string automaton where Q is the set of states, the set of input symbols, : Q! P(Q) the transition function, q 0 Q the initial state, and Q d Q the set of nal states. In the usual way, the transition function is extended to the function ~ : P(Q)! P(Q), and the language accepted by M is the language L(M) fw j ~(fq 0 g; w) \ Q d 6 ;g. In [Sch6], nite state string automata were generalized by associating with every transition an element a A of a semiring (A; +; ; 0; 1) assumed to be the cost (or multiplicity) of this transition (also cf. [Eil74, SS78, BR88, KS86]). In this approach, the transition function turns into a (Q Q)- matrix (Ahhii) QQ where Ahhii is the set of formal power series (in non-commuting variables), i.e., mappings of type! A. Intuitively, p;q () A is the cost of making a transition from state q to state p while reading (note the order of the indices p and q). The usual automaton (without cost functions) is reobtained by considering the boolean semiring. Then can be used to dene the behaviour of a nite state automaton M with cost functions such that it is a formal power series M Ahh ii obtained by iterating the transition matrix. This approach was generalized in [Kui97a] to tree automata over semirings (for a theoretical investigation on formal tree series, i.e., mappings of the type T! A, cf. [BR8]; in that paper, A is a vector space). Since the input symbols now have ranks, the transition matrix has to be replaced by a family ( k j k 0) of transition matrices k (Ahh(X k )ii) QQk where (X k ) f(x 1 ; : : : ; x k ) j (k) g. Intuitively, if the tree automaton has already accepted the input trees t 1 ; : : : ; t k in the states q 1 ; : : : ; q k, respectively, then ( k ) q;q1:::q k ((x 1 ; : : : ; x k )) A is the cost for making a transition from q 1 ; : : :; q k to q while reading the k-ary symbol. The semantics of a tree automaton M with cost functions is a formal tree series M AhhT ii. We mention that tree automata with cost functions have been used in order to select code in compilers on the basis of certain cost criteria [FSW94] (also cf. [Sei9]). In [Kui99] such tree automata with cost functions have been equipped with output and thereby the concept of tree transducer with cost functions has been introduced. In order to make this concept understandable on the background of the development explained above, let us reconsider the tree automata with cost functions. Such an automaton can be considered as a generating device: the trees together with their costs are built up and maintained in the entries of a matrix of type (AhhT ii) Q1. An alternative point of view is to formalize a tree automaton with cost functions as a transducer which computes the partial identity. More precisely, it computes the function M : T! AhhT ii such that the formal tree series M (s) has the value 0 for every tree t such that t 6 s. Then, the transitions are specied by a family ( k j k 0) of mappings k : (k)! (Ahh(X k )ii) QQk and the behaviour of the automaton can be specied by means of the initial homomorphism h : T! (AhhT ii) Q1. Now the tree transducers with cost functions as they are dened in [Kui99] (called: tree transducers there) can be understood as a modication of this alternative approach in the sense that the input tree is not reproduced, but another output tree is produced. Then the transitions are dened by a family ( k j k 0) of mappings k : (k)! (AhhT (X k )ii) QQk for some output alphabet. On the basis of these tree representations, in [Kui99] restricted types of top-down tree transducers with cost functions have been introduced and studied; the underlying transducer model is called nondeterministically simple top-down tree transducer in which, roughly speaking, to every subtree at most one state is applied (cf. [GS84]). Then a tree transducer M with cost functions computes the function M : T! AhhT ii. This nishes the short review about the origins of our investigation. In the present paper, we take a more liberal point of view on tree representations. We consider tree Q(Q(Xk )) representations of the form k : k! (AhhT (X)ii) such that only for nitely many indices (q; w) Q (Q(X k )), k () q;w 6 ~0, where ~0 is the formal tree series which maps every tree to 0. The following table summarizes the explained development of the function.

3 type of concept transition function computed function nite state automaton : Q! P(Q) L(M) string automaton with cost functions (Ahhii) QQ M Ahh ii tree automaton with cost functions k (Ahh(X k)ii) QQ k M AhhT ii tree automaton with cost functions k() (Ahh(Xk)ii) QQ k M : T! AhhT ii (alternative point of view) (partial identity) tree transducers with cost functions k() (AhhT(X QQ k k)ii) M : T! AhhT ii (in the sense of [Kui99]) tree series transducers k() (AhhT(X)ii) Q(Q(X k)) M : T! AhhT ii (in the present paper) Next we instantiate these tree representations in two dierent ways thereby obtaining the bottomup and the top-down case. In the bottom-up case we additionally require that, for every index w, if w 6 q 1 (x 1 ) : : : q k (x k ) for some q 1 ; : : : ; q k Q, then k () q;w ~0. Moreover, if k () q;w 6 ~0, then k () q;w AhhT (X k )ii. Then we call a bottom-up tree representation. To show an example, let us consider the two rules (q 1 (x 1 ); q (x ); q 3 (x 3 ))! q((x ; (x 3 ; x ))) j p((x 1 ; ((x 3 ); x 1 ))) of some bottom-up tree transducer, where,, are input/output symbol of rank 3,, and 1, respectively, q 1 ; q ; q are states, and j delimits the right-hand sides of the two rules. Let us assume that the costs for these two rules are a and b, respectively, where a; b A. Then these rules are represented by 3 () q;w ((x ; (x 3 ; x ))) a 3 () p;w ((x 1 ; ((x 3 ); x 1 ))) b where w q 1 (x 1 )q (x )q 3 (x 3 ). We observe that the structure of w is very restricted: w is the left hand side of the rule from which the input symbol (and the commas) have been stripped of. On the other hand, the structure of the trees (x ; (x 3 ; x )) and (x 1 ; ((x 3 ); x 1 )) to which the mapping 3 () q;w is applied, is liberal in the sense that variables of the left hand side of the rule may occur an arbitrary number of times. In the top-down case we require that, if k () q;w 6 ~0, then k () q;w Ahhc T (X l )ii, where l jwj and c T (X l ) denotes the set of all trees t over [ X l such that, for every 1 i k, the variable x i occurs exactly once in t and, reading the leaves of t from left to right, the variables occur in the order x 1 < x < : : : < x n. Then is a top-down tree representation. As an example, let us consider the two rules q((x 1 ; x ; x 3 ))! (p(x ); (q(x 1 ); p(x 1 )) j ((q(x )); p(x )) with costs c and d, respectively. These rules are represented by 3 () q;v ((x 1 ; (x ; x 3 ))) c 3 () q;v 0(((x 1 )); x )) d where v p(x )q(x 1 )q(x 1 ) and v 0 q(x )p(x ). We observe that now v and v 0 are more exible: they are obtained by scanning the frontier of the right hand side of the rules and listing all objects of the form p(x i ). On the other hand, the trees (x 1 ; (x ; x 3 ))) and ((x 1 )); x ) to which the mappings 3 () q;v and 3 () q;v 0 are applied, respectively, have a much more restricted form than in the bottom-up case: they are linear in the variables and the variables occur ordered. The bottom-up tree representation and the top-down tree representation dene the bottom-up tree series transducer and the top-down tree series transducer, respectively, via the concept of the initial homomorphism h : T! (AhhT ii) Q1. With every tree series transducer, we associate two semantics: the tree to tree series semantics (t-ts semantics) which is a mapping of type T! AhhT ii, and the tree series to tree series semantics (ts-ts semantics), which is a mapping of type AhhT ii! AhhT ii. As in 3

4 the case of tree transducers (cf. [Eng75]) we dene certain important subclasses, like linear, relabeling, and homomorphism tree series transducers. The contents of this paper is as follows. In Section we develop the notions and basic denitions of bottom-up and top-down tree transducers, semirings, formal tree series; we tried to make this paper self contained. In Section 3 we introduce the concepts of bottom-up tree series transducers and of top-down tree series transducers, and we discuss an example in detail. In Section 4 we show that, using the boolean semiring, there is a one-to-one correspondence between tree series transducers and tree transducers (of the same type). In Section 5 we prove two characterization results: 1) polynomial bottom-up tree series transducers can be characterized by the composition of nite state relabeling tree series transducers and homomorphism tree series transducers, and ) deterministic top-down tree series transducers can be characterized by the composition of homomorphism tree series transducers and linear deterministic top-down tree series transducers. The proofs of the correctness of constructions is usually done by an induction on the tree structure. In order to save space, we only show the induction step, and not the induction base. Preliminaries.1 Sets and relations Let H be a set. Then P(H) is the powerset of H. The empty set is denoted by ;. Let F, G, and H be sets and let % F G and G H be two binary relations. The composition of % and is the binary relation % f(a; c) F H j there is a b G such that (a; b) % and (b; c) g. The notion of composition is extended to classes of relations. For two classes C 1 and C of relations we dene C 1 C f% j % C 1 and C g.. Strings If is an alphabet, then denotes the set of strings over, while the empty string is denoted by ". The length of a string w is dened in a standard way and denoted by jwj..3 Trees and tree transducers Let be a ranked alphabet. By (k) we denote the set of all symbols of which have rank k. Moreover, let H be a set disjoint with. The set of (nite, labeled and ordered) trees over indexed by H, denoted by T (H), is the smallest subset T of ( [ H [ f(; )g [ f; g), such that (i) H T and (ii) if (k) with k 0 and s 1 ; : : : ; s k T, then (s 1 ; : : : ; s k ) T. In case k 0, we identify ( ) with. Moreover, T (;) is denoted by T. It should be clear that T ; if and only if (0) ;. Since we are not interested in this particular case, we assume that (0) 6 ; for every ranked alphabet appearing as input or output ranked alphabet of some tree transducer in this paper. We will need the set X fx 1 ; x ; : : :g of variable symbols. For every k 0, we dene X k fx 1 ; : : : ; x k g, thus X 0 ;. We use the variables to occur in trees, so we will frequently consider the sets T (X), T (X k ), etc. of trees where is a ranked alphabet. The tree substitution is dened as follows. Let t T (X k ) for some ranked alphabet and k 0. Moreover, let t 1 ; : : : ; t k be also trees over (maybe other) ranked alphabets. Then t[x 1 t 1 ; : : : ; x k t k ], or shortly just t[t 1 ; : : : ; t k ], stands for the tree which is obtained from t by substituting, for every 1 i k, the tree t i for each occurrence of x i. A tree t T (X k ) is called linear if every variable x i, 1 i k occurs at most once in it. We distinguish a subset c T (X k ) of T (X k ) as follows. Let a tree t T (X k ) be in c T (X k ) if for every 1 i k, the variable x i occurs exactly once in t and, reading the leaves of t from left to right, the variables occur in the order x 1 < x < : : : < x n. Note that all elements of c T (X k ) are linear. Let t T (H). The linearization of t with respect to H, denoted by lin H (t), is dened as the unique pair (t 0 ; w) where t 0 c T (X k ) and w a 1 : : : a k H such that t t 0 [x 1 a 1 ; : : : ; x k a k ]. If Q is a unary ranked alphabet, i.e., the rank of all symbols in Q is 1, then Q(X k ) stands for the set fq(x i ) j q Q and 1 i kg. A string w (Q(X k )) is called linear if every variable x i, 1 i k occurs at most once in it. Next we shortly recall the concept of the bottom-up tree transducer and the top-down tree transducer. For more terminology and details about tree transducers the reader is advised to consult [Eng75]. 4

5 A bottom-up tree transducer is a tuple M (Q; ; ; Q d ; R) where Q is a unary ranked alphabet (of states), and are ranked alphabets (of input symbols and output symbols, resp.), Q d Q is the set (of nal states), and R is a nite set of rules of the form (q 1 (x 1 ); : : : ; q k (x k ))! q(t), where k 0, (k), q 1 ; : : : ; q k ; q Q, and t T (X k ). The derivation relation (induced by M) is the binary relation ) M T Q[[ T Q[[ dened by ' ) M i there is a c T (X 1 ), there is a rule (q 1 (x 1 ); : : : ; q k (x k ))! q(t) in R, there are t 1 ; : : : ; t k T such that ' [x 1 (q 1 (t 1 ); : : : ; q k (t k ))] and [x 1 q(t[x 1 t 1 ; : : : ; x k t k ])]. The tree transformation computed by M is the set M f(s; t) j s ) M q(t) for some q Q dg. The class of all tree transformations computed by bottom-up tree transducers is denoted by BOT tt. A top-down tree transducer is a tuple M (Q; ; ; Q d ; R) where Q is a unary ranked alphabet (of states), and are ranked alphabets (of input symbols and output symbols, resp.), Q d Q is the set (of initial states), and R is a nite set of rules of the form q((x 1 ; : : : ; x k ))! t, where q Q, k 0, (k), and t T (Q(X k )). The derivation relation (induced by M) is the binary relation ) M T Q[[ T Q[[ dened by ' ) M i there is a c T (X 1 ), there is a rule q((x 1 ; : : :; x k ))! t in R, there are s 1 ; : : :; s k T such that ' [x 1 q((s 1 ; : : : ; s k ))] and [x 1 t[x 1 s 1 ; : : : ; x k s k ]]. The tree transformation computed by M is the set M f(s; t) j q(s) ) M t for some q Q dg. The class of all tree transformations computed by top-down tree transducers is denoted by T OP tt..4 Semirings For a survey paper on the relevance of semirings and formal power series to formal languages and automata cf. [Kui97b]. A semiring is an algebraic structure A (A; +; ; 0; 1) with two operations sum + and product such that (A; +; 0) is a commutative monoid, (A; ; 1) is a monoid, and for every a; b; c A the following laws hold: (a + b) c a c + b c, a (b + c) a b + a c, and a 0 0 a 0. Whenever the operations (+ and ) and the neutral elements (0 and 1) are clear from the context, then we denote the semiring (A; +; ; 0; 1) just by A. We give some examples of semirings. The boolean semiring is the semiring IB (f0; 1g; _; ^; 0; 1) with disjunction and conjunction as sum and product, resp. Let be an alphabet. The semiring of formal languages (over ), denoted by Lang(), is the semiring (P( ); [; ; ;; f"g) is a semiring where is the usual concatenation of string languages. The semiring of natural numbers is the semiring N at (IN; +; ; 0; 1) with the usual addition and multiplication operations. Let (A; +; ; 0; 1) be a semiring. A is commutative if, for every a; b A, the equation a b b a holds. A is naturally ordered if the binary relation v on A is a partial order on A, where v is dened by a v b i there is a c A such that a + c b. A is complete if it is possible to dene sums for all families (a i j i I) of elements of A, where I is an arbitrary index set, such that the following three conditions are satised: (i) P i; a i 0, P ifjg a i a j, P ifj;kg a i a j + a k for j 6 k. (ii) P jj (P ii j a i ) P ii a i, if S jj I j I and I j \ I k ; for j 6 k. (iii) P ii (c a i) c ( P ii a i), P ii (a i c) ( P ii a i) c. A is!-continuous if the following three conditions holds. (i) A is naturally ordered. (ii) A is complete. (iii) If P 0in a i v c for all n IN, then P iin a i v c for all sequences (a i j i IN) in A and all c A. 5

6 .5 Matrices We recall some notions concerning matrices (cf. [Kui97b]). Let I; J and S be arbitrary sets. An I J matrix over S is a mapping M : I J! S. The set of all I J matrices over S is denoted by S IJ. An element M(i; j) S is also denoted by M i;j. If I (or J) is a singleton, then we identify it by f1g and we write just 1 for I (or J). Moreover in that case an element of the form M i;1 (or M 1;j ) is written just M i (or M j ). Let A be some structure in which the operations (innite) sum and multiplication are dened. Let M : I J! A be a matrix and a A. Then am is the I J matrix over A such that for every (i; j) I J, (am) i;j am i;j where is the multiplication of A. Moreover, for a family (M (k) j k K) of I J matrices over A we dene the I J matrix P kk M (k) by X X ( M (k) ) i;j kk kk(m (k) ) i;j :.6 Formal tree series Let be ranked alphabet and let (A; +; ; 0; 1) be a semiring. A formal tree series (over and A) is a mapping r : T! A. For every t T, the element r(t) A is called coecient of t and it is also denoted by (r; t); then the mapping itself is written as the formal sum ^PtT (r; t)t. The set of all formal tree series over and A is denoted by AhhT ii. Note that in this paper we make a distinction between the formal sum ^P which denotes a formal tree series, and the sum P which is an operation over some domain. Example.1 Let be a ranked alphabet and let mx maxfj j (j) 6 ;g. Moreover, let N fi j 0 i mxg. Consider the mapping occ : T! P(N ) which maps every tree to its set of occurrences. More precisely, for every t (0), occ(t) f"g, and for every t (t 1 ; : : : ; t k ) where (k), k 1 and s 1 ; : : :; s k T, occ(t) f"g [ S 1ik fig occ(t i). For example, if f () ; (0) g and t ((; ); ), then occ(t) f"; 1; 11; 1; g. Hence, occ is a formal tree series over the semiring Lang(N). In the rest of this paper A stands for a commutative and continuous semiring. Let a A and let r 1 ; r AhhT ii. The formal tree series a r 1 is the formal tree series ^PtT (a (r 1 ; t))t. Moreover, r 1 + r denotes the formal tree series ^PtT ((r 1 ; t) + (r ; t))t and, for every family (r i j i I) of formal tree series over and A, P ii r i ^PtT ( P ii (r i; t); t). Let r : T! A be a formal tree series. If there is an a A such that for every t T we have (r; t) a, then r is called the constant a and is denoted by ~a. The support of r is dened as the set supp(r) ft T j (r; t) 6 0g. Moreover, the formal tree series r is called a polynomial if supp(r) is nite and is called a singleton if supp(r) is a singleton. Hence a singleton is a polinomial. The set of all polynomials over and A is denoted by AhT i. A polynomial r AhT i is written as a 1 t 1 ^+ : : : ^+a n t n, where supp(r) ft 1 ; : : : ; t n g and, for every 1 i n, a i (r; t i ). Moreover, if a i 1, then a i t i is written as t i. Hence for t T either t denotes the tree in T or it denotes the formal tree series r : T! A such that (r; t) 1 and (r; s) 0 for every s T? ftg. However, it will always be clear from the context which meaning is intended. Let be a ranked alphabet. We dene the binary relation pick f(r; t) j r IBhT i; t supp(r)g IBhhT ii T. Note that pick is a set of pairs and not a set of coecients in IB. The class of relations pick for an arbitrary ranked alphabet is denoted by P ICK. Now we dene the substitution of formal tree series. This is a straightforward generalization of the IO-substitution of languages [ES77, ES78, EV85]. Let t T (X l ) and ~s (s 1 ; : : : ; s l ) (AhhT (X)ii) l. Dene t (i) If t x j, 1 j l, then t ~s s j. (ii) If t (0), then t ~s. (iii) If t 6 (0) [ X l, then t ~s ^Pt 1;:::;t lt(x) (s 1; t 1 ) : : :(s l ; t l )(t[t 1 ; : : : ; t l ]). ~s AhhT (X)ii as follows: Note that, for every t T, t ( ) t. Moreover, if s i ~0 for some 1 i l, then also t ~s ~0 no matter whether x i occurs in t or not. 6

7 Now P let r AhhT (X l )ii and ~s (s 1 ; : : : ; s l ) (AhhT (X)ii) l. Dene r ~s AhhT (X)ii by r ~s tt (X l) (r; t)(t ~s). Certainly ~0 ~s ~0. If the semiring A is the boolean semiring IB, then the substitution of formal tree series is the same as the IO-substitution on formal languages as dened in [ES77, ES78]. We note that in [Kui99] an OI-substitution of formal tree series is dened. The reason why we work with IO-substitution will be discussed at the end of Section 4..7 Tree series transformations A tree to tree series transformation (for short: t-ts transformation) is a mapping : T! AhhT ii. A tree series to tree series transformation (for short: ts-ts transformation) is a mapping : AhhT ii! AhhT ii. Every t-ts transformation can be extended in a natural way to a ts-ts transformation. Let : T! AhhT ii be a t-ts transformation. The extension of is the mapping e : AhhT ii! AhhT ii such that for every r AhhT ii, e(r) P st ((r; s)(s)). We also extend a mapping : T! AhhT ii IJ to e : AhhT ii! AhhT ii IJ. In this case for an r AhhT ii the extension e(r) is dened by the same equation but now it will be an I J matrix over AhhT ii. Let 1 : T! AhhT ii and : T! AhhT? ii be two t-ts transformations. Then dene the composition of 1 and, denoted by 1 ~, as the t-ts transformation 1 ~ : T! AhhT? ii where ( 1 ~ )(s) e ( 1 (s)) for every s T. We note that ~ is not a composition of relations in the sense we dened it in Subsection.1. The composition of two classes C 1 and C of t-ts transformations, denoted by C 1 ~ C is the class f 1 ~ j i C i ; 1 i g of t-ts transformations. There is a close relationship between the composition of t-ts transformations and ts-ts transformations which will be made clear in the next lemma. Lemma. Let 1 : T! AhhT ii, : T! AhhT? ii, and : T! AhhT? ii be three t-ts transformations. Then, if 1 ~, then e 1 e e : Proof. Let r AhhT ii. Then e(r) P st (r; s)(s) P st (r; s)( 1 ~ )(s) P st (r; s)e ( 1 (s)) P st (r; s)( P tt ( 1 (s); t) (t)) P st PtT ((r; s) 1 (s); t) (t) P tt ( P st (r; s) 1 (s); t) (t) P tt (e 1 (r); t) (t) e (e 1 (r)) (e 1 e )(r). For every ranked alphabet there is a particular tree series transformation : T! AhhT ii dened by (s) 1 s for every s T. We call the identity tree series transformation over and A. 3 Tree series transducers In this section we dene the notion of a bottom-up tree series transducer and top-down tree series transducer which are generalizations of the notions of bottom-up tree transducers and top-down tree transducers, resp., as they occur in the literature (cf. e.g. [Eng75]). The generalization consists of allowing the transducers to compute a formal tree series as output rather than only trees. Also, a top-down tree series transducer generalizes the concept of the tree transducer as it is dened in [Kui99], because the latter can only cover the nondeterministically simple case of top-down tree transducers [GS84] (cf. Fig. 1 where going north-east along an edge means to generalize from the boolean semiring to an arbitrary semiring, and going north-west along an edge means to generalize from the nondeterministically simple case to the arbitrary nondeterministic case). 7

8 term rewriting systems bottom-up trtr [Eng75] formal tree series bottom-up tree series transducers * [present paper] top-down tree series transducers * [present paper] } top-down trtr [Tha68, Eng75] * } nondeterministically simple top-down trtr (syntactical sublcass of top-down trtr) [GS84] tree transducers [Kui99] Figure 1: Generalization diagram. In the following let and be two ranked alphabets, and Q be a unary ranked alphabet. Moreover, let (A; +; ; 0; 1) be a commutative continuous semiring. Denition 3.1 A tree representation (over Q,,, and A) is a family ( k j k 0) of mappings k : (k) Q(Q(Xk ))! (AhhT (X)ii) such that only for nitely many indices (q; w) Q (Q(X k )), k () q;w 6 ~0. A tree representation is bottom-up if the following additional conditions hold. For every (k), q Q, and w (Q(X k )), if w 6 q 1 (x 1 ) : : : q k (x k ) for some q 1 ; : : :; q k Q, then k () q;w ~0. Moreover, if k () q;w 6 ~0, then k () q;w AhhT (X k )ii, A tree representation is top-down if the following additional condition holds. For every (k), q Q, and w (Q(X k )), if k () q;w 6 ~0, then k () q;w Ahhc T (X l )ii, where l jwj. Denition 3. Let be either a bottom-up tree representation or a top-down tree representation. For every k 0 and (k), induces the mapping k () : (AhhT (X)ii) Q1 : : : (AhhT (X)ii) Q1! (AhhT (X)ii) Q1 with k arguments. The mapping k () is dened for every P 1 ; : : : ; P k (AhhT (X)ii) Q1 and q Q as: k ()(P 1 ; : : :; P k ) q X wq 1(x i1 ):::q l(xil )(Q(X k)) k () q;w ((P i1 ) q1 ; : : : ; (P il ) ql ): Note that, for k 0, (Q(X k )) f"g. Hence, according to our convention about matrix notation, in case (0) we write 0 () q;" as 0 () q. Since ((AhhT (X)ii) Q1 ; ( k () j k 0; (k) )) and also ((AhhT ii) Q1 ; ( k () j k 0; (k) )) are -algebras, there is a unique homomorphism dened for every (k), t 1 ; : : : ; t k T by h : T! (AhhT ii) Q1 h ((t 1 ; : : : ; t k )) k ()(h (t 1 ); : : : ; h (t k )): The mapping h extends to f h : AhhT ii! (AhhT ii) Q1, for the denition see Subsection.7. 8

9 Denition 3.3 A bottom-up tree series transducer (over A) is a tuple M (Q; ; ; A; Q d ; ) where Q is a unary ranked alphabet (of states), and are ranked alphabets (of input symbols and output symbols, resp.), A is a semiring, Q d Q (set of nal states), and is a bottom-up tree representation over Q,,, and A. Denition 3.4 A top-down tree series transducer (over A) is a tuple M (Q; ; ; A; Q d ; ) where Q is a unary ranked alphabet (of states), and are ranked alphabets (of input symbols and output symbols, resp.), A is a semiring, Q d Q (set of initial states), and is a top-down tree representation over Q,,, and A. We associate with every bottom-up (and top-down) tree series transducer M two transformations, the t-ts transformation computed by M and the ts-ts transformation computed by M. These are dened as follows. Denition 3.5 Let M (Q; ; ; A; Q d ; ) be a bottom-up tree series transducer or a top-down tree series transducer. 1. The t-ts transformation computed by M is the mapping M : T! AhhT ii which is dened by M (s) P qq d h (s) q for every s T.. The ts-ts transformation computed by M is the extension f M : AhhT ii! AhhT ii of M, cf. Subsection.7. Note that f M (s) P qq d f h (s) q for every s T. Sometimes we use the notion of t-ts semantics (and ts-ts semantics) of M to speak about the t-ts transformation (and ts-ts transformation, respectively) computed by M. Now we give an example for a bottom-up tree series transducer. In the theory of term rewriting systems one is interested among others in sucient conditions for the property of termination (cf. e.g. [Jan97, BN98]). One such condition is expressed in the following implication. Let R be a term rewriting system, then R is terminating if, for every derivation t 1 ) R t ) R t 3 : : :, there is no pair i; j with i < j such that t i can be embedded homeomorphically into t j. For two trees s and t, t can be embedded homeomorphically into s (for short: s emb t) if t is obtained from s by dropping an arbitrary number of occurrences of symbols in s (cf. Denition 5.4. of [BN98]). Example 3.6 For the particular ranked alphabet f () ; (1) ; (0) g and s; t T, we have s emb t if t is obtained from s by dropping an arbitrary number of occurrences of 's or 's. The relation emb, as tree transformation, can be computed by the bottom-up tree transducer M drop (Q; ; ; Q d ; R drop ), i.e., Mdrop emb, where Q Q d fg and R drop consists of the following rules: ((x 1 ); (x ))! ((x 1 ; x )) ((x 1 ); (x ))! (x 1 ) ((x 1 ); (x ))! (x ) ((x 1 ))! ((x 1 )) ((x 1 ))! (x 1 )! (): Clearly, for a given tree s T there are (nitely) many trees t T such that s emb t. Also it is clear that, for a given tree t T, there are (innitely) many trees s T such that s emb t. Now we might be interested in the following question: given two trees s; t T, how many possibilities are there to embed t into s? For instance, consider the trees s (; (())) and t (; ()). Hence, s emb t. One way of obtaining t from s is to drop the at occurrence of s, another way to obtain t is to drop the at occurrence 1. As another example, consider t 0 s. Then s emb t 0 and there is one way of obtaining t 0 from s, viz. dropping no occurrence of a symbol. Finally, consider t 00 ((); ). Clearly, s 6 emb t 00 and hence, the number of dierent ways in which t 00 can be obtained from s is zero. Thus, we dene the t-ts transformation #emb : T! N athht ii such that, for every s; t T, n if s emb t and there are n dierent ways to embed t into s (#emb(s); t) 0 otherwise 9

10 Note that, (#emb(s); t) 6 0 i s emb t. Thus, e.g., #emb((; (()))) (; (())) ^+(; ()) ^+(; ) ^+(()) ^+() ^+: Now, based on M drop, we construct a bottom-up tree series transducer M # drop (Q; ; ; N at; Q d; ) such that M # drop #emb, where is dened as follows. () () ;(x1)(x ) (x 1 ; x ) ^+x 1 ^+x () 1 () ;(x1) (x 1 ) ^+x 1 () 0 () ;" Now let us compute M # drop (s) for our example tree s (; (())). M # (s) P qq d h (s) q P qfg h (s) q h (s). Since h is dened by induction on the structure of its argument, we rst compute h (), second h (()), third h ((())), and nally h ((; (()))). h () 0 ()() P w" 0() ;w () 0 () ;" () h (()) P 1 ()(h ()) w(x 1) 1() ;w (h () ) 0 () ;(x1) (h () ) ((x P 1 ) ^+x 1 ) () tt (X ((x 1) 1) ^+x 1 ; t)(t ()) ((x 1 ) ()) + (x 1 ()) ( ^PsT (; s)((x (X) 1)[s])) + ( ^PsT (; s)(x (X) 1[s])) () ^+ h ((()))... ((x 1 ) ^+x 1 ) (() ^+) tt (X ((x 1) 1) ^+x 1 ; t)(t (() ^+)) ((x 1 ) (() ^+)) + (x 1 (() ^+)) ( ^PsT (() (X) ^+; s)((x 1 )[s])) + ( ^PsT (() (X) ^+; s)(x 1 [s])) ((()) ^+) + (() ^+) (()) ^+ () ^+ h ((; (()))) ()(h (); h ((()))) w(x 1)(x ) () ;w (h () ; h ((())) ) () ;w (h () ; h ((())) ) ((x 1 ; x ) ^+x 1 ^+x ) (; (()) ^+ () ^+ ) tt (X ((x ) 1; x ) ^+x 1 ^+x ; t)(t (; (()) ^+ () ^+ )) 10

11 ((x 1 ; x ) (; (()) ^+ () ^+ )) + (x 1 (; (()) ^+ () ^+ )) + (x (; (()) ^+ () ^+ )) ^Ps 1;s T (X) (; s 1)((()) ^+ () ^+ ; s )((x 1 ; x )[s 1 ; s ]) + ^Ps 1;s T (X) (; s 1)((() ^+ () ^+ ; s )(x 1 [s 1 ; s ]) + ^Ps 1;s T (X) (; s 1)((()) ^+ () ^+ ; s )(x [s 1 ; s ]) ((; (()) ^+(; ()) ^+(; )) + + ((()) ^+() ^+) (; (())) ^+ (; ()) ^+ (; ) ^+(()) ^+() ^+ Thus, we have veried that M # (s) #emb(s) for the tree s (; (())). drop Next we dene some restricted versions of bottom-up tree series transducers. These are the same restrictions which are imposed on tree transducers, see e.g. [Eng75, Bak79, GS84]. Denition 3.7 Let M (Q; ; ; A; Q d ; ) be a bottom-up tree series transducer. 1. M is deterministic if for every k 0, (k), q 1 ; : : :; q k Q, there is at most one q Q, such that k () q;w 6 ~0 where w q 1 (x 1 ) : : : q k (x k ), and if k () q;w 6 ~0 for some q and w, then k () q;w is a singleton.. M is total if Q d Q and for every k 0, (k), q 1 ; : : : ; q k Q, there is at least one q Q such that k () q;w 6 ~0. 3. M is a bottom-up homomorphism tree series transducer if it is total deterministic and the set Q of states is a singleton. 4. M is linear if for every k 0, (k), q; q 1 ; : : :; q k Q, every t supp( k () q;q1(x 1):::q k(xk) ) is linear. 5. M is a bottom-up nite state relabeling tree series transducer if for every k 0, (k) and q; q 1 ; : : : ; q k Q, every t supp( k () q;q1(x 1):::q k(xk) ) has the form (x 1 ; : : : ; x k ) for some (k). 6. M is polynomial if for every k 0, (k), q; q 1 ; : : : ; q k Q, k () q;q1(x 1):::q k(xk) AhT (X)i. Notice that bottom-up nite state relabeling tree series transducers are polynomial. The class of t-ts transformations induced by bottom-up tree series transducers is denoted by BOT t?ts. The classes of t-ts transformations which correspond to the syntactic subclasses of Denition are denoted by d-bot t?ts, t-bot t?ts, h-bot t?ts, l-bot t?ts, r-bot t?ts, and p-bot t?ts, resp. If more than one syntactic restriction holds, then BOT t?ts is prexed by the string of corresponding letters, e.g. ltd-bot t?ts denotes the class of t-ts transformations induced by linear, total, deterministic bottom-up tree series transducers. If all the considered bottom-up tree series transducers are over the same semiring A, then the corresponding class of t-ts transformations is indexed by A as e.g. in ltd-bot t?ts (A). The class of ts-ts transformations induced by bottom-up tree series transducers is denoted by BOT ts?ts. The denotation of the classes of ts-ts transformations computed by subclasses of bottom-up tree series transducers is obtained from the denotation of the corresponding classes of t-ts transformations by replacing t? ts by ts? ts. Note that the bottom-up tree series transducer M # drop considered in Example 3.6 is in fact a bottomup homomorphism tree series transducer. In what follows, like in case of M # drop, we will denote the only state of a homomorphism tree series transducer by. Now we introduce the corresponding restrictions for top-down tree series transducers. Denition 3.8 Let M (Q; ; ; A; Q d ; ) be a top-down tree series transducer. 1. M is deterministic if the following conditions hold. Q d is a singleton. For every k 0, (k), q Q, there is at most one w (Q(X k )) such that k () q;w 6 ~0, and if k () q;w 6 ~0 for some q and w, then k () q;w is a singleton.. M is total if for every k 0, (k), q Q, there is at least one w (Q(X k )) such that k () q;w 6 ~0, and Q d 6 ;. 11

12 3. M is a top-down homomorphism tree series transducer if it is total deterministic and the set Q of states is a singleton. 4. M is linear if for every k 0, (k), q Q, w (Q(X k )), if k () q;w 6 ~0, then w is linear. 5. M is a top-down nite state relabeling tree series transducer if for every k 0, (k), q Q, w (Q(X k )), if k () q;w 6 ~0, then w q 1 (x 1 ) : : : q k (x k ) and every t supp( k () q;w ) has the form (x 1 ; : : : ; x k ) for some (k). 6. M is polynomial if for every k 0, (k), q Q, w (Q(X k )), if k () q;w AhT (X)i. Again like in the bottom-up case, top-down nite state relabeling tree series transducers are polynomial. The class of t-ts transformations induced by top-down tree series transducers is denoted by T OP t?ts. The classes of t-ts transformations which correspond to the syntactic subclasses of Denition are denoted by d-t OP t?ts, t-t OP t?ts, h-t OP t?ts, l-t OP t?ts, r-t OP t?ts, and p-t OP t?ts, resp. For combinations of syntactic restrictions and for the restriction to one semiring A we use a notation similar to that of the bottom-up case. The class of ts-ts transformations induced by top-down tree series transducers is denoted by T OP ts?ts and also here we use the same notation system as in the bottom-up case. We note that a tree transducer in the sense of [Kui97a] is a top-down tree series transducer in which the elements k of the tree representation have the particular form k : k! (AhhT (X k )ii) QQk where Q k Q(fx 1 g) : : : Q(fx k g). Finally, we show that, like in case of conventional tree transducers, the dierence between the bottom-up and the top-down approaches disappears for certain basic classes of tree series transducers (cf. Lemma 3. of [Eng75] and the remark after Denition 3.14 of [Eng75]). Lemma 3.9 1) h-t OP t?ts h-bot t?ts and h-t OP ts?ts h-bot ts?ts. ) r-t OP t?ts r-bot t?ts and r-t OP ts?ts r-bot ts?ts Proof. First we show that 1) holds. We prove only the rst statement because the second one follows immediately from the denition of the ts-ts semantics.. Let M (fg; ; ; A; fg; ) be a top-down homomorphism tree series transducer. We construct the bottom-up homomorphism tree series transducer M 0 (fg; ; ; A; fg; 0 ) in the following way. For every k 0, (k), and t T (X k ), let lin X (t) (t 0 ; x i1 : : : x il ). Now let us dene ( 0 k () ;(x 1):::(x k) ; t) ( k () ;(xi1 ):::(x il ); t 0 ). It is easy to see that M M 0, which proves the inclusion h-t OP t?ts h-bot t?ts. To show the converse of the inclusion, let us take a bottom-up homomorphism tree series transducer M (fg; ; ; A; fg; ). Now we construct the top-down homomorphism tree series transducer M 0 (fg; ; ; A; fg; 0 ) as follows. For every k 0, (k), sequence (x i1 ) : : :(x il ) where 1 i 1 ; : : : ; i l k, and tree t 0 c T (X l ), let t t 0 [x 1 x i1 ; : : : ; x l x il ]. Now let us dene ( 0 k () ;(x i1 ):::(x il ); t 0 ) ( k () ;(x1):::(x k) ; t). Again it is easy to see that M M 0, which proves h-bot t?ts h-t OP t?ts. The proof of ) is left to the reader, cf. the corresponding statement for tree transducers in [Eng75]. From now on we call a bottom-up homomorphism tree series transducers as well as a top-down homomorphism tree series transducer a homomorphism tree series transducers. Similarly we drop the notions "bottom-up" and "top-down" from bottom-up and top-down nite state relabeling tree series transducers. Let us denote the classes h-t OP t?ts and r-t OP t?ts (and h-t OP ts?ts and r-t OP ts?ts ) by HOM t?ts and QREL t?ts (HOM ts?ts and QREL ts?ts ), respectively. It is an exercise to show that for every ranked alphabet and semiring A is in any of the classes of tree to tree series transformations dened in this section. We will use this fact in Section 5. 4 Tree series transducers over IB In this section we show that, under the t-ts semantics, every bottom-up tree transducer can be simulated by a polynomial bottom-up tree series transducer over IB and vice versa, every polynomial bottom-up tree series transducer over IB can be simulated by a bottom-up tree transducer. The same correspondence holds for the top-down case. 1

13 4.1 The bottom-up case Denition 4.1 Let M (Q; ; ; Q d ; R) be a bottom-up tree transducer and M 0 (Q; ; ; IB; Q d ; ) be a polynomial bottom-up tree series transducer over IB. M and M 0 are related if the following equivalence is true: for every k 0, (k), t T (X k ), and q; q 1 ; : : : ; q k Q, ( k () q;q1(x 1):::q k(xk) ; t) 1 i there is a rule (q 1 (x 1 ); : : : ; q k (x k ))! q(t) in R: Lemma 4. Let M be a bottom-up tree transducer with and as input and as output ranked alphabet, respectively, and M 0 be a polynomial bottom-up tree series transducer such that M and M 0 are related. Then, M M 0 pick. Proof. Let M and M 0 be specied as in Denition 4.1. First, we can prove by induction on s that, for every s T, t T, and q Q the following equivalence holds: (h (s) q ; t) 1 i s ) M q(t). s (s 1 ; : : : ; s k ) with k 1: s ) M q(t) i (s 1 ; : : : ; s k ) ) M q(t) i there are q 1 ; : : : ; q k Q and t 1 ; : : : ; t k T such that (s 1 ; : : : ; s k ) ) M (q 1(t 1 ); : : : ; q k (t k )) ) M q(t) i there are q 1 ; : : : ; q k Q, t 1 ; : : : ; t k T, t 0 T (X k ), and there is a rule (q 1 (x 1 ); : : : ; q k (x k ))! q(t 0 ) in R such that i s i ) M q i(t i ) and t t 0 [t 1 ; : : : ; t k ] there are q 1 ; : : : ; q k Q, t 1 ; : : : ; t k T, t 0 T (X k ) such that ( k () q;q1(x1):::q k(xk) ; t 0 ) 1 and (h (s i ) qi ; t i ) 1 and t t 0 [t 1 ; : : : ; t k ] P i (( wq 1(x 1):::q k(xk) Pt 0 T (X k) ( k () q;w ; t 0 )( ^Pt 1;:::;t kt (h (s 1 ) q1 ; t 1 ) : : :(h (s k ) qk ; t k )t 0 [t 1 ; : : :; t k ])) q ; t) 1 i (( wq 1(x 1):::q k(xk) Pt 0 T (X k) i ( k () q;w ; t 0 )(t 0 (h (s 1 ) q1 ; : : : ; h (s k ) qk ))) q ; t) 1 (( wq 1(x 1):::q k(xk) k () q;w (h (s 1 ) q1 ; : : : ; h (s k ) qk )) q ; t) 1 i ( k ()(h (s 1 ); : : : ; h (s k )) q ; t) 1 i (h (s) q ; t) 1. Second, we can prove that M M 0 pick. Let s T and t T. (s; t) M i 9q Q d : s ) M q(t) i 9q Q d : (h (s) q ; t) 1 i 9q Q d : (h (s) q ; t) pick i ( qq d h (s) q ; t) pick i ( M 0(s); t) pick i (s; t) M 0 pick Lemma 4.3 For every bottom-up tree transducer M there is a polynomial bottom-up tree series transducer M 0 such that M and M 0 are related. Proof. Let M (Q; ; ; Q d ; R) be a bottom-up tree transducer. Construct the bottom-up tree series transducer M 0 (Q; ; ; IB; Q d ; ) with the bottom-up tree representation over Q,,, and IB as follows: for every k 0, (k), t T (X k ), and q; q 1 ; : : : ; q k Q, ( k () q;q1(x 1):::q k(xk) ; t) 1 if there is a rule (q 1 (x 1 ); : : : ; q k (x k ))! q(t) in R, and ( k () q;q1(x 1):::q k(xk) ; t) 0 if there is no such rule. Obviously, M 0 is a polynomial, and M and M 0 are related. Example 4.4 (Cf. Example.1 of [Eng75].) Consider the bottom-up tree transducer M (Q; ; ; Q d ; R) where Q fqg, f (1) ; (1) ; (0) g, f () ; (1) 1 ; (1) ; (0) g, Q d fqg, and R consists of the following rules: 13

14 (q(x 1 ))! q((x 1 ; x 1 )) (q(x 1 ))! q( 1 (x 1 )) (q(x 1 ))! q( (x 1 ))! q() This is an example of Property (B1) in [Eng75]: a bottom-up tree transducer is able to process a subtree of an input tree in a nondeterministic way and copy the results of this process. Using Lemma 4.3 we construct the bottom-up tree series transducer M 0 (Q; ; ; IB; Q d ; ) as follows: 1 () q;q(x1) (x 1 ; x 1 ) 1 () q;q(x1) 1 (x 1 ) ^+ (x 1 ) 0 () q;" Then, according to Lemmas 4.3 and 4., M 0 is the bottom-up tree series transducer such that M M 0pick. Note that M 0 is not deterministic, hence is not a homomorphismtree series transducer. Let us compute now in detail h (), h (), and h (), and eventually M 0(r) for the polynomial formal tree series r. h () q 0 ()() q 0 () q;" h () q 1 ()(h ()) q P wp 1(x i1 ):::p l(xil )Q(fx 1g) 1() q;w (h () p1 ; : : : ; h () pl ) 1 () q;q(x1) (h () q ) ( 1 (x 1 ) ^+ (x 1 )) () 1 () ^+ (): h () q 1 ()(h ()) q 1 () q;q(x1) (h () q ) (x 1 ; x 1 ) ( 1 () ^+ ()) ^Pt 1T (X) ( 1() ^+ (); t 1 )(x 1 ; x 1 )[t 1 ] ( 1 ; 1 ) ^+( ; ). Finally, M 0() P qq d h () q h () q ( 1 ; 1 ) ^+( ; ). Lemma 4.5 For every polynomial bottom-up tree series transducer M over IB there is a bottom-up tree transducer M 0 such that M 0 and M are related. Proof. Let M (Q; ; ; IB; Q d ; ) be a polynomial bottom-up tree series transducer. Construct the bottom-up tree transducer M 0 (Q; ; ; Q d ; R) as follows. For every k 0, (k), t T (X k ), and q; q 1 ; : : :; q k Q, the rule (q 1 (x 1 ); : : :; q k (x k ))! q(t) is in R if ( k () q;q1(x 1):::q k(xk) ; t) 1. Note that, since M is polynomial, R is a nite set. Obviously, M and M 0 are related. From Lemmas 4., 4.3, and 4.5 immediately a characterization of bottom-up tree transducers in terms of polynomial bottom-up tree series transducers follows. Theorem 4.6 p-bot t?ts (IB) P ICK BOT tt. 14

15 4. The top-down case Next we show that top-down tree transducers (as they occur in the literature) are characterized by polynomial top-down tree series transducers over the boolean semiring IB. Denition 4.7 Let M (Q; ; ; Q d ; R) be a top-down tree transducer and let M 0 (Q; ; ; IB; Q d ; ) be a polynomial top-down tree series transducer over IB. M and M 0 are related if the following equivalence is true: for every k 0, (k), q Q, w (Q(X k )), t 0 c T (X l ) where l jwj, ( k () q;w ; t 0 ) 1 i there is a rule q((x 1 ; : : : ; x k ))! t in R such that lin Q(X) (t) (t 0 ; w): Lemma 4.8 Let M be a top-down tree transducer with and as input and as output ranked alphabets, respectively, and M 0 be a polynomial top-down tree series transducer such that M and M 0 are related. Then, M M 0 pick holds. Proof. Let M and M 0 be specied as in Denition 4.7. We can prove that, for every s T, t T, and q Q the following equivalence holds: (h (s) q ; t) 1 i q(s) ) M t. s (s 1 ; : : : ; s k ) with k 1: q(s) ) M t i there are q 1 ; : : : ; q l Q, i 1 ; : : :; i l f1; : : :; kg, t 1 ; : : : ; t l T, and t 0 T c(x l ) such that q((s 1 ; : : : ; s k )) ) M t 0 [q 1 (s i1 ); : : : ; q l (s il )] ) M t0 [t 1 ; : : : ; t l ] t i there are q 1 ; : : : ; q l Q, i 1 ; : : :; i l f1; : : :; kg, t 1 ; : : : ; t l T, and t 0 T c(x l ) such that q((x 1 ; : : : ; x k ))! t 0 [q 1 (x i1 ); : : : ; q l (x il )] is in R and q j (s ij ) ) M t j and t 0 [t 1 ; : : :; t l ] t i there are q 1 ; : : : ; q l Q, i 1 ; : : :; i l f1; : : :; kg, t 1 ; : : : ; t l T, and t 0 T c(x l ) such that ( P k () q;q1(xi1 ):::q l(xil ); t 0 ) 1 and (h (s ij ) qj ; t j ) 1 and t 0 [t 1 ; : : : ; t l ] t P i ( wq 1(x i1 ):::q l(xil ) t 0 T (X l) ( k () q;w ; t 0 ) ^Pt P 1;:::;t (h lt(x) (s i1 ) q1 ; t 1 ) : : :(h (s il ) ql ; t l )t 0 [t 1 ; : : : ; t l ]; t) 1 i ( P P wq 1(x i1 ):::q l(xil ) t 0 T (X l) ( k () q;w ; t 0 )(t 0 (h (s i1 ) q1 ; : : : ; h (s il ) ql ); t) 1 i ( wq 1(x i1 ):::q l(xil ) k() q;w (h (s i1 ) q1 ; : : : ; h (s il ) ql ); t) 1 i ( k ()(h (s 1 ); : : : ; h (s k )) q ; t) 1 i (h (s) q ; t) 1. At the step which is marked by we have used the fact that the top-down tree representation ( k () q;q1(x i1 ):::q l(xil ); t 0 ) 0 if t 0 6 c T (X l ). The equation M M 0 pick can be veried in the same way as in the proof of Lemma 4.. Lemma 4.9 For every top-down tree transducer M there is a polynomial top-down tree series transducer M 0 such that M and M 0 are related. Proof. Let M (Q; ; ; Q d ; R) be a top-down tree transducer. Construct the top-down tree series transducer M 0 (Q; ; ; IB; Q d ; ) with the top-down tree representation over Q,,, and IB as follows: for every k 0, (k), q Q, w (Q(X k )), t 0 T c (X l ) where l jwj, dene ( k () q;w ; t 0 ) 1 if there is a rule q((x 1 ; : : : ; x k ))! t in R such that lin Q(X) (t) (t 0 ; w), otherwise ( k () q;w ; t 0 ) 0. Obviously, M 0 is and polynomial. Moreover, M and M 0 are related. Example 4.10 (cf. Example. of [Eng75].) Consider the top-down tree transducer M (Q; ; ; Q d ; R) where Q fqg, f (1) ; (1) ; (0) g, f () ; (1) 1 ; (1) ; (0) g, Q d fqg, and R consists of the following rules: q((x 1 ))! (q(x 1 ); q(x 1 )) q((x 1 ))! 1 (q(x 1 )) q((x 1 ))! (q(x 1 )) q()! 15

16 This exhibits an example of the property (T) of [Eng75]: a top-down tree transducer is able to copy a subtree of its input tree and process the copies dierently. By Lemma 4.9 we construct the top-down tree representation over Q,,, and IB as follows: 1 () q;q(x1)q(x 1) (x 1 ; x ) 1 () q;q(x1) 1 (x 1 ) ^+ (x 1 ) 0 () q;" Then, according to Lemmas 4.8 and 4.9, M 0 (Q; ; ; IB; Q d ; ) is the top-down tree series transducer such that M M 0 pick. Let us again compute in detail h (), h (), and h (), and eventually M 0(r) for the polynomial formal tree series r. h () q 0 ()() q 0 () q;" h () q 1 ()(h ()) q P wp 1(x i1 ):::p l(xil )Q(fx 1g) 1() q;w (h () p1 ; : : : ; h () pl ) 1 () q;q(x1) (h () q ) ( 1 (x 1 ) ^+ (x 1 )) () P tt (X 1) ( 1(x 1 ) ^+ (x 1 ); t) t () ( 1 (x 1 ) ()) ^+ ( (x 1 ) ()) 1 () ^+ (): h () q 1 ()(h ()) q P wp 1(x i1 ):::p l(xil )Q(fx 1g) 1() q;w (h () p1 ; : : : ; h () pl ) 1 () q;q(x1)q(x 1) (h () q ; h () q ) (x 1 ; x ) ( 1 () ^+ (); 1 () ^+ ()) ^Pt 1;t T (X) ( 1() ^+ (); t 1 )( 1 () ^+ (); t )(x 1 ; x )[t 1 ; t ] ( 1 ; 1 ) ^+( 1 ; ) ^+( ; 1 ) ^+( ; ). Finally, M 0() P qq d h () q h () q ( 1 ; 1 ) ^+( 1 ; ) ^+( ; 1 ) ^+( ; ). Lemma 4.11 For every polynomial top-down tree series transducer M over IB there is a top-down tree transducer M 0 such that M and M 0 are related. Proof. Let M (Q; ; ; IB; Q d ; ) be a polynomial top-down tree series transducer. Construct the top-down tree transducer M 0 (Q; ; ; Q d ; R) as follows. Let R be the smallest set of rules satisfying the condition that for every k 0, (k), q Q, w q 1 (x i1 ) : : : q l (x il ) (Q(X k )), and t 0 c T (X l ) with ( k () q;w ; t 0 ) 1, the rule q((x 1 ; : : : ; x k ))! t is in R, where t t 0 [x 1 q 1 (x i1 ); : : : ; x l q l (x il )]. (Note that lin Q(X) (t) (t 0 ; w).) Again, since M is polynomial, R is a nite set. Obviously, M and M 0 are related. Thus, it follows from Lemmas 4.8, 4.9, and 4.11 that also for the top-down case we obtain a characterization of top-down tree transducers in terms of polynomial top-down tree series transducers. Theorem 4.1 p-t OP t?ts (IB) P ICK T OP tt. We would like to nish this section by comparing the embedding of top-down tree transducers into top-down tree series transducers as done in our Lemma 4.9 with the embedding of nondeterministically simple root-to-frontier tree transducers into tree transducers as done on page 139 of [Kui99]. This comparison is reasonable, because nondeterministically simple root-to-frontier transducers are particular top-down tree transducers, and tree transducers of [Kui99] are particular top-down tree series transducers. 16