Unifying Synchronous Tree-Adjoining Grammars and Tree Transducers via Bimorphisms

Transcription

1 Unifying ynchronous Tree-Adjoining Grmmrs nd Tree Trnsducers vi Bimorphisms turt M. hieer Division of Engineering nd Applied ciences Hrvrd University Cmridge, MA, UA turt M. hieer Unifying ynchronous Tree-Adjoining Grmmrs nd Tree Trnsducers vi Bimorphisms. In Proceedings of the 11th Conference of the Europen Chpter of the Assocition for Computtionl Linguistics (EACL-06), Trento, Itly, 3 7 April. Astrct We plce synchronous tree-djoining grmmrs nd tree trnsducers in the single overrching frmework of imorphisms, continuing the unifiction of synchronous grmmrs nd tree trnsducers initited y hieer (2004). Along the wy, we present new definition of the tree-djoining grmmr derivtion reltion sed on novel direct inter-reduction of TAG nd mondic mcro tree trnsducers. Tree trnsformtion systems such s tree trnsducers nd synchronous grmmrs hve seen renewed interest, sed on perceived relevnce to new pplictions, such s importing syntctic structure into sttisticl mchine trnsltion models or founding formlism for speech commnd nd control. The exct reltionship mong vriety of formlisms hs een uncler, with lrge numer of seemingly unrelted formlisms eing independently proposed or chrcterized. An initil step towrd unifying the formlisms ws tken (hieer, 2004) in mking use of the formllnguge-theoretic device of imorphisms, previously used to chrcterize the tree reltions definle y tree trnsducers. In prticulr, the tree reltions definle y synchronous tree-sustitution grmmrs (TG) were shown to e just those definle y liner complete imorphisms, therey providing for the first time cler reltionship etween synchronous grmmrs nd tree trnsducers. In this work, we show how the imorphism frmework cn e used to cpture more powerful formlism, synchronous tree-djoining grmmrs, providing further uniting of the vrious nd disprte formlisms. After some preliminries (ection 1), we egin y reclling the definition of tree-djoining grmmrs nd synchronous tree-djoining grmmrs (ection 2). We turn then to set of known results relting context-free lnguges, tree homomorphisms, tree utomt, nd tree trnsducers to extend them for the tree-djoining lnguges (ection 3), presenting these in terms of restricted kinds of functionl progrms over trees, using simple grmmticl nottion for descriing the progrms. This llows us to esily express generliztions of the notions: mondic mcro tree homomorphisms, utomt, nd trnsducers, which er (t lest some of) the sme interreltionships tht their trditionl simpler counterprts do (ection 4). Finlly, we use this chrcteriztion to plce the synchronous TAG formlism in the imorphism frmework (ection 5), further unifying tree trnsducers nd other synchronous grmmr formlisms. We lso, in pssing, provide new chrcteriztion of the reltion etween TAG derivtion nd derived trees, nd new simpler nd more direct proof of the equivlence of TALs nd the output lnguges of mondic mcro tree trnsducers. 1 Preliminries We will notte sequences with ngle rckets, e.g.,,,c, or where no confusion results, simply s c, with the empty string written ε. Trees will hve nodes leled with elements of RANKED ALPHABET, set of symols F, ech with non-negtive integer RANK or ARITY ssigned to it, determining the numer of children for nodes so leled. To emphsize the rity of symol, we will write it s prenthesized superscript, for instnce f (n) for symol f of rity n. Anlogously, we write F (n) for the set of symols in F with rity n. ymols with rity zero (F (0) ) re clled NULLARY symols or CON-

2 TANT. The set of nonconstnts is written F ( 1). To express incomplete trees, trees with holes witing to e filled, we will llow leves to e leled with vriles, in ddition to nullry symols. The set of TREE OVER A RANKED AL- PHABET F AND VARIABLE X, notted T(F,X), is the smllest set such tht (i) f T(F,X) for ll f F (0) ; (ii) x T(F,X) for ll x X; nd (iii) f (t 1,...,t n ) T(F,X) for ll f F ( 1), nd t 1,...,t n T(F,X). We revite T(F, /0), where the set of vriles is empty, s T(F), the set of GROUND TREE over F. We will lso mke use of the set of n numericlly ordered vriles X n = {x 1,...,x n }, nd write x, y, z s synonyms for x 1, x 2, x 3, respectively. Trees cn lso e viewed s mppings from TREE ADDREE, sequences of integers, to the lels of nodes t those ddresses. The ddress ε is the ddress of the root, 1 the ddress of the first child, 12 the ddress of the second child of the first child, nd so forth. We will use the nottion t/p to pick out the sutree of the node t ddress p in the tree t. Replcing the sutree of t t ddress p y tree t, written t[p t ] is defined s (using for the insertion of n element on list) t[ε t ] = t f (t 1,...,t n )[(i p) t ] = f (t 1,...,t i [p t ],...,t n ) for 1 i n. The HEIGHT of tree t, notted height(t), is defined s follows: height(x) = 0 for ll x X nd height( f (t 1,...,t n )) = 1+mx n i=1 height(t i) for ll f F. We cn use trees with vriles s CONTEXT in which to plce other trees. A tree in T(F,X n ) will e clled context, typiclly denoted with the symol C. For context C T(F,X n ) nd sequence of n trees t 1,...,t n T(F), the UBTITU- TION OF t 1,...,t n INTO C, notted C[t 1,...,t n ], is defined inductively s follows: ( f (u 1,...,u m ))[t 1,...,t n ] = f (u 1 [t 1,...,t n ],...,u m [t 1,...,t n ]) x i [t 1,...,t n ] = t i. A tree t T(F,X) is LINEAR if nd only if no vrile in X occurs more thn once in t. We will use nottion kin to BNF to specify equtions defining functionl progrms of vrious sorts. As n introduction to the nottion we will use, here is grmmr defining trees over rnked lphet nd vriles (essentilly identiclly to the definition given ove): f (n) F (n) x X ::= x 0 x 1 x 2 t T(F,X) ::= f (m) (t 1,...,t m ) x The nottion llows definition of clsses of expressions (e.g., F (n) ) nd specifies metvriles over them ( f (n) ). These clsses cn e primitive (F (n) ) or defined (X), even inductively in terms of other clsses or themselves (T(F,X)). We use the metvriles nd suscripted vrints on the right-hnd side to represent n ritrry element of the corresponding clss. Thus, the elements t 1,...,t m stnd for ritrry trees in T(F,X), nd x n ritrry vrile in X. Becuse numericlly suscripted versions of x pper explicitly on the right hnd side of the rule defining vriles, numericlly suscripted vriles (e.g., x 1 ) on the right-hnd side of ll rules re tken to refer to the specific elements of x, wheres otherwise suscripted elements (e.g., x i ) re tken genericlly. 2 Tree-Adjoining Grmmrs Tree djoining grmmr (TAG) is tree grmmr formlism distinguished y its use of tree djunction opertion. Trditionl presenttions of TAG, which we will ssume fmilirity with, tke the symols in elementry nd derived trees to e unrnked; nodes leled with given nonterminl symol my hve differing numers of children. (Joshi nd ches (1997) present good overview.) For exmple, foot nodes of uxiliry trees nd sustitution nodes hve no children, wheres the similrly leled root nodes must hve t lest one. imilrly, two nodes with the sme lel ut differing numers of children my mtch for the purpose of llowing n djunction (s the root nodes of α 1 nd β 1 in Figure 1). In order to integrte TAG with tree trnsducers, however, we move to rnked lphet, which presents some prolems nd opportunities. (In some wys, the rnked lphet definition of TAGs is slightly more elegnt thn the trditionl one.) Although the ulk of the lter discussion integrting TAGs nd trnsducers ssumes (without loss of expressivity (Joshi nd ches, 1997, fn. 6)) limited form of TAG tht includes djunction ut not sustitution, we define the more complete form here. We will thus tke the nodes of TAG trees to e leled with symols from rnked lphet F; given symol then hs fixed rity nd fixed

3 α 1 : α 2 : T β 1 : /0 β 2 : /0 α 1 : 1 β 1 : /0 β 2 : /0 ε : T c T 1 1 c Figure 1: mple TAG for the copy lnguge {wcw w {,} }. Figure 2: mple core-restricted TAG for the copy lnguge {wcw w {,} }. numer of children. However, in order to mintin informtion out which symols my mtch for the purpose of djunction nd sustitution, we tke the elements of F to e explicitly formed s pirs of n unrnked lel e nd n rity n. (For nottionl consistency, we will use e for unrnked nd f for rnked symols.) We will notte these elements, using nottion, s e (n), nd mke use of function to unrnk symols in F, so tht e (n) = e. To hndle foot nodes, for ech non-nullry symol e (i) F ( 1), we will ssocite new nullry symol e, which one cn tke to e the pir of e nd ; the set of such symols will e notted F. imilrly, for sustitution nodes, F will e the set of nullry symols e for ll e (i) F ( 1). These dditionl symols, since they re nullry, will necessrily pper only t the frontier of trees. Finlly, to llow null djoining constrints, for ech f F (i), we introduce symol f /0 lso of rity i, nd tke F /0 to e the set of ll such symols. We will extend the function to provide the unrnked symol ssocited with these symols s well, so e = e = e (i) /0 = e. A TAG is then qudruple F,,I,A, where F is rnked lphet; F is distinguished initil symol; I is the set of initil trees, finite suset of T(F F /0 F ); nd A is the set of uxiliry trees, finite suset of T(F F /0 F F ). An uxiliry tree β whose root is leled f must hve exctly one node leled with f F nd no other nodes leled in F ; this node is its foot node, its ddress notted foot(β). In Figure 1, α 1 nd α 2 re initil trees; β 1 nd β 2 re uxiliry trees. In order to llow reference to prticulr tree in the set P, we ssocite with ech tree in P unique index, conventionlly notted with suscripted α or β for initil nd uxiliry trees respectively. This further llows us to hve multiple instnces of tree in I or A, distinguished y their index. (We will use nottion y using the index nd the tree tht it nmes interchngly.) The trees re comined y two opertions, sustitution nd djunction. Under sustitution, node leled e (t ddress p) in tree α cn e replced y n initil tree α with the corresponding lel f t the root when f = e. The resulting tree, the sustitution of α t p in α, is α[p α ]. Under djunction, n internl node of α t p leled f F is split prt, replced y n uxiliry tree β rooted in f when f = f. The resulting tree, the djunction of β t p in α, is α[p β[foot(β) α/p]]. This definition (y requiring f to e in F, not F or F ) mintins the stndrd convention, without loss of expressivity, tht djunction is disllowed t foot nodes nd sustitution nodes. The TAG in Figure 1 genertes tree set whose yield is the non-context-free copy lnguge {wcw w {,} }. The rities of the nodes re suppressed, s they re cler from context. A derivtion tree D records the opertions over the elementry trees used to derive given derived tree. Ech node in the derivtion tree specifies n elementry tree α, the node s child sutrees D i recording the derivtions for trees tht re djoined or sustituted into tht tree. A method is required to record t which node in α the tree specified y child sutree D i opertes. For trees recording derivtions in context-free grmmrs, there re exctly s mny sustitution opertions s nonterminls on the right-hnd side of the rule used. Thus, child order in the derivtion tree cn e used to record the identity of the sustitution node. But for TAG trees, opertions occur throughout the tree, nd some, nmely djunctions, cn e optionl, so simple convention using child order is not possile. Trditionlly, the rnches in the derivtion tree hve een notted with the ddress of the node in the prent tree t which the child node opertes. Figure 4 presents derivtion tree () using this nottion, long with the corresponding derived tree () for the string c. For simplicity elow, we use stripped down TAG formlism, one tht loses no expressivity in wek genertive cpcity ut is esier for nlysis purposes. First, we mke ll djunction oligtory, in the

4 B /0 1 3 A B A 2 B cn define trnsducer s kind of functionl progrm, tht is, set of equtions chrcterized y the following grmmr for equtions Eqn. (The set of sttes is conventionlly notted Q, with memers notted q. One of the sttes is distinguished s the INITIAL TATE of the trnsducer.) 1 Figure 3: mple TAG tree mrked with dicritics to show the permuttion of operle nodes. sense tht if node in tree llows djunction, n djunction must occur there. To get the effect of optionl djunction, for instnce t node leled B, we dd vestigil tree of single node ε B = B, which hs no djunction sites nd does not itself modify ny tree tht it djoins into. It thus founds the recursive structure of derivtions. econd, now tht it is determinte whether n opertion must occur t node, the numer of children of node in derivtion tree is determined y the elementry tree t tht node; it is just the numer of djunction or sustitution nodes in the tree, the OPERABLE NODE. All tht is left to determine is the mpping etween child order in the derivtion tree nd node in the elementry tree leling the prent, tht is, permuttion π on the operle nodes (or equivlently, their ddresses), so tht the i-th child of node leled α in derivtion tree is tken to specify the tree tht opertes t the node π i in α. This permuttion cn e thought of s specified s prt of the elementry tree itself. For exmple, the tree in Figure 3, which requires opertions t the nodes t ddresses ε, 12, nd 2, my e ssocited with the permuttion 12,2,ε. This permuttion cn e mrked on the tree itself with numeric dicritics i, s shown in the figure. Finlly, s mentioned efore, we eliminte sustitution (Joshi nd ches, 1997, fn. 6). With these chnges, the smple TAG grmmr nd derivtion tree of Figures 1 nd 4() might e expressed with the core TAG grmmr nd derivtion tree of Figures 2 nd 4(c). 3 Tree Trnsducers, Homomorphisms, nd Automt 3.1 Tree Trnsducers Informlly, TREE TRANDUCER is function from T(F) to T(G) defined such tht the symol t the root ofthe input tree nd current stte determines n output context in which the recursive imges of the sutrees re plced. Formlly, we q Q f (n) F (n) g (n) G (n) x i X ::= x 0 x 1 x 2 Eqn ::= q( f (n) (x 1,...,x n )) = τ (n) τ (n) R (n) ::= g (m) (τ (n) 1,...,τ(n) m ) q j (x i ) where 1 i n Intuitively speking, the expressions in R (n) re right-hnd-side terms using vriles limited to the first n. For exmple, the grmmr llows definition of the following set of equtions defining tree trnsducer: 2 q( f (x)) = g(q (x),q(x)) q() = q ( f (x)) = f (q (x)) q () = This trnsducer llows for the following derivtion: q( f ( f ()) = g(q ( f (),q( f ()))) = g( f (q ()),g(q (),q())) = g( f (),g(,)) The reltion defined y tree trnsducer with initil stte q is { t,u q(t) = u}. By virtue of nondeterminism in the equtions, multiple equtions for given stte q nd symol f, tree trnsducers define true reltions rther thn merely functions. TREE HOMOMORPHIM re sutype of tree trnsducers, those with only single stte, hence essentilly stteless. Other sutypes of tree trnsducers cn e defined y restricting the trees τ 1 trictly speking, wht we define here re nondeterministic top-down tree trnsducers. 2 Full definitions of tree trnsducers typiclly descrie trnsducer in terms of set of sttes, n input nd output rnked lphet, nd n initil stte, in ddition to the set of trnsitions, tht is, defining equtions. We will leve off these detils, in the expecttion tht the sets of sttes nd symols cn e inferred from the equtions, nd the initil stte determined under convention tht it is the stte defined in the textully first eqution. Note lso tht we vil ourselves of consistent renming of the vriles x 1, x 2, nd so forth, where convenient for redility.

5 tht form the right-hnd sides of equtions, the elements of R (n) used. A trnsducer is LINEAR if ll such τ re liner; is COMPLETE if τ contins every vrile in X n ; is ε-free if τ X n ; is YMBOL-TO-YMBOL if height(τ) = 1; nd is DELABELING if τ is complete, liner, nd symolto-symol. Another sucse is TREE AUTOMATA, tree trnsducers tht compute prtil identity function; these re deleling tree trnsducers tht preserve the lel nd the order of rguments. Becuse they compute only the identity function, tree utomt re of interest for their domins, not the mppings they compute. Their domins define tree lnguges, in prticulr, the so-clled REGU- LAR TREE LANGUAGE. 3.2 The Bimorphism Chrcteriztion of Tree Trnsducers Tree trnsducers cn e chrcterized directly in terms of equtions defining simple kind of functionl progrm, s ove. There is n elegnt lterntive chrcteriztion of tree trnsducers in terms of constelltion of elements of the vrious sutypes of trnsducers homomorphisms nd utomt we hve introduced, clled imorphism. A imorphism is triple L,h i,h o, consisting of regulr tree lnguge L (or, equivlently, tree utomton) nd two tree homomorphisms h i nd h o. The tree reltion defined y imorphism is the set of tree pirs tht re generle from elements of the tree lnguge y the homomorphisms, tht is, L( L,h i,h o ) = { h i (t),h o (t) t L}. We cn limit ttention to imorphisms in which the input or output homomorphisms re restricted to certin type, liner (L), complete (C), epsilonfree (F), symol-to-symol (), deleling (D), or unrestricted (M). We will write B(I, O) where I nd O chrcterize suclss of homomorphisms for the set of imorphisms for which the input homomorphism is in the suclss indicted y I nd the output homomorphism is in the suclss indicted y O. Thus, B(D,M) is the set of imorphisms for which the input homomorphism is deleling ut the output homomorphism cn e ritrry. The tree reltions definle y tree trnsducers turn out to e exctly this clss B(D,M) (Comon et l., 1997). The imorphism notion thus llows us to chrcterize the tree trnsductions purely in terms of tree utomt nd tree homomorphisms. We hve shown (hieer, 2004) tht the tree reltions defined y synchronous tree-sustitution grmmrs were exctly the reltions B(LC, LC). Intuitively speking, the tree lnguge in such imorphism represents the set of derivtion trees for the synchronous grmmr, nd ech homomorphism represents the reltion etween the derivtion tree nd the derived tree for one of the projected tree-sustitution grmmrs. The homomorphisms re liner nd complete ecuse the tree reltion etween tree-sustitution grmmr derivtion tree nd its ssocited derived tree is exctly liner complete tree homomorphism. To chrcterize the tree reltions defined y synchronous tree-djoining grmmr, it similry suffices to find simple homomorphism-like chrcteriztion of the tree reltion etween TAG derivtion trees nd derived trees. In ection 5 elow, we show tht liner complete emedded tree homomorphisms, which we introduce next, serve this purpose. 4 Emedded Tree Trnsducers Emedded tree trnsducers re generliztion of tree trnsducers in which sttes re llowed to tke single dditionl rgument in restricted mnner. They correspond to restrictive sucse of mcro tree trnsducers with one recursion vrile. We use the term emedded tree trnsducer rther thn the more cumersome mondic mcro tree trnsducer for revity nd y nlogy with emedded pushdown utomt (ches nd Vijy-hnker, 1990), nother utomt-theoretic chrcteriztion of the tree-djoining lnguges. We modify the grmmr of trnsducer equtions to dd n extr rgument to ech occurrence of stte q. To highlight the specil nture of the extr rgument, it is written in ngle rckets efore the input tree rgument. We uniformly use the otherwise unused vrile x 0 for this rgument in the left-hnd side, nd dd x 0 s possile right-hnd side itself. Finlly, right-hnd-side occurrences of sttes my e pssed n ritrry further righthnd-side tree in this rgument. q Q f (n) F (n) x i X ::= x 0 x 1 x 2 Eqn ::= q [x 0 ] ( f (n) (x 1,...,x n )) = τ (n) τ (n) R (n) ::= f (m) (τ (n) 1,...,τ(n) m ) x 0 q j τ (n) j (x i ) where 1 i n

6 Emedded trnsducers re strictly more expressive thn trditionl trnsducers, ecuse the extr rgument llows unounded communiction etween positions unoundedly distnt in depth in the output tree. For exmple, simple emedded trnsducer cn compute the reversl of string, e.g., 1(2(2(nil))) reverses to 2(2(1(nil))). (This is not computle y trditionl tree trnsducer.) It is given y the following equtions: r (x) = r nil (x) r x 0 (nil) = x 0 r x 0 (1(x)) = r 1(x 0 ) (x) r x 0 (2(x)) = r 2(x 0 ) (x) (1) This is, of course, just the norml ccumulting reverse functionl progrm, expressed s n emedded trnsducer. The dditionl power of emedded trnsducers is, we will show in this section, exctly wht is needed to chrcterize the dditionl power tht TAGs represent over CFGs in descriing tree lnguges. In prticulr, we show tht the reltion etween TAG derivtion tree nd derived tree is chrcterized y deterministic liner complete emedded tree trnsducer (DL- CETT). The reltion etween tree-djoining lnguges nd emedded tree trnsducers my e implicit in series of previous results in the forml-lnguge theory literture. 3 For instnce, Fujiyoshi nd Ksi (2000) show tht liner, complete mondic context-free tree grmmrs generte exctly the tree-djoining lnguges vi norml form for spine grmmrs. eprtely, the reltion etween context-free tree grmmrs nd mcro tree trnsducers hs een descried, where the reltionship etween the mondic vrints of ech is implicit. Thus, tken together, n equivlence etween the tree-djoining lnguges nd the imge lnguges of mondic mcro tree trnsducers might e pieced together. In the present work, we define the reltion etween tree-djoining lnguges nd liner complete mondic tree trnsducers directly, simply, nd trnsprently, y giving explicit constructions in oth directions, crefully hndling the distinction etween the unrnked trees of tree-djoining grmmrs nd the rnked trees of mcro tree trnsducers nd other importnt issues of detil in the constructions. The proof requires reductions in oth directions. First, we show tht for ny TAG we cn construct DLCETT tht specifies the tree reltion etween the derivtion trees for the TAG nd the derived 3 We re indeted to Uwe Mönnich for this oservtion. trees. Then, we show tht for ny DLCETT we cn construct TAG such tht the tree reltion etween the derivtion trees nd derived trees is relted through simple homomorphism to the DL- CETT tree reltion. 4.1 From TAG to Trnsducer Given n elementry tree α with the lel A t its root, let the sequence π = π 1,...,π n e permuttion on the nodes in α t which djunction occurs. (We use this ordering y mens of the dicritic representtion elow.) Then, if α is n uxiliry tree, construct the eqution q A x 0 (α(x 1,...,x n )) = α nd if α is n initil tree, construct the eqution q A (α(x 1,...,x n )) = α where the right-hnd-side trnsformtion is defined y 4 A /0 (t 1,...,t n ) = A( t 1,..., t n ) k A(t 1,...,t n ) = q A A /0 (t 1,...,t n ) (x k ) A = x 0 = (2) Note tht the equtions re liner nd complete, ecuse ech vrile x i is generted once s the tree α is trversed, nmely t position π i in the trversl (mrked with i ), nd the vrile x 0 is generted t the foot node only. Thus, the generted emedded tree trnsducer is liner nd complete. Becuse only one eqution is generted per tree, the trnsducer is trivilly deterministic. By wy of exmple, we consider the core TAG grmmr given y the following trees: α : 1 A(e) β A : A /0 ( 1 B(), 2 C( 3 D(A ))) β B : 1 B(,B ) ε B : B ε C : C ε D : D 4 It my seem like trickery to use the dicritics in this wy, s they re not relly components of the tree eing trversed, ut merely reflexes of n extrinsic ordering. But their use is enign. The sme trnsformtion cn e defined, it more cumersomely, keeping the permuttion π seprte, y trcking the permuttion nd the current ddress p in revised trnsformtion π,p defined s follows: A /0 (t 1,...,t n ) π,p = A( t 1 π,p 1,..., t n π,p n ) A(t 1,...,t n ) π,p = q A A /0 (t 1,...,t n ) π,p (x π 1 (p) ) A π,p = x 0 π,p = We then use α π,ε for the trnsformtion of the tree α.

7 α 2 1 α 1 ε β 1 β 2 2 T c α 1 β 1 β 2 () () (c) Figure 4: Derivtion nd derived trees for the smple grmmrs: () derivtion tree for the grmmr of Figure 1; () corresponding derived tree; (c) corresponding derivtion tree for the core TAG version of the grmmr in Figure 2. trting with the uxiliry tree β A = A /0 ( 1 B(), 2 C( 3 D(A ))), the djunction sites, corresponding to the nodes leled B, C, nd D t ddresses 1, 2, nd 21, hve een ritrrily given preorder permuttion. We therefore construct the eqution s follows: q A x 0 (β A (x 1,x 2,x 3 )) = A /0 ( 1 B(), 2 C( 3 D(A ))) = A( 1 B(), 2 C( 3 D(A )) ) = A(q B B /0 () (x 1 ), 2 C( 3 D(A )) ) = A(q B B( ) (x 1 ), 2 C( 3 D(A )) ) = = A(q B B() (x 1 ),q C C(q D D(x 0 ) (x 3 )) (x 2 )) imilr derivtions for the remining trees yield the (deterministic liner complete) emedded tree trnsducer defined y the following set of equtions: q A (α(x 1 )) = q A A(e) (x 1 ) q A x 0 (β A (x 1,x 2,x 3 )) = A(q B B() (x 1 ),q C C(q D D(x 0 ) (x 3 )) (x 2 )) q B x 0 (β B (x 1 )) = q B B(,x 0 ) (x 1 ) q B x 0 (ε B ()) = x 0 q C x 0 (ε C ()) = x 0 q D x 0 (ε D ()) = x 0 We cn use this trnsducer to compute the derived tree for the derivtion tree α(β A (β B (ε B ),ε C,ε D )). q A (α(β A (β B (ε B ),ε C,ε D ))) = q A A(e) (β A (β B (ε B ),ε C,ε D )) = A( q B B() (β B (ε B )), q C C(q D D(A(e)) (ε D )) (ε C )) = A(q B B(,B()) (ε B ),C(q D D(A(e)) (ε D ))) = A(B(,B()),C(D(A(e)))) ε As finl step, useful lter for the imorphism chrcteriztion of synchronous TAG, it is strightforwrd to show tht the trnsducer so constructed is the composition of regulr tree lnguge nd liner complete emedded tree homomorphism. 4.2 From Trnsducer to TAG Given liner complete emedded tree trnsducer, we construct corresponding TAG s follows: For ech rule of the form q i [x 0 ] ( f (m) (x 1,...,x m )) = τ we uild tree nmed q i, f,τ. Where this tree ppers is determined solely y the stte q i, so we tke the root node of the tree to e the stte. Any foot node in the tree will lso need to e mrked with the sme lel, so we pss this informtion down s the tree is uilt inductively. The tree is therefore of the form q i /0( τ i ) where the right-hnd-side trnsformtion i constructs the reminder of the tree y the inductive wlk of τ, with the suscript noting tht the root is leled q i. f (t 1,...,t m ) i = f /0 ( t 1 i,..., t m i ) q j τ (x k ) i = k q j ( τ i ) x 0 i = q i i = Note tht t x 0, foot node is generted of the proper lel. (Becuse the eqution is liner, only one foot node is generted, nd it is leled ppropritely y construction.) Where recursive processing of the input tree occurs (q j τ (x l )), we generte tree tht dmits djunctions t q j. The role of the dicritic k is merely to specify the permuttion of operle nodes for interpreting derivtion trees; it sys tht the k-th child in derivtion tree rooted in the current elementry tree is tken to specify djunctions t this node. The trees generted y this TAG re intended to correspond to the outputs of the corresponding tree trnsducer. Becuse of the more severe constrints on TAG, in prticulr tht ll comintoril limittions on putting sutrees together must e mnifest in the lels in the trees themselves, the outputs ctully contin more structure thn the corresponding trnsducer output. In prticulr, the stte-leled nodes re merely for ookkeeping. A homomorphism removing these nodes gives the desired trnsducer output. Most importntly, then, the wek genertive cpcity of TAGs nd LCETTs re identicl.

8 ome exmples my clrify the construction. Recll the reversl emedded trnsducer in (1) ove. The construction ove genertes TAG contining the following trees. We hve given them indictive nmes rther thn the cumersome ones of the form q i, f,τ. α : r /0 (1 : r (nil)) β nil : r /0(r ) β 1 : r /0(1 : r (1 /0 (r ))) β 2 : r /0(1 : r (2 /0 (r ))) It is simple to verify tht the derivtion tree derives the tree α(β 1 (β 2 (β 2 (β nil )))) r(r 6 (2(r (2(r (1(r (nil)))))))) imple homomorphisms tht extrct the input function symols on the input nd drop the ookkeeping sttes on the output reduce these trees to 1(2(2(nil))) nd 2(2(1(nil))) respectively, just s for the corresponding tree trnsducer. 5 ynchronous TAGs s Bimorphisms The mjor dvntge of chrcterizing TAG derivtion in terms of tree trnsducers (vi the compiltion (2)) is the integrtion of synchronous TAGs into the imorphism frmework. A synchronous TAG (hieer, 1994) is composed of set of triples t L,t R, where the two trees t L nd t R re elementry trees nd is set of links specifying pirs of linked operle nodes from t L nd t R. Without loss of generlity, we cn stipulte tht ech operle node in ech tree is impinged upon y exctly one link in. (If node is unlinked, the triple cn never e used; if overlinked, set of replcement triples cn e multiplied out.) In this cse, projection of the triples on first or second component, with permuttion defined y the corresponding projections on the links, is exctly TAG s defined ove. Thus, derivtions proceed just s in single TAG except tht nodes linked y some link in re simultneously operted on y pired trees derived y the grmmr. In order to model synchronous grmmr formlism s imorphism, the well-formed derivtions of the synchronous formlism must e chrcterizle s regulr tree lnguge nd the reltion etween such derivtion trees nd ech of the pired derived trees s homomorphism of some sort. For synchronous tree-sustitution grmmrs, derivtion trees re regulr tree lnguges, nd the mp from derivtion to ech of the pired derived trees is liner complete tree homomorphism. Thus, synchronous tree-sustitution grmmrs fll in the clss of imorphisms B(LC,LC). The other direction cn e shown s well; ll imorphisms in B(LC, LC) define tree reltions expressile y n TG. A similr result follows immeditely for TAG. Crucilly relying on the result ove tht the derivtion reltion is DLCETT, we cn use the method of hieer (2004) directly to chrcterize the synchronous TAG tree reltions s just B(ELC, ELC). We hve thus integrted synchronous TAG with the other trnsducer nd synchronous grmmr formlisms flling under the imorphism umrell. Acknowledgements We wish to thnk Mrk Drs, Uwe Mönnich, Reecc Nesson, Jmes Rogers, nd Ken hn for helpful discussions on the topic of this pper. This work ws supported in prt y grnt II from the Ntionl cience Foundtion. References H. Comon, M. Duchet, R. Gilleron, F. Jcquemrd, D. Lugiez,. Tison, nd M. Tommsi Tree utomt techniques nd pplictions. Aville t: tt. Relese of Octoer 1, A. Fujiyoshi nd T. Ksi pinl-formed context-free tree grmmrs. Theory of Computing ystems, 33: Arvind Joshi nd Yves ches Treedjoining grmmrs. In G. Rozenerg nd A. lom, editors, Hndook of Forml Lnguges, volume 3, pges pringer, Berlin. Yves ches nd K. Vijy-hnker Deterministic left to right prsing of tree djoining lnguges. In Proceedings of the 28th Annul Meeting of the Assocition for Computtionl Linguistics, pges , Pittsurgh, Pennsylvni, 6 9 June. turt M. hieer Restricting the wekgenertive cpcity of synchronous tree-djoining grmmrs. Computtionl Intelligence, 10(4): , Novemer. Also ville s cmp-lg/ turt M. hieer ynchronous grmmrs s tree trnsducers. In Proceedings of the eventh Interntionl Workshop on Tree Adjoining Grmmr nd Relted Formlisms (TAG+7), pges 88 95, Vncouver, Cnd, My