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U.S .AA b s t r a c t W e h a v e e m b e d d ed T r e e A d j o in i n g G r a m m a r s ( T A G ) i n a fe a-t u r e s t r u c t u r e b a s e d u n i f i c a t i o n s y s t e m . T h e r e s u l t i n g s y s te m , F e a t u r eS t r u c t u r e b a s e d T r e e A d j o i n i n g G r a m m a r s ( F T A G ) , c a p t u r e s t h e p r i n c i -p l e of f a c t o r i n g d e p e n d e n c i e s a n d r e c u r si o n , f u n d a m e n t a l t o T A G ' s . W cs h o w t h a t F T A G h a s a n e n h a n c e d d e s c r i p t i v e c a p a c i t y c o m p a r e d t o T A Gf o r m a l is n r . W e c o n s i d e r s o m e r e s t r i c t e d v e r s i o n s o f t h i s s y s t e m a n d s o m ep o s s i b l e l i n g u i s t i c s t i p u l a t i o n s t h a t c a n b e m a d e . W e b r i e f l y d e s c r i b e ac a l c u l u s t o r e p r e s e n t t h e s t r u c t u r e s u s e d b y t h i s s y s t e m , e x t e n d i n g o nth e wo r k o f R o u n d s , an d Kasp er [ R o u n d s e t a l . 1 9 8 6, Kasp er e t a l . 1 9 8 6]i n v o l v i n g th e l o g i c a l f o r m u l a t i o n o f f e a t u r e s t r u c t u r e s .

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1 In t ro d u c t io nT r e e A d j o i n i n g G r a m m a r s ( T A G ) w e r e f i r s t i n t r o d u c e d b y J o s h i , L e v y ,a n d T a k a l m s h i [ J o s h i e t a l . 1 9 7 5] . T h e f i r s t s t u d y o f t h i s s y s t e m , f r o mt h e p o i n t o f v i e w o f i t s f o r m a l p r o p e r t i e s a n d l i n g u i s t i c a p p li c a b i l i t y , w a sc a r r i e d o u t b y J o s h i i n [ J o sh i 1 9 85 ]. T A G ' s h a v e b e e n u s e d i n p r o v i d i n gl i n g u i s t i c a n a l y s e s ; a d e t a i l e d s t u d y o f t h e l i n g u i s t i c re l e v a n c e w a s d o n eb y Kr o ch an d J o sh i in [ Kr o ch e t a l . 1 9 8 5 ] .

I n t h i s p a p e r , w e s h o w l m w T A G ' s c a n b e e m b e d d e d i n a f e a t u r e s t r u c -t u r e b a s e d f ra m e w o r k . F e a t u r e s t r u c t u r e b a s e d T r e e A d j o i n i n g G r a m m a r s( FTA G) a r e in t r o d u ced in Se c t io n 2 , an d i s f6 1 1 owed b y a co m p ar s io n o ft h e d e s c r i p t i v e c a p a c i t y o f F T A G a n d T A G . A r e s t r i c t e d v e r s io n o f F T A Gi s p r o p o se d a n d s o m e p o s s i b l e l i n g u i s t i c s t i p u l a t i o n s a r e c o n s i d e r e d . I nSec t io n 3 , we in t r o d u ce a ca lcu lu s , wh ich i s an ex ten s io n o f th e lo g ica lca lcu lu s o f R o u n d s an d K asp er [ R o u n d s e t a l . 1 9 8 6, Kasp er e t a l . 1 9 8 6 ]a l l o w i n g A - a b s tr a c t i o n a n d a p p l i c a t i o n , i n o r d e r t o d e s c r i b e t h e s t r u c t u r e su s e d i n F T A G ' s . F i n a l l y , i n S e c t i o n 4, w e s u m m a r i z e t h e w o r k p r e s e n t e di n t h i s p a p e r .1 . 1 I n t r o d u c t i o n t o T r e e A d j o i n i n g G r a m m a r sT r e e A d j o i n i n g G r a m m a r s ( T A G ) , u n l i k e o t h e r g r a m m a t i c a l s y s t e m s u se di n c o m p u t a t i o n a l l i n g u i s t i c s , i s a t r e e r e w r i t i n g s y s t e m . U n l i k e t h e s t r i n gr e w r i t i n g f o r m a l i s m s w h i c h w r i t e s r e c u r s i o n i n t o t h e r u l e s t h a t g e n e r a t et h e p h r a s e s t r u c t u r e , a T A G f a c t o r s r e e u rs i o n a n d d e p e n d e n c i e s i n t o af i n i t e s e t o f e l e m e n t a r y t r e e s . T h e e l e m e n t a r y t r e e s in a T A G c o r r e s p o n dto m i n i m a l l i n g u i s t i c s t r u c t u r e s t h a t l o c a l i z e t h e d e p e n d e n c i e s s u c h a sa g r e e m e n t , s u b c a t e g o r i z a t i o n , a n d f i l l e r -g a p . T h e r e a r e t w o k i n d s o f e l -e n r e n t a r y t re e s : t h e i n i t i a l t re e s a n d a u x i l i a ry t re e s . T h e i n i t i a l t r e e sr o u g h l y ( F i g u r e 1 ) c o r r e s p o n d t o s i m p l e s e nt e n c e s . T h u s , t h e r o o t o f a ni n i t i a l tr c e i s la b e l l e d b y t h e s y m b o l S . T h e y a r e r e q u i r e d t o h a v e af r o n t i e r m a d e u p o f t e r m i n a l s .

T h e a u x i l i a r y t r e e s ( F i g u r e 2 ) c o r r e s p o n d r o u g h l y t o m i n i m a l r e c u r -s i v e c o n s t ru c t i o n s . T h u s , i f t h e r o o t o f a n a u x i l i a r y t re e i s l a b e l l e d b y an o n t e r m i n a l s y m b o l , X , t h e n t h e r e i s a n o d e ( c a l l e d t h e f o ot n o d e ) i n t h ef r o n t i e r o f t h i s t r e e w h i c h i s l a b e l l e d b y X . T h e r e s t o f t h e n o d e s i n t h ef r o n t i e r a r e l a b e l l e d b y t e r m i n a l s y m b o l s .

1 T h i s w o r k w a s p a r t ia l l y s u p p o r t e d b y N S F g r a n t s M C S - 8 2 - 1 9 1 1 6 - C E R , D C R - 8 4 -1 0 4 1 3 , A R O f f r a n t D A A 2 9 - 8 4 - 9 - ~ 0 2 7 , a n d D A R P A g r a n t N 0 0 1 4 - 8 5 - K 0 0 1 8

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a n o d e l a b e l l e d b y X . L e t f l b e a n a u x i l i a r y t r e e , w h o s e r o o t a n d f o o tn o d e a r e a l s o l a b e l l e d b y X . T h e n , a d j o i n i n g / 3 a t t h e n o d e l a b e l l e d b yX i n 7 w i l l re s u l t i n t b e t r e e i l l u s t r a t e d i n F i g u r e 3 . I n F i g u r e 3 , w e a l s o

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F i g u r e 3 : T h e o p e r a t i o n o f a d j o i n fn gs h o w t l~ e r e s u l t o f a d j o i n i n g t h e a u x i l i a r y t re e f l l a t t h e s u b j e c t N P n o d eo f t h e i n i t i a l tr e e a l .

S o f a r , t h e o n l y r e s t r i c t i o n w e h a v e p l a c e d o n t h e s e t o f a u x i l i a r y t r e e st h a t c a n b e a d j o i n e d a t a n o d e i s t h a t t h e l a b e l o f t h e n o d e m u s t b e t h e

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same as the label of tile root (and the foot) node of the auxiliary tree.Fm'ther restriction on this set of auxiliary trees is done by enumeratingwith each node the subset of anxiliary trees which can be adjoined at thatnode. This specification of a set of auxiliary trees, which can be adjoinedat a node, is called the S e l e c t i v e A d j o i n i n g (SA) constraints. In tim casewhere we specify the empty set, we say that the node has a N a i l A d j o i n i n g( N A ) constraint:~. It is possible to insist that adjunction is man dato ry ata node. In such a case, wc say that the node has an O b l i g a t o r y A d j o i n i n g(OA) constraint.A more detailed description of TAG's and their linguistic relevancemay be found in [Kroe h et al. 1985].1 . 2 F e a t u r e S t r u c t u r e B a s e d G r a m m a t i c a l S y s t e m sSeveral different approaches to natural language granunars have devel-oped the notion of f e a t u r e s t r u c t u r e s to describe linguistic objects. Inorder to captu re certain linguistic pheno mena such as agreeme nt, subcat-egorization, cte., a number of. recent grammatical systems have added,on top of a CFG skclcton, a feature based informatioual element. Ex-ampl e or" sncb syst ems (see [Shieber 1985a]) include Gen eraliz ed Phras eStructure G ramma rs (GPSG), Lexical functional Grammars (LFG), andtIead-driven Phrase Structure Grammars (IIPSG). A feature structure(as given below) is essentially a set of attribute-value pairs where valuesmay be atomic ~*ymbols or an other fe ature str ucture.

c a t : Sca t

[ : [ ] a g rc a t :

2 : agr :s u b j e c t

53" 1][]Tim notation of the co-indexing box ([] in this example) is used to ex-press the f;~ct that the values of two subfeatures are t he stone. Featurestructur es wit h co-indexing boxes have also been called r e e n t r a n t featurestructures in the literature.

We can define a partial ordering, E, on a set of feature structuresusing tbe notion of s u b s n m p t i o n (carries l e s s i n / o r m a l i o n or is m o r e g e n -eral). Unification of two feat, re st ructures (if it is defined) correspondsto the feature ~;tructure that has all the information contained in theoriginal two feal;nre structu res and nothing more. We will not describefeat ure structur,~s any fnrth er (see [Shieber 1985a] for more details on fea-turc structures and an introduction to the unification based approach togrammars).2 Featm'e Structure Based Tree Adjoining

G r a m m a r s (FTAG)The linguistic theory underlying TAG's is centered around the factor-ization of reeursion and localization of dependencies into the elementarytrees. The "de penden t" items usually belong to the same elementa rytree2. Thus, for example, the predicate and its arguments will be in thesame tree, as will the filler and the gap. Our main goal in embeddingTAG's in an unificational framework is to capture this localization of de-pendencies. Therefore, we would like to associate feature structures withthe elementary trees (rather than break these trees into a CFG-like rulebased systems, and then use some mechan ism to ensure only the treesprodnced by the "lAG itself are generateda)~ In tbd':'feature structures

2It is eometime~possible for "dependent" iterem to belong to an elementary treeand the immediate auxiliary tree that is adjoined in it.aSuch a scheme wotdd be an alternate way of embeddingTAG's in an unifieationalframework. IIowever, t does not capture the linguistic intuitions tmderlying TAG's,and losc~ the attract ive feature of localizing depende~tcles.

associated with the elementary trees, we can state the constraints amongthe depen dent nodes dircctly. IIence, in an initial tree corresponding toa simple sentence, wc can state that the main verb and the subject NP(which are part of the same initial tree) share the agree ment feature.Thus, such checking, in many cases, can be precompiled (of course onlyafter lexical insertion) and need not be done dynamically.

2 . 1 Genera l SchemaIll unification grammars, a feature structure is associated with a nodein a derivation tree in order to describe that node and its realtion tofeatnres of other nodes in tile derivation tree. In a TAG, any node in anelementary tree is related to the other nodes in that trec in two ways.Feature structures written in FTAG using the standard matrix notation,describing a node, ~h can be made on the basis of:

1. the rela tion of I1 to its supertrce, i.e., tile view of the uode from thetop. Let us call this feature stru cture as t,~.

2. the rclati on to its descend ants, i.e., the view from below. Thisfeature stru cture is called bo.

Note tha t b oth the t,~ and b,~ featu re struc ture hol d of the nod e r . Ina derivation tree of a CFG based unification system, wc associate onefeatnre structure with a node (the unification of these two structures)since both the statements, t and b, together hold for the node, and uofurther nodes are introduced between the node's supertrce and subtrec.This property is not trne in a TAG. On adjunction, at a node there is~o longer a single node; ra the r ~ul auxiliary tr ec replaces the nod e. Wcbelieve that this approach of ~sociating two statements with a node inthe auxiliary tree is iu the spirit of TAG's because of the O A constraintsin TAG's . A node with O A constraints c a n n o t bc viewed as a singlenode and must be considered as something that has to be replaced byan auxiliary tree. t and b axe restrictions about tile auxiliary tree thatmust be adjoined at this node. Note that if the node does not have O Aconstraint t hen we should expect t a nd b to be compatible. For example,in the final sentential tree, this node will be viewed as a single entity.

Thus, in general, with every internal node, ~, (i.e., where adjunctioncould take place), we associate two structures, tn and bn. With eachterminal node, we would associate only one structure 4,

x t r o o tf o ' ~ tX__ _ - ~b f o o tFigure 4: Feature structures and adjunction

4It is posslblc to allow adjunctlons at nodes corresponding o pre-lexlcalitems.For example, we may wish to obtain verb-clustersby adiunctions at nodes which arelabelled~s verbs. In such a c~se, we will have to associate two feature structures withpre.lexical nodes too.

715

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Let Us now consider the case when adjoining takes place as showu inthe figure 4. The no tation we use is to write alongside each node, thet and b statements, with the t statement written above the b statement.Let us say that t~oot,b~oot aud t loo~,b/oo~ are the t and b statements ofthe root and foot nodes of the auxiliary tree used for adjunction at thenode r/. Based on what t and b stand for, it is obvious that on adjnnctiontim statements t,~ and troot hold of the node corresponding to the root ofthe anxifiary tree. Similarly, the statements b, and b/oot hold of the nodecorresponding to the foot of the auxiliary tree. Thus, ou adjunction, weunify t, with t~oot, and b,~ wit h b/oot. In fact, this adj unetion is permissibleonly if t ,.oot and t o are cmnpati ble as are b/oo~ and b,. If we do not adjoinat the node, 0, then we unify t s With b,. A t the end of a derivation, thetree generated must not have any nodes with O A constraints. We cheekthat by unifying the t and b feature stru ctures of every node. More detailsof the definition of FTAG m ay be found in [Vijayashanker 1987].

We now give an example of an initial tree and an auxiliary tree. Wewould like to note that, just as in a TAG, the elementary trees whichare the domain of co-occurenee restrictions is available as a single unitduring each step of the deriva tion. Thus, most of these co-occurenceconstraints can be eheckcd e ven before the tree is used in a derivation,and this checking need not be linked to th e derivation process.2 . 2 Uni f i ca t i on and Cons t r a in t sSince we expect that there are linguistic reasons determining why someauxiliary tree can be adjoined at a tree and why some cannot, or why somenodes have O A constraint, we would like to express these constraints inthe feature structm:es associated with nodes. Further, as described inSection 2.1, adjunc tions will be allowed only if the app ropriate fe aturestructure s can be unified. Thus, we expect to impleme nt the adjoiningconstraints of TAG's simply by making declarative statements made inthe feature structures associated with the nodes to ensure that only theappropriate trees get adjoined at a node.

The adjoining con straints are i mplemented in FTA G as follows. No-tice, from Figure 4, t~ and troot, and b, and b.toa must be compatible foradjunction to occur. We hope to specify some feature-values in these t, bstatements to specify the local constraints so that

1. if some auxiliary tree should not adjoined at a node (because of itsS A constraint ) then some unification involved (tu with troop, or b/oo~with b,~) in our a tte mpt to adjoin this auxiliary tree will fail, and

2. if a node has O A constraint, we should ensure that an appropriateauxiliary tree does get adjoined at that node. This is ensured if t,is incompatible with b,.

The exa mple, given in Figure 7, illustrates the imple mentatio n of boththe O A and S A constraint. The view of the root node of a from below.suggests tha t b stat emen t for this node makes the assertion that t he valueof the t e n s e attri bute is - (or untensed). However, the t state ment shouldassert t e n s e : + (since every complete sentence must be telised) 5. Thus ,an auxiliary tree whose root node will correspond to a tensed sentence andwhose foot node will dominate an untensed sentence can be adjoined atthis node. Therefore, only those auxiliary trees whose main verb subcate-

5t s t a t e m e n t i s more complicated than just " v i e w f r o m t h e t o p " , t ~ t a t e m e n t i sa statement a b o u t t h e node wlfile viewing h e n o d e f r o m t h e top, and hence is astatement eoncenfing he entire subtree below this node (i.e., including the part d ueto an auxiliary tree adjoined at the node), and how it constrains the derivation ofthe nodes wlfich are its siblings alld ancestors, b remains the sa me a s before, andis the statement about this node and the subtree below it, without considering headjunctlon at t his node.7 1 6

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Figure 6: Illustration of implementation of SA and QA constraintsgorizes for an untensed sentence (or an infinitival clause) can be adjoinedat the root node of this initial tree. This shows why only auxiliary treesuch as fl can be adjoined, whereas an auxiliary tree corresponding toJo h n t h i n ks S can not be adjoined since the verb t h i n ks subcategories fora tensed sentence. The example also serves to illustrate the implementa-tion of OA constraint at the root of a, since the t and b feature structuresfor this node are not unifiable.

2 .2 . 1 Comments on the Implemen ta t ion of Cons t r ain t s in FTAGIn the TA G formalism, local constraints are specified by enumeration.However, specification by enumeration is not a linguistically attractivesolution. In FTAG we associate with each node two feature structur eswhich are declarations of linguistic facts about the node. The fact tha tonly appropriate trees get adjoined is a corollary of the fact that onlytrees consistent with these declarations are acceptable trees in FTAG. Asa result, in a FTAG, constraints are dynamica lly instantiat ed and arenot pre-slpecified as in a TAG. This ca n be ad vmlta geou s and u seful foreconomy of gra mmar specification. For example, consider the derivationof the sentence

W h a t d o yo u t h i n k M a ry t h o u g h t Jo h n sa wIn the TAG formalism, we are forced to replicate some auxiliary trees.Consider the auxiliary tree fll in the TA G fragment in Figure 7. Sincethe intermediate phrase w h a t M a ry t h o u g h t Jo h n sa w is not a completesentence, we will have to use O A constraints at the root of the auxiliarytree ill. However, tlfis root node should not have O A constraints when itis used in some other context; as in the case of the derivation of

M a r y t h o u g h t J o h n s a w P e t e rWe will need another auxiliary tree, fs, with exactly the same tree struc-ture as fll except that the root of/32 will not have an O A constraint.Further, the root nodes in c~1 and c~2 have S A constraints that allowfor adjunetion only by fll and f~2 respectively: As seen in the Figure 8,corresponding to the FTAG fragment, we can make use of the fact thatconstraints are dynamically inatantiated and give only one specificationof ill. When used in the deriv ation of

W h a t d o yo u t h i n k M a ry t h o u g h t Jo h n sa wtroot inherits the feature i n v e r t e d : + which it otherwise does not have,and broot inherits the feature i n v e r t e d : -. Thus, the node which corre-sponds to root of ill, by the d ynamic instantiati on of the feature structu re,gets an O A constraint. Note that there will not be any O A eoustraint innodes of the final tree corresponding to

W h a t d o yo u t h i n k M a ry t h o u g h t Jo h n sa w .

Also, the root of the auxiliary tree, corresponding to M a r y t h o u g h t S ,does not get O A constraint, when this tree is used in the derivation ofthe sentence

M a r y t h o u g h t J o h n s a w P e t e r .

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2. 3 S o m e P o s s i b l e L i n g u i s t i c S t i p u l a t i o n s i n F T A Gh i t h i s s e c t i o n , w e w i l l d i s c u s s s o m e p o s s i b l e s t i p u l a t i o n s fo r a F T A Gg r a n m m r , t I o w e v e r , a t t h i s s t a g e , w e d o n o t w a n t t o c o n s i d e r t h e s e s t i p -u l a t i o n s a s a p a r t o f t h e f o r m a l i s m o f F T A G . F i r s t , s o m e o f t h e l i n g u i s t i ci s s u e s p e r t a i n i n g t o t h e s e s t i p u l a t i o n s h a v e n o t y e t b e e n s e t t l e d . S e c-o n d ly , o u ~ ~ ir n a r y o ~ c e r n ~ ' t o s p ~ i f y / t l , FTA o r m a l is m . ~ , t h e r ,i f th e f o r m * l im a h aS t~) in co r p o r a te ~h e ie 4 t ip ~u la t ib n s , i t ( can b e d o n e so ,w i t b o n t , l t ~ , n g t b e ~ o c h an l sm s ,g ~ m 0 ~ n ly .

T h e c u r r e n t l i n g u i st i c t h e o ry u ~ d e rl y i n g T A G ' s . . . . . . t h a t e v e ryf o o t n o d e h as * ~ N A c o n s t r a i n t . T h e j u s t i f i c a t i o n o f t h i s s t i p u l a t i o n i si s i nf i l ar t o t h e p r o j e c t i o n p r i n c i p l e i n C h o m s k y ' s ~ r a n s f o r m a t i o n t h e o ry .! It i s a p p e a l i n g t o s t a t e t h a t t h e a d j u n e t i o n .o p e r a t io n d o e s n o t a l t e r t h e. g r a r m n a t i c a l r e l a t i o n s d e fi n e d b y t h e i n t e r m e d i a t e t r e e s t r u c t u r e s . F o r~ e xa m p le , c o n s id e r t h e f o l l o w i ng d e r i v a t i o n o f t h e ~ n t e n c e

M a ~ y t h o u g h t J o h n s a w B i l l h i t J i l l .I f t h e d e r i v a t i o n r e s u l t s i n t h e i n t e r m e d i a t e t r e e c o r r e s p o n d i n g t o M a r yt h o u g h t B i l l h i t J i l l , t h e n w e w o f i l d e x p e c t t o o b t a i n ' t h e r e l a t i o n o f M a r yt h i n k i n g t h a t " B i l l h i t J i l l " . T h i s r e l a t i o n i s a l t e r e d b y t h e a d j u n c t i o n a tt h e n o d e c o r r e s p o n d i n g t o t h e f o o t n o d e o f t h e ' a u x i l i a r y t r e e c o r r e s p o n d -i n g t o M a r y t h o u g h t S .

f f w e w i s h t o i m p l e m e n t t h i s s t i p u l a t i o , o n e s o l u t i o n i s t o i n s i s t t h a to n l y o n e F - V s t a t e m e n t i s m a d e w i t h t h e f o o t n o d e , i . e , t h e tloo~ a n d

b l o o t a r e c o m b i n e d . T h e d e f i n i t io n o f a d j u n c t i o n c a n b e s u i t a b l y a l t e r e d .T h e s e c o n d s t i p u l a t i o n i n v o l v e s t h e c o m p l e x i t y of t h e f e a t u r e s t r u c t u r e

a s s o c i a t e d w i t h t h e n o d e s . S o f a r, w e h a v e n o t p l a c e d a n y r e s t r i c t i o n s o nt h e g r o w t h o f t h e se f e a t u r e s t r u c t u r e s . O n e o f t h e p o s s i b l e s t i p u l a t i o n st h a t a r e b e i n g c o n s i d e r e d f r o m t h e p o i n t o f v i e w o f l i n g u i s t i c r e l e v a n c ei s t o p u t a b o u n d o n t h e i n f o r m a t i o n c o n t e n t i n t h e s e f e a t u r e s t r u c t u re s .T h i s r e s u l t s i n a b o u n d o n t h e s i z e o f f e a t u r e s t r u c t u r e s a n d h e n c e o nt h e n u m b e r o f po s s i b l e f e a t u r e s t r u c t u r e s t h a t c a n b e a s s o c i a t e d w i t h an o d e . A n F T A G g r a m m a r , w h i c h i n c o r p o r a t e s t h i s s t i p u l a t i o n , w i l l b ee q u i v a l e n t t o a T A G f r o m t h e p o i n t o f v i ew o f g e n e r a t iv e c a p a c i t y b u to n e w i t h a n e n h a n c e d d e s c r i p t i v e c a p a c i t y .

U n b o u n d e d f e a t u r e s t r u c t u r e s h a v e b e e n u s e d t o c a p t u r e t h e s u b e a t -~ e go ri za ti on p h e n o m e n o n b y h a v i n g f e a t u r e s t r u c t u r e s t h a t a c t l i k e s t a c k s( a n d h e n c e u n b o u n d e d i n s iz e ) , l l o w e v e r , in T A G ' s , t h e e l e m e n t a r y t r e e sg i v e t h e s u b e a t e g o r i z a t i o n ( I o m a in . A s n o t e d e a r l i e r , t h e e l e m e n t s s u b -c a t e g o r i z e d b y t h e m a i n v e rt ~ i n a n e l e m e n t a r y t r e e a r e p a r t o f t h e s a m ee l e m e n t a r y tr e e . T h u s , w i t h t h e f e a t u re s t r u c t u r e s a s s o c i a t e d w i t h t h ee l e m e n t a r y t r e e s w e c a n j u s t p o i n t t o t h e s u b c a t e g o r i z e d e l e m e n t s a n d d on o t n e e d a n y f u r t h e r d e v ic e s . N o t e , t h a t a n y s t a c k b a s e d m e c h a n i s m t h a tm i g h t b e n e e d e d f o r s u b e a t e g o r i z a t i o n i s p r o v i d e d b y t h e T A G f o r m a l i s mi t s e l f, i n w h i c h t h e t r e e s e t s g e n e r a t e d b y T A G ' s h a v e c o n t e x t f r e e p a t h s( u n l i k e C F G ' s w h i c h h a v e re g u l a r p a t h s ) . T h i s a d d i t i o n a l p ow e r p r o v i d e db y t h e T A G f o r m a l i s m h a s b e e n u s e d t o a n a d v a n t a g e i n g i v i n g a n a c c o u n to f W e s t G e r m a n i c v e r b - r a i s i n g [ S a n t o r i n i 1 9 86 ].

3 A C a l c u l u s t o R e p r e s e n t F T A G G r a m -m a r s

W e w i l l n ow c o n s i d e r a c a l c u l u s t o r e p r e s e n t F T A G ' s b y e x t e n d i n g o n th el l o g ie a l o r m u l a t i o n o f t b a t u r e s t r u c t u r e s g i v e n b y R o u n d s a n d K a s p e r [ R o uK a s p e r e t a l . 1 9 8 6 ]. F e a t u r e s t r u c t u r e s i n t h i s l o g i c ( h e n c e f o r th c a l l e d l t -! K lo g ic ) a r e r ep r e sen ted as f o r mu lae . Th e se t o f wel l - f o r med f o r mu lae inth is lo g ic is r ecu r s iv e l y d ef in ed as f o l lo ws .e : : = N I L

T O PaI : e le l A e2e~. V e2{ p l . . . . . . }

w h e r e a i s a n a t o m i c v a l u e , e l , e 2 a r e w e l l - f o r m e d f o r m u l a e . N I L a n d( T O P c l )n v ey "n o in ( o r m at io n " an d " in co n s is ten t in f o r ma t io n " r esp ec-! ~ iv e ly . ~aeh p l r ep r esen ts a p a th o f th e f o r m l i,1 : l i , z . . . . : l i ,m re -i spec t iv e l~y. Th i s f o r mu la i s in te r p r e te d as P t . . . . = p , , an d i s u sed toiex p r ess r een t r an cy .

O u r r e p r e s e n t a t i o n o f f e a t u r e s t r u c t u r e s s i m i l a r t o t h e I / - K l o g i e ' s: r ep r esen ta t io n o f f ea tu r e s t r u c tu r es a n d d i f f e r s o n ly in th e c lau se f or r een -] tr an e y. G i v e n t h a t w e w a n t t o r e p r e s e n t t h e g r a m m a r i t s e l f i n o u r c M c u -l u s , w e c a l l n o t r e p r e s e n t r e e n t r a n c y b y a f i n i t e s e t o f p a t h s . F o r e x a m p l e ,s u p p o s e w e w i s h to m a t e t h a t a g r e e m e n t f e a t u r e s o f a v e r b m a t c h e s w i t h, t h a t o f i t s s u b j e c t ( n o t e i n a T A G t h e v e r b a n d i t s s u b j e c t a r e i n t h e s a m ee l e m e n t a r y t r e e ) , t i l e t w o p a t h s t o b e i d e n t i fi e d c a n n o t b e s t a t e d u n t i lw e o b t a i n t h e f i n a l d e r i v e d t re e . T o a v o i d t h i s p r o b l e m , w e u s e a s e t o fe q u a t i o n s t o s p e c i f y t h e r e e n t r a n c y . T h e s e t o f e q u a t i o n s h a v e t h e f o r mg i v e n b y x i = e i f o r 1 < i < n , w h e r e ~ 1 , . . . , x n a r e v a r i ab l e s , e l , . . . , e n! ar e f o r m u l a e w h i c h c o u l d i n v o l v e t h e s e v a r i a b l e s .

7 1 7

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F o r e x a m p l % t h e r e e n t r a n t f e a t u r e s t r u c t u r e u s e d i n S e c t i o n 1 . 2 , i sr e p r e s e n t e d b y t h e s e t o f e q u a t i o n s

z = e a t : S h l : y A 2 : ( ea t : V P h ag e : z A s ub j e c t : y )y = c a t : N P A a g r : z

W e r e p r e s e n t a s e t o f e q u a t i o n s , x i = e i f o r 1 < : i < n a sr e c ( Z h . . . , X n > ~ ( e l , . . . , e n ~ .

L e t u s n o w c o n s i d e r t h e r e p r e s e n t a t i o n o f t r e e s i n F T A G a n d t h ef e a t u r e s t r u c t u r e s t h a t a r e a ~ so ' c ia t e d w i t h t h e n o d e s . T h e e l e m e n t a r yf e a t u r e s t r u c t u r e a s s o c i a t e d w i t h e a c h e l e m e n t a r y t r e e e n c o d e s c e r t a i nr e l a t i o n s h i p s b e t w e e n t h e n o d e s . I n c l u d e d a m o n g t h e s e r e l a t i o n s h i p s a ret h e s i b l i n g a n d a n c e s t o r / d e s c e n d e n t r e l a t i o n s h i p s ; in s h o r t , t h e a c t u a ls t r u c t u r e o f t h e t r e e . T h u s , a s s o c i a t e d w i t h e a c h n o d e i s a f e a t u r e s t r u c -t u r e w h i c h e n c o d e s t h e s u b t r e e b e l o w i t . W e u s e t h e a t t r i b u t e s i E . hf t od en o t e th e i ~h ch i ld o f a n o d e .

T o u n d e r s t a n d t h e r e p r e s e n t a t i o n o f t h e a d j u n c t i o n p r o c e s s , c o n s i d e rt h e t r e e s g i v e n i n F i g u r e 4 , a n d i n p a r t i c u l a r , t h e n o d e y . T h e f e a t u r es t r u c t u r e a s s o c i a t e d w i t h t h e n o d e w h e r e a d j u n c t i o n t a k e s p l a c e s h o u l dr e f le c t t h e f e a t u r e s t r u c t u r e a f t e r a d j u n c t i o n a n d a s w e l l a s w i t h o u t a d -

j u n c t i o n ( i f t h e c o n s t r a i n t i s n o t o b l i g a t o ry ) . F u r t h e r , t h e f e a t u r e s t r u c -t u r e ( c o r r e s p o n d i n g t o t h e t r e e s t r u c t u r e b e l o w i t ) t o b e a s s o c i a t e d w i t ht h e f o o t n o d e i s n o t k n o W n b n t g e t s s p e c if i e d u p o n a d j u n e t i o n . T h u s , t h eb o t t o m f e a t u r e s t r u c t u r e a s s o c i a t e d w i t h t h e f o o t n o d e , w h i c h i s b l oo t be -f o re a d j u n c t i o n , i s i n s t a n t i a t e d o n a d j u n c t i o n b y u n i f y i n g i t w i t h a f e a t u r es t r u c t u r e f o r t h e t r e e t h a t w i l l f i n a l l y a p p e a r b e l o w t h i s n o d e . P r i o r t oa d j u n c t i o n , s i n c e t h i s f e a t u r e s t r u c t u r e i s n o t k n o w n , w e w i l l t r e a t i t a s ia v a r i a b l e ( t h a t g e t s i n s t a n t i a t e d o n a d j u n c t i o n ) . T h i s t r e a t m e n t c a n b e!o b t a i n e d i f w e t h i n k o f t h e a u x i l i a r y t r e e a s c o r r e s p o n d i n g t o f u n c t io n a lo v e r f e a t u r e s t r u c t u r e s ( b y A - a b s t r a c t i n g t h e v a r i a b l e c o r r e s p o n d i n g t ot h e f e a t u r e s t r u c t u r e f o r t h e t r e e t h a t w i l l a p p e a r b e l o w t h e f o o t n o d e ) .A d j u n c t i o n c o r r e p o n d s t o a p p l y i n g t h i s f u n c t i o n t o t h e fe a tu re s t r u c t u r ec o r r e s p o n d i n g t o t h e s u b t r e e b e l o w t h e n o d e w h e r e t a k e s p l a c e .

W e w i l l f o r m a l i z e r e p r e s e n t a t i o n o f F T A G a s f o ll o w s . I f w e d o n o ttco n s id er ad jo in in g a t th e n o d e y , th e f o r m u la f o r "y w i l l b e o f th e f o r m

( . . . t , 1 A b , A . . . /S u p p o s e t h e f o r m u l a f o r t h e a u x i l i a r y t r e e # i s o f th e f o r m

( t ~ o o ~ A . . . b s o o , )

t i m t r e e o b t a i n e d a f t e r a d j u n c t i o n a t t h e n o d e r w i l l t h e n b e r e p r e s e n t e ~b y t h e f o r m u l a

( . . . t , A ( t , , A . . . b s o o , ) A N A . . . )W e w o u l d l i k e t o s p e c i f y o n e f o r m u l a w i t h t h e t r e e % a n d u s e a p p r o p r i -a t e o p e r a t i o n c o r r e s p o n d i n g t o a d j u n c t i o n b y ~ o r t h e c a s e w h e r e w e d on o t a d j o i n a t ~ . I m a g i n i n g a d j u n c t i o n a s f u n c t io n a p p l i c a t i o n w h e r e we~c o n s i d e r a u x i l i a r y t r e e s a s f u n c t i o n s, t h e r e p r e s e n t a t i o n o f / 3 i s a fu n c t i o ns a y f z , o f t h e f o r m

~ f . ( t , o o , A . . . ( b l o , , ^ f ) )T o a l l o w t i l e a d j u n e t i o n o f ~ a t t h e n o d e ~ , w e h a v e t o r e p r e s e n t T b y

( . . . t , A f # ( b s ) ^ . . . )T h e n , c o r r e s p o n d i n g t o a d j u n c t i o n , w e u s e f u n c t i o n a p p l i c a t i o n t o o b t a i nt h e r e q u i r e d f o r m u l a . B u t n o t e t h a t i f w e d o n o t a d i o i n a t ~ l , w e w o u l dl ik e to r e p r esen t ")" b y th e f o r mu la

( . . . t , A b , A ~ . . )w h i c h c a n b e o b t a i n e d b y r e p r e s e n t i n g T b y718

( . . . t , ~ A Z ( b , ~ ) A . . . )w h e r e I i s t h e i d e n t i t y f u n c t i o n . S i m i l a r l y , w e i n a y ha v e t o a t t e m p t a d -j u n c t i o n a t ~ b y a n y a u x i l i a r y t r e e ( S A c o n s t r a i n t s a r e h a n d l e d b y s u c c e s so r f a i l u r e o f u n i f ic a t i o n ) . T h u s , i f / 3 1 , . . . , / 3 , f o r m t h e s e t o f a u x i l i a r y t r e e ,w e h a v e a f u n c t i o n , F , g i v e n b y

V = A L ( I m ( I ) v . . . v / ~ . ( / ) V I ( I ) ) = ~ f . ( l m ( f ) V . . . V l ~ ( I ) v f )

a n d r e p r e s e n t 7 b y ( . . . t , A F ( b , ) A . . . )I l l t h i s w a y, w e c a n r e p r e s e n t t i l e e l e m e n t a r y t r e e s ( a n d h e n c e t i l e g r a m -m a r ) i n a n e x t e n d e d v e r s i o n o f r t - K l o g i c ( t o w h i c h w e a d d A - a b s t r a c t io na n d a p p l i c a t i o n ) .3 , 1 R e p r e s e n t i n g T r e e A d jo i n in g G r a m m a r sW e w i l l n o w t u r n o u r a t t e n t i o n t o t h e a c t u a l r e p r e s e n t a t i o n o f a n F T A Gg r a m m a r , h a v i n g c o n s i d e r e d h o w t h e i n d i v i d u a l e l e m e n t a r y t r e e s a r e r ep -r e s e n t e d . A c c o r d i n g t o o u r d i s c u s s i o n in t h e p r e v i o u s s e c t i o n , t h e a u x i l i a r yt r e e s a r e r e p r e s e n t e d a s f u n c t i o n s o f t h e f o r m A z . e w h e r e e i s a t e r m i nF S T R w h i c h i n v o lv e s t h e v a r i a b l e ~ . I f / 3 1 , . . . , # n a r e t h e a u x i l i a r y t re e so f a F T A G , G , t h e n w e h a v e e q u a t i o n s o f t h e f o r m

fl = ~ x . e lf . = Ax.e,~

e l , . . . , e ~ a r e e nc o d i n gs of a ux i l i a ry t r e e s # h . . . , f l , a s d i sc u s se d ab o v e .Th e se ex p r ess io n s o b ey th e sy n t ax wh ich i s d ef in ed ~ccu r s iv e ly as f o llo ws.

e : := N I L:: = T O P

: :~ C l A e 2: : ~ e I V g 2::---- f ( e )

w h e r e x j s a v a r i a b l e o v e r f e a t u r e s t r u c t u r e s a n d f i s a f u n c t i o n v a r ia b l e .

I n a d d i t i o n , a s d i s c u s s e d a b o v e , w e h a v e a n o t h e r e q u a t i o n g i v e n b yf o = A x . / I ( x ) V . . . V f n ( ~ )

T h e i n i t i a l t r e e s a r e r e p r e s e n t e d b y a s e t o f e q u a t i o n s o f t h e f o r m

!xrn ~ ~raw h e r e e ~ , . . ', e m a r e e x p r e s s i o n s w h i c h d e s c r i b e t h e i n i t i a l t r e e s a t , . . . , ~ nN o t e t h a t i n th e e x p r e s s i o n s e l , . . . , e , , e ~ , . . . , e , , , w h e r e v e r a d j u n c t i o n i sp o s s i b l e , w e u s e t h e f u n c t i o n v a r i a b l e f 0 a s d e s c r i b e d a bo v e . T h e g r a m -m a r i s c h a r a c t e r i z e d b y t h e s t r u c t u r e s d e r i v a b l e f r o m a n y o n e o f t h e i n i t i a lt r e e s . T h e r e f o r e , w e a d d

~0 ----Z t V... V ~ t nA s s u m i n g t h a t w e s p e c i f y r e e n t r a n c y u s i n g t h e V a r i a bl e s Y l , . . . ~ Y k a n de q u a t i o n s Yt : e~' f o r 1 _ i < k , a n F T A G g r a m m a r i s t h u s re p r e s e n t e db y t h e s e t o f e q u a t i o n s o f t h e f o r m

. f i r s t ( r e e ( x o , x l . . . . x , ~ , Y t . . . . , Y k , f o , 1 1 . . . . , / , )( e o , e ~ , . . . ' 1 1 e " l, e r a , e l , " ' , k , g . . . . , g , ) )

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a.2 Semantics of FTACIS o f a r , w e h a v e o n l y c o n s i d e r e d o n l y t h e s y n t a x o f t h e c a l c n l u s u s e d t b rr e p r e s e n t i n g f c a t n r e s t r u c t u r e s a n d F T A G g r a m m a r s . I a t h is s e e @ m , w ec o n s i d e r t h e m a t h e m a t i c a l m o d e l l i n g o f t h e c a l c u l u s . T h i s c a n b e u s e d t os h o w t h a t t h e s e t o f e q u a t i o n s d e s c r i b i n g a g r a m m a r w i l l a l w a y s h a v e as o l u t i o n , w h i c h w e c a n c o n s i d e r a s t h e d e n o t a t i o n o f t h e g r a m m a r .

T i r e m o d e l t h a t w e p r e s e n t h e r e i s b a s e d o n t h e w o r k b y l l x n m d s a n dK s s p e r [ P m u n d , e t a l . 1 9 86 ] a n d i n p a r t i c u l a r t h e i r n o t i o n o f s a t i s f i a b i l i t yo f f o r m u l a e . [ , s t I " b e t h e s p a c e o f p a r t i a l f l m e t i o n s ( w i th t h e p a r L ia lo r d e r i n g E , t h e s t a n d a r d o r d e r i n g o n p a r t i a l f u n c t i o n s ) d ef i n e d b y / " =(L . - ~ F) + A w h e r e A i s s e t o f a t o m s a n d L i s s e t o f l a be l s . T h i s s p a c eh a s b e e n c h a r a c t e r i z e d b y P e r e i r a a n d S h e i b e r [ P e r e i r a c t a l . 1 9 8 4 ]. A n ye x p r e s s i o n e ( w h i c h i s n o t a h m c t i o n ) c a n b e t h o u g h t w~ u p w a r d c l o s e ds u b s e t o f F ( t h e s e t o f p a r t i a l f u n c t i o n s w h i c h s a t i s f y t h e d e s c r i p t io nel . N o t e t h a t i f n p a r t i a l f i m e t i o n s a t i s t i e s a d e s c r i p t i o n t h e n s o w i l la n y f u n c t i o n a b o v e i t . W e l e t U ( F ) s t m ] d f o r t h e c o l le c t i o n o f u p w a r dc l o s ed s u b s e t s o f F . E x p r e s s i o n s a r e i n t e r p r e t e d r e l a t i v e t o a n e n v i r m n n e n t( s i n c e w e h a v e v a r i a b l e s a s c x p r e s s i o n s , w c n e e d t o c o n s i d e r e n v i r o n m e n t sw h i c h m a p e r a ' t a b l e s t o a m e m b e r o f U ( F ) ) . F u n c t i m m g e t i n t e r p r e t e d a sc o n t i n u o u s f u n c t i o n s i n t i m s p a c e U ( / ; ') - ~ U(F' ) , w i t h t h e e n v i r o m n c n tm a p p i n g f i m e t i o n v a r i a b l e s t o f i m c t i o n s o n U ( P ) . N o t e t h a t t h e o r d e r i n go n U ( F ) i s t h e i n v e r s e o f s e t i n c l u s i o n , s i nc e m o r e f u n c t i o n s s a t i s f y t h ed e s c r i p t i o n o f a m o r e g e n e r a l f e a t n r e s t r u c t u r e .

B e c a u s e o f s p a c e l i m i t a t i o n s , w e c a n n o t g o i n t o t h e d e t a i l s o f t h ei n t e r p r e t a t i o n s f u n c t i o n . [ { o n g h l y , t h e i n t e r p r e t a t i o n i s a s fo l l o w s . W ei n t e r p r e t t h e e x p r e s s i o n a a s t h e s e t c o n t a i n i n g j u s t t h e a t o m "a"; th ee x p r e s s i o u 1 : e i s i n t e r l) r e t e d a s t i r e s e t o f f n n c t i o n s w h i c h m a p / t o a ne l e m e n t i u t h e . ': at d e n o t e d b y e ; e o n j m m t i o n a n d d i s j u n e t i o n a r e t r e a t e da s i n t e r s e c ti o n s n d u n i o n r e s p e c t i v e l y e x c e p t t h a t w e h a v e t o e n s u r e t h a tr e l y v a l u e a ss i g n e d t