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<Paper uid="C88-2147">
  <Title>Feature Structures Based Tree Adjoining Grammars 1</Title>
  <Section position="3" start_page="714" end_page="716" type="metho">
    <SectionTitle>
2 Featm'e Structure Based Tree Adjoining
Grammars (FTAG)
</SectionTitle>
    <Paragraph position="0"> The linguistic theory underlying TAG's is centered around the factorization of reeursion and localization of dependencies into the elementary trees. The &amp;quot;dependent&amp;quot; items usually belong to the same elementary tree 2. Thus, for example, the predicate and its arguments will be in the same tree, as will the filler and the gap. Our main goal in embedding TAG's in an unificational framework is to capture this localization of dependencies. Therefore, we would like to associate feature structures with the elementary trees (rather than break these trees into a CFG-like rule based systems, and then use some mechanism to ensure only the trees prodnced by the &amp;quot;lAG itself are generateda)~ In tbd':'feature structures 2It is eometime~ possible for &amp;quot;dependent&amp;quot; iterem to belong to an elementary tree and the immediate auxiliary tree that is adjoined in it.</Paragraph>
    <Paragraph position="1"> aSuch a scheme wotdd be an alternate way of embedding TAG's in an unifieational framework. IIowever, it does not capture the linguistic intuitions tmderlying TAG's, and losc~ the attractive feature of localizing depende~tcles.</Paragraph>
    <Paragraph position="2"> associated with the elementary trees, we can state the constraints among the dependent nodes dircctly. IIence, in an initial tree corresponding to a simple sentence, wc can state that the main verb and the subject NP (which are part of the same initial tree) share the agreement feature.</Paragraph>
    <Paragraph position="3"> Thus, such checking, in many cases, can be precompiled (of course only after lexical insertion) and need not be done dynamically.</Paragraph>
    <Section position="1" start_page="714" end_page="715" type="sub_section">
      <SectionTitle>
2.1 General Schema
</SectionTitle>
      <Paragraph position="0"> Ill unification grammars, a feature structure is associated with a node in a derivation tree in order to describe that node and its realtion to featnres of other nodes in tile derivation tree. In a TAG, any node in an elementary tree is related to the other nodes in that trec in two ways.</Paragraph>
      <Paragraph position="1"> Feature structures written in FTAG using the standard matrix notation, describing a node, ~h can be made on the basis of:  1. the relation of I 1 to its supertrce, i.e., tile view of the uode from the top. Let us call this feature structure as t,~.</Paragraph>
      <Paragraph position="2"> 2. the rclation to its descendants, i.e., the view from below. This  feature structure is called bo.</Paragraph>
      <Paragraph position="3"> Note that both the t,~ and b,~ feature structure hold of the node r l. In a derivation tree of a CFG based unification system, wc associate one featnre structure with a node (the unification of these two structures) since both the statements, t and b, together hold for the node, and uo further 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; rather ~ul auxiliary trec replaces the node. Wc believe that this approach of ~sociating two statements with a node in the auxiliary tree is iu the spirit of TAG's because of the OA constraints in TAG's. A node with OA constraints cannot bc viewed as a single node and must be considered as something that has to be replaced by an auxiliary tree. t and b axe restrictions about tile auxiliary tree that must be adjoined at this node. Note that if the node does not have OA constraint then we should expect t and 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 adjunction could take place), we associate two structures, tn and b n. With each terminal node, we would associate only one structure 4,  For example, we may wish to obtain verb-clusters by adiunctions at nodes which are labelled ~s verbs. In such a c~se, we will have to associate two feature structures with pre.lexical nodes too.</Paragraph>
      <Paragraph position="4">  Let Us now consider the case when adjoining takes place as showu in the figure 4. The notation we use is to write alongside each node, the t and b statements, with the t statement written above the b statement. Let us say that t~oot,b~oot aud tloo~,b/oo~ are the t and b statements of the root and foot nodes of the auxiliary tree used for adjunction at the node r/. Based on what t and b stand for, it is obvious that on adjnnction tim statements t,~ and troot hold of the node corresponding to the root of the anxifiary tree. Similarly, the statements b, and b/oot hold of the node corresponding to the foot of the auxiliary tree. Thus, ou adjunction, we unify t, with t~oot, and b,~ with b/oot. In fact, this adjunetion is permissible only if t,.oot and t o are cmnpatible as are b/oo~ and b,. If we do not adjoin at the node, 0, then we unify t s With b,. At the end of a derivation, the tree generated must not have any nodes with OA constraints. We cheek that by unifying the t and b feature structures of every node.* More details of the definition of FTAG may be found in \[Vijayashanker 1987\].</Paragraph>
      <Paragraph position="5"> We now give an example of an initial tree and an auxiliary tree. We would like to note that, just as in a TAG, the elementary trees which are the domain of co-occurenee restrictions is available as a single unit during each step of the derivation. Thus, most of these co-occurence constraints can be eheckcd even before the tree is used in a derivation, and this checking need not be linked to the derivation process.</Paragraph>
    </Section>
    <Section position="2" start_page="715" end_page="716" type="sub_section">
      <SectionTitle>
2.2 Unification and Constraints
</SectionTitle>
      <Paragraph position="0"> Since we expect that there are linguistic reasons determining why some auxiliary tree can be adjoined at a tree and why some cannot, or why some nodes have OA constraint, we would like to express these constraints in the feature structm:es associated with nodes. Further, as described in Section 2.1, adjunctions will be allowed only if the appropriate feature structures can be unified. Thus, we expect to implement the adjoining constraints of TAG's simply by making declarative statements made in the feature structures associated with the nodes to ensure that only the appropriate trees get adjoined at a node.</Paragraph>
      <Paragraph position="1"> The adjoining constraints are implemented in FTAG as follows. Notice, from Figure 4, t~ and troot, and b, and b.toa must be compatible for adjunction to occur. We hope to specify some feature-values in these t, b statements to specify the local constraints so that  1. if some auxiliary tree should not adjoined at a node (because of its SA constraint) then some unification involved (tu with troop, or b/oo~ with b,~) in our attempt to adjoin this auxiliary tree will fail, and 2. if a node has OA constraint, we should ensure that an appropriate  auxiliary tree does get adjoined at that node. This is ensured if t, is incompatible with b,.</Paragraph>
      <Paragraph position="2"> The example, given in Figure 7, illustrates the implementation of both the OA and SA constraint. The view of the root node of a from below .suggests that b statement for this node makes the assertion that the value of the tense attribute is - (or untensed). However, the t statement should assert tense : + (since every complete sentence must be telised) 5. Thus, an auxiliary tree whose root node will correspond to a tensed sentence and whose foot node will dominate an untensed sentence can be adjoined at this node. Therefore, only those auxiliary trees whose main verb subcate5t statement is more complicated than just &amp;quot;view from the top&amp;quot;, t ~tatement is a statement about the node wlfile viewing the node from the top, and hence is a statement eoncenfing the entire subtree below this node (i.e., including the part due to an auxiliary tree adjoined at the node), and ho w it constrains the derivation of the nodes wlfich are its siblings alld ancestors, b remains the same as before, and is the statement about this node and the subtree below it, without considering the adjunctlon at this node.</Paragraph>
      <Paragraph position="3">  gorizes for an untensed sentence (or an infinitival clause) can be adjoined at the root node of this initial tree. This shows why only auxiliary tree such as fl can be adjoined, whereas an auxiliary tree corresponding to John thinks S can not be adjoined since the verb thinks subcategories for a tensed sentence. The example also serves to illustrate the implementation of OA constraint at the root of a, since the t and b feature structures for this node are not unifiable.</Paragraph>
      <Paragraph position="4">  In the TAG formalism, local constraints are specified by enumeration.</Paragraph>
      <Paragraph position="5"> However, specification by enumeration is not a linguistically attractive solution. In FTAG we associate with each node two feature structures which are declarations of linguistic facts about the node. The fact that only appropriate trees get adjoined is a corollary of the fact that only trees consistent with these declarations are acceptable trees in FTAG. As a result, in a FTAG, constraints are dynamically instantiated and are not pre-slpecified as in a TAG. This can be advmltageous and useful for economy of grammar specification. For example, consider the derivation of the sentence What do you think Mary thought John saw In the TAG formalism, we are forced to replicate some auxiliary trees. Consider the auxiliary tree fll in the TAG fragment in Figure 7. Since the intermediate phrase what Mary thought John saw is not a complete sentence, we will have to use OA constraints at the root of the auxiliary tree ill. However, tlfis root node should not have OA constraints when it is used in some other context; as in the case of the derivation of Mary thought John saw Peter We will need another auxiliary tree, fs, with exactly the same tree structure as fll except that the root of/32 will not have an OA constraint. Further, the root nodes in c~1 and c~2 have SA constraints that allow for 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 that constraints are dynamically inatantiated and give only one specification of ill. When used in the derivation of What do you think Mary thought John saw troot inherits the feature inverted : + which it otherwise does not have, and broot inherits the feature inverted : -. Thus, the node which corresponds to root of ill, by the dynamic instantiation of the feature structure, gets an OA constraint. Note that there will not be any OA eoustraint in nodes of the final tree corresponding to What do you think Mary thought John saw.</Paragraph>
      <Paragraph position="6"> Also, the root of the auxiliary tree, corresponding to Mary thought S, does not get OA constraint, when this tree is used in the derivation of</Paragraph>
    </Section>
    <Section position="3" start_page="716" end_page="716" type="sub_section">
      <SectionTitle>
2.3 Some Possible Linguistic Stipulations in FTAG
</SectionTitle>
      <Paragraph position="0"> hi this section, we will discuss some possible stipulations for a FTAG granmmr, tIowever, at this stage, we do not want to consider these stipulations as a part of the formalism of FTAG. First, some of the linguistic issues pertaining to these stipulations have not yet been settled. Secondly, ou~ ~irnary C/o~cern ~'to sp~ify/tl,C/ FTA 9 formalism. ~,ther, if the form*lima haS t~) incorporate ~heie 4tip~ulatibns, it( can be done so, witbont ,lt~,ng tbe ~ochanlsm s,g~m0~n ly.</Paragraph>
      <Paragraph position="1"> The current linguistic theory u~derlying TAG's ...... that every foot node has *~ NA constraint. The justification of this stipulation is isinfilar to the projection principle in Chomsky's ~ransformation theory.</Paragraph>
      <Paragraph position="2"> !It is appealing to state that the adjunetion .operation does not alter the .grarmnatical relations defined by the intermediate tree structures. For ~example, consider the following derivation of the ~ntence Ma~y thought John saw Bill hit Jill.</Paragraph>
      <Paragraph position="3"> If the derivation results in the intermediate tree corresponding to Mary thought Bill hit Jill, then we wofild expect to obtain 'the relation of Mary thinking that &amp;quot;Bill hit Jill&amp;quot;. This relation is altered by the adjunction at the node corresponding to the foot node of the'auxiliary tree corresponding to Mary thought S.</Paragraph>
      <Paragraph position="4"> ff we wish to implement this stipulatio a, one solution is to insist that only one F-V statement is made with the foot node, i.e, the tloo~ and bloot are combined. The definition of adjunction can be suitably altered.</Paragraph>
      <Paragraph position="5"> The second stipulation involves the complexity of the feature structure associated with the nodes. So far, we have not placed any restrictions on the growth of these feature structures. One of the possible stipulations that are being considered from the point of view of linguistic relevance is to put a bound on the information content in these feature structures.</Paragraph>
      <Paragraph position="6"> This results in a bound on the size of feature structures and hence on the number of possible feature structures that can be associated with a node. An FTAG grammar, which incorporates this stipulation, will be equivalent to a TAG from the point of view of generative capacity but one with an enhanced descriptive capacity.</Paragraph>
      <Paragraph position="7"> Unbounded feature structures have been used to capture the subeat~egorization phenomenon by having feature structures that act like stacks (and hence unbounded in size), llowever, in TAG's, the elementary trees give the subeategorization (Iomain. As noted earlier, the elements sub-categorized by the main vert~ in an elementary tree are part of the same elementary tree. Thus, with the feature structures associated with the elementary trees we can just point to the subcategorized elements and do not need any further devices. Note, that any stack based mechanism that might be needed for subeategorization is provided by the TAG formalism itself, in which the tree sets generated by TAG's have context free paths (unlike CFG's which have regular paths). This additional power provided by the TAG formalism has been used to an advantage in giving an account of West Germanic verb-raising \[Santorini 1986\].</Paragraph>
    </Section>
  </Section>
  <Section position="4" start_page="716" end_page="718" type="metho">
    <SectionTitle>
3 A Calculus to Represent FTAG Gram-
</SectionTitle>
    <Paragraph position="0"> mars We will now consider a calculus to represent FTAG's by extending on the llogieal formulation oftbature structures given by Rounds and Kasper \[Rou Kasper et al. 1986\]. Feature structures in this logic (henceforth called lt!K logic) are represented as formulae. The set of well-formed formulae in this logic is recursively defined as follows.</Paragraph>
    <Paragraph position="1">  where a is an atomic value, el,e2 are well-formed formulae. NIL and (TOP cl)nvey &amp;quot;no in(ormation&amp;quot; and &amp;quot;inconsistent information&amp;quot; respec!~ively. ~aeh pl represents a path of the form li,1 : li,z .... : li,m reispectivel~y. This formula is interpreted as Pt .... = p,, and is used to iexpress reentrancy.</Paragraph>
    <Paragraph position="2"> Our representation of feature structures similar to the I/-K logie's :representation of feature structures and differs only in the clause for reen\]traney. Given that we want to represent the grammar itself in our cMculus, we call not represent reentrancy by a finite set of paths. For example, suppose we wish to mate that agreement features of a verb matches with ,that of its subject (note in a TAG the verb and its subject are in the same elementary tree), tile two paths to be identified can not be stated until we obtain the final derived tree. To avoid this problem, we use a set of equations to specify the reentrancy. The set of equations have the form given by xi = ei for 1 &lt; i &lt; n, where ~1,... ,xn are variables, el,... ,en !are formulae which could involve these variables.</Paragraph>
    <Paragraph position="3">  For exampl% the reentrant feature structure used in Section 1.2, is represented by the set of equations z = eat : S h l : y A 2 : (eat : VP h age : z A subject : y) y = cat : N P A agr : z We represent a set of equations, xi = ei for 1 &lt;: i &lt; n as rec ( Zh...,Xn &gt;~( el,...,en ~.</Paragraph>
    <Paragraph position="4"> Let us now consider the representation of trees in FTAG and the feature structures that are a~so'ciated with the nodes. The elementary feature structure associated with each elementary tree encodes certain relationships between the nodes. Included among these relationships are the sibling and ancestor/descendent relationships; in short, the actual structure of the tree. Thus, associated with each node is a feature structure which encodes the subtree below it. We use the attributes i E .hf to denote the i ~h child of a node.</Paragraph>
    <Paragraph position="5"> To understand the representation of the adjunction process, consider the trees given in Figure 4, and in particular, the node y. The feature structure associated with the node where adjunction takes place should reflect the feature structure after adjunction and as well as without adjunction (if the constraint is not obligatory). Further, the feature structure (corresponding to the tree structure below it) to be associated with the foot node is not knoWn bnt gets specified upon adjunetion. Thus, the bottom feature structure associated with the foot node, which is bloot before adjunction, is instantiated on adjunction by unifying it with a feature structure for the tree that will finally appear below this node. Prior to adjunction, since this feature structure is not known, we will treat it asi a variable (that gets instantiated on adjunction). This treatment can be! obtained if we think of the auxiliary tree as corresponding to functional over feature structures (by A-abstracting the variable corresponding to i the feature structure for the tree that will appear below the foot node). Adjunction correponds to applying this function to th e *feature structure corresponding to the subtree below the node where takes place.</Paragraph>
    <Paragraph position="6"> We will formalize representation of FTAG as follows. If we do nott consider adjoining at the node y, the formula for &amp;quot;y will be of the form (...t, 1 Ab, A.../ Suppose the formula for the auxiliary tree # is of the form (t~oo~ A . . . bsoo,) tim tree obtained after adjunction at the node r I will the n be represente~ by the formula (...t, A (t,degdeg, A... bsoo,) A N A . . .) We would like to specify one formula with the tree % and use appropriate operation corresponding to adjunction by ~ or the case where we do not adjoin at ~. Imagining adjunction as function application where we~ consider auxiliary trees as functions, the representation of/3 is a function i say fz, of the form ~f.(t,oo, A...(blo,, ^ f)) To allow tile adjunetion of ~ at the node ~, we have to represent T by (...t, A f#(bs) ^...) Then, corresponding to adjunction, we use function application to obtain the required formula. But note that if we do not adioin at ~l, we would like to represent &amp;quot;)&amp;quot; by the formula (...t, A b, A ~..) which can be obtained by representing T by</Paragraph>
    <Paragraph position="8"> where I is the identity function. Similarly, we inay have to attempt adjunction at ~ by any auxiliary tree (SA constraints are handled by success or failure of unification). Thus, if/31,...,/3, form the set of auxiliary tree, we have a function, F, given by V = AL(Im(I) v... v/~.(/) V I(I)) = ~f.(lm(f) V... V l~(I) v f) and represent 7 by (...t, A F(b,) A...) Ill this way, we can represent tile elementary trees (and hence tile grammar) in an extended version of rt-K logic (to which we add A-abstraction and application).</Paragraph>
    <Section position="1" start_page="718" end_page="718" type="sub_section">
      <SectionTitle>
3,1 Representing Tree Adjoining Grammars
</SectionTitle>
      <Paragraph position="0"> We will now turn our attention to the actual representation of an FTAG grammar, having considered how the individual elementary trees are represented. According to our discussion in the previous section, the auxiliary trees are represented as functions of the form Az.e where e is a term in FSTR which involves the variable ~. If/31,..., #n are the auxiliary trees of a FTAG, G, then we have equations of the form</Paragraph>
      <Paragraph position="2"> el,...,e~ are encodings of auxiliary trees #h...,fl, as discussed above.</Paragraph>
      <Paragraph position="3"> These expressions obey the syntax which is defined ~ccursively as follows.</Paragraph>
      <Paragraph position="4">  where x js a variable over feature structures and f is a function variable. In addition, as discussed above, we have another equation given by</Paragraph>
      <Paragraph position="6"> The initial trees are represented by a set of equations of the form ! xrn ~ ~ra where e~,.. ' ., e m are expressions which describe the initial trees at ,..., ~n Note that in the expressions el,..., e,, e~,.. ., e,,, wherever adjunction is possible, we use the function variable f0 as described above. The grammar is characterized by the structures derivable from any one of the initial trees. Therefore, we add ~0 ---- Zt V... V ~tn Assuming that we specify reentrancy using the Variables Yl,...~ Yk and equations Yt : e~' for 1 _ i &lt; k, an FTAG grammar is thus represented by the set of equations of the form .first (ree(xo, xl .... x,~, Yt .... , Yk, fo, 11 .... ,/,) (eo,e~,.. . ' 11 e&amp;quot; l ,era,el,&amp;quot;', k,g .... ,g,))</Paragraph>
    </Section>
    <Section position="2" start_page="718" end_page="718" type="sub_section">
      <SectionTitle>
a.2 Semantics of FTACI
</SectionTitle>
      <Paragraph position="0"> So far, we have only considered only the syntax of the calcnlus used tbr representing fcatnre structures and FTAG grammars. Ia this see@m, we consider the mathematical modelling of the calculus. This can be used to show that the set of equations describing a grammar will always have a solution, which we can consider as the denotation of the grammar.</Paragraph>
      <Paragraph position="1"> Tire model that we present here is based on the work by llxnmds and Kssper \[Pmund, et al. 1986\] and in particular their notion ofsatisfiability of formulae. \[,st I&amp;quot; be the space of partial flmetions (with the parLial ordering E, the standard ordering on partial functions) defined by /&amp;quot; = (L .-~ F) + A where A is set of atoms and L is set of labels. This space has been characterized by Pereira and Sheiber \[Pereira ctal. 1984\]. Any expression e (which is not a hmction) can be thought w~ upward closed subset of F (the set of partial functions which satisfy the description el. Note that if n partial fimetion satisties a description then so will any function above it. We let U(F) stm\]d for the collection of upward closed subsets of F. Expressions are interpreted relative to an envirmnnent (since we have variables as cxpressions, wc need to consider environments which map era'tables to a member of U(F)). Functimm get interpreted as continuous functions in tim space U(/;') -~ U(F'), with the enviromncnt mapping fimetion variables to fimctions on U(P). Note that the ordering on U(F) is the inverse of set inclusion, since more functions satisfy the description of a more general featnre structure.</Paragraph>
      <Paragraph position="2"> Because of space limitations, we cannot go into the details of the interpretations function. \[{onghly, the interpretation is as follows. We interpret the expression a as the set containing just the atom &amp;quot;a&amp;quot;; the expressiou 1 : e is interl)reted as tire set of fnnctions which map / to an element iu the .':at denoted by e; eonjmmtion and disjunetion are treated as intersection snd union respectively except that we have to ensure that rely value assigned t&lt;) a wtriable in one of the eonjunets is the same as the valne assigned to the same variable in the other conjnncg.</Paragraph>
      <Paragraph position="3"> Since the grammar is given by a set of equation;;, the denotation is given by tim least solution. This is obtaiued by considering the fimctiou corresponding to the set of equations in the standard way, and obtaining its least fixpoint. Details of these issues rnay be found in \[Vij ayashaaker i 9 In \[Vijayashanker 1987\], we have shown that any set of equations has a solution. Thus, we can Live semantics for recursivc set of eqnatkms which may be used to describe cyclic feature structure. For example, we give the solution for equations such as x:: f :xAg:a As shown in \[V \]ayas ran mr 1987\], we can obtain the least lixedopoint by assuming the le~rst vahm for x (which is the cntirc set of partial fnnetions or the intcrl)retatkm of NIL) mrd obtaining better and better approxima-, lions. The least npper bound of these approximations (which will give the least fixed-point) corresponds to the reqnired cyclic structure, ;is desired.</Paragraph>
    </Section>
  </Section>
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