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<?xml version="1.0" standalone="yes"?> <Paper uid="J92-4004"> <Title>Using Descriptions of Trees in a Tree Adjoining Grammar</Title> <Section position="5" start_page="495" end_page="500" type="metho"> <SectionTitle> 3 A complement tree (for example, the tree f12 in Figure 2) is one where the foot node corresponds to One </SectionTitle> <Paragraph position="0"> of the arguments required by the anchor of the tree.</Paragraph> <Paragraph position="1"> K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar adjoined at this pair (as determined by the associated feature structures). Unlike the previous case, adjunction is not barred per se. Instead, attempting to adjoin at such a pair will never yield well-formed structures. This is because of the nature of such a pair and of the auxiliary quasi-trees in the given grammar. In the TAG formalism, both these interpretations are captured by the same operational mechanism.</Paragraph> <Paragraph position="2"> The first kind of NA constraint is easily stated. According to this interpretation, for each pair of quasi-nodes with an &quot;NA constraint,&quot; the two quasi-nodes are indeed the same node (since we are stating that there is no possible separation). Since the two quasi-nodes are to be identified, the feature structure associated with the resulting quasi-node must reflect both the relationship of the quasi-node with its ancestor (which we assume stands for the top feature structure) as well as its relationship with its descendants (the bottom feature structure).</Paragraph> <Paragraph position="3"> Earlier we had stated that the target of an adjunction operation must be a pair of quasi-nodes that have not been identified (i.e., merged). Suppose that a pair of quasi-nodes (71,72) were merged. Let the quasi-root and quasi-foot of some auxiliary quasi-trees fl be given by r and f. Adjoining fl at the pair given by 71 and ~2 (after they have been identified) will result in the identification of 71 with r and 72 with f and thus r with f. If we stipulate that in all auxiliary quasi-trees, the quasi-root and quasi-foot do not refer to the same node (i.e., the quasi-root properly dominates the quasi-foot), then no adjunction can occur at a pair of quasi-nodes that have been identified. Thus the identification of a pair of quasi-nodes captures &quot;NA constraints&quot; of the first kind. As far as the second kind of &quot;NA constraints&quot; is concerned, we note that it is only a specific case of &quot;SA constraints.&quot; Therefore, given a pair of quasi-nodes, if the associated feature structures are such that no auxiliary quasi-tree can be adjoined at this pair then it has an &quot;NA constraint&quot; (of the second kind). However, because of the nature of feature structures (in that they capture only partial information), it is hard to detect if a pair of quasi-nodes has such an &quot;NA constraint.&quot; In Section 3.4, we will consider such an example.</Paragraph> <Section position="1" start_page="496" end_page="500" type="sub_section"> <SectionTitle> 3.4 Comparing the Implementation of Adjoining Constraints </SectionTitle> <Paragraph position="0"> In the TAG formalism, selective adjoining constraints are specified by enumeration, and hence are stipulations stating which trees can be adjoined at a node. Hence, specifying adjoining constraints in such a way is not a linguistically appealing solution.</Paragraph> <Paragraph position="1"> Obviously, such stipulations are needed because the information content of the labels of nodes in a TAG is often insufficient to determine the trees that can be adjoined at various nodes. In the case of FFAG, labeling of quasi-nodes by symbols such as NP, S is only a part of information contained in the feature structures associated with them. We associate with a pair of quasi-nodes feature structures that describe the features of the top and bottom quasi-nodes. The fact that only appropriate quasi-trees get adjoined is a corollary of the fact that only those consistent with these declarations are acceptable.</Paragraph> <Paragraph position="2"> Additionally, in a FTAG, &quot;adjoining constraints&quot; can be dynamically instantiated and are not pre-specified as in a TAG.</Paragraph> <Paragraph position="3"> We will now point out some differences between the implementation of adjoining constraints in TAG and b-TAG that arise because of different methods adopted in adjoining constraint specification. Of course, if the constraints are prespecified as in TAG, then little work has to be done (say by a parser) to verify whether an auxiliary tree can be adjoined at a node during the derivation process. This is not the case in FTAG, because of dynamic instantiation of &quot;constraints&quot; in b-TAG. For example, instead of f17 (Figure 10), suppose we consider/39 shown in Figure 12. The result of adjoining f19 at the pair (S1~$2) of O~ 9 is ')'7. There is a pair of quasi-nodes, ($3~$4) , in &quot;~7 with values of - and + for the tense attribute (thus giving rise to &quot;OA constraints&quot;).</Paragraph> <Paragraph position="5"> Comparison of adjunction constraints--Example 1.</Paragraph> <Paragraph position="6"> In a TAG grammar, the SA constraints at the root of tree corresponding to a9 would be given to disallow this adjunction. In the case of FTAG, as shown in Figure 12, this adjunction is allowed, because the associated unifications did not fail. Now suppose (as one might expect) the auxiliary quasi-trees in the grammar were such that none of them had their quasi-root with a feature structure compatible with tense: - and quasi-foot with a feature structure compatible with tense: +. In this case, although the adjunction of/39 was permitted, no tree can ever be derived from the result of adjunction. In fact, until we try all possible adjunctions at the node ~ in 77, we cannot realize that adjunction of f19 at the root of a9 can result in a final acceptable tree. Thus, the pair (s3, s4) has an NA constraint of the second kind.</Paragraph> <Paragraph position="7"> Now we will consider an example where specification of constraints in TAG suffers in comparison with the implementation of &quot;constraints&quot; in FTAG. Consider the following well-formed sentences (1) Who did John see? (2) Who did Peter think John saw? (3) I wonder who John saw.</Paragraph> <Paragraph position="8"> (4) I wonder who Peter thought John saw.</Paragraph> <Paragraph position="9"> (5) Peter thought John saw Mary.</Paragraph> <Paragraph position="10"> and the following, which are not well-formed sentences.</Paragraph> <Paragraph position="11"> (6) Who John saw? We will first consider a TAG account (in traditional style). The trees (without considering adjoining constraints) given in Figure 13 have been suggested in literature to account for the well-formed sentences above. We have drawn these trees accounting for substitution at the NP nodes.</Paragraph> <Paragraph position="12"> From the well-formedness of (1) and ill-formedness of (6) it follows that the node 7/of eel 0 must have an OA constraint with fl12 in its SA constraint. On the other hand, from the well-formedness of (3) and ill-formedness of (6) it follows that the root of ~10 must have an OA constraint with fit0 in its SA constraint. However, the requirement of an OA constraint on these two nodes in ~10 is mutually exclusive. Because of this, a TAG grammar that accounts for the sentences above must have two trees, that have exactly the same tree structure but only differ in the adjoining constraints attached at the nodes.</Paragraph> <Paragraph position="13"> Now, from the well-formedness of (5), which can be derived by adjoining/311 at the root of c~11, we can conclude that there need not be an OA constraint on the root of /311. However, suppose we adjoin flu at the node ~ in ~10 such that the frontier matches with (8). From the ill-formedness of (8) and the well-formedness of (2) we realize that there must be an OA constraint on the root of fin with/312 in its OA constraint. Thus, Computational Linguistics Volume 18, Number 4 again we will need two trees (corresponding to fl11), with identical tree structure but differing in the adjoining constraints.</Paragraph> <Paragraph position="14"> We will see that such replication of tree structure is not necessary. Now consider the FTAG fragment (inspired by similar treatment in Abeille \[1991\]) given in Figure 14. If the feature structures of sl and s2 quasi-nodes of c~12 are unified then the other pair</Paragraph> <Paragraph position="16"> Comparison of adjunction constraints--Example 2.</Paragraph> <Paragraph position="17"> K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar of quasi-nodes labeled S will obtain an &quot;OA constraint&quot; and vice versa as required (hence (6) can not be derived). In fact, if fl13 were adjoined at the root of Oq2 (and thus showing (3) is well formed) then it will no longer be possible to derive (7). Likewise, by adjoining ills at the pair of s3 and s4 quasi-nodes in a~2, we can derive (1) but will no longer be able to derive (7).</Paragraph> <Paragraph position="18"> Proceeding in this manner we can show the well-formedness of (1)-(5) and the ill-formedness of (6)-(8). Thus we have shown that if appropriate assertions can be stated about the individual nodes then a more succinct grammar can be given: one that does not require replication of tree structures, due to the fact that adjoining constraints are not pre-specified as in a TAG.</Paragraph> </Section> </Section> <Section position="6" start_page="500" end_page="505" type="metho"> <SectionTitle> 4. A Logical Formulation </SectionTitle> <Paragraph position="0"> A central theme in our definition of FTAG has been the view that the objects manipulated by a grammar are descriptions of trees (rather than trees). This separation of descriptions of trees from the trees (models) derived has been crucial in embedding TAG in the unification framework. The question of which language to use to describe trees (together with its semantics) arises. We have used quasi-trees (as the descriptions themselves) in order to focus on TAG, and have not introduced some general formal framework for describing trees. The discussion below does not constitute a suggestion about how such general descriptions may be given, but is one way to specify an FTAG that will be convenient for our purposes here.</Paragraph> <Paragraph position="1"> In this section, we describe a logical formulation of the unification-based approach to TAGs. The purpose of providing a logical formulation of FTAG is so that we can find the denotation of an FTAG grammar (the set of structures generated) as well as contrast it with context-free grammar-based unification grammars. To define the denotation of an FTAG grammar, we will first describe how an FTAG grammar can be represented. This representation uses the logical formulation of feature structures as given by Kasper and Rounds (1986) and Johnson (1988) and is similar in approach to the logical formulation of Functional Unification Grammar (FUG) given by Rounds and Manaster-Ramer (1987).</Paragraph> <Paragraph position="2"> In the framework of Rounds and Manaster-Ramer (1987), an FUG (or any context-free grammar with associated unification equations as in, say PATR-II) can be represented by means of a set of equations, using the formulae of Kasper-Rounds to represent feature structures. For example, a context-free grammar rule S --* NP VP can be represented as s ::= CAT : S A 1 : np A 2 : vp. Here s, np, and vp are type variables. The attributes 1 and 2 are used to indicate the first and second children respectively.</Paragraph> <Paragraph position="3"> Using standard techniques to derive fixed points from a set of recursive rules, the denotation of type variables are obtained. The denotation of the type variables gives the set of structures derived from the corresponding nonterminals.</Paragraph> <Paragraph position="4"> Now suppose we wish to express reentrancy in feature structures by using variables; it is clear that we have to use individual variables and not type variables. As in Johnson (1988), we use individual variables and equalities to express reentrancy. The syntax we adopt to describe attribute-value structures is as follows. Firstly, the set of terms is defined as</Paragraph> <Paragraph position="6"> where a is an atomic value where x is an individual variable where 1 is a label (or attribute) and tl is a term. where tl,t2 are terms where 01,02 are formulae where 01~ 02 are formulae.</Paragraph> <Paragraph position="7"> For example, (l(x) = y) A (h(x) = z) A (g(y) = z) A (z = a) describes (among others) the following feature structure.</Paragraph> <Paragraph position="9"> Note that individual variables (that stand for individual feature structures) are being used to capture reentrancy, whereas typed variables play a role analogous to the role of nonterminals in grammars (such as CFGs) and stand for a set of feature structures. For the purpose of describing an FTAG, we need individual variables to specify reentrancy (as well as to refer to quasi-nodes) and &quot;typed&quot; variables to denote the set of structures derived from elementary quasi-trees. To distinguish between these two kinds of variables, in our framework, we will use monadic predicate instead of typed variables.</Paragraph> <Section position="1" start_page="501" end_page="503" type="sub_section"> <SectionTitle> 4.1 Expressing an FTAG </SectionTitle> <Paragraph position="0"> Firstly, we note that quasi-initial trees are analogous to nonterminals in CFGs. Thus, as indicated above, quasi-initial trees will be represented by monadic predicates. If a is a quasi-initial tree, then we will use a predicate symbol ~ to represent this quasi-tree. If a structure ,,4 is derivable in the grammar starting from a then we would like to have .4 to belong to the set denoted by ~. For example, any structure described by Ot14 can be assumed to satisfy the requirements on the variable x in cat(x) ~ S A Dom(x,y) A cat(y) ~ S A l(y) ~ z A count(y) ~ zero A cat(z) ~, c.</Paragraph> <Paragraph position="1"> This description is intended to not only describe the features of nodes, but also the structure of the subtrees rooted at each node (with attributes 1,2,... used to specify the first, second, ... child of a node). In the formula given above, x represents the quasi-root node. Therefore, we will define ~14 by ~14(X) K===~ cat(x) ~ S A Dom(x,y) A cat(y) ~ S A l(y) ~ z A count(y) ,~ zero A cat(z) ~ c. In this case, Dom(x, y) is used to indicate that the quasi-root (x) dominates the associated bottom quasi-node (given by y).</Paragraph> <Paragraph position="2"> Now if we view the definition of 0t14 independent of the rest of the grammar, then Dom(x,y) represents domination in any arbitrary manner. However, the rest of the grammar specifies the constraints on the domination relation by defining the actual possibilities for the domination. This is because a pair of quasi-nodes (say as given by x and y in c~14) is intended to mean that either they are the same objects or are different nodes that are related by proper domination. In our definition, the separation can take place only by adjunction. So given a grammar, we can specify that the domination relationship is actually defined by Dom(x,y) 4==~ x ~ y V fll(x~y) V ... V -'fin(X~y) where {ill,..., fin} are the quasi-auxiliary trees in the grammar. Here we assume that fl captures the (domination) relationship between its quasi-root and quasi-foot nodes K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar of the quasi-auxiliary tree ft. Since the actual definition of the domination between a pair of quasi-nodes is determined by the quasi-trees of the grammar, it is appropriate to consider fixed-point semantics to define the denotation of a grammar.</Paragraph> <Paragraph position="3"> Before we discuss the fixed point we will complete our discussion about how we can specify a grammar. Let us define another monadic predicate Inittree by Inittree(x) 4=~ ~l(x) V ... V ~m(X) where {al,... ,OLm} is the set of initial trees. If we further wish to stipulate that a structure is derived in a FTAG if it is derived from some quasi-initial tree and is rooted in S we can define</Paragraph> <Paragraph position="5"> Note that for a quasi-node (referred to as x) where substitution can take place, we can specify Inittree(x) to specify the substitution.</Paragraph> <Paragraph position="6"> We will now illustrate the representation of an FTAG grammar, shown pictorially in Figure 15. This grammar contains c~14 and fl16. Apart from the cat information, the only other attribute used in the feature structures are count (counts the number of adjoining operations used in deriving a tree), one (used in counting), and attributes 1,2, 3 (which are used for specifying the children of a node).</Paragraph> <Paragraph position="7"> To compare our representational scheme for FTAG with that for FUG given by Rounds and Manaster-Ramer (1987), we have used predicate symbols instead of type variables. The use of monadic predicates alone is sufficient to represent FUG (or actually a CFG-based unification grammar) since only &quot;substitution&quot; is used. Binary</Paragraph> <Paragraph position="9"> Example: An FTAG grammar and its representation.</Paragraph> <Paragraph position="10"> Computational Linguistics Volume 18, Number 4 predicates are used to capture adjunction (which is defined as a pair of substitutions) in FTAG.</Paragraph> </Section> <Section position="2" start_page="503" end_page="505" type="sub_section"> <SectionTitle> 4.2 Fixed-Point Semantics (Denotation of an FTAG Grammar) </SectionTitle> <Paragraph position="0"> As mentioned before, the set of terms is defined recursively as</Paragraph> <Paragraph position="2"> where a is an atomic value where x is an individual variable where I is a label and tl is a term. However the set of formulae is now defined by</Paragraph> <Paragraph position="4"> where tl,. *., tn are terms and P is a n-ary predicate symbol where C/1, C/2 are formulae where C/1, C/2 are formulae.</Paragraph> <Paragraph position="5"> From the discussion given in the previous section any FTAG can be stated as</Paragraph> <Paragraph position="7"> where C/1,..., Cn are formulae and t1,1,.. * tm,1, tl,n * .., tm,n are terms such that for 1 _< i,j < n, if i C/ j then the symbol Pi ~ Pj. Of course for describing an FFAG, monadic and binary predicates are enough.</Paragraph> <Paragraph position="8"> The structures that terms denote are the finite state automata (actually equivalence classes containing such automata; for details, we refer to Moshier \[1988\] for a discussion about these structures) as defined by Kasper and Rounds (1986) and used in defining the satisfiability of formulae in their logic. We can give a fixed point semantics of a grammar in the standard way.</Paragraph> <Paragraph position="9"> Definition Let p be a function that maps each variable to an automaton. We define a Value function as a partial function that returns the denotation of a term (an automaton) relative to an environment (mapping variables to automata)*</Paragraph> <Paragraph position="11"> corresponds to the atom a.</Paragraph> <Paragraph position="12"> * Valuep(l(t)) -- fit~l, if fit/l is defined, where 1 is an attribute, t is a term and Valuep(t) = fit. If Valuep(t) is not defined or Valuep(t) = fit but fit/l is not defined then Valuep(l(t)) is not defined.</Paragraph> <Paragraph position="13"> Let p be an environment function and I be an interpretation mapping predicate symbols to their denotations, i.e*, if P is a n-ary predicate symbol then I maps P to some set of n-tuples of automata* Given an interpretation function I and an environment p we define ~ in the following way.</Paragraph> <Paragraph position="14"> We now define a transformation function mapping interpretations in the following way. For some m ~ 1, let Pi(ti,l,..., ti,ni) 4==~ dpi (1 < i < m) be the grammar specification. We define the transformation function, To, such that given an interpretation,/, Tc returns an interpretation TG (I) given by Definition For all substitutions, p, where Valuep(ti,j) is defined for 1 <_ j <_ ni, (Valuen(tia),..., Valuep(ti,nl)) E TG(I)(Pi) iff (I,p) ~ C/5i.</Paragraph> <Paragraph position="15"> Ordering relations We use the ordering relationship, f-, as defined by Rounds and Kasper (1986) i.e., `41 ___ `42 iff there is a homomorphism mapping the states of Jt I to the states of -/~2 that preserves the transition and output functions. We extend this ordering relation to an ordering on n-tuples and state that (.41,... ,.An) u (131,... ~13nl iff for 1 < i < n `4/ __13/.</Paragraph> <Paragraph position="16"> Among pairs of sets of n-tuples of automata, say 191,/92, we use the same ordering as that used by Rounds and Manaster-Ramer (1987) and state that/91 G /92 iff/91 G /92. The least element among the sets of n-tuples of automata is the empty set. The ordering among interpretation functions is defined as h G /2 iff for all predicate symbols P, h(P) f-/2(P), i.e., h(P) c_ I2(P).</Paragraph> <Paragraph position="17"> Lemma 4.1.</Paragraph> <Paragraph position="18"> If/1 _/2, then for all environments, p, and formulae, q~, if (11, p) ~ ~ then (/2, P) ~ q~. This can be easily shown by using induction on the structure of the formula ~. Theorem 4.1.</Paragraph> <Paragraph position="19"> The transformation function is monotonic.</Paragraph> <Paragraph position="20"> Let/1 _/2. We have to show for all P that Tc(I1)(P) C Tc(I2)(P). Let P(h,..., tn) ~ C/3 be a part of the grammar specification and let (`41,...,.An) E TG(I1)(P). Thus, for any environment p such that (h, p) ~ ~b and for 1 < i < n we have Valuep(ti) = `4i. By the above lemma, we also have I2,p ~ 4 and hence (`41,... ,An) C Tc(I2)(P). Thus, T~(I1)(P) C TG(I2)(P) and To(h) _E To(/2).</Paragraph> <Paragraph position="21"> We will call an interpretation, L finite if for all predicate symbols, P, I(P) is a finite set.</Paragraph> <Paragraph position="22"> Lemma 4.2.</Paragraph> <Paragraph position="23"> For all environments, p, and interpretations, L if (I, p) interpretation I0 such that I0 u I and (I, p) ~ ~b.</Paragraph> <Paragraph position="24"> ~b then there is a finite This can be shown by a straightforward induction on the structure of ~b, and by constructing I0 in the obvious manner.</Paragraph> <Paragraph position="25"> Computational Linguistics Volume 18, Number 4 Theorem 4.2.</Paragraph> <Paragraph position="26"> The transformation function is continuous.</Paragraph> <Paragraph position="27"> This can be easily established using Lemma 4.1 and Lemma 4.2.</Paragraph> <Paragraph position="28"> Since Tc is continuous, the least fixed point of T6 can be obtained as</Paragraph> <Paragraph position="30"> where I+- is the least interpretation function and is given by I_L(P) the empty set for all predicate symbols P. Let Ic be the fixed point of TG. Then the set of structures derived by a grammar G is given by It(Grammar), where Grammar is the distinguished predicate symbol as defined earlier.</Paragraph> </Section> <Section position="3" start_page="505" end_page="505" type="sub_section"> <SectionTitle> 4.3 Some Remarks </SectionTitle> <Paragraph position="0"> The logical formulation of FTAG given above is similar to the formulation of FUG and the associated semantics given by Rounds and Manaster-Ramer. This logical formulation of FUG essentially captures CFG-based unification grammars where substitution (and associated unifications) is the operation used for composition. This can be seen from their semantic treatment where type variables are repeatedly substituted for. Rather than using type variables for &quot;nonterminals,&quot; in our formulation predicate symbols represent the nonterminals. Although &quot;substitution&quot; at frontier nodes can be effectively captured by Rounds-Manaster-Ramer calculus, we found it less cumbersome to express adjunction operation and FTAG in the above DCG-like style.</Paragraph> <Paragraph position="1"> The domination relation and adjunction operation are easily captured by using binary predicates and their substitutions. Despite these syntactic differences, the presentation of the semantics is essentially the same traditional fixed-point semantics. Not only do we capture the substitution operation, as was done in the Rounds-Manaster-Ramer calculus, but we are also able to contrast FUG (and CFG-based unification grammars) with FTAG by capturing adjoining as a pair of substitutions.</Paragraph> </Section> </Section> <Section position="7" start_page="505" end_page="514" type="metho"> <SectionTitle> 5. Some Consequences of the New Interpretation </SectionTitle> <Paragraph position="0"> So far we have concerned ourselves with an interpretation of TAG that is compatible with the constraint-based approach to grammars. We will now briefly discuss some possible implications that this new interpretation may have on design or development of TAG grammars. The point of this section is simply to raise certain possibilities and questions. Providing definitive answers and solutions involves exploring linguistic issues that are beyond the scope of this work.</Paragraph> <Section position="1" start_page="505" end_page="513" type="sub_section"> <SectionTitle> 5.1 Adjoining, Multi-Component Adjoining, and Substitution </SectionTitle> <Paragraph position="0"> We defined the adjoining operation as an operation that fits a structure in the gap between a pair of associated quasi-nodes. Although the nature of the adjoining operation itself has not been examined in much detail in this paper (apart from defining it in terms of quasi-nodes in a manner such that it is similar to the traditional definition), questions that arise from this work are: how different is the adjoining operation from the more commonly used substitution operation; and whether the definition of adjoining itself (as stated here) follows from some more fundamental linguistic assumptions.</Paragraph> <Paragraph position="1"> To motivate our arguments, we start by considering an example using the so-called multi-component adjoining.</Paragraph> <Paragraph position="2"> K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar Consider the derivation of: (1) Which picture did you buy a copy of? (2) Which picture did you buy a photograph of a copy of? This form of long-distance dependency cannot be localized in a TAG if we wish to localize the predicate-argument dependencies as well (for details, see Kroch \[1987\]). On the other hand, an analysis has been given using a version of multi-component adjoining. Multi-Component Tree Adjoining Grammar (MCTAG) differs from (the traditional definition of) TAG in that the elementary objects of the grammar are sets of trees rather than trees, and multi-component adjoining involves the composition of these elementary sets of trees 4 (rather than elementary trees). See Joshi (1987) for more details on Multi-Component Tree Adjoining Grammars (MCTAG).</Paragraph> <Paragraph position="3"> The multi-component sets, given in Figure 16, may be used to give an account for sentences (1) and (2). Obtained by adjoining the two components of fl. 17 in C~lS, % can be used for sentence (1) (Figure 17).</Paragraph> <Paragraph position="4"> The need for introducing multi-component sets and multi-component adjoining (in this case, at least) arises because of the decision in traditional TAGs to compose trees (rather than descriptions of trees, i.e., quasi-trees). In particular, the domination relations allow us to give partial descriptions of trees such as oL16 (in Figure 18) that captures the same information as in the multi-component set fl17 (in Figure 16). Note that OL16 can be described by using the same principles that relate (X13 and O~12 (see Figure 14). If, for a moment, we consider c~15 to be an auxiliary quasi-tree (rather than an initial quasi-tree) and use it for &quot;adjoining&quot; (treating the n2 quasi-node as the quasi-foot) then we obtain the same structure as &quot;Y8 (Figure 17).</Paragraph> <Paragraph position="5"> Two issues can be raised with respect to this example. Firstly we can question whether such uses of multi-component adjoining (and where the foot node of one component dominates the root of the other components in the eventual structure 5) can be considered to be adjoining in the quasi-tree framework; and secondly whether these operations can be thought of as essentially the substitution operation when viewed in this framework (that uses quasi-trees rather than trees). However, c~15 would normally be called an initial quasi-tree, and we would have considered substitution at the 1&quot;/2 quasi-node rather than treating C~lS as an auxiliary quasi-tree and the n2 quasi-node as the quasi-foot. Nevertheless, this &quot;adjunction&quot; of oL15 seems to be really playing the role of substitution (with a sub-quasi tree though).</Paragraph> <Paragraph position="6"> Addressing the first issue, in the case of the multi-component adjoining example used here, we believe the need for multi-component adjoining arises from the fact that objects being composed were defined to be trees. Even in the previous version of FTAG, it was assumed that the objects being composed were trees despite the fact that two feature structures were associated with each node. These top and bottom feature structures associated with a node were supposed to account for a view of that node from two different perspectives (from the top and from below). However, 4 There are three different definitions of multi-component adjoining that have been proposed. The version considered here is the simplest kind: one where a set of trees are simultaneously adjoined into a single tree. This version leads to a system weakly equivalent to TAG. The other definitions include the case where sets of trees are adjoined simultaneously into nodes in trees that belong to another set and finally where a set of auxiliary trees are adjoined simultaneously without any restriction on the adjoining sites, 5 Although not a part of the definition of multi-component adjoining, in all analyses we are aware of, it is the case that the foot node of one component dominates the root of the other component in the eventual structure.</Paragraph> <Paragraph position="7"> A multi-component tree adjoining grammar.</Paragraph> <Paragraph position="8"> because one was dealing with a single node, it was taken for granted that the two feature structures associated with the single node would assign the node the same label (S, NP,...), no matter which perspective (viewing a node from above or from below) one took. That is, we could not consider the possibility of a node whose top and bottom parts were labeled by S and NP. Therefore, instead of using one quasi-tree (c~16), a multi-component set, fl~7, composed of two trees is used. Assuming the possibility of stating domination between quasi-nodes with different labels (as in c~16), we can similarly extend the definition of &quot;auxiliary quasi-trees&quot; to allow for the quasi-root and quasi-foot nodes being labeled differently. This is the assumption we made when we &quot;adjoined&quot; the quasi-tree c~lS to capture the effect of multi-component adjoining. Assuming that c~15 is an auxiliary structure points out the similarities between multi-component adjoining and adjoining. However, it is more natural to assume it is an initial quasi-tree and use substitution at the object NP node (rather than call c~15 an auxiliary quasi-tree and n2 quasi-node, without any justification, a quasi-foot). A similar situation arises when we consider the so-called complement auxiliary trees (see Kroch \[1987\]). fl19, an elementary quasi-tree anchored by a verb such as &quot;think,&quot; would be defined to be a complement auxiliary quasi-tree because the quasi-foot is present due to the subcategorization requirements of the anchor. In general, in the lexicalized approach to TAG, it is assumed that such an argument node is expanded Derivations in MCTAG.</Paragraph> <Paragraph position="9"> by substitution. This is consistent with Figure 19 where we could call fl19 an initial quasi-tree and substitute a17 at the supposed quasi-foot (sz) to derive a structure for Peter thinks John saw Mary. However, a quagi-tree such as fl19 must be treated as an auxiliary structure in order that we could use it for adjoining so that it can be adjoined in a18 (see Figure 18) at the pair (s3, s4} to derive a structure for who did Peter think John saw.</Paragraph> <Paragraph position="10"> The question about the basis of deciding when one should call an elementary structure auxiliary or initial remains. It is hard to justify that s2 quasi-node of fl19 is the quasi-foot on the basis of factoring of recursion (the original reason for introducing auxiliary structures). However, while developing a grammar, the s2 node in fl19 is not expanded further because we wish to factor recursion, but because it is required by the subcategorization of the anchor and such nodes are expanded as a result of a derivation step. Among the quasi-nodes that appear in the frontier of flw, the s2 quasi-node is called the quasi-foot because extraction cannot occur from a tree that can appear below the subject NP quasi-node, whereas it can in the case of s2 quasi-node.</Paragraph> <Paragraph position="11"> However, on this basis, one could also call a15 an auxiliary quasi-tree and state that the n2 quasi-node is the quasi-foot.</Paragraph> <Paragraph position="12"> Structures such as fl19 and alS (of Figure 20) raise the question of whether there is an essential difference between initial and (complement) auxiliary quasi-trees, and whether adjoining is only a special form of substitution. It appears that in the case of the two examples above, we came to the situation of calling certain structures auxiliary Initial or auxiliary? structures solely for the purpose of using the adjoining operation. If we wish to claim that there is no essential difference between initial and auxiliary structures (at least of the complement auxiliary tree variety), then we must account for the apparent difference between substitution and adjoining operations. We argue now that it may not be necessary to make this distinction if we take a closer look at the adjoining and substitution operations.</Paragraph> <Paragraph position="13"> Recall that the substitution operation was defined by the identification of two quasi-nodes. So far this has been illustrated by identifying a quasi-node that appears in the frontier of a quasi-tree with the quasi-root of another quasi-tree. However, now consider fl19 (see Figure 21) and &quot;substitution&quot; at s2 by the subtree rooted at s4 (i.e., Adjoining.</Paragraph> <Paragraph position="14"> identify the nodes referred by s2 and s4). If we insist that the resulting structure must describe a tree, then we must have either sl dominate s3 or s3 dominate s~. Now suppose there are some fundamental linguistic principles (perhaps those principles that govern the makeup of elementary structures and hence also the characteristics of the domination link between paired quasi-nodes) that determine that it is the case that s 3 must dominate sl and not vice versa. In this case we obtain 79 (as shown in Figure 21), a structure obtained by &quot;adjoining&quot; fl19. In fact, that s3 must dominate Sl must be derivable from any reasonable linguistic theory that is used to produce the elementary structures concerned (for otherwise a wrong sequence of words would be predicted)* One possible explanation of why s3 dominates Sl could be given by importing a device like the functional uncertainty machinery (Kaplan and Maxwell 1988) used in LFG* The treatment used in LFG, when imported here, would suggest that zero or more structures of the form given by fl19 would fit in the gap specified by the domination link between s3 and s4. Thus when the identification of s2 and s4 takes place, s3 must dominate sl and again zero or more structures of the form of fl19 could fit between s3 and sl now (see Joshi and Vijay-Shanker \[1989\]) for a discussion of the treatment of long-distance dependency in TAG and LFG). Another way to explain the domination of s3 over sl could be done by using the notions of maximal government domains discussed by Kroch (1989) and using it now to define the characteristics of the domination links such as that between s3 and s4. Note that once the nature of the adjoining operation has been derived, one can pre-compile out the linguistic principles and machinery used to express it. Thus even if one uses, say, the functional uncertainty machinery or maximal government domains, these additional devices (used during the developmental stages) of the grammar need not be used again during the derivation process once we have derived the adjoining operation* This is analogous to the situation with elementary structures. Some linguistic theory will be involved in defining the elementary structures of a TAG. However, once the grammar has been developed, these principles are no longer directly involved during the derivation phase* This is because the principles have been pre-compiled into the elementary structures built* Figure 22 describes the general situation that may be used to contrast substitution, adjoining and multi-component adjoining* As usual, the identification of the bl and b2 quasi-nodes defines the substitution of the O~21 at the bl quasi-node of c~20. Now suppose instead of considering a (quasi) root such as the one named b2 we consider a pair of Multi-component adjoining.</Paragraph> <Paragraph position="15"> quasi-nodes, such as Cl and b3, that are interior quasi-nodes. Now suppose we unify the b~ and b3 quasi-nodes. Since we will assume that the resulting structure must be a description of a tree, we must have the al quasi-node dominate Cl quasi-node or vice versa. If the cl quasi-node dominates al (as in &quot;710), we have a structure that appears like the one obtained by adjoining. Suppose there is some principle that predicts this situation to occur when substitution takes place; then we can conclude that adjoining is not a fundamental operation in itself but rather a derived operation. Trying to capture the above-mentioned principle would involve specifying the characteristics of the domination link between pairs of quasi-nodes such as that specified by Cl and b3 and the makeup of elementary structures of a grammar.</Paragraph> <Paragraph position="16"> Let us now consider the other case. Suppose we substitute at the bl node with the quasi-tree rooted by b3; there is no reason to assume that cl must dominate al.</Paragraph> <Paragraph position="17"> Consider the case when al dominates cl. In this case, the structure ~20 must be spliced into two ({~0 and 0PS~) as indicated in Figure 23. There are several possibilities. First, c~0 may appear above all of 0PS22 as indicated by &quot;711. This appears to correspond to the version of multi-component adjoining where different components of a set ({{~0, ~}) are adjoined simultaneously into another multi-component set, ({0PS~2, ~})&quot; Other possibilities include 0PS20 and O~2 splintered into some number of pieces (depending on the domination links found in them) and interleaved in a more complex fashion.</Paragraph> <Paragraph position="18"> To summarize, when we substitute at bl by identifying it with a quasi-root of another structure, we have the standard substitution. On the other hand, when we substitute at bl by identifying bl and b3, if cl dominates al then the resulting structure appears to be the one formed after adjunction. When al dominates Cl the situation seems to be comparable with that of multi-component adjoining, where (~20 and 0PS22 are multi-component sets made up of 0PS20,' c~20&quot; and 0~22 ,' 0PS22 ,'' respectively. Such multi-component adjoining has been used previously in providing linguistic analyses. Since both cases occur (al dominates cl or vice versa), we believe it only further justifies our claim that in situations where we consider substitutions as above, whether we have cl dominating al (adjoining) or not (multi-component adjoining) depends on the linguistic principles being instantiated during the development of elementary structures (and hence also determining the nature of domination links). Thus, this raises the question that although adjoining is used in defining the TAG formalism, could it too (like the elementary structures) be precompiled from some more fundamental principles?</Paragraph> </Section> <Section position="2" start_page="513" end_page="514" type="sub_section"> <SectionTitle> 5.2 Describing the Elementary Objects of a Grammar </SectionTitle> <Paragraph position="0"> In this section we show that the new interpretation of the TAG formalism allows the possibility of representing a grammar in a more compact fashion. This is illustrated by means of an example.</Paragraph> <Paragraph position="1"> The structure named &quot;/12 (Figure 24) pictorially represents the normal (or default) tree structure that can be associated with any verb, whereas 3'13 will be used specifically in the case of a simple transitive verb. The default structure associated with a simple transitive verb can be obtained by considering the description illustrated pictorially by 3'13 and inheriting the description (3'12) that is common for all verbs. Now since the Vl and v2 nodes have to be identified, we have the following.</Paragraph> <Paragraph position="2"> * The domination link between vpl and Vl quasi-nodes indicates a path length greater than or equal to 0. However, in this case since the labels of these quasi-nodes are different, they cannot refer to the same node. Thus, in this case we have a path length that is greater than 0.</Paragraph> <Paragraph position="3"> * vp2 quasi-node immediately dominates the v2 quasi-node (i.e., path length=l).</Paragraph> <Paragraph position="4"> * Since Vl and v2 quasi-nodes are identified and since we insist on a tree structure, we have vpl and vp2 quasi-nodes in the domination relation. In fact vpl quasi-node must dominate vp2 quasi-node in the resulting structure by a path of length 0 or more (from the two observations above).</Paragraph> <Paragraph position="5"> Thus we get the structure given by c~2a as desired. Rogers and Vijay-Shanker (1992) describe a proof system that can be used to perform the type of reasoning involved in constructing the structure o~23 as described above.</Paragraph> <Paragraph position="6"> In the manner described above we can build the default structure for every sub-categorization frame. Such structures will be specified in any lexicalized TAG; the difference (in the envisaged specification method) is that we no longer precompile out K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar all possibilities (thus repeating the structure ~'12 in all structures associated with every type of verb). To complete the description of the rest of the elementary quasi-trees one would have to use transformations, meta-rules, or lexical rules to specify the structures for passivization, wh-movement, topicalization, etc. Work along this direction is being carried out (Vijay-Shanker and Schabes 1992).</Paragraph> </Section> </Section> class="xml-element"></Paper>