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<Paper uid="P89-1027">
  <Title>TREATMENT OF LONG DISTANCE DEPENDENCIES IN LFG AND TAG: FUNCTIONAL UNCERTAINTY IN LFG IS A COROLLARY IN TAG&amp;quot;</Title>
  <Section position="5" start_page="220" end_page="220" type="metho">
    <SectionTitle>
T TOPIC -T COMPdegOBJ
</SectionTitle>
    <Paragraph position="0"> The functional uncertainty device restricts the labels (such as COMP deg) to be drawn from the class of regular expressions. The definition of f-structures is extended to allow such equations \[2, 3\]. Informally, this definition states that if f is a f-structure and a is a regular set, then (fa) = v holds if the value of f for the attribute s is a f-structure fl such that (flY) -- v holds, where sy is a string in a, or f = v and e E a.</Paragraph>
    <Paragraph position="1"> The functional uncertainty approach may be characterized as a localization of the long distance dependencies; a localization at the level of f-structures rather than at the level of c-structures.</Paragraph>
    <Paragraph position="2"> This illustrates the fact that if we use CFG-like rules to produce the surface structures, it is hard to state some generalizations directly; on the other hand, f-structures or elementary trees in TAGs (since they localize the predicate argument dependencies) are appropriate domains in which to state these generalizations. We show that there is a direct link between the regular expressions used in LFG and the elementary trees of TAG.</Paragraph>
  </Section>
  <Section position="6" start_page="220" end_page="220" type="metho">
    <SectionTitle>
I.I OUTLINE OF THE PAPER
</SectionTitle>
    <Paragraph position="0"> In Section 2, we will define briefly the TAG formalism, describing some of the key points of the linguistic theory underlying it. We will also describe briefly Feature Structure Based Tree Adjoining Grammars (FTAG), and show how some elementary trees (auxiliary trees) behave as func: tions over feature structures. We will then show how regular sets over labels (such as COMP deg) can also be denoted by functions over feature structures. In Section 3, we will consider the example of topicalization as it appears in Section 1 and show that the same statements are made by the two formalisms when we represent both the elementary trees of FTAG and functional uncertainties in LFG as functions over feature structures. We also point out some differences in the two analyses which arise due to the differences in the formalisms. In Section 4, we point out how these similar statements are stated differently in the two formalisms. The equations that capture the linguistic generalizations are still associated with individual rules (for the c-structure) of the grammar in LFG. Thus, in order to state generalizations for a phenomenon that is not localized in the cstructure, extra machinery such as functional uncertainty is needed. We show that what this extra machinery achieves for CFG based systems follows as a corollary of the TAG framework. This results from the fact that the elementary trees in a TAG provide an extended domain of locality, and factor out recursion and dependencies. A computational consequence of this result is that we can obtain these analyses without going outside the power of TAG and thus staying within the class of constrained grammatical formalisms characterized as mildly context.sensitive (Joshi \[1\]). Another consequence of the differences in the representations (and localization) in the two formalisms is as follows. In a TAG, once an elementary tree is picked, there is no uncertainty about the functionality in long distance dependencies. Because LFG relies on a CFG framework, interactions between uncertainty equations can arise; the lack of such interactions in TAG can lead to simpler processing of long distance dependencies. Finally, we make some remarks as to the linguistic significance of restricting the use of regular sets in the functional uncertainty machinery by showing that the linguistic theory instantiated in TAG can predict that the path depicting the &amp;quot;movement&amp;quot; in long distance dependencies can be characterized by regular sets.</Paragraph>
  </Section>
  <Section position="7" start_page="220" end_page="221" type="metho">
    <SectionTitle>
2 INTRODUCTION TO TAG
</SectionTitle>
    <Paragraph position="0"> Tree Adjoining Grammars (TAGs) are tree rewriting systems that are specified by a finite set of elementary trees. An operation called adjoining ~ is used to compose trees. The key property of the linguistic theory of TAGs is that TAGs allow factoring of recursion from the domain of dependencies, which are defined by the set of elementary trees. Thus, the elementary trees in a TAG correspond to minimal linguistic structures that localize the dependencies such as agreement, subcategorization, and filler-gap. There are two kinds of elementary trees: the initial trees and auxiliary trees. The initial trees (Figure 1) roughly correspond to &amp;quot;simple sentences&amp;quot;. Thus, the root of an initial tree is labeled by S or ~. The frontier is all terminals.</Paragraph>
    <Paragraph position="1"> The auxiliary trees (Figure 1) correspond roughly to minimal recursive constructions. Thus, if the root of an auxiliary tree is labeled by a non-terminal symbol, X, then there is a node (called the foot node) in the frontier which is labeled by X. The rest of the nodes in the frontier are labeled by terminal symbols.</Paragraph>
    <Paragraph position="2"> 2We do not consider lexicalized TAGs (defined by Schabes, Abeille, and Joshi \[7\]) which allow both adjoining and sub6titution. The ~uhs of this paper apply directly to them. Besides, they are formally equivalent to TAGs.  We will now define the operation of adjoining.</Paragraph>
    <Paragraph position="3"> Consider the adjoining of/~ at the node marked with * in a. The subtree of a under the node marked with * is excised, and/3 is inserted in its place. Finally, the excised subtree is inserted below the foot node of w, as shown in Figure 1.</Paragraph>
    <Paragraph position="4"> A more detailed description of TAGs and their linguistic relevance may be found in (Kroch and ao hi \[51).</Paragraph>
  </Section>
  <Section position="8" start_page="221" end_page="222" type="metho">
    <SectionTitle>
2.1 FEATURE STRUCTURE BASED
TREE ADJOINING GRAMMARS
(FTAG)
</SectionTitle>
    <Paragraph position="0"> In unification grammars, a feature structure is associated with a node in a derivation tree in order to describe that node and its relation to features of other nodes in the derivation tree. In a FTAG, with each internal node, T/, we associate two feature structures (for details, see \[9\]). These two feature structures capture the following relations  (Figure 2) 1. The relation ofT/to its supertree, i.e., the view of the node from the top. The feature structure that describes this relationship is called ~.</Paragraph>
    <Paragraph position="1">  Note that both the t, and b, feature structures hold for the node 7. On the other hand, with each leaf node (either a terminal node or a foot node), 7, we associate only one feature structure (let us call it t,3).</Paragraph>
    <Paragraph position="2"> Let us now consider the case when adjoining takes place as shown in the Figure 2. 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 troo~,broot and tloot= bLoo~ are the t and b statements of the root and foot nodes of the auxiliary tree used for adjoining at the node 7. Based on what t and b stand for, it is obvious that on adjoining the statements t, and troot hold for the node corresponding to the root of the auxiliary tree. Similarly, the statements b, and b/oo~ hold for the node corresponding to the foot of the auxiliary tree. Thus, on adjoining, we unify t, with troot, and b, with b/oot. In fact, this adjoining-is permissible only if t.oo~ and t. are compatible and so are b/oot and b~. If we do not adjoin at the node, 7, then we unify t, with b,. More details of the definition of FTAG may be found in \[8, 9\].</Paragraph>
    <Paragraph position="3"> We now give an example of an initial tree and an auxiliary tree in Figure 3. We have shown only the necessary top and bottom feature structures for the relevant nodes. Also in each feature structure  shown, we have only included those feature-value pairs that are relevant. For the auxiliary tree, we have labeled the root node S. We could have labeled it S with COMP and S as daughter nodes. These details are not relevant to the main point of the paper. We note that, just as in a TAG, the elementary trees which are the domains of dependencies are available as a single unit during each step of the derivation. For example, in al the topic and the object of the verb belong to the same tree (since this dependency has been factored into al) and are coindexed to specify the movemeat due to topicalization. In such cases, the dependencies between these nodes can be stated directly, avoiding the percolation of features during the derivation process as in string rewriting systems. Thus, these dependencies can be checked locally, and thus this checking need not be linked to the derivation process in an unbounded manner.</Paragraph>
    <Paragraph position="5"> to adjoining, since this feature structure is not known, we will treat it as a variable that gets instantiated on adjoining. This treatment can be formalized by treating the auxiliary trees as functions over feature structures (by A-abstracting the variable corresponding to the feature structure for the tree that will appear below the foot node).</Paragraph>
    <Paragraph position="6"> Adjoining corresponds to applying this function to the feature structure corresponding to the subtree below the node where adjoining takes place.</Paragraph>
    <Paragraph position="7"> Treating adjoining as function application, where we consider auxiliary trees as functions, the representation of/3 is a function, say fz, of the form (see Figure 2) ~f.($roo, A ...(broot A f)) If we now consider the tree 7 and the node T?, to allow the adjoining of/3 at the node ~, we must</Paragraph>
    <Paragraph position="9"> Note that if we do not adjoin at ~7, since t, and /3, have to be unified, we must represent 7 by the formula (...~Ab~A...) which can be obtained by representing 7 by</Paragraph>
  </Section>
  <Section position="9" start_page="222" end_page="223" type="metho">
    <SectionTitle>
2.2 A CALCULUS TO REPRESENT
FTAG
</SectionTitle>
    <Paragraph position="0"> In \[8, 9\], we have described a calculus, extending the logic developed by Rounds and Kasper \[4, 6\], to encode the trees in a FTAG. We will very briefly describe this representation here.</Paragraph>
    <Paragraph position="1"> To understand the representation of adjoining, consider the trees given in Figure 2, and in particular, the node rl. The feature structures associated with the node where adjoining takes place should reflect the feature structure after adjoining and as well as without adjoining. Further, the feature structure (corresponding to the tree structure below it) to be associated with the foot node is not known prior to adjoining, but becomes specified upon adjoining. Thus, the bottom feature structure associated with the foot node, which &amp;quot;is b foot before adjoining, is instantiated on adjoining by unifying it with a feature structure for the tree that will finally appear below this node. Prior</Paragraph>
    <Paragraph position="3"> where I is the identity function. Similarly, we must allow adjoining by any auxiliary tree adjoinable at 7/(admissibility of adjoining is determined by the success or failure of unification). Thus, if /31,... ,/3, form the set of auxiliary trees, to allow for the possibility of adjoining by any auxiliary tree, as well as the possibility of no adjoining at a node, we must have a function, F, given by F = Af.(f~x(f) V... V f:~(f) V f) and then we represent 7 by (. ..t, A F(b,) A .. .).</Paragraph>
    <Paragraph position="4"> In this way, we can represent the elementary trees (and hence the grammar) in an extended version of K-K logic (the extension consists of adding A-abstraction and application).</Paragraph>
  </Section>
  <Section position="10" start_page="223" end_page="223" type="metho">
    <SectionTitle>
3 LFG AND TAG ANALYSES
FOR LONG DISTANCE DE-
PENDENCIES
</SectionTitle>
    <Paragraph position="0"> We will now relate the analyses of long distance dependencies in LFG and TAG. For this purpose, we will focus our attention only on the dependencies due to topicalization, as illustrated by sentences 1, 2, and 3 in Section 1.</Paragraph>
    <Paragraph position="1"> To facilitate our discussion, we will consider regular sets over labels (as used by the functional uncertainty machinery) as functions over feature structures (as we did for auxiliary trees in FTAG).</Paragraph>
    <Paragraph position="2"> In order to describe the representation of regular sets, we will treat all labels (attributes) as functions over feature structures. Thus, the label COMP, for example, is a function which given a value feature structure (say v) returns a feature structure denoted by COMP : v. Therefore, we can denote it by Av.COMP : v. In order to describe the representation of arbitrary regular sets we have to consider only their associated regular expressions. For example, COMP deg can be represented by the function C* which is the fixed-point 4</Paragraph>
    <Paragraph position="4"> structure will have a general form described by TOPIC : v A COMP : COMP : ... OBJ : v.</Paragraph>
    <Paragraph position="5"> Consider the FTAG fragment (as shown in Figure 3) which can be used to generate the sentences 1, 2, and 3 in Section 1. The initial tree al will be represented by cat : &amp;quot;~ A F(topic : v A F(pred : telephonedAobj : v)). Ignoring some irrelevant details (such as the possibility of adjoining at nodes other than the S node), we cnn represent ax as al = topic : v A F(obj : v) Turning our attention to /~h let us consider the bottom feature structure of the root of/~1. Since its COMP ~ the feature structure associated with the foot node (notice that no adjoining is allowed at the foot node and hence it has only one feature structure), and since adjoining can take place at the root node, we have the representation of 81 as tin \[8\], we have established that the fixed-point exists. aWe use the fact that R&amp;quot; = R'RU {e}.</Paragraph>
    <Paragraph position="7"> where F is the function described in Section 2.2.</Paragraph>
    <Paragraph position="8"> From the point of view of the path from the root to the complement, the NP and VP nodes are irrelevant, so are any adjoinings on these nodes.</Paragraph>
    <Paragraph position="9"> So once again, if we discard the irrelevant information (from the point of view of comparing this analyses with the one in LFG), we can simplify the representation of 81 as</Paragraph>
    <Paragraph position="11"> As explained in Section 2.2, since j31 is the only auxiliary tree of interest, F would be defined as F = a/.Zl(/)v/. Using the definition of/~1 above, and making some reductions we have</Paragraph>
    <Paragraph position="13"> This is exactly the same analysis as in LFG using the functional uncertainty machinery. Note that the fixed-point of F isC,. Now consider al. Obviously any structure derived from it can now be represented as topic : v A C * (obj : v) This is the same analysis as given by LFG.</Paragraph>
    <Paragraph position="14"> In a TAG, the dependent items are part of the same elementary tree. Features of these nodes can be related locally within this elementary tree (as in a,). This relation is unaffected by any adjoinings on nodes of the elementary tree. Although the paths from the root to these dependent items are elaborated by the adjoinings, no external device (such as the functional uncertainty machinery) needs to be used to restrict the possible paths between the dependent nodes. For instance, in the example we have considered, the fact that TOPIC = COMP : COMP... : OBJ follows from the TAG framework itself. The regular path restrictions made in functional uncertainty statements such as in TOPIC = COMP*OBJ is redundant within the TAG framework.</Paragraph>
  </Section>
  <Section position="11" start_page="223" end_page="224" type="metho">
    <SectionTitle>
4 COMPARISON OF THE TWO
FORMALISMS
</SectionTitle>
    <Paragraph position="0"> We have compared LFG and TAG analyses of long distance dependencies, and have shown that what functional uncertainty does for LFG comes out as a corollary in TAG, without going beyond the power of mildly context sensitive grammars.</Paragraph>
    <Paragraph position="1">  Both approaches aim to localize long distance dependencies; the difference between TAG and LFG arises due to the domain of locality that the formalisms provide (i.e., the domain over which statements of dependencies can be stated within the formalisms).</Paragraph>
    <Paragraph position="2"> In the LFG framework, CFG-like productions are used to build the c-structure. Equations are associated with these productions in order to build the f-structure. Since the long distance dependencies are localized at the functional level, additional machinery (functional uncertainty) is provided to capture this localization. In a TAG, the elementary trees, though used to build the &amp;quot;phrase structure&amp;quot; tree, also form the domain for localizing the functional dependencies. As a result, the long distance dependencies can be localized in the elementary trees. Therefore, such elementary trees tell us exactly where the filler &amp;quot;moves&amp;quot; (even in the case of such unbounded dependencies) and the functional uncertainty machinery is not necessary in the TAG framework. However, the functional uncertainty machinery makes explicit the predictions about the path between the &amp;quot;moved&amp;quot; argument (filler) and the predicate (which is close to the gap). In a TAG, this prediction is not explicit.</Paragraph>
    <Paragraph position="3"> Hence, as we have shown in the case of topicalization, the nature of elementary trees determines the derivation sequences allowed and we can confirm (as we have done in Section 3) that this prediction is the same as that made by the functional uncertainty machinery.</Paragraph>
  </Section>
  <Section position="12" start_page="224" end_page="224" type="metho">
    <SectionTitle>
4.1 INTERACTIONS AMONG UNCER-
TAINTY EQUATIONS
</SectionTitle>
    <Paragraph position="0"> The functional uncertainty machinery is a means by which infinite disjunctions can be specified in a finite manner. The reason that infinite number of disjunctions appear, is due to the fact that they correspond to infinite number of possible derivations. In a CFG based formalism, the checking of dependency cannot be separated from the derivation process. On the other hand, as shown in \[9\], since this separation is possible in TAG, only finite disjunctions are needed. In each elementary tree, there is no uncertainty about the kind of dependency between a filler and the position of the corresponding gap. Different dependencies correspond to different elementary trees. In this sense there is disjunction, but it is still only finite. Having picked one tree, there is no uncertainty about the grammatical function of the filler, no matter how many COMPs come in between due to adjoining. This fact may have important consequences from the point of view of relative efficiency of processing of long distance dependencies in LFG and TAG. Consider, for example, the problem of interactions between two or more uncertainty equations in LFG as stated in \[2\]. Certain strings in COMP deg cannot be solutions for (f TOPIC) = (.f COMP&amp;quot; GF) when this equation is conjoined (i.e., when it interacts) with (f COMP SUBJ NUM) = SING and (f TOPIC NUM) = PL. In this case, the shorter string COMP SUBJ cannot be used for COMP&amp;quot; GF because of the interaction, although the strings COMP i SUB J, i &gt;_ 2 can satisfy the above set of equations. In general, in LFG, extra work has to be done to account for interactions.</Paragraph>
    <Paragraph position="1"> On the other hand, in TAG, as we noted above, since there is no uncertainty about the grammatical function of the filler, such interactions do not arise at all.</Paragraph>
  </Section>
  <Section position="13" start_page="224" end_page="225" type="metho">
    <SectionTitle>
4.2 REGULAR SETS IN FUNCTIONAL
UNCERTAINTY
</SectionTitle>
    <Paragraph position="0"> From the definition of TAGs, it can be shown that the paths are always context-free sets \[11\]. If there are linguistic phenomena where the uncertainty machinery with regular sets is not enough, then the question arises whether TAG can provide an adequate analysis, given that paths are context-free sets in TAGs. On the other hand, if regular sets are enough, we would like to explore whether the regularity requirement has a linguistic significance by itself. As far as we are aware, Kaplan and Zaenen \[3\] do not claim that the regularity requirement follows from the linguistic considerations. Rather, they have illustrated the adequacy of regular sets for the linguistic phenomena they have described. However, it appears that an appropriate linguistic theory instantiated in the TAG framework will justify the use of regular sets for the long distance phenomena considered here.</Paragraph>
    <Paragraph position="1"> To illustrate our claim, let us consider the elementary trees that are used in the TAG analysis of long distance dependencies. The elementary trees, Sl and/31 (given in Figure 3), are good representative examples of such trees. In the initial tree, C/zt, the topic node is coindexed with the empty NP node that plays the grammatical role of object. At the functional level, this NP node is the object of the S node of oq (which is captured in the bottom feature structure associated with the S node). Hence, our representation of  at (i.e., looking at it from the top) is given by topic : v A F(obj : v), capturing the &amp;quot;movement&amp;quot; due to topicalization. Thus, the path in the functional structure between the topic and the object is entirely determined by the function F, which in turn depends on the auxiliary trees that can be adjoined at the S node. These auxiliary trees, such as/~I, are those that introduce complementizer predicates. Auxiliary trees, in general, introduce modifiers or complementizer predicates as in/~1. (For our present discussion we can ignore the modifier type auxiliary trees). Auxiliary trees upon adjoining do not disturb the predicate argument structure of the tree to which they are adjoined. If we consider trees such as/~I, the complement is given by the tree that appears below the foot node. A principle of a linguistic theory instantiated in TAG (see \[5\]), similar to the projec~ion principle, predicts that the complement of the root (looking at it from below) is the feature structure associated with the foot node and (more importantly) this relation cannot be disrupted by any adjoinings. Thus, if we are given the feature structure, f, for the foot node (known only after adjoining), the bottom feature structure of the root can be specified as comp : jr, and that of the top feature structure of the root is F(comp : f), where F, as in a,, is used to account for adjoinings at the root.</Paragraph>
    <Paragraph position="2"> To summarize, in al, the functional dependency between the topic and object nodes is entirely determined by the root and foot nodes of auxiliary trees that can be adjoined at the S node (the effect of using the function F). By examining such auxiliary trees, we have characterized the latter path as Af.F(comp : f). In grammatical terms, the path depicted by F can be specified by right-linear productions F -* F comp : / I Since right-linear grammars generate only regular sets, and TAGs predict the use of such right-linear rules for the description of the paths, as just shown above, we can thus state that TAGs give a justification for the use of regular expressions in the functional uncertainty machinery.</Paragraph>
  </Section>
  <Section position="14" start_page="225" end_page="225" type="metho">
    <SectionTitle>
4.3 GENERATIVE CAPACITY AND
LONG DISTANCE DEPENDENCY
</SectionTitle>
    <Paragraph position="0"> We will now show that what functional uncertainty accomplishes for LFG can be achieved within the FTAG framework without requiring power beyond that of TAGs. FTAG, as described in this paper, is unlimited in its generative capacity. By placing no restrictions on the feature structures associated with the nodes of elementary trees, it is possible to generate any recursively enumerable language. In \[9\], we have defined a restricted version of FTAG, called RFTAG, that can generate only TALs (the languages generated by TAGs). In RFTAG, we insist that the feature structures that are associated with nodes are bounded in size, a requirement similar to the finite closure membership restriction in GPSG. This restricted system will not allow us to give the analysis for the long distance dependencies due to topicalization (as given in the earlier sections), since we use the COMP attribute whose value cannot be bounded in size. However, it is possible to extend RFTAG in a certain way such that such analysis can be given. This extension of RFTAG still does not go beyond TAG and thus is within the class of mildly context-sensitive grammar formalisms defined by Joshi \[1\]. This extension of RFTAG is discussed in \[10\].</Paragraph>
    <Paragraph position="1"> To give an informal idea of this extension and a justification for the above argument, let us consider the auxiliary tree,/~1 in Figure 3. Although we coindex the value of the comp feature in the feature structure of the root node of/~1 with the feature structure associated with the foot node, we should note that this coindexing does not affect the context-freeness of derivation. Stated differently, the adjoining sequence at the root is independent of other nodes in the tree in spite of the coindexing. This is due to the fact that as the feature structure of the foot of/~1 gets instantiated on adjoining, this value is simply substituted (and not unified) for the value of the comp feature of the root node. Thus, the comp feature is being used just as any other feature that can be used to give tree addresses (except that comp indicates dominance at the functional level rather than at the tree structure level). In \[10\], we have formalized this notion by introducing graph adjoining grammars which generate exactly the same languages as TAGs. In a graph adjoining grammar, /~x is represented as shown in Figure 4. Notice that in this representation the comp feature is like the features 1 and 2 (which indicate the left and right daughters of a node) and therefore not used explicitly.</Paragraph>
  </Section>
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