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<Paper uid="C96-2103">
  <Title>Coordination in Tree Adjoining Grammars: Formalization and Implementation*</Title>
  <Section position="5" start_page="0" end_page="610" type="metho">
    <SectionTitle>
3 Trees as Structured Categories
</SectionTitle>
    <Paragraph position="0"> In (Joshi and Schabes, 1991) elementary trees as well as derived trees in an LTAG were considered as structured categories defined as a 3-tuple of an elementary or derived tree, the string it spanned  and the fnnctional type of the tree, e.g (cq, ll, Vl) in Fig. 2. Functional types for trees could be thought of us defining un-Curried flmctions col rcsponding to the Curried CCG counterpart. A fimctional type was given to sequences of lexical items in trees even when they were not contiguous; i.e. discontinuous constituents were also assigned types. They were, however, barred from coordinating. null  Coordination of two structured categories cq,a2 succeeded if tile lexieai strings of both categories were contiguous, the functional types were identical, and the least nodes dominating tile strings spanned by the component tree have the same label. For example, in Fig. 3 the tree corresponding to eats cookies and drinks beer would be obtained by: \]. equating the NP nodes I in cq and t72, preserving the linear precedence of the argu- null ments.</Paragraph>
    <Paragraph position="1"> 2. coordinating the VP nodes, which are the least nodes dominating tile two contiguous strings.</Paragraph>
    <Paragraph position="2"> 3. collapsing the supertrees above the VP node. 4. selecting the leftmost NP as the lexical site for the argument, since precedence with the verb is maintained by this choice.</Paragraph>
    <Paragraph position="4"> The process of coordination built a new derived structure given previously built pieces of 1This notion of sharing should not be confilscd with a deletion type an;dysis of coordination. The scheme presented in (Joshi attd Schabes, 1991) as well ~*s the analysis presented in this paper are not deletion analyses.</Paragraph>
    <Paragraph position="5"> derived structure (or perhaps elementary structures). There is no clear notion of a derivation structure for this process.</Paragraph>
  </Section>
  <Section position="6" start_page="610" end_page="612" type="metho">
    <SectionTitle>
4 Coordination in TAG
</SectionTitle>
    <Paragraph position="0"> An account for coordination in a standard LTAG cannot be given without introducing a notion of sharing of arguments in tile two lexically anchored trees because of the not;ion of localily of argurnents in IPSFAG. In (1) for instance, the NP the beans in I;he &amp;quot;right node raising&amp;quot; construction has to bc shared by the two eleinentary trees (anchored by cooked and ate respectively).</Paragraph>
    <Paragraph position="1"> (l) (((Harry cooked) and (Mary ate)) the beans) We introduce a notation that will enable us to talk about this more formally. In Fi R. 1 the notation ~ denotes that a node is a non-terminal and hence expects a substitution operation to oc~ cur. The notation , marks tile foot node of an auxiliary tree. Making this explicit we c~m view an elementary tree as a ordered pair of the tree structure ~n(l a ordered set 2 of such nodes fi'om its frontier 3, e.g. the tree for cooked will be represented ~s (~(cooked), {\],2.2}}. Note that this representation is not required by the LTAG formalism. The second projection of this ordered pair is used here for ease of explication. Let the second projection of the pair minus the foot nodes be (;he substitution set. We will occasionally use the first projection of the elementary tree to refer to the ordered pair.</Paragraph>
    <Paragraph position="2"> Setting up Contractions. We introduce an operation called build-contraction that takes an elementary tree, places a subset from its second projection into a contraction set and assigns the difference of the set in the second projection of the original elementary trec and the contraction set to the second projection of the new elementary tree.</Paragraph>
    <Paragraph position="3"> The contents of the contraction set of a tree can be inferred from the contents of the set in the second projection of the elementary tree. Hence, while we refer to the contraction set of an elementary tree, it does not have to bc stored along with its representation.</Paragraph>
    <Paragraph position="4"> Fig. 4 gives some examples; each node in the contraction set is circled in the figure. In the tree (a(cooked), {1,2.2}) application of the operation on the NP node at address 2.2 gives us a tree with the contraction set {2.2}. The new tree is denoted by {a(eookcd){u.2}, {1}), or o~(cooked)D.2 } for short. Placing the NP nodes at addresses 1 and 2.2 of the tree a(cooked) into the contraction set gives us a(cooked)tl,2~ ).</Paragraph>
    <Paragraph position="5"> 2'Fhc ordering is given by the fact that the elements of the set ~re Gorn ~tddresses.</Paragraph>
    <Paragraph position="6"> 3We sh~ll assume there are no adjunction constraints in this paper.</Paragraph>
    <Paragraph position="7">  We assume ~hat the anchor cannot be involved in a build-contraction. This assumption needs to be revised when gapping is considered in this framework (SS5).</Paragraph>
    <Paragraph position="8"> The Coordination Schema. We use the standard notion of coordination shown in Fig. 5 which maps two constituents of like type, but with different interpretations, into a constituent of the same</Paragraph>
    <Paragraph position="10"> We add a new rewriting operation to the LTAG formalism called conjoin 5. While substitution and adjunction take two trees to give a derived tree, conjoin takes three trees and composes them to give a derived tree. One of the trees is always the tree obtained by specializing the schema in Fig. 5 for a particular category 6.</Paragraph>
    <Paragraph position="11"> Informally, the conjoin operation works as follows: The two trees being coordinated are substituted into the conjunction tree. This notion of substitution differs from the traditional LTAG substitution operation in the following way: In LTAG substitution, always the root node of the tree being substituted is identified with the substitution site. In the conjoin operation however, the node substituting into the conjunction tree is given by an algorithm, which we shall call FindRoot that takes into account the contraction sets of the two trees. FindRoot returns the lowest node that dominates all nodes in the substitution set of the elementary tree 7, e.g. FindRoot(a(cooked){2.2}) will return the root node, i.e. corresponding to the S 4In this paper, we do not consider coordination of unlike categories, e.g. Pat is a Republican and proud of it. (Sarkar and Joshi, 1996) discusses such cases, following Jorgensen and Abeill6 (1992).</Paragraph>
    <Paragraph position="12"> SLater we will discuss an alternative which replaces this operation by the traditional operations of substitution and adjunction.</Paragraph>
    <Paragraph position="13">  dress 2.1, corresponding to the V conj Vinstantiation. null The conjoin operation then creates a contraction between nodes in the contraction sets of the trees being coordinated. The term contraction is taken from the graph-theoretic notion of edge contraction. In a graph, when an edge joining two vertices is contracted, the nodes are merged and the new vertex retains edges to the union of the neighbors of the merged vertices s. The conjoin operation supplies a new edge between each corresponding node in the contraction set and then contracts that edge. As a constraint on the application of the conjoin operation, the contraction sets of the two trees must be identical.</Paragraph>
    <Paragraph position="14"> Another way of viewing the conjoin operation is as the construction of an auxiliary structure fi'om an elementary tree. For example, from the elementary tree (a(drinks), {1, 2.2}), the conjoin operation would create the auxiliary structure (fl(drinks){1}, {2.2}) shown in Fig. 6. The adjunction operation would now be responsible for creating contractions between nodes in the contraction sets of the two trees supplied to it. Such an approach is attractive for two reasons. First, it uses only the traditional operations of substitution and adjnnction. Secondly, it treats conj X as a kind of &amp;quot;modifier&amp;quot; on the left conjunct X. We do not choose between the two representations but continue to view the conjoin operation as a part of our formalism.</Paragraph>
    <Paragraph position="15">  For example, applying conjoin to the trees Conj(and), a(eats){1} and c~(drinks){l} gives us tile derivation tree and derived structure for the constituent in (2) shown in Fig. 7.</Paragraph>
    <Paragraph position="16"> (2) ... eats cookies and drinks beer.</Paragraph>
    <Paragraph position="17"> In Fig. 7 the nodes (~(eats){1} and a(drinks)\[ll signify an operation left incomplete at address 1. in (Joshi and Schabes, 1991). A coordinated node will never dominate multiple foot nodes. Such a case occurs, e.g., two auxiliary trees with substitution nodes at the same tree address are coordinated with only the substitution nodes in the contraction set.</Paragraph>
    <Paragraph position="18"> SMerging in the graph-theoretic definition of contraction involves the identification of two previously distinct nodes. In the process of contraction over nodes in elementary trees it is the operation on that node (either substitution or adjunction) that is identified. null</Paragraph>
    <Paragraph position="20"> The Effects of Contraction. One of the effects of contraction is that; the notion of a derivation tree for the 12FAG formalism has to be extended to an acyclic derivation graph 9. Simultaneous substitution or adjunction modifies a derivation tree into a graph as can be seen in Fig. 8.</Paragraph>
    <Paragraph position="21"> If a contracted node in a tree (after the conjoin  operation) is a substitution node, then the argument is recorded as a substitution into the two elementary trees ms for example in the sentences (3) and (4).</Paragraph>
    <Paragraph position="22"> (3) Chapman eats cookies and drinks beer.</Paragraph>
    <Paragraph position="23"> (4) Keats steals and Chapman eats apples.</Paragraph>
    <Paragraph position="24">  Fig. 8 contains the derivation and derived structures for (3) and Fig. 9 for (4). In Fig. 9 the d(,riw~tion graph for sentence (4) accounts \['or the coordinations of the traditionM nonconstituent &amp;quot;Keats steals&amp;quot; by carrying out the coordination at the root, i.e. S conj S. No constituent corresponding to &amp;quot;Keats steals&amp;quot; is created in the process of co-</Paragraph>
    <Paragraph position="26"> and drinks beer.</Paragraph>
    <Paragraph position="27"> The derived structures in Figs. 8 and 9 are diff, cult to reconcile with traditional notions of phrase structure 1deg. However, the derivation structure gives us all the information about dependency degWe shall use the general notation derivation structure to refer to both derivation trees and derivation graphs.</Paragraph>
    <Paragraph position="28"> tdegMcCawley (1982) rMsed the heterodox view that a discontinuous constituent structure should be given for right node raising cases, having the same notion of constituency as our approach. IIowever, no conditions on the construction of such a structm'e was given. In fact, his mechanism also covered cases of parenthetical placement, scrambling, relative clause extraposition  man eats apples.</Paragraph>
    <Paragraph position="29"> that we need about the constituents. The derivation encodes exactly how particular elementary trees are put together. Obtaining a tree structure fi:om a derived structure built by the conjoin operation is discussed in (Sarkar and Joshi, 1996). Considerations of the locality of movement phenomena and its representation in the LTAG tbrrealism (Kroch and Joshi, 1986) can also now expkdn constraints on coordinate structure, such as across-the-board exceptions to the well known co-ordinate structure constraint, see Fig. 10. Also in eases of unbounded right node raising such as Keats likes and Chapman thinks Mary likes beans, Chapman thinks simply adjoins into the right conjunct of the coordinate structure 11.</Paragraph>
  </Section>
  <Section position="7" start_page="612" end_page="613" type="metho">
    <SectionTitle>
5 Contractions on Anchors
</SectionTitle>
    <Paragraph position="0"> An LTAG along with the operations of substitution and adjnnction also has tile implicit operation of lexical insertion (represented as the diamond mark in Fig. 11). Under this view, the and heavy NP shift.</Paragraph>
    <Paragraph position="1"> 11A eomparision of this paper's approach with the derivational machinery in CCG and the devices of 3-D coordination is done in (Sarkar and Joshi, 1996).</Paragraph>
    <Paragraph position="2">  LTAG trees are taken to be templates. For example, the tree in Fig. 11 is now represented as &lt;~(eat), {1, 2.1, 2.2\]).</Paragraph>
    <Paragraph position="3">  If we extend the notion of contraction in the conjoin operation together with the operation of lexical insertion we have the following observations: The two trees to be used by the conjoin operation are no longer strictly lexicalized as the label associated with the diamond mark is a preterminal. Previous uses of conjoin applied to two distinct trees. If the lexicalization operation is to apply simultaneously, the same anchor projects two elementary trees from the lexicon. The process of contraction ensures that the anchor is placed into a pair of LTAG tree templates with a single lexical insertion.</Paragraph>
    <Paragraph position="4"> Gapping. Using this extension to conjoin, we can handle sentences that have the &amp;quot;gapping&amp;quot; construction like sentence (5).</Paragraph>
    <Paragraph position="5"> (5) John ate bananas and Bill strawberries.</Paragraph>
    <Paragraph position="6"> The conjoin operation applies to copies of the same elementary tree when the lexical anchor is in the contraction set. For example, let o~(eats) be the tree selected by cats. The coordination of o~(cats){2.l} with a copy of itself and the subsequent derivation tree is depicted in Fig. 1212 .  An extension of the approach here will be to permit the conjoin operation to create contractions on all the nodes in contraction sets that it a2In English, following Ross (1970), the anchor goes to the left conjunct.</Paragraph>
    <Paragraph position="7"> dominates during a derivation, allowing us to recognize cases of gapping such as: John wants Penn to win and Bill, Princeton. and John wants to try to see Mary and Bill, Susan.</Paragraph>
    <Paragraph position="8"> Coordinating Ditransitive verbs. In sentence (6) if we take the position that the string Mary a book is not a constituent (i.e. give has a structure as in Fig. 13), then we can use the notion of contraction over the anchor of a tree to derive the sentence in (6). The structure we derive is shown in Fig. 14.  contractions on the anchor allow the derivation of sluicing structures such as (7) (where the conjunct Bill too can be interpreted as \[John loves\] Bill too or as Bill \[loves Mary\] too 13.</Paragraph>
    <Paragraph position="9"> (7) John loves Mary and Bill too.</Paragraph>
  </Section>
  <Section position="8" start_page="613" end_page="614" type="metho">
    <SectionTitle>
6 Parsing Issues
</SectionTitle>
    <Paragraph position="0"> This section discusses parsing issues that arise in the modified TAG formalism that we have presented. We do not discuss general issues in parsing TAGs, rather we give the appropriate modifications that are needed to the existing Earley-type parsing algorithm for TAGs due to Schabes and Joshi (1988).</Paragraph>
    <Paragraph position="1"> The algorithm relies on a tree traversal that scans the input string from left to right while recognizing the application of the conjoin operation. The nodes in the elementary trees are visited in a top-down left to right manner (Fig. 15). F, ach dot in Fig. 15 divides the tree into a left context and a 13Whether this should be derived syntactically is controversial, for example, see (Steedman, 1990).</Paragraph>
    <Paragraph position="2">  right context, enabling the algorithm to scan the elementary tree while trying to recognize possible applications of the conjoin operation.</Paragraph>
    <Paragraph position="3">  3'he derived structure corresponding to a coordination is a compositc structure built by applying the conjoin operation to two elementary trees and an instantiation of the coordination schema. The algorithm never builds derived structures. It builds the derivation by visiting the appropriate ,,()des during its tree traversal it, the following order (see Fig. 16).</Paragraph>
    <Paragraph position="4"> I 2.-.3 d...5 6...2' 7'...'( 4'...5/ 6 I.-.78 The algorithm must also compute the correct span of the string for the nodes that have been identified via a contraction. Fig. 16 gives the possible scenarios tbr the position of nodes that have been linked by a contraction. Whet, loot nodes undergo contraction, the algorithm has to ensure that both the foo~ nodes share the sub-tree pushed under them, e.g. 9. * * 10 and 9 ~. * * 10 ~ in l,'ig. 16(a). Similarly, when substitution nodes undergo contraction, the algorithm has to ensure that the tree recognized dile by pr('.dicting a substitution is shared by the nodcs, e.g. 11 * * * 12 and l l'... 12' in Figs. 16(b) and 16(c). '1'he traversals 9 ... 10 should st)an the same length of the intro, as 9'...10', similarly for 11... 12 and 11'...12'. Various positions for such traversals is shown in Fig. 116. A derivation is valid if the input string is accepted and each i, ode in a contraction sl)ans a valid subs,ring in the inI)ut. 'rite complete and formal (leseription of the l)arsing algorithm is given in (Sarkar and Joshi, 1996).</Paragraph>
  </Section>
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