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<Paper uid="P00-1057">
  <Title>Multi-Component TAG and Notions of Formal Power</Title>
  <Section position="4" start_page="2" end_page="2" type="metho">
    <SectionTitle>
3 Linguistic Applications
</SectionTitle>
    <Paragraph position="0"> In cases where TAG models dependencies correctly, the use of R-MCTAG is straightforward: when an auxiliary tree adjoins at a site pair which is just a single node, it looks just like conventional adjunction. However, in problematiccases we can use the extra expressivepower of R-MCTAG to model dependencies correctly. Two such cases are discussed below.</Paragraph>
    <Section position="1" start_page="2" end_page="2" type="sub_section">
      <SectionTitle>
3.1 Bridge and Raising Verbs
</SectionTitle>
      <Paragraph position="0"> Consider the case of sentences which contain both bridge and raising verbs, noted by Rambow et al. #281995#29. In most TAG-based analyses, bridge verbs adjoin at S #28or C  #29, and raising verbs adjoin at VP #28or I  #29. Thus the derivation for a sentence like #281#29 John thinks that Mary seems to sleep.</Paragraph>
      <Paragraph position="1"> will have the trees for thinks and seems simultaneously adjoining into the tree for like, which, when interpreted, gives an incorrect dependency structure.</Paragraph>
      <Paragraph position="2"> But under the present view we can analyze sentences like #281#29 with derivations mirroring dependencies. The desired trees for #281#29 are shown in Figure 7. Since the tree for that seems can meta-adjoin around the subject, the tree for thinks correctly adjoins into the tree for seems rather than eat.</Paragraph>
      <Paragraph position="3"> Also, although the above analysis produces the correct dependency links, the directions are inverted in some cases. This is a disadvantage compared to, for example, DSG; but since the directions are consistently inverted, for applications like translation or statistical modeling, the particular choice of direction is usually immaterial.</Paragraph>
    </Section>
    <Section position="2" start_page="2" end_page="2" type="sub_section">
      <SectionTitle>
3.2 More on Raising Verbs
</SectionTitle>
      <Paragraph position="0"> Tree-local MCTAG is able to derive #282a#29, but unable to derive #282b#29 except by adjoining the auxiliary tree for to be likely at the foot of the auxiliary tree for seem #28Frank et al., 1999#29.</Paragraph>
      <Paragraph position="1"> #282#29 a. Does John seem to sleep? b. Does John seem to be likely to sleep? The derivation structure of this analysis does not match the dependencies, however|seem adjoins into to sleep.</Paragraph>
      <Paragraph position="2"> DSG can derive this sentence with a derivation matching the dependencies, but it loses some of the advantage of TAG in that, for example, cases of super-raising #28where the verb is raised out of two clauses#29 must be explicitly ruled out by subsertion-insertion constraints. Frank et al. #281999#29 and Kulick #282000#29 give analyses of raising which assign the desired derivation structures without running into this problem. It turns out that the analysis of raising from the previous section, designed for a translation problem, has both of these properties as well. The grammar is shown back in Figure 4.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="2" end_page="6" type="metho">
    <SectionTitle>
4 A Parser
</SectionTitle>
    <Paragraph position="0"> Figure 8 shows a CKY-style parser for our restrictionof MCTAG as a system of inference rules. It is limited to grammars whose trees are at most binary-branching.</Paragraph>
    <Paragraph position="1"> The parser consists of rules over items of one of the following forms, where w</Paragraph>
    <Paragraph position="3"> specify nodes of the grammar; i, j, k, and l are integers between 0 and n inclusive; and code is either + or ,: #0F #5B#11;code;i;,;,;l;,;,#5D and #5B#11;code;i;j;k;l;,;,#5D function as in a CKY-style parser for standard TAG #28Vijay-Shanker, 1987#29: the subtree rooted by #11 2 T derives a tree whose fringe is w</Paragraph>
    <Paragraph position="5"> if T is the lower auxiliary tree of a set and F is the label of its foot node. In all four item forms,</Paragraph>
    <Paragraph position="7"> the label of #11 l .Intuitively, it means that a potential site-segment has been recog-</Paragraph>
    <Paragraph position="9"> longs to the upper tree of a set, that the subtree rooted by #11, the segment</Paragraph>
    <Paragraph position="11"> i, and the lower tree concatenated together derive a tree whose fringe is</Paragraph>
    <Paragraph position="13"> , where F is the label of the lower foot node. Intuitively,it means that a tree set has been partially recognized, with a site-segment inserted between the two components.</Paragraph>
    <Paragraph position="14"> The rules which require di#0Ber from a TAG parser and hence explanation are Pseudopod, Push, Pop, and Pop-push. Pseudopod applies to any potential lower adjunction site and is so called because the parser essentially views every potential site-segment as an auxiliary tree #28see Section 2.3#29, and the Pseudopod axiom recognizes the feet of these false auxiliary trees.</Paragraph>
    <Paragraph position="15"> The Push rule performs the adjunction of one of these false auxiliary trees|that is, it places a site-segmentbetween the two trees of an elementary tree set. It is so called because the site-segmentissaved in a #5Cstack&amp;quot; so that the rest of its elementary tree can be recognized later. Of course, in our case the #5Cstack&amp;quot; has at most one element.</Paragraph>
    <Paragraph position="16"> The Pop rule does the reverse: every completed elementary tree set must contain a site-segment, and the Pop rule places it back where the site-segment came from, emptying the #5Cstack.&amp;quot; The Pop-push rule performs set-local adjunction: a completed elementary tree set is placed between the two trees of yet another elementary tree set, and the #5Cstack&amp;quot; is unchanged.</Paragraph>
    <Paragraph position="17"> Pop-push is computationally the most expensive rule; since it involves six indices and three di#0Berent elementary trees, its running  #29.</Paragraph>
    <Paragraph position="18"> It was noted in #28Chiang et al., 2000#29 that for synchronous RF-2LTAG, parse forests could not be transferred in time O#28n  #29. This fact turnsout to be connected to several properties of RF-TAG #28Rogers, 1994#29.</Paragraph>
    <Paragraph position="19">  Thanks to Anoop Sarkar for pointing out the #0Crst The CKY-style parser for regular form TAG described in #28Rogers, 1994#29 essentially keeps track of adjunctions using stacks, and the regular form constraint ensures that the stack depth is bounded. The only kinds of adjunction that can occur to arbitrary depth are root and foot adjunction, which are treated similarly to substitution and do not a#0Bect the stacks. The reader will note that our parser works in exactly the same way.</Paragraph>
    <Paragraph position="20"> A problem arises if we allow both root and foot adjunction,however. It iswell-known that allowing both types of adjunction creates derivationalambiguity#28Vijay-Shanker, 1987#29:  would. The problem is not the ambiguity per se, but that the regular form TAG parser, unlike a standard TAG parser, does not always distinguish these multiple derivations, because root and foot adjunction are both performed by the same rule #28analogous to ourPop-push#29.Thusfora given application of this rule, it is not possible to say which tree is adjoining into which without examining the rest of the derivation.</Paragraph>
    <Paragraph position="21"> But this knowledge is necessary to perform certain tasks online: for example, enforcing adjoining constraints, computing probabilities #28and pruning based on them#29, or performing synchronous mappings. Therefore we arbitrarily forbid one of the two possibilities.  The parser given in Section 4 already takes this into account.</Paragraph>
  </Section>
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