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<Paper uid="P00-1057">
  <Title>Multi-Component TAG and Notions of Formal Power</Title>
  <Section position="3" start_page="1" end_page="2" type="intro">
    <SectionTitle>
2 Formalism
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="1" end_page="1" type="sub_section">
      <SectionTitle>
2.1 Restricting set-local MCTAG
</SectionTitle>
      <Paragraph position="0"> The way we propose to deal with multi-component adjunction is #0Crst to limit the number of components to two, and then, roughly speaking, to treat two-component adjunction as one-component adjunction by temporarily removing the material between the two adjunction sites. The reasons behind this scheme will be explained in subsequent sections, but we mention it now because it motivates the somewhat complicated restrictions on possible adjunction sites:  #0F One adjunction site must dominate the other. If the two sites are #11 h and #11 l , call the set of nodes dominated by one node but not strictly dominated by the other  is itself the foot node.</Paragraph>
      <Paragraph position="1"> #0F If two tree sets adjoin into the same tree, the two site-segments must be simultaneously removable. That is, the two site-segments must be disjoint, or one must contain the other.</Paragraph>
      <Paragraph position="2"> Because of the #0Crst restriction, we depict tree sets with the components connected by a dominance link #28dotted line#29, in the manner of #28Becker et al., 1991#29. As written, the above rules only allow tree-local adjunction; we can generalize them to allow set-local adjunction by treating this dominance link like an ordinary arc. But this would increase the weak generative capacity of the system. For present purposes it is su#0Ecient just to allow one type of set-local adjunction: adjoin the upper tree to the upper foot, and the lower tree to the lower root #28see Figure 5#29. This does not increase the weak generative capacity, as will be shown in Section 2.3. Observe that the set-local TAG given in Figure 5 obeys the above restrictions.</Paragraph>
    </Section>
    <Section position="2" start_page="1" end_page="2" type="sub_section">
      <SectionTitle>
2.2 2LTAG
</SectionTitle>
      <Paragraph position="0"> For the following section, it is useful to think of TAG in a manner other than the usual.</Paragraph>
      <Paragraph position="1"> Instead of it being a tree-rewriting system whose derivation history is recorded in a derivation tree, it can be thought of as a set of trees #28the `derivation' trees#29 with a yield function #28here, reading o#0B the node labels of derivation trees, and composing corresponding elementary trees by adjunction or substitution as appropriate#29 applied to get the TAG trees. Weir #281988#29 observed that several TAGs could be daisy-chained into a multi-level TAG whose yield function is the composition of the individual yield functions.</Paragraph>
      <Paragraph position="2"> More precisely: a 2LTAG is a pair of</Paragraph>
      <Paragraph position="4"> the meta-level grammar. The object-level grammar is a standard TAG: #06 and NT are its terminal and nonterminal alphabets, I and A are its initial and auxiliary trees, and S 2 I contains the trees which derivations may start with.</Paragraph>
      <Paragraph position="5"> The meta-level grammar G  is de#0Cned so that it derives trees that look like derivation trees of G: #0F Nodes are labeled with #28the names of#29 elementary trees of G.</Paragraph>
      <Paragraph position="6"> #0F Foot nodes have no labels.</Paragraph>
      <Paragraph position="7"> #0F Arcs are labeled with Gorn addresses.  The Gorn address of a root node is #0F;ifanodehas Gorn address #11, then its ith child has Gorn address</Paragraph>
      <Paragraph position="9"/>
      <Paragraph position="11"> #0F Whena tree #0C isadjoinedat a node #11, #11 is rewritten as #0C, and the foot of #0C inherits the label of #11.</Paragraph>
      <Paragraph position="12"> The tree set of hG;G  are combined to form a derived tree, which is then interpreted as a derivation tree for G, which gives instructions for combining elementary trees of G into the #0Cnal derived tree.</Paragraph>
      <Paragraph position="13"> It was shown in Dras #281999#29 that when the meta-level grammar is in the regular form of Rogers #281994#29 the formalism is weakly equivalenttoTAG. null  g adjoining into an initial tree #0B #28Figure 6#29. Recall that we de#0Cned a site-segment of a pair of adjunction sites to be all the nodes which are dominated by the upper site but not the lower site. Imagine that the  , and so on, and then #0C #0B adjoins between the last pair of trees. This will produce the same result as a series of set-local adjunctions. More formally: 1. Fuse all the elementary tree sets of the grammar by identifying the upper foot #11 #01 i.</Paragraph>
      <Paragraph position="14"> with the lower root. Designate this fused node the meta-foot.</Paragraph>
      <Paragraph position="15"> 2. For each tree, and for every possiblecom null bination of site-segments, excise all the site-segments and add all the trees thus produced #28the excised auxiliary trees and the remainders#29 to the grammar.</Paragraph>
      <Paragraph position="16"> Now that our grammar has been smashed to pieces, we must make sure that the right pieces go back in the right places. We could do this using features, but the resultinggrammar would only be strongly equivalent, not derivationally equivalent, to the original. Therefore we use a meta-level grammar instead: 1. For each initial tree, and for every possible combination of site-segments, construct the derivation tree that will reassemble the pieces created in step #282#29 above and add it to the meta-level grammar. null 2. For each auxiliarytree, andfor every possible combination of site-segments, construct a derivation tree as above, and for the node which corresponds to the piece containing the meta-foot, add a child, label its arc with the meta-foot's address #28within the piece#29, and mark it a foot node. Addthe resulting#28meta-level#29 auxiliary tree to the meta-level grammar. Observe that set-local adjunction corresponds to meta-level adjunction along the #28meta-level#29 spine. Recall that we restricted set-local adjunction so that a tree set can only adjoin at the foot of the upper tree and the root of the lower tree. Since this pair of nodes corresponds to the meta-foot, we can restate our restriction in terms of the converted grammar: no meta-level adjunction is allowed along the spine of a #28meta-level#29 auxiliary tree except at the #28meta-level#29 foot. Then all meta-level adjunction is regular adjunction in the sense of #28Rogers, 1994#29. Therefore this converted 2LTAG produces derivation tree sets which are recognizable, and therefore our formalism is strongly equivalenttoTAG. null Note that this restriction is much stronger than Rogers' regular form restriction. This was done for two reasons. First, the de#0Cnition of our restriction would have been more complicated otherwise; second, this restriction overcomes some computational di#0Eculties with RF-TAG whichwe discuss below.</Paragraph>
    </Section>
  </Section>
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