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<Paper uid="P92-1010">
  <Title>Reasoning with Descriptions of Trees *</Title>
  <Section position="8" start_page="79" end_page="79" type="concl">
    <SectionTitle>
7 Extracting Trees from Quasi-trees
</SectionTitle>
    <Paragraph position="0"> Having derived some quasi-tree satisfying a set of relationships, we would like to produce a &amp;quot;minimal&amp;quot; representative of the trees it characterizes. In section 3.1 we define the conditions under which a quasi-tree is a tree. Working from those conditions we can determine in which ways a quasi-tree M may fail to be a tree, namely: , (~oM)* is a proper subset of:D M, * L M and/or 7) M may be partial, ie: for some t,u, U ~: (t -~ uV-~t -~ u) or U ~ (t ,~* u V -~t ,~* u).</Paragraph>
    <Paragraph position="1"> The case of partial L: M is problematic in that, while it is possible to choose a unique representative, its choice must be arbitrary. For our applications this is not significant since currently in TAGs left-of is fully specified and in parsing it is always resolved by the input. Thus we make the assumption that in every quasi-tree M from which we need to extract a tree, left-of will be complete. That is, for all terms t,u: M ~ t -~ uV-~t -~ u. Thus M ~ t ~* u V-~t ~* u ::v M ~ u ~* t.</Paragraph>
    <Paragraph position="2"> Suppose M ~ u ,~* t and M ~: (t 4&amp;quot; u V-~t ,~* u), and that zM(u) = x and zM(t) = y. In D-theory, this case never arises, since proper domination, rather than domination, is primitive. It is clear that the TAG applications require that x and y be identified, ie: (y, x) should be added to/)m. Thus we choose to complete 7) M by extending it. Under the assumption that /: is complete this simply means: if M ~ -~t ,~* u, 7) M should be extended such that M ~ t ,~* u. That M can be extended in this way consistently follows from lemma 3. That the result of completing ~)M in this way is unique follows from the fact that, under these conditions, extending &amp;quot;D M does not extend either ,A M or ~M. The details can be found in (Rogers &amp; Vijay-Shanker, 1992). In the resulting quasi-tree domination has been resolved into equality or proper domination. To arrive at a tree we need only to expand pM such that (,pM)* .: ~)M. In the proof of lemma 4 we show that this will be the case in any quasi-tree T closed under:</Paragraph>
    <Paragraph position="4"> The second of these conditions is our mechanism for completing/)M. The first amounts to taking immediate domination as the parent relation -precisely the mechanism for finding the standard referent. Thus the tree we extract is both the circumscriptive reading of (Vijay-Shanker, 1992) and the standard referent of (Marcus, Hindle &amp; Fleck, 1983).</Paragraph>
  </Section>
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