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<Paper uid="P90-1014">
  <Title>Free Indexation: Combinatorial Analysis and A Compositional Algorithm*</Title>
  <Section position="6" start_page="108" end_page="109" type="concl">
    <SectionTitle>
5 Conclusions
</SectionTitle>
    <Paragraph position="0"> This paper has shown that free indexation produces an exponential number of indexings per phrase structure. This implies that all algorithms that compute free indexation, that is, assign indices, must also take at least exponential time. In this section, we will discuss whether it is possible for a principle-based parser to avoid the combinatorial 'blow-up' predicted by analysis.</Paragraph>
    <Paragraph position="1"> First, let us consider the question whether the 'full power' of the free indexing mechanism is necessary for natural languages. Alternatively, would it be possible to 'shortcut' the enumeration procedure, that is, to get away with producing fewer than B, indexings? After all, it is not obvious that a sentence with a valid interpretation can be constructed for every possible indexing. However, it turns out (at least for small values of n; see Figures 5 and 6 below) that language makes use of every combination predicted by analysis. This implies, that all parsers must be capable of producing every indexing, or else miss valid interpretations for some sentences.</Paragraph>
    <Paragraph position="2"> There are B3 = 5 possible indexings for three noun phrases. Figure 5 contains example sentences for each possible indexing. 9 Similarly, there are fifteen possible indexings for four noun phrases. The corresponding examples are shown in Figure 6.</Paragraph>
    <Paragraph position="3"> Although it may be the case that a parser must be capable of producing every possible indexing, it does not necessarily follow that a parser must enumerate every indexing when parsing a parlicular sentence. In fact, for many cases, it is possible to avoid exhaustively exploring the search space of possibilities predicted by combinatorial analysis. To do this, basically we must know, a priori, what classes of indexings are impossible for a given sentence. By factoring in knowledge about restrictions on the locality of reference of the items to be indexed (i.e. binding principles), it is possible to explore the space of indexings in a controlled fashion. For example, although free indexation implies that there are five indexings for &amp;quot;John thought \[s Tom forgave himself \] &amp;quot;, we can make use of the fact that &amp;quot;himself&amp;quot; must be coindexed with an element within the subordinate clause to avoid gen-STo make the boundary cases match, just define c(0, 0) to be 1, and let c(0, m) = 0 and c(n, 0) = 0 for m &gt; 0 and n &gt; 0, respectively.</Paragraph>
    <Paragraph position="4">  crating indexings in which &amp;quot;Tom&amp;quot; and &amp;quot;himself&amp;quot; are not coindexed. 1deg Note that the early elimination of ill-formed indexings depends crucially on a parser's ability to interleave binding principles with structure building. But, as discussed in Section 4, the interleaving of binding principles logically depends on the ability to interleave free indexation with structure building. Hence the importance of an formulation of free indexation, such as the one introduced in Section 4, which can be effectively interleaved.</Paragraph>
  </Section>
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