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<Paper uid="P99-1060">
  <Title>An Earley-style Predictive Chart Parsing Method for Lambek Grammars</Title>
  <Section position="6" start_page="470" end_page="471" type="concl">
    <SectionTitle>
5 Efficiency and Complexity
</SectionTitle>
    <Paragraph position="0"> The method is shown to be non-polynomial by considering a simple class of examples of the form X1,...Xa-I,a =~ a, where each Xi is a/(a/(a\a)). Each such Xi gives a hypothetical whose dependency is encoded by a multiset index. Examination of the chart reveals spans for which there are multiple edges, differing in their 'initial' multiset (and other ways), there being  one for edge for each subset of the indices deriving from the antecedents XI,... Xn-2, i.e. giving 2 ('~-2) distinct edges. This non-polynomial number of edge results in non-polynomial time for the completer step, and in turn for the algorithm as a whole. Hence, this approach does not resolve the open question of the polynomial time parsability of the Lambek calculus. Informally, however, these observations are suggestive of a possible locus of difficulty in achieving such a result. Thus, the hope for polynomial time parsability of the Lambek calculus comes from it being an ordered 'list-like' system, rather than an unordered 'bag-like' system, but in the example just discussed, we observe 'bag-like' behaviour in a compact encoding (the multiset) of the dependencies of hypothetical reasoning.</Paragraph>
    <Paragraph position="1"> We should note that the DTG parsing method of (Rambow et al., 1995), from which the current approach is derived, is polynomial time. This follows from the fact that their compilation applies to a preset DTG, giving rise to a fixed maximal set of distinct indices in the LPMG that the compilation generates. This fixed set of indices gives rise to a very large, but polynomial, worst-case upper limit on the number of edges in a chart, which in turn yields a polynomial time result. A key difference for the present approach is that our task is to parse arbitrary initial sequents, and hence we do not have the fixed initial grammar that is the basis of the Rambow et al. complexity result.</Paragraph>
    <Paragraph position="2"> For practical comparison to the previous Lambek chart methods, consider the highly ambiguous artificial example shown in Figure 5, (which has six readings). KSnig (1994) reports that a Prolog implementation of her method, running on a major workstation produces 300 edges in 50 seconds. A Prolog implementation of the current method, on a current major work station, produces 75 edges in less than a tenth of a second. Of course, the increase in computing power over the years makes the times not strictly comparable, but still a substantial speed up is indicated. The difference in the number of edges suggests that the KSnig method is sub-optimal in its compaction of alternative derivations. null</Paragraph>
  </Section>
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