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<Paper uid="P98-2157">
  <Title>Prefix Probabilities from Stochastic Tree Adjoining Grammars*</Title>
  <Section position="7" start_page="958" end_page="958" type="concl">
    <SectionTitle>
5 Complexity and concluding
</SectionTitle>
    <Paragraph position="0"> remarks We have presented a method for the computation of the prefix probability when the underlying model is a Tree Adjoining Grammar. Function P,p,t is the core of the method. Its equations can be directly translated into an effective algorithm, using standard functional memoization or other tabular techniques. It is easy to see that such an algorithm can be made to run in time O(n6), where n is the length of the input prefix.</Paragraph>
    <Paragraph position="1"> All the quantities introduced in SS4.3 (Ht, Lt,t,, etc.) are independent of the input and should be computed off-line, using the system of equations that can be derived as indicated. For quantities Ht we have a non-linear system, since equations (2) to (6) contain quadratic terms.</Paragraph>
    <Paragraph position="2"> Solutions can then be approximated to any degree of precision using standard iterative methods, as for instance those exploited in (Stolcke, 1995). Under the hypothesis that the grammar is consistent, that is Pr(L(G)) = 1, all quantities H~ and H~ evaluate to one. For quantities Lt,t, and the like, SS4.3 provides linear systems whose solutions can easily be obtained using standard methods. Note also that quantities La,t, are only used in the off-line computation of quantities Lt,t,, they do not need to be stored for the computation of prefix probabilities (compare equations for Lt,t, with (31) and (32)).</Paragraph>
    <Paragraph position="3"> We can easily develop implementations of our method that can compute prefix probabilities incrementally. That is, after we have computed the prefix probability for a prefix al ... an, on input an+l we can extend the calculation to prefix al&amp;quot;&amp;quot;anan+l without having to recompute all intermediate steps that do not depend on an+l.</Paragraph>
    <Paragraph position="4"> This step takes time O(n5).</Paragraph>
    <Paragraph position="5"> In this paper we have assumed that the parameters of the stochastic TAG have been previously estimated. In practice, smoothing to avoid sparse data problems plays an important role. Smoothing can be handled for prefix probability computation in the following ways. Discounting methods for smoothing simply produce a modified STAG model which is then treated as input to the prefix probability computation. Smoothing using methods such as deleted interpolation which combine class-based models with word-based models to avoid sparse data problems have to be handled by a cognate interpolation of prefix probability models.</Paragraph>
  </Section>
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