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<Paper uid="J91-3004">
  <Title>Computation of the Probability of Initial Substring Generation by Stochastic Context-Free Grammars</Title>
  <Section position="4" start_page="321" end_page="322" type="concl">
    <SectionTitle>
5. Conclusion
</SectionTitle>
    <Paragraph position="0"> While the LRI algorithm together with formulas (19) and (20) constitutes the solution to the stated problem, its practicality is limited to grammars whose total number of nonterminals is sufficiently limited so as to allow the calculation of the inverse \[I - PL\] -1 The algorithm itself has exactly twice the complexity of the Inside Algorithm computing P(Hli , i + hi) by formula (15), and is thus of order n 3. In fact, once all the probabilities required for the computation of P(s ~ 1, k) are computed, to get the next probability of interest, P(s ~ 1, k + 1), one needs to compute the following quantities:  1. The probabilities P(Gli, kl) for i = k, k- 1,..., 1, in that order.</Paragraph>
    <Paragraph position="1"> 2. The probabilities P(H KK i, k + 1) for i = k + 1, k .... ,2, in that order. 3. The probability P(s ~ 1, k + 1).</Paragraph>
    <Paragraph position="2">  Let us finally recall that the language model of speech recognition provides to the recognizer the probability P(Wk = v \[ wlw2...Wk-1) for all possible words v, and that we therefore must be able to compute the probability P(s --+ wlw2... Wk-lV...) for all N words v of the vocabulary. Fortunately, this does not mean carrying out the LRI algorithm N times for each word position k, but only M times, where M is the number of nonterminals of the grammar.</Paragraph>
    <Paragraph position="3"> In fact, a simple modification of the algorithm allows one to compute the probabilities of P(s --* wlw2... Wk-1 gi...) where gi is an element of the set of nonterminals</Paragraph>
    <Paragraph position="5"> in the algorithm of Section 3. Our desired LRI probabilities can then be computed by the formula</Paragraph>
    <Paragraph position="7"> This modification is particularly practical when the size of the vocabulary greatly exceeds the number of nonterminals in the grammar.</Paragraph>
    <Paragraph position="8">  Jelinek and Lafferty Probability of Initial Substring Generation</Paragraph>
  </Section>
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