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<Paper uid="C00-2091">
  <Title>Generation, Lambek Calculus, Montague's Semantics and Semantic Proof Nets</Title>
  <Section position="6" start_page="631" end_page="632" type="evalu">
    <SectionTitle>
5 Example
</SectionTitle>
    <Paragraph position="0"> Let us process on an example the previous results.</Paragraph>
    <Paragraph position="1"> We still use the lexicon of table 3, and we want to generate (if possible) a sentence whose meaning is given by the proof net of figure 3(b).</Paragraph>
    <Paragraph position="2"> We first need to associate every atom with an index (in the figures, we indicate a number i beside the atom). Of course, we have to know how to recognize the atoms that are the same in U (figure 5(b)) and in I\[ (figure 5(a)). This can be done by looking at the typed constants decorating the input conclusions (for the moment, we don't have a general procedure). null We also assume in this numbering that we know which of the atoms in ~(F) is linked to t + (the unique output). In our case where C/4 = 0, it is not a problem to make such a statement. In other cases, the complexity would increase at most polynomially. null Then, the given  We can add this matching to the syntactic forest of figure 6(a) (do not forget that the link between S + alld c~,- is in U1 and not in U3, and that U:3 represents edges between atoms with i E \[17, 22\]) and obtain on F the matching of figure 6(b).</Paragraph>
    <Paragraph position="3"> We still have to ensure the correctness of this proof net (because we add all the tensor and par links), but it has a quadratic complexity (less than the matricial computation). In this case, it is correct. null Actually, this only gives us the axiom links. It still requires to compute the word order to have no crossing axiom link. This can be done from the axiom links easier than quadratic time (it is a bracketing problem).</Paragraph>
  </Section>
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