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<Paper uid="E93-1032">
  <Title>Towards efficient parsing with proof-nets</Title>
  <Section position="2" start_page="0" end_page="0" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> In this paper, we present a method for parsing Lambek grammars based on graph-theoretic properties. We expect that it may be done efficiently by an algorithm (time-polynomial even in the worst case) which aims at finding an alternating spanning tree in a graph. We do not give the explicit formulation of such an algorithm in this paper: we will only give an idea and an illustration of it. This paper is thus mostly devoted to the properties on which the method is based. We call connection graphs the special kind of proof-nets we explore, just in order to make explicit some difference with the usual method of proof-nets, as it can be found in \[Roorda, 1991; 1992\] and \[Moortgat 1992\], but the two concepts are very similar. In many respects, connection graphs are a mere conservative extension of the earlier method of syntactic connection, discovered by Ajduckiewicz \[1935\]. The method amounts to link the nodes of an ordered sequence of trees in such a way that properties of connexion, &amp;quot;non overlapping&amp;quot;, acyclicity and &amp;quot;strong connectivity&amp;quot; are verified. Connection graphs are simpler than proof-nets in that they loose some information. As they are here formulated, they are only convenient for the associative version of the product-free Lambek calculus. One of their advantages lies in the geometrical viewpoint they provide on the proofs of a sequent. By means of this viewpoint, questions of provability may be reduced to graph-theoretical problems. For instance, every reading of a sentence is provided by an alternating spanning tree.</Paragraph>
    <Paragraph position="1"> In many aspects, the method resembles the well known method of chart-parsing. Ktnig \[1991, 1992\] was the first to apply chart-parsing to Lambek calculus.</Paragraph>
    <Paragraph position="2"> Hepple \[1992\] has recently improved this application.</Paragraph>
    <Paragraph position="3"> An obvious difference with the method proposed here lies in the fact that, in ours, words are points and intervals between them are edges instead of the contrary in chart-parsing. In both cases, computational advantages are found by keeping correct partial analyses after they have been obtained. A chart is actually used in both methods.</Paragraph>
  </Section>
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