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<Paper uid="P89-1033">
  <Title>PARSING AS NATURAL DEDUCTION</Title>
  <Section position="2" start_page="0" end_page="272" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> Sophisticated techniques have been developed for the implementation of parsers for (augmented) context-free grammars. \[Pereira/Warren 1983\] gave a characterization of these parsers as being resolution based theorem provers. Resolution might be taken as an instance of Hilbert-style theorem proving, where there is one inference rule (e.g. Modus Ponens or some other kind of Cul Rule) which allows for deriving theorems from a set of axioms.</Paragraph>
    <Paragraph position="1"> In the case of parsing, the grammar rules and the lexicon would be the axioms.</Paragraph>
    <Paragraph position="2"> When categorial grammars were discovered for computational linguistics, the most obvious way to design parsers for categorial grammars seemed 1 &amp;quot;natural deduction&amp;quot; is used here in its broad sense, i.e. natural deduction as opposed to Hilbert-style deduction to apply the existing methods: The few combination rules and the lexicon constitute the set of axioms, from which theorems are derived by a resolution rule. However, this strategy leads to unsatisfactory results, in so far as extended categorial grammars, which make use of combination rules like functional composition and type raising, provide for a proliferation of derivations for the same reading of a sentence. This phenomenon has been dubbed the spurious ambiguity problem \[Pareschi/Steedman 1987\]. One solution to this problem is to describe normal forms for equivalent derivations and to use this knowledge to prune the search space of the parsing process \[Hepple/Morrill 1989\].</Paragraph>
    <Paragraph position="3"> Other approaches to cope with the problem of spurious ambiguity take into account the pecularities of categorial grammars compared to grammars With &amp;quot;context-free skeleton&amp;quot;. One characteristic of categorial grammars is the shift of information from the grammar rules into the lexicon: grammar rules are mere combination schemata whereas syntactic categories do not have to be atomic items as in the &amp;quot;context-free&amp;quot; formalisms, but can also be structured objects as well.</Paragraph>
    <Paragraph position="4"> The inference rule of a Hilbert-style deduction system does not refer to the internal structure of the propositions which it deals with. The alternative to Hilbert-style deduction is natural deduction (in the broad sense of the word) which is &amp;quot;natural&amp;quot; in so far as at least some of the inference rules of a natural deduction system describe explicitly how logical operators have to be treated. Therefore natural deduction style proof systems are in principle good candidates to function as a framework for categorial grammar parsers. If one considers categories as formulae, then a proof system would have to refer to the operators which are used in those formulae.</Paragraph>
    <Paragraph position="5">  The natural deduction approach to parsing with categorial grammars splits up into two general mainstreams both of which use the Gentzen sequent representation to state the corresponding calculi. The first alternative is to take a general purpose calculus and propose an adequate translation of categories into formulae of this logic. An example for this approach has been carried out by Pareschi \[Pareschi 1988\], \[Pareschi 1989\]. On the other hand, one might use a specialized calculus. Lambek proposed such a calculus for categorial grammar more than three decades ago \[Lambek 1958\].</Paragraph>
    <Paragraph position="6"> The aim of this paper is to describe how Lambek Calculus can be implemented in such a way that it serves as an efficient parsing mechanism. To achieve this goal, the main drawback of the original Lambek Calculus, which consists of a version of the &amp;quot;spurious ambiguity problem&amp;quot;, has to be overcome. In Lambek Calculus, this overgeneration of derivations is due to the fact that the calculus itself does not giye enough constraints on the order in which the inference rules have to be applied.</Paragraph>
    <Paragraph position="7"> In section 2 of the paper, we present Lambek Calculus in more detail. Section 3 consists of the proof for the existence of normal form proof trees relative to the readings of a sentence. Based on this result, the parsing mechanism is described in section 4.</Paragraph>
    <Paragraph position="8"> head of a complex category is the head of its value category. The category in the succedens of a sequent is called goal category. The category which is &amp;quot;decomposed&amp;quot; by an inference rule application is called current functor.</Paragraph>
    <Paragraph position="10"> the president of Iceland np/n, n, (n\n)/np, np --* np n, (n\n)/np, np --. n np --* np</Paragraph>
  </Section>
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