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<Paper uid="P98-1088">
  <Title>Memoisation for Glue Language Deduction and Categorial Parsing</Title>
  <Section position="9" start_page="542" end_page="543" type="concl">
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7 Application ~1: Categorial
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    <Section position="1" start_page="542" end_page="543" type="sub_section">
      <SectionTitle>
Parsing
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      <Paragraph position="0"> The associative Lambek calculus (Lambek, 1958) is perhaps the most familiar representative of the class of categorial formalisms that fall within the 'type-logical' tradition. Recent work has seen proposals for a range of such systems, differing in their resource sensitivity (and hence, implicitly, their underlying notion of 'linguistic structure'), in some cases combining differing resource sensitivities in one system, s Many of SSee, for example, the formalisms developed in (Moortgat et al., 1994), (Morrill, 1994), (Hepple, 1995). these proposals employ a 'labelled deductive system' methodology (Gabbay, 1996), whereby types in proofs are associated with labels which record proof information for use in ensuring correct inferencing. A natural 'base logic' on which to construct such systems is the multiplicative fragment of linear logic, since (i) it stands above the various categorial systems in the hierarchy of substructural logics, and (ii) its operators correspond to precisely those appearing in any standard categorial logic. The key requirement for parsing categorial systems formulated in this way is some theorem proving method that is sufficient for the fragment of linear logic employed (although some additional work will be required for managing labels), and a number of different approaches have been used, e.g.</Paragraph>
      <Paragraph position="1"> proof nets (Moortgat, 1992), and SLD resolution (Morrill, 1995). Hepple (1996) introduces first-order compilation for implicational linear logic, and shows how that method can be used with labelling as a basis parsing implicational categorial systems. No further complications arise for combining the extended compilation approach described in this paper with labelling systems as a basis for efficient, non-redundant parsing of categorial formalisms in the core multiplicative fragment. See (Hepple, 1996) for a worked example.</Paragraph>
      <Paragraph position="2"> 8 Application ~2: Glue Language Deduction In a line of research beginning with Dalrymple et al. (1993), a fragment of linear logic is used as a 'glue language' for assembling sentence meanings for LFG analyses in a 'deductive' fashion (enabling, for example, an direct treatment of quantifier scoping, without need of additional mechanisms). Some sample expressions: hates: VX, Y.(s ~t hates(X, Y) )o-( (f .,., eX) (r) (g&amp;quot;-% Y) ) everyone: VH, S.(H-,-*t every(person, S) ) o-(Vx.(H x)) The operator ~ serves to pair together a 'role' with a meaning expression (whose semantic type is shown by a subscript), where a 'role' is essentially a node in a LFG f-structure. For our purposes roles can be treated as if they were just atomic symbols. For theorem proving purposes, the universal quantifiers above can be deleted: the uppercase variables can be treated  as Prolog-like variables, which become instantiated under matching during proof construction; the lowercase variables can be replaced by arbitrary constants. Such deletion leaves a residue that can be treated as just expressions of multiplicative linear logic, with role/meaning pairs serving as 'basic formulae'. 9 An observation contrasting the categorial and glue language approaches is that in the categorial case, all that is required of a deduction is the proof term it returns, which (for 'linguistic derivations') provides a 'semantic recipe' for combining the lexical meanings of initial formulae directly. However, for the glue language case, given the way that meanings are folded into the logical expressions, the lexical terms themselves must participate in a proof for the semantics of a LFG derivation to be produced.</Paragraph>
      <Paragraph position="3"> Here is one way that the first-order compilation approach might be used for glue language deduction (other ways are possible). Firstly, we can take each (quantifier-free) glue term, replace each role/meaning pair with just the role component, and associate the resulting formula with a unique semantic variable. The set of formulae so produced can then undergo the first-order compilation procedure. Crucially for compilation, although some of the role expressions in the formulae may be ('Prolog-like') variables, they correspond to atomic formulae (so there is no 'hidden structure' that compilation cannot address). A complication here is that occurrences of a single role variable may end up in different first-order formulae. In any overall deduction, the binding of these multiple variable instances must be consistent, but we cannot rely on a global binding context, since alternative proofs will typically induce distinct (but internally consistent) bindings. Hence, bindings must be handled locally (i.e. relative to each database formula) and combinations will involve merging of local binding contexts. Each proof term that tabular deduction returns corresponds to a natural deduction proof over the precompilation formulae. If we mechanically mirror this pattern of proof over the original glue terms (with meanings, but quantifier-free), a role/meaning 9See (Fry, 1997), who uses a proof net method for glue language deduction, for relevant discussion. This paper also provides examples of glue language uses that require a full deductive system for the multiplicative fragment. pair that provides a reading of the original LFG derivation will result.</Paragraph>
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