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<Paper uid="E93-1034">
  <Title>Tuples, Discontinuity, and Gapping in Categorial Grammar*</Title>
  <Section position="2" start_page="0" end_page="287" type="intro">
    <SectionTitle>
1 Introduction
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    <Paragraph position="0"> In \[Lambek, 1958\] the suggestive recursive fractional categorial notations of \[Ajdukiewicz, 1935\] and \[Bar-Hillel, 1953\] were provided with a foundational setting in mathematical logic. This takes the form of a model theory interpreting category formulas in algebraic structures. A Gentzen style sequent proof theory for which there is a Cut-elimination result means that a decision procedure is provided on the basis of sequent calculus.</Paragraph>
    <Paragraph position="1"> The category formulas are freely generated from atomic category formulas (e.g. N for referring nominals, S for sentences, CN for common nouns, ...) by binary operators \ ('under'), / ('over') and * ('product'). The interpretation is in semigroups, i.e. algebras (L, +) where + is a binary operation satisfying the associativity axiom sl + (s~ + s3) = (sl + s2) + s3. (In the non-associative formulation of \[Lambek, 1961\], this condition is withdrawn.) We may in particular consider the algebra obtained by taking the set V* of strings over a vocabulary V; then L is V* - {t} where t is the empty string. Each category formula A is interpreted as a subset D(A) of L. Given such a mapping for atomic category formulas it is extended to the compound category formulas thus:</Paragraph>
    <Paragraph position="3"> In general we may define L in terms of a semigroup algebra (L*, +, t) where f E L is an identity element, i.e. an element such that s + t = t + s -- s; then L is L* - {t}. In the sequent calculus of \[Lambek, 1958\] a sequent is of the form A1,..., An =~ A where n &gt; 0,1 and is read as asserting that for any elements  sl,..., Sn in A1,. * *, An respectively, sl + ... + Sn is in A. Thus the relevant prosodic operations are encoded by the linear ordering of antecedents in the sequent, and structural rules of permutation, contraction, and weakening are not valid. The calculus is as follows. The notation P(A) represents an antecedent containing a subpart A.</Paragraph>
    <Paragraph position="5"> As is normal in sequent calculus, each operator has a L(eft) rule of use and a R(ight) rule of proof. Cut-free backward chaining proof search is terminating since in every proof step going from conclusion to premises, the total number of operator occurrences is reduced by one.</Paragraph>
    <Paragraph position="6"> The original development of categorial grammar grew from semantic concerns, and as is well known, the formalism embraces compositional type-logical semantics. In particular, division categories A\B and B/A can be seen as Fregean functors: incomplete Bs the meanings of which are abstracted over A argument meanings. Complete (or: saturated) expressions bearing primary meanings belong to atomic categories. Given some basic semantic domains (e.g.</Paragraph>
    <Paragraph position="7"> truth values {0, 1}, a set of entities E, ...) a hierarchy of spaces for a type-logicai semantics may be generated by such operations as function formation (r\[2: the set of all functions from r2 into rl) and pair formation (rl x r~: the set of all ordered pairs comprising a rl followed by a r2). Each category formula A is associated with a semantic domain T(A). Such a type map T for atomic category formulas (e.g. T(N) = E,T(S) = {0, 1},T(CN) = {0, 1} E) is extended to compound category formulas by T(A\B) = T(B/A) = T(B) T(A) and T(A*B) = T(A) x T(B). Each category formula A is now interpreted as a set of two dimensional 'signs': a subset D(A) of L x T(A). Such an interpretation for atomic category formulas is extended to one for compound A =~ A to =~ A/A which as a theorem would assert that the identity element t is a member of each category of the form A/A (similarly for A\A). Since we have defined categories to be interpreted as subsets of a set L which does not necessarily contain an identity element, such a theorem would not be valid, and it is prevented by defining sequents as having at least one antecedent formula. category formulas by: 2</Paragraph>
    <Paragraph position="9"> Proofs can be annotated to associate typed semantic lambda terms with each theorem \[Moortgat, 1988\].</Paragraph>
    <Paragraph position="10"> A sequent has the form zi:A1,...,xn:An ::~ C/:A where n &gt; 0, no semantic variable is associated with more than one category formula, and C/ is a typed lambda term over (free) variables {xl,..., x,}. It is to be read as stating that the result of applying the prosodic operation implicit in the ordering, and the semantic operation represented explicitly by C/, to the prosodic and semantic components of any signs in At,..., An yields a sign in A. This system is understood as observing the type map in the obvious way, and is an instance of the Curry-Howard correspondence between (intuitionistic) proofs and typed lambda terms. It was first employed in relation to categorial grammar in \[van Benthem~ 1983\]; for generalisation to other connectives see \[Morrill, 1990b; Morrill, 1992a\]</Paragraph>
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