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<Paper uid="P96-1001">
  <Title>Higher-Order Coloured Unification and Natural Language Semantics</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 Higher-Order Unification and NL
</SectionTitle>
    <Paragraph position="0"> semantics The basic idea underlying the use of HOU for NL semantics is very simple: the typed A-calculus is used as a semantic representation language while semantically under-specified elements (e.g. anaphors and ellipses) are represented by free variables whose value is determined by solving higher-order equations. For instance, the discourse (la) has (lb) as a semantic representation where the value of R is given by equation (lc) with solutions (ld) and (le).  (1) a. Dan likes golf. Peter does too.</Paragraph>
    <Paragraph position="1">  b. like(dan, golf)AR(peter) c. like(dan,golf) = R(dan) d. R = Ax. like(x, golf) e. R = Ax. like(dan,golf)  The process of solving such equations is traditionally called unification and can be stated as follows: given two terms M and N, find a substitution of terms for free variables that will make M and N equal. For first order logic, this problem is decidable and the set of solutions can be represented by a single most general unifier. For the typed A-calculus, the problem is undecidable, but there is an algorithm which - given a solvable equation - will enumerate a complete set of solutions for this equation (Huet, 1975).</Paragraph>
    <Paragraph position="2"> Note that in (1), unification yields a linguistically valid solution (ld) but also an invalid one: (le). To remedy this shortcoming, DSP propose an informal restriction, the Primary Occurrence Restriction: null In what follows, we present a unification framework which solves both of these problems.</Paragraph>
  </Section>
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