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<Paper uid="P06-1019">
  <Title>Sydney, July 2006. c(c)2006 Association for Computational Linguistics Partially Specified Signatures: a Vehicle for Grammar Modularity</Title>
  <Section position="4" start_page="0" end_page="145" type="intro">
    <SectionTitle>
2 Type signatures
</SectionTitle>
    <Paragraph position="0"> We assume familiarity with theories of (typed) unification grammars, as formulated by, e.g., Carpenter (1992) and Penn (2000). The definitions in this section set the notation and recall basic notions. For a partial function F, 'F(x)|' means that F is defined for the value x.</Paragraph>
    <Paragraph position="1"> Definition 1 Given a partially ordered set &lt;P,[?]&gt; , the set of upper bounds of a subset S [?] P is the set Su = {y [?] P  |[?]x [?] S x [?] y}.</Paragraph>
    <Paragraph position="2"> For a given partially ordered set &lt;P,[?]&gt; , if S [?] P has a least element then it is unique.</Paragraph>
    <Paragraph position="3"> Definition 2 A partially ordered set &lt;P,[?]&gt; is a bounded complete partial order (BCPO) if for every S [?] P such that Su negationslash= [?], Su has a least element, called a least upper bound (lub).</Paragraph>
    <Paragraph position="4"> Definition 3 A type signature is a structure &lt;TYPE,[?], FEAT,Approp&gt; , where:  2. FEAT is a finite set, disjoint from TYPE.</Paragraph>
    <Paragraph position="5"> 3. Approp : TYPExFEAT - TYPE (the appro- null priateness specification) is a partial function such that for every F [?] FEAT: (a) (Feature Introduction) there exists a type Intro(F) [?] TYPE such that Approp(Intro(F),F)|, and for every</Paragraph>
    <Paragraph position="7"> and s [?] t, then Approp(t,F)  |and Approp(s,F) [?] Approp(t,F).</Paragraph>
    <Paragraph position="8"> Notice that every signature has a least type, since the subset S = [?] of TYPE has the non-empty set of upper bounds, Su = TYPE, which must have a least element due to bounded completeness. Definition 4 Let &lt;TYPE,[?]&gt; be a type hierarchy and let x,y [?] TYPE. If x [?] y, then x is a supertype of y and y is a subtype of x. If x [?] y, x negationslash= y and there is no z such that x [?] z [?] y and z negationslash= x,y then x is an immediate supertype of y and y is an immediate subtype of x.</Paragraph>
  </Section>
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