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<Paper uid="P99-1038">
  <Title>Two Accounts of Scope Availability and Semantic Underspecification</Title>
  <Section position="4" start_page="0" end_page="294" type="intro">
    <SectionTitle>
2 Underspecification
</SectionTitle>
    <Paragraph position="0"> A recent area of interest has been with under-specified representations of an ambiguous sentence's meaning, for example, Quasi-Logical  ory (UDRT) (Reyle, 1995). We shall characterise the desirable properties of an underspecified meaning representation as:  1. the meaning of a sentence should be represented in a way that is not committed to any one of the possible (intended) meanings of the sentence, and 2. it should be possible to incrementally intro null duce partial information about the meaning, if such information is available, and without the need to undo work that has already been done.</Paragraph>
    <Paragraph position="1"> A principal aim of systems providing an underspecified representation of quantifier scope is the ability to represent partial scopings. That is, it should be possible to state that some of the quantifiers have some scope relative to each other, while remaining uncommitted to the relative scope of the remaining quantifiers. However, representations which simply allow partial scopes to be stated without further analysis do not adequately capture the behaviour of quantitiers in a sentence. Consider the sentence Every representative of a company saw most samples, represented in the style of QLF: _:see(&lt;+i every x _:rep.of(x, &lt;+j exists y co(y)&gt;)&gt;, &lt;+k most z sample(z)&gt;) A fully scoped logical form of this QLF is:</Paragraph>
    <Paragraph position="3"> where the list of quantifier labels indicates the relative scope of qnantifiers at that point in the sentence.</Paragraph>
    <Paragraph position="4"> Although this formula is well formed in the QLF language, it does not correspond to a well formed sentence of logic, seeming closer to the formula: every (x, rep. of (x, y), most (z, sample (z), exists(y, co(y), see(x, z)))) where the variable y does not appear in the scope of its quantifier. A language such as QLF will generally allow this scoping to be expressed, even though it does not correspond to a reading available to a speaker. In QLF semantics, a scoping which does not give rise to any well formed readings is considered &amp;quot;uninterpretable&amp;quot;; ie. there is no interpretation in which an evaluation function maps the QLF onto a truth value.</Paragraph>
    <Paragraph position="5"> Our aim is to present a system in which there is a straightforward computational test of whether a well-formed reading of a sentence exists in which a partial scoping is satisfied, without requiring recourse to the final logical form. The language CLLS (Egg et al., 1998) has recently been developed which correctly generates the well-formed readings by using dominance constraints over trees. Readings of a sentence can be represented using a tree, where dominance represents outscoping, and quantifiers are represented using binary trees whose daughters correspond to the quantifiers' restriction and scope. So for the current example, Every representative of a company saw most samples, the reading: every(x, a(y, co(y), rep.o f ( x, y ) ), most(z, sample(z), see(x, z) ) ) can be represented by the tree in figure 1, where the restrictions of a and most have been omitted for clarity. Domination in the tree represents outscoping in the logical form.</Paragraph>
    <Paragraph position="6">  Underspecification can be captured by defining dominance constraints between nodes representing the quantifiers and relations in a sentence. Readings of the sentence with a free variable are avoided by asserting that each relation containing a variable must be dominated by that variable's quantifier, and an available reading of the sentence is represented by a tree in which all the dominance constraints are satisfied. So the ill-formed readings of the sentence can be avoided by stating that the relation rep.of is dominated by the restriction of every and the scope of a, while see is dominated by the scopes of both a and most. This is represented in figure 2, where the dominance constraints are illustrated by dotted lines.</Paragraph>
    <Paragraph position="7"> Further partial scope information can be introduced with additional dominance constraints. So the partial scope requirement that</Paragraph>
    <Paragraph position="9"> most should outscope every would be captured by a constraint stating that the node representing most should dominate the node representing every in the constraints' solution.</Paragraph>
    <Paragraph position="10"> It is has been shown (Koller et al., 1998) that determining the consistency of these constraints is NP-hard. In the rest of this paper, we show that an alternative theory of scope availability yields a constraint system that can be solved in polynomial time.</Paragraph>
  </Section>
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