<?xml version="1.0" standalone="yes"?>
<Paper uid="C90-3034">
  <Title>A Quantifier Scoping Algorithm without A Free Variable Constraint</Title>
  <Section position="2" start_page="0" end_page="190" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> \[Hobbs and Shieber 1987\] presented an algorithm to generate quantifier scopings from a representation of &amp;quot;predicate-argument relations and the relations of grammatical subordination&amp;quot; (pg 49).</Paragraph>
    <Paragraph position="1"> This representation is successively modified by a recursive algorithm until all the quantifiers present in the input have been dealt with and given scope over some part of the output. A sample input representation is, i. Ioves(&lt;a x woman(x)&gt; &lt;every y man(y)&gt;) where representations of quantified noun phrases, called complex terms, are left as arguments to  the verb. A sample output is 2. (a x woman(x) (every y man(y) lovesCx,Y)))  which uses a four-part quantifier notation, and in which no complex terms are present. In converting 1) into 2) the recursive procedure may be called upon representations of intermediate format, eg (a woman( )loves( &lt;every y man(y)&gt;)) where a four part quantifier phrase has an embedded complex term.</Paragraph>
    <Paragraph position="2"> The algorithm is claimed to be more successful than previous accounts in dealing with complex noun phrases such as &amp;quot;every representative of a company&amp;quot; and in coping with certain 'opaque' predicates such as negation. 1 Two properties of an algorithm which Hobbs and Shieber (H&amp;S) approve of are completeness and soundness. An algorithm with these properties might be used as a benchmark for other algorithms designed for efficiency or the use of heuristics governing the plausibility of the various interpretations. Unfortunately, demonstrating that H&amp;S's algorithm is sound requires a semantics for the input language and the intermediate forms. That is not straightforward. I present a modified algorithm which avoids such intermediate forms. The input to the algorithm consists of English syntax. The steps of the algorithm retrace steps through a truth definition for the input language. Clearly, the algorithm is sound and complete with respect to that. The algorithm is also sound and complete with respect to English, if you agree that the input  language fairly represents the actual language of English speakers. Furthermore, the algorithm is somewhat simpler than H&amp;S's algorithm and makes no appeal to logical syntax. There is a Prolog implementation of the algorithm.</Paragraph>
  </Section>
class="xml-element"></Paper>