<?xml version="1.0" standalone="yes"?>
<Paper uid="E91-1044">
  <Title>AN ALGORITHM FOR GENERATING NON-REDUNDANT QUANTIFIER SCOPINGS</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
INTRODUCTION
</SectionTitle>
    <Paragraph position="0"> Natural language sentences like the notorious (1) Every man loves a woman, are usually regarded to be scope ambiguous. There have been two ways to attack this problem: To generate the most probable scoping and ignore the rest, or to generate all theoretically possible scopings.</Paragraph>
    <Paragraph position="1"> Choosing the first alternative is actually not a bad solution, since any sample piece of tex t usually contains few possibilities for (real) scope ambiguity, and since reasonable heuristics in most cases pick out the intended reading. However, there are cases which seem to be genuinely ambiguous, or where the selection of the intended reading requires extensive world knowledge.</Paragraph>
    <Paragraph position="2"> If the second alternative is chosen, there are basically two possible approaches: To integrate the generation of scopings into the grammar (like e.g. in Johnson and Kay (90) or Halvorsen and? Kaplan (88)), or to devise a procedure that generates the scopings from the parse output (like in Hobbs and Shieber (87)). In both cases, only structurally impossible scopings are ruled out, like the reading of (2) Every representative of a company saw most samples in which &amp;quot;most samples&amp;quot; is outscoped by &amp;quot;every representative&amp;quot; but outscopes &amp;quot;a company&amp;quot; (Hobbs and Shieber (87)).</Paragraph>
    <Paragraph position="3"> Logically equivalent readings are not ruled out on either of these proposals. Hobbs and Shieber argue that &amp;quot;When we move beyond the two first-order quantifiers to deal with the so-called generalized quantifiers, such as &amp;quot;most&amp;quot;, these logical redundancies become quite rare&amp;quot;.</Paragraph>
    <Paragraph position="4"> Theoretically, they become rare. But it may very well be that sentences with several occurrences of non-first-order generalized quantifiers are not very commonly used. On the other hand, sentences with several occurrences of existential or universal quantifiers may be quite common. What kinds of expressions that really resemble first-order quantifiers is of course a controversial question. But working natural language systems, with inference mechanisms that are based on f'trst-order logic, often have to simplify the interpretation process by interpreting broad classes of expressions as plain universal or existential quantifiers. Thus, the gain of generating only non-equivalent scopings may be quite significant in practical systems.</Paragraph>
    <Paragraph position="5"> Ordering of the scopings according to preference is also not treated on approaches like that of Hobbs &amp; Shieber (87) or Johnson &amp; Kay (90). Hobbs &amp; Shieber (87) are quite aware of this, and give some suggestions on how to build ordering heuristics into the algorithm. On the approach of Johnson &amp; Kay (90), scopings are generated with a DCG grammar augmented with procedure calls for &amp;quot;shuffling&amp;quot; and applying the quantifiers 1. The program will return new scopings by backtracking. Because of the recursive inside-out nature of the algorithm, it seems difficult to preserve generation-by-backtracking if one wants to order the scopings.</Paragraph>
    <Paragraph position="6"> IThe quantifier shuffling method is essentially the same as  in Pereira &amp; Shieber (87), but correctly avoids the &amp;quot;structurally impossible&amp;quot; seopings mentioned above. - 251 -Scope islands: In English, only existential quantitiers may be extracted out of relative clauses. Notice the difference between (3a) An owner of every company attended the meeting.</Paragraph>
    <Paragraph position="7"> (3b) A man who owns every company attended  the meeting.</Paragraph>
    <Paragraph position="8"> A scoping algorithm must take this into account, since it will be very difficult to filter out such readings at a later stage. In the algorithm of Johnson &amp; Kay (90), adding such a mechanism seems to be quite easy, since the shuffling and application of quantifiers are handled in the: grammar rules. In the algorithm of Hobbs &amp; Shieber (87), it is a bit more difficult, since the language of the input forms does not distinguish between relative clauses and other kinds of NP modifiers.</Paragraph>
    <Paragraph position="9"> In general, any working scoping algorithm should meet as many linguistic constraints on scope generation as possible.</Paragraph>
    <Paragraph position="10"> Modularity: The main concern of Johnson &amp; Kay (90) is to build a grammar that is independent of semantic formalism. This is done by a DCG grammar using &amp;quot;curly bracket notation&amp;quot; to include calls to formalism-dependent constructor functions.</Paragraph>
    <Paragraph position="11"> It is tempting to take this approach one step further, and let the generation Of scopings be: independent on both the syntactic and semantic theory chosen. null A MODULAR APPROACH The algorithm I propose provides solutions to the four problems mentioned above simultaneously. It is an extension and generalisation of the algorithm presented in Vestre (87)2.</Paragraph>
    <Paragraph position="12"> In the following I will make the (commonly made) assumption that quantified formulas are 4part objects. I will occasionally use a simple language of generalized quantifiers, where the formula format is DET(x,~(x .... ),Y=(x .... )) for determiners DET and formulas ~, ~. DET will be referred to as the determiner of the quantifier, x is its variable, ~ its restriction, and V is its scope. The term quantifier will usually refer to the determiner with variable and restriction.</Paragraph>
    <Paragraph position="13"> 2This paper is in Norwegian, I'm afraid. An English overview of the work is included in Fenstad, Langholm and Vestre (89), but the details of the seoping algorithm are not described there.</Paragraph>
    <Paragraph position="14"> Treating quantifiers in this way, it is easy to rule * out the &amp;quot;structurally impossible&amp;quot; scopings mentioned above because the formulas corresponding to the &amp;quot;impossible scopings&amp;quot; will contain free variables. For instance, in sentence (2), the variable of &amp;quot;a company&amp;quot; (say, y) will also occur in the restrictor of &amp;quot;every representative&amp;quot;. So in order to avoid an unbound occurrence of that variable, &amp;quot;a company&amp;quot; must either have wider scope than &amp;quot;every representative&amp;quot; or be bound inside its restrictor. The algorithm presupposes that a few access functions are included for the type of input structure s used. Further, a:few constructor functions must be included to define the format of the logical forms generated.</Paragraph>
    <Paragraph position="15"> The role of the main access function, getquants, is to pick out the parts of the input structure that are quantifiers, and to return them as a list, where the list order gives the default quantification order. There are almost no limits to what kinds of input structures that may be used, but the quantifiers that are returned by the access functions must contain their restri0tors as a substructure. Of course, using input structures that already contain such lists of quantifiers as substructures will make the implementation of get.-.quants almost trivial.</Paragraph>
    <Paragraph position="16"> In the following, I will give some rather informal descriptions of the main functions involved. The algorithm has been implemented in Common Lisp.</Paragraph>
  </Section>
class="xml-element"></Paper>