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<Paper uid="W00-1424">
  <Title>Generating Vague Descriptions</Title>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
1 Introduction: Vague properties
</SectionTitle>
    <Paragraph position="0"> and Gradable Adjectives Some properties can apply to an object to a greater or lesser degree. Such continuous, or vague properties, which can be expressed by, among other possibilities, gradable adjectives (e.g., 'small', 'large', e.g. Quirk et al. 1972 sections 5.5 and 5.39), pose a difficult challenge to existing semantic theories, theoretical as well as computational. The problems are caused partly by the extreme context-dependence of the expressions involved, and partly by the resistance of vague properties to discrete mathematical modeling (e.g., Synthese 1975, Pinkal 1995). The weight of these problems is increased by fact that vague expressions are ubiquitous in many domains.</Paragraph>
    <Paragraph position="1"> The present paper demonstrates how a Natural Language Generation (NLG) program can be enabled to -generate uniquely referring descriptions containing one gradable adjective, despite the vagueness of the adjective. Having presented a semantic analysis for such vague descriptions, we describe the semantic core of an NLG algorithm that has numerical data as input and vague (uniquely referring) descriptions as output.</Paragraph>
    <Paragraph position="2"> One property setting our treatment of vagueness apart from that in other NLC programs-(e.g. Goldberg 1994) is that it uses **vague properties for an exact task, namely the ruling out of distractors in referring expressions (Dale and Reiter 1995). Another distinctive property is that our account allows the 'meaning' of vague expressions to be determined by a combination of linguistic context (i.e., the Common Noun following the adjective) and nonlinguistic context (i.e., the properties of the elements in the domain).</Paragraph>
  </Section>
  <Section position="4" start_page="0" end_page="179" type="metho">
    <SectionTitle>
2 The Meaning of Vague
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="0" end_page="179" type="sub_section">
      <SectionTitle>
Descriptions
</SectionTitle>
      <Paragraph position="0"> Several different analyses are possible of what it means to be, for example, 'large': larger than average, larger than most, etc. But there is not necessrily just one correct analysis. Consider a domain of four mice, sized 2,5,7, and 10cm. 1 In this case, for example, one can speak of  1. The large mouse (= the one whose size is lOcm), and of 2. The two large mice  (= the two whose sizes are 7 and lOcm). Clearly, what it takes to be large has not been written in stone: the speaker may decide that 7cm is enough (as in (2)), or she may set the standards higher (as in (1)). A numeral (explicit, or implicit as in (1)), allows the reader to make inferences about the standards employed by the speaker3 More precisely, it appears that in a definite description, the absolute form of the adjective is semantically equivalent with the superlative form: The n large mice - The largest n mice The large mice - The largest mice The large mouse - The largest mouse.</Paragraph>
      <Paragraph position="1"> 1For simplicity, the adjectives involved will be assumed to be one-dimensional. Note that the degree of precision reflected by the units of measurement affects the descriptions generated, and even the objects (or sets) that can be described, since it determines which objects count as having the same size.</Paragraph>
      <Paragraph position="2"> 2Thanks are due to Matthew Stone for this observation.  This claim, which has been underpinned by a small experiment with human subjects (see Appendix), means that if a sentence containing .one element of a pair is true then so is the corresponding sentence containing the other. There are bound to be differences between the two forms, but these will be taken to be of a pragmatic nature, having to do with felicity rather than truth (see section 5.2).</Paragraph>
      <Paragraph position="3"> An important qualification must be made with respect to the analysis that we propose: to simplify matters, we assume that the entire domain of relevant individuals is: available -and ~ha'g it-is-:this d~: main alone which is taken into account when the adjective is applied. In the case of the example above, this means that all mice are irrelevant except the four that are mentioned: no other knowledge about the size of mice is assumed to be available. 3</Paragraph>
    </Section>
    <Section position="2" start_page="179" end_page="179" type="sub_section">
      <SectionTitle>
2.1 A Formal Semantics for Vague
Descriptions
</SectionTitle>
      <Paragraph position="0"> Let us be more precise. In our presentation, we will focus on the adjective 'large', without intended loss of generality. For simplicity, 'large' will be treated as semantically one-dimensional.</Paragraph>
      <Paragraph position="1"> i. 'The largest n mouse/mice'. Imagine a set C of contextually relevant animals. Then the NP 'The largest n mouse/mice' (n &gt; 0) presupposes that there is an S C_ C that contains n elements, all of which are mice, and such that (1) C - S C/ (r) and (2) every mouse in C - S is smaller than every mouse in S. If such a set S exists then the NP denotes S. The case where n = 1, realized as 'The \[Adj\]-est \[CN~g\]' (sg = singular), falls out automatically. null ii. 'The largest mice'. This account can be extended to cover cases of the form 'The \[Adj\]-est \[CNpt\]' (pl = plural), where the numeral n is suppressed: these will be taken to be ambiguous between all expressions of the form 'The \[Adj\]-est n \[CN\]' where n &gt; 1. Thus, in a domain where there are five mice, of sizes 4,4,4,5,6 cm, the only possible value of n. is 2, causing the NP to denote the two mice of 5 and 6 cm size.</Paragraph>
      <Paragraph position="2"> iii. 'The n large mouse/mice'. We analyse 'The n \[Adj\] \[CN\]' (n &gt; 0) as semantically equivalent with the corresponding NP of the form 'The \[Adj\]-est n \[CN\]'. 'The two large mice', for example, denotes a set of two mice, each of which is bigger than all other contextually relevant mice.</Paragraph>
      <Paragraph position="3"> iv. 'The large mice'. Expressions of this form can be analysed as being of the form 'The n \[Adj\] \[CN\]' for some value of n. In other words, we will take aln other words, only perceptual context-dependence is taken into account, as opposed to no,'maltve or functional context-dependence Ebeling and Gehnan (1994).</Paragraph>
      <Paragraph position="4"> them to be ambiguous or unspecific - the difference will not matter for present purposes - between 'The .2 large mice', 'The 3. large mice', etc.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="179" end_page="180" type="metho">
    <SectionTitle>
3 Generation of Crisp Descriptions
</SectionTitle>
    <Paragraph position="0"> Generation of descriptions covers a number of tasks, one of which consists of finding a set L of properties which allows a reader to pick out a given unique individual or set of individuals. The state of the art is discussed in Dale and Reiter (1995), who present a computationally tractable algorithm for character:. izing~i~dividuods.x This,algorithm-(henceforth_D&amp;R), deals with vague properties, such as size, to some extent, but these are treated as if they were contextindependent: always applying to the same sets of objects.</Paragraph>
    <Paragraph position="1"> In many cases, generating vague descriptions involves generating a plural and no generally accepted account of the generation of plural descriptions has been advanced so far. In the following section, therefore, a generalization or D&amp;R will be offered, called D&amp; RPlur, which focuses on sets of individuals. Characterization of an individual will fall out as a special case of the algorithm.</Paragraph>
    <Section position="1" start_page="179" end_page="180" type="sub_section">
      <SectionTitle>
3.1 Plural Descriptions: Dale and Reiter
</SectionTitle>
      <Paragraph position="0"> generalized The properties which form the basis of D&amp;Rpt~r are modeled as pairs of the form {Attribute,Value). In our presentation of the algorithm, we will focus on complete properties (i.e., (Attribute,Value) pairs) rather than attributes, as in Dale and Reiter (1995), since this facilitates the use of set-theoretic terminology. Suppose S is the 'target' set of individuals (i.e., the set of individuals to be characterized) and C (where S C_ C) is the set of individuals from which S is to be selected. 4 Informally - and forgetting about the special treatment of head nouns what happens is the following: Tile algorithm iterates through a list P in which the properties appear in order of 'preference'; for each attribute, it checks whether specifying a value for that attribute would rule out at least one additional member of C; if so, the attribute is added to L, with a suitable value.</Paragraph>
      <Paragraph position="1"> (The value can be optimized using some further constraints but these will be disregarded here.) Individuals that are ruled out by a property are removed from C. The process of expanding L and contracting C continues until C = S. The properties in L can be used by a linguistic realization module to produce NPs such as 'The white mice', 'The white mice * that arepregnant', etc. Schematically, the algorithm goes as follows: (Notation: Given a property Q, the set of objects that have the property Q is denoted \[\[o\]\].) * 1Note that C contains r, unlike Dale and Reiter's 'contrast set' C, which consists of those elements of the domain from which r is set apart.</Paragraph>
      <Paragraph position="3"> would remove distractors from C #} then do</Paragraph>
      <Paragraph position="5"> \[\[Pi\]\] are removed from C #} If C = S then Return L {# Success #} Return Failure-'{S,d: All-properties in Phave been tested, yet C -7= S ~ } of one vague property. Case i of section 2.1, 'The largest n chihuahuas' will be discussed in some detail. All the others are minor variations.</Paragraph>
      <Paragraph position="6"> 'Success' means that the properties in L are sufficient to characterize S. Thus, ~{\[\[Pi\]\] : Pie L} = S. The case in which S is a singleton set amounts to the generation of a singular description: D~RPIur becomes equivalent to D&amp;R (describing the individual r) when S in D&amp;aPlur is replaced by {r}.</Paragraph>
      <Paragraph position="7"> D&amp;RPlu r uses hill climbing: an increasingly good approximation of S is achieved with every contraction of C. Provided the initial C is finite, D&amp;apt~,finds a suitable L if there exists one. Each property is considered at most once, in order of 'preference'.</Paragraph>
      <Paragraph position="8"> As a consequence, L can contain semantically redundant properties - causing the descriptions to become more natural, of. Dale and Reiter 1995 - and the algorithm is polynomial in the cardinality of P.</Paragraph>
      <Paragraph position="9"> Caveats. D&amp;RPtur does not allow a generator to include collective properties in a description, as in 'the two neighbouring houses', for example. Furthermore, D~l-tPlur cannot be employed to generate conjoined NPs: It generates NPs like 'the large white mouse' but not 'tile black cat and the large white mouse'.</Paragraph>
      <Paragraph position="10"> From a general viewpoint of generating descriptions, this is an important limitation which is, moreover, difficult to overcome in a computationally tractable account. In the present context, however, the limitation is inessential, since what is crucial here is the interaction between an Adjective and a (possibly complex) Common Noun following it: in more complex constructs of the form 'NP and the Adj CN', only CN affects the meaning of Adj. 5 There is no need for us to solve the harder problem of finding an efficient algorithm for generating NPs uniquely describing arbitrary sets of objects, but only the easier problem of doing this whenever a (nonconjunctive) NP of the form 'tile Adj CN' is possible.</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="180" end_page="181" type="metho">
    <SectionTitle>
4 Generation of Vague Descriptions
</SectionTitle>
    <Paragraph position="0"> \Ve nOw turn our attention to extensions of D&amp;RPlur that generate descriptions containing the expression ~\[n &amp;quot;The elephant and the big mous(,', for example, the mouse does not have to be bigger than any elephant.</Paragraph>
    <Paragraph position="1"> Superlative adjectives. First, 'The largest chihuahua'. We will assume that size is stored (in the KB that forms the input to the generator) as an attribute with exact numerical values. We will take them to be of the form n crn, where n is a positive natural number. For example,</Paragraph>
    <Paragraph position="3"> With this KB as input, D~R allows us to generate NPs based on L = {yellow,chihuahua,9~n}, for example, exploiting the number-valued attribute size.</Paragraph>
    <Paragraph position="4"> The result could be the NP 'The 9cm yellow chihuahua', for example. The challenge, however, is to generate superlatives like 'The largest yellow chihuahua' instead.</Paragraph>
    <Paragraph position="5"> There are several ways in which this challenge may be answered. One possibility is to replace an exact value like 9cm, in L, by a superlative value whenever all distractors happen to have a smaller size. The result would be a new list L = {yellow,chihuahua,largestl}, where 'largestt' is the property 'being the unique largest element of C'.</Paragraph>
    <Paragraph position="6"> This list can then be realized as a superlative NP.</Paragraph>
    <Paragraph position="7"> We will present a different approach that is more easily extended to plurals, given that a plural description like 'the 2 large mice' does not require the two mice to have the same size.</Paragraph>
    <Paragraph position="8"> Suppose size is the only vague property in the KB.</Paragraph>
    <Paragraph position="9"> Vague properties are less 'preferred' (in the sense of section 3.1) than others (Krahmer and Theune 1999).6 As a result, when they are taken into consideration, all tile other relevant properties are already in L. For instance, assume that this is the KB, and that the object to be described is c4:</Paragraph>
    <Paragraph position="11"> At this point, inequalities of tile form size(x) &gt; m cm are added to the KB. For every value of ,the form n ~n oecuring in-the oldKB, all..inequatities of the form size(x) &gt; n an are added whose truth follows from the old I&lt;B. Inequalities are more 6Note, by contrast, that vague properties tend to be realized first (Greenbaum et al. 1985, Shaw and Hatzivassiloglou 1999). Surface realization, however, is not the topic of lids paper.</Paragraph>
    <Paragraph position="12">  preferred than equalities, while logicaUy stronger in-</Paragraph>
    <Paragraph position="14"> The first property that makes it into L is 'chihuahua', which removes Ps but not ca from the context set. (Result: C = {cl,...,c4}.) Now size is taken into account, and the property size(x) &gt; 8cm singles out c4..The .resulting.listA s L =,~cchihuahua , &gt; 8cm}. This implies that c4 is the only chihuahua in the KB that is greater than 8cm and consequently, the property size(x) &gt; 8cm can be replaced, in L, by the property of 'being larger than all other elements of C'. The result is a list that may be written as L = {chihuahua, largesh }, which can be employed to generate the description 'the largest chihuahua'.</Paragraph>
    <Paragraph position="15"> Plurals can be treated along analogous lines. Suppose, for example, the facts in the KB are the same as above and the target set S is {ca, c4}. Its two elements share the property size(x) &gt; 5cm. This prop-erty is exploited by n&amp;Rm~ to construct the list L = {chihuahua,&gt;5cm}. Analogous to the singular case, the inequality can be replaced by the property 'being a set all of whose elements are larger than all other elements of C' (largestm for short), leading to NPs such as 'the largest chihuahuas'. Optionally, the numeral may be included in the NP ('the two largest chihuahuas').</Paragraph>
    <Paragraph position="16"> - 'Absolute' adjectives. The step from the superlative descriptions of case i to the analogous 'absolute' descriptions is a small one. Let us first turn to case iii, 'The n large mouse/mice'. Assuming the correctness of the semantic analysis in section 2, the NP 'The n large mouse/mice' is semantically equivalent to the one discussed under i. Consequently, an obvious variant of the algorithm that was just described can be used for generating it. (For pragmatic issues, see section 5.2) Finally. case iv, 'The large mice'. Semantically, this does not introduce an 3&amp;quot; new problems, since it is to case iii what case ii is to case i. According to the semantic analysis of section 2.1 'The large mice' should be analysed just like 'The n large mouse/mice', except that the muneral n is suppressed. This means that a simplified version (i.e., without a cardinality check) of the algorithm that takes care of case iii will be sufificient to generate descriptions of this kind.</Paragraph>
    <Paragraph position="17"> rE.g, size(x) &gt; m is preferred over sZze(x) &gt; n iff m &gt; n. The preference for inequalities causes the generator to avoid the mentioning of measurements unless they are needed for the identification ~ff the target object.</Paragraph>
  </Section>
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