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<Paper uid="E93-1041">
  <Title>A Tradeoff between Compositionality and Complexity in the Semantics of Dimensional Adjectives</Title>
  <Section position="2" start_page="0" end_page="348" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> In the past decade, the field of knowledge representation (KR) has seen impressive growth of sophistication in the representation of uncertain quantitative knowledge about physical properties in common-sense reasoning and qualitative physics. The input to most of these systems is entered by hand, but some of them, especially those with commonsense domains involving spatial and temporal knowledge, are amenable to interaction by means of a natural language interface. Linguistic access to knowledge about properties such as durations, rates of change, distances, the sizes of the symmetry axes of objects, and so on, is provided by dimensional adjectives (e.g. long-short in the spatial and temporal senses, fast-slow, near-far, tall-short). In this paper, I will investigate two aspects of their semantics that have an impact on the quality of a KR system with an NL interface. One aspect is the complexity of reasoning entailed by their semantic interpretations. As an example, suppose that we have a text about the installation of new kitchen appliances that contains the following sentences:  (1) a. The refridgerator is about 60 cm wide.</Paragraph>
    <Paragraph position="1"> b. The cupboard is about as deep as the refridgerator is wide.</Paragraph>
    <Paragraph position="2"> c. The kitchen table is about 5 cm longer  than the cupboard is deep.</Paragraph>
    <Paragraph position="3"> d. The oven is about twice as high as the table is long.</Paragraph>
    <Paragraph position="4"> We may view the relations expressed by these sentences as constraints on the measurements of the object axes (the width of the fridge, the depth of the cupboard, and so on), which are represented as parameters in a constraint system. Then constraint propagation, along with some knowledge about the  sizes that are typical for object categories, should allow us to derive the following sentences (among others) from (1): (2) a. The cupboard is about 60 cm deep.</Paragraph>
    <Paragraph position="5"> b. The kitchen table is longer than the refridgerator is wide.</Paragraph>
    <Paragraph position="6"> c. The kitchen table is short (for a kitchen table).</Paragraph>
    <Paragraph position="7"> d. The oven is about 70 cm higher than the cupboard is deep.</Paragraph>
    <Paragraph position="8"> e. The oven is high (for an oven).</Paragraph>
    <Paragraph position="9"> The inferences from (1) to (2) are rather simple, but reasoning can become very complicated if a large number of parameters and constraints must be accounted for. As we will see below, the computational properties of this kind of reasoning are dependent on the types of relations that appear in the knowledge base. Thus in the present paper, I investigate the kinds of relations that appear in formal theories of the meanings of the following morphosyntactic constructions of dimensional adjectives:  (3) a. Positive The board is long/short.</Paragraph>
    <Paragraph position="10"> b. Comparative  The board is (6 cm) longer/shorter than the table is wide.</Paragraph>
    <Paragraph position="11"> c. Equative The board is (three times) as long as the table is wide.</Paragraph>
    <Paragraph position="12"> d. Measurement The board is 50 cm long.</Paragraph>
    <Paragraph position="13"> This brings us to the second issue: the compositionality of meaning representations proposed for the sentences in (3). It is appealing from the view-point of theoretical linguistics to regard each of the morphosyntactic categories (positive, etc.) as lexical items with their own semantics, and to assume that the semantics of each sentence in (3) is a compositional function of the semantics of the morphosyntactic category and the semantics Of the adjective stem. Compositional meaning representations may also be computationally more advantageous, since they can be computed very efficiently from syntactic representations (e.g. in unification-based formalisms). Most formal theories of the meanings of adjectives attempt to fulfill this criterion of compositionality, but as we will see, they differ on a more far-reaching criterion: whether the meaning of the differential comparative (6 cm shorter than) and the equative with factor term (three times as long as) is a compositional function of the meanings the difference and factor terms (6 cm and three times) and the meanings of the simple comparative and equative, respectively. Although compositionality is generally regarded as a virtue in and of itself, some authors (\[Pinkal, 1990\], \[Klein, 1991\]) have objected to compositional treatments of difference and factor terms on the grounds that they introduce an excessive amount of mathematical structure into our linguistic models. In section 3, I will compare semantic representations that do and do not foresee a compositional treatment of difference and factor terms by analyzing the complexity of reasoning that they entail. In particular, I will investigate the complexity of constraint propagation in a system where the meaning representations appear as constraints. In this paradigm, uncertain quantitative knowledge is accounted for with real-valued intervals, a popular choice in KR systems, and constraint propagation is performed by the Waltz algorithm (which gets its name from David Waltz \[1975\]). Ernest Davis \[1987\] shows in his detailed analysis that the Waltz algorithm is one of the best choices for this task, for reasons that I will explain in section 3.1 It turns out that the constraint propagation with the Waltz algorithm under the compositional approach is more complex; thus, we apparently face a tradeoff between compositionality and complexity. I argue in section 4 that this is indeed a tradeoff, since the non-compositional formation of meaning representations may be expensive, and the increased complexity of the compositional approach may be manageable, especially if certain assumptions can be made about the domain of physical properties being represented.</Paragraph>
  </Section>
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