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<Paper uid="E93-1013">
  <Title>LFG Semantics via Constraints</Title>
  <Section position="2" start_page="0" end_page="98" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> In languages like English, the substantial scaffolding provided by surface constituent structure trees is often a useful guide for semantic composition, and the h-calculus is a convenient formalism for assembling the semantics along that scaffolding \[Montague, 1974\]. This is because the derivation of the meaning of a phrase can often be viewed as mirroring the surface constituent structure of the English phrase.</Paragraph>
    <Paragraph position="1"> The sentence Bill kissed Hillary has the surface constituent structure indicated by the bracketing in 1:</Paragraph>
    <Paragraph position="3"> The verb is viewed as bearing a close syntactic relation to the object and forming a constituent with it; this constituent then combines with the subject of the sentence. Similarly, the meaning of the verb can be viewed as a two-place function which is applied first to the object, then to the subject, producing the meaning of the sentence.</Paragraph>
    <Paragraph position="4"> However, this approach is not as natural for languages whose surface structure does not resemble that of English. For instance, a problem is presented by VSO languages such as Irish \[McCloskey, 1979\].</Paragraph>
    <Paragraph position="5"> To preserve the hypothesis that surface constituent structure provides the proper scaffolding for semantic interpretation in VSO languages, one of two assumptions must be made. One must assume either that semantic composition is nonuniform across languages (leading to loss of explanatory power), or that semantic composition proceeds not with reference to surface syntactic structure, but instead with reference to a more abstract (English-like) constituent structure representation. This second hypothesis seems to us to render vacuous the claim that surface constituent structure is useful in semantic composition. null Further problems are encountered in the semantic analysis of a free word order language such as Warlpiri \[Simpson, 1983; Simpson, 1991\], where surface constituent structure does not always give rise to units that are semantically coherent or useful. Here, an argument of a verb may not even appear as a single unit at surface constituent structure; further,  arguments of a verb may appear in various different places in the string. In such cases, the appeal to an order of composition different from that of English is particularly unattractive, since different orders of composition would be needed for each possible word order sequence.</Paragraph>
    <Paragraph position="6"> The observation that surface constituent structure does not always provide the optimal set of constituents or hierarchical structure to guide semantic interpretation has led to efforts to use a more abstract, cross-linguistically uniform structure to guide semantic composition. As originally proposed by Kaplan and Bresnan \[1982\] and Halvorsen \[1983\], the functional structure or f-structure of LFG is a representation of such a structure. However, as noted by Halvorsen \[1983\] and Reyle \[1988\], the A-calculus is not a very natural tool for combining meanings of f-structure constituents. The problem is that the subconstituents of an f-structure are not assumed to be ordered, and so the fixed order of combination of a functor with its arguments imposed by the A-calculus is no longer an advantage; in fact, it becomes a disadvantage, since an artificial ordering must be imposed on the composition of meanings. Furthermore, the components of the f-structure may be not only complements but also modifiers, which contribute to the final semantics in a very different way.</Paragraph>
    <Paragraph position="7"> Related approaches. In an effort to solve the problem of the order-dependence imposed by standard versions of the A-calculus, Reyle \[1988\] proposes to extend the A-calculus to reduce its sequential bias, assembling meanings by an enhanced application mechanism. However, it is not clear how Reyle's system can be extended to treat modification or complex predicates, phenomena which our use of linear logic allows us to handle.</Paragraph>
    <Paragraph position="8"> Another means of overcoming the problem of the order-dependence of the A-calculus is to adopt semantic terms whose structure resembles f-structures \[Fenstad et al., 1985; Pollard and Sag, 1987; Halvorsen and Kaplan, 1988\]. On these approaches, attribute-value matrices are used to encode semantic information, allowing the syntactic and semantic representations to be built up simultaneously and in the same order-independent manner. However, when expressions of the A-calculus are replaced with attribute-value matrices, other problems arise: in particular, it is not clear how to view such attribute-value matrices as formulas, since issues such as the representation of variable binding and scope are not treated precisely.</Paragraph>
    <Paragraph position="9"> These problems have been noted, and remedies have been proposed. Sometimes, for example, an algorithm is given which globally examines a semantic attribute-value matrix representation to construct a sentence in a well-defined logic; for instance, Halvorsen \[1983\] presents an approach in which attribute-value matrices are translated into formulas of intensional logic. However, the compuration involved is concerned with manipulating these representations in procedural ways: it is hard to see how these procedural mechanisms translate to meaning preserving manipulations on the formulas that the matrices represent. In sum, such approaches tend to sacrifice the semantic precision and declarative simplicity of logical approaches (e.g. A-calculus based approaches), and seem difficult to extend generally or motivate convincingly.</Paragraph>
    <Paragraph position="10"> Our approach. Our approach shares the order-independent features of approaches that represent semantic information using attribute-value matrices, while still allowing a well-defined treatment of variable binding and scope. We do this by identifying (1) a language of meanings and (2) a language for assembling meanings.</Paragraph>
    <Paragraph position="11"> In principle, (1) can be any logic (e.g., Montague's higher-order logic); for the purposes of this paper all we need is the language of first-order terms. Because we assemble the meaning out of semantically precise components, our approach shares the precision of the A-calculus based approaches. For example, the assembled meaning has precise variable binding and scoping.</Paragraph>
    <Paragraph position="12"> We take (2) to be a fragment of first-order (linear) logic carefully chosen for its computational properties, as discussed below. In contrast to using the h-calculus to combine fragments of meaning via ordered applications, we combine fragments of meaning through unordered conjunction, and implication. Rather than using )~-reduction to simplify meanings, we rely on deduction, as advocated by Pereira \[1990; 1991\].</Paragraph>
    <Paragraph position="13"> The elements of the f-structure provide an unordered set of constraints, expressed in the logic, governing how the semantics can fit together. Constraints for combining lexically-provided meanings can be encoded in lexical items, as instructions for combining several arguments into a result. 1 In effect, then, our approach uses first order logic as the 'glue' with which semantic representations are assembled. Once all the constraints are assembled, deduction in the logic is used to infer the meaning of the entire structure. Throughout this process we maintain a sharp distinction between assertions about the meaning (the glue) and the meaning itselfi To better capture some linguistic properties, we make use of first order linear logic as the glue with which meanings are assembled \[Girard, 1987\]. ~ One a Constraints may also be provided as rules governing particular configurations. Such rules are applicable when properties not of individual lexical items in the construction but of the construction as a whole are responsible for its interpretation; these cases include the semantics of relative clauses. We will not discuss examples of configurationally-defined rules in this paper.</Paragraph>
    <Paragraph position="14"> 2Specifically, we make use only of the tensor\]ragment of linear logic. The fragment is closed under conjunction, universal quantification and implication (with atomic an- null way of thinking about linear logic is that it introduces accounting of premises and conclusions, so that deductions consume their premises to generate their conclusions. It turns out that this property of linear logic nicely captures the LFG requirements of coherence and consistency, and additionally provides a natural way to handle modifiers: a modifier consumes the unmodified meaning of the structure it modifies and produces from it a new, modified meaning. null In the following, we first illustrate our approach by discussing a simple example, and then present more complex examples showing how modifiers and valence changing operations are handled.</Paragraph>
  </Section>
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