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<Paper uid="P90-1022">
  <Title>EXPRESSING DISJUNCTIVE AND NEGATIVE FEATURE CONSTRAINTS WITH CLASSICAL FIRST-ORDER LOGIC.</Title>
  <Section position="2" start_page="0" end_page="174" type="intro">
    <SectionTitle>
ABSTRACT
</SectionTitle>
    <Paragraph position="0"> In contrast to the &amp;quot;designer logic&amp;quot; approach, this paper shows how the attribute-value feature structures of unification grammar and constraints on them can be axiomatized in classical first-order logic, which can express disjunctive and negative constraints. Because only quantifier-free formulae are used in the axiomatization, the satisfiability problem is NPcomplete. null INTRODUCTION.</Paragraph>
    <Paragraph position="1"> Many modern linguistic theories, such as  Structure Grammar \[18\], replace the atomic categories of a context-free grammar with a &amp;quot;feature structure&amp;quot; that represents the.syntactic and semantic properties of the phrase. These feature structures are specified in terms of constraints that they must satisfy. Lexical entries constrain the feature structures that can be associated with terminal nodes of the syntactic tree, and phrase structure rules simultaneously constrain the feature structures that can be associated with a parents and its immediate descendants. The tree is well-formed if and only if all of these constraints are simultaneously satisfiable. Thus for the purposes of recognition a method for determining the satisfiability of such constraints is required: the nature of the satisfying feature structures is of secondary importance.</Paragraph>
    <Paragraph position="2"> Most work on unification-based grammar (including the references cited above) has adopted a type of feature structure called an attribute-value structure. The elements in an attribute-value structure come in two kinds: constant elements and complex elements. Constant elements are atomic entities with no internal structure: i.e. they have no attributes. Complex elements have zero or more attributes, whose  values may be any other element in the structure (including a complex element) and ally element can be the value of zero, one or several attributes. Attributes are partial: it need not be the case that every attribute is d ef!ned for every complex element. The set of attribute-value structures partially ordered by the subsumption relation (together with all additional entity T that every attribute-value structure subsumes) forms a lattice, and the join operation on this lattice is called the unification operati(m 119\]. Example: (from \[16\]). The attribute-value structure (1) has six complex elements labelled el ... e6 and two corastant elements, singular and third. The complex element el has two attributes, subj and pred, the value of which are the complex elements e 2 and e 3 respectively.</Paragraph>
    <Paragraph position="3">  The unification of elements el of(l) and e7 of(2) results in the attribute-value structure (3), the minimal structure in the subsumption lattice</Paragraph>
    <Paragraph position="5"> ...~e 5 el0 agr~agr number person singular third If constraints on attribute-value structures are restricted to conjunctions of equality constraints (i.e. requirements that the value of a path of attributes is equal to a constant or the value of another path) then the set of satisfying attribute-value structures is the principal filter generated by the minimal structure that satisfies the constraints. The generator of the satisfying principal filter of the conjunction of such constraints is the unification of the generators of the satisfying principal filters of each of the conjuncts. Thus the set of attribute-value structures that simultaneously satisfy a set of such constraints can be characterized by computing the unification of the generators of the corresponding principal filters, and the constraints are satisfiable iff the resulting generator is not &amp;quot;T (i.e. -T- represents unification failure). Standard t, nification-based parsers use unification in exactly this way.</Paragraph>
    <Paragraph position="6"> When disjunctions and negations of constraints are permitted, the set of satisfying attribute-value structures does not always form a principal filter \[11\], so the simple unification-based technique for determining the satisfiability of feature structure constraints must be extended. Kasper and Rounds \[11\] provide a formal framework for investigating such constraints by reviving a distinction originally made (as far as I am aware) by Kaplan and Bresnan \[10\] between the language in which feature structure constraints are expressed and the structures that satisfy these constraints.</Paragraph>
    <Paragraph position="7"> Unification is supplanted by conjunction of constraints, and disjunction and negation appear only in the constraint language, not in the feature structures themselves (an exception is \[3\] and \[2\], where feature bundles may contain negative arcs).</Paragraph>
    <Paragraph position="8"> Research in this genre usually follows a general pattern: an abstract model for feature structures and a specialized language for expressing constraints on such structures are &amp;quot;custom-crafted&amp;quot; to treat some problematic feature constraint (such as negative feature constraints). Table 1 sketches some of the variety of feature structure models and constraint types that previous analyses have used.</Paragraph>
    <Paragraph position="9"> This paper follows Kasper and Rounds and most proposals listed in Table 1 by distinguishing the constraint language from feature structures, and restricts disjunction and negation to the constraint language alone. It  (A1) For all Constants c and attributes a, a(c) = 3-. (A2) For all distinct pairs of constants Cl, c2, Cl ~ c2. (A3) For all attributes a, a(3-) = +-.</Paragraph>
    <Paragraph position="10"> (A4) For all constants c, c ~ +-.</Paragraph>
    <Paragraph position="11"> (A5) For all terms u, v, U = V ~-~ ( U = V A U # +- )  differs by not proposing a custom-built &amp;quot;designer logic&amp;quot; for describing feature structures, but instead uses standard first-order logic to axiomatize attribute-value structures and express constraints on them, including disjunctive and negative constraints. The resulting system is a simplified version of Attribute-Value Logic \[9\] which does not allow values to be used as attributes (although it would be easy to do this). The soundness and completeness proofs in \[9\] and other papers listed in Table 1 are not required here because these results are well-known properties of first-order logic.</Paragraph>
    <Paragraph position="12"> Since both the axiomatizion and the constraints are actually expressed in a decidable class of first-order formulae, viz. quantifier-free formulae with equality, 1 the decidability of feature structure constraints follows trivially. In fact, because the satisfiability problem for quantifier-free formulae is NP-complete \[15\] and the relevant portion of the axiomatization and translation of constraints can be constructed in polynomial time, the satisfiability problem for feature constraints (including negation) is also NP-complete.</Paragraph>
  </Section>
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