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<Paper uid="J91-2001">
  <Title>Features and Formulae</Title>
  <Section position="8" start_page="147" end_page="148" type="concl">
    <SectionTitle>
3. Conclusion
</SectionTitle>
    <Paragraph position="0"> The general approach adopted here of separating the feature structures and the constraints that they must satisfy is used in most accounts of feature structures. The novel aspect of this work is that feature structures are axiomatized in and the constraints on feature structures are expressed in a decidable class of first-order logic, so important results such as decidability and compactness follow directly. The Sch6nfinkel-Bernays class of formulae used in this paper are sufficiently expressive so that &amp;quot;set-valued&amp;quot; features can be axiomatized quite directly.</Paragraph>
    <Paragraph position="1"> We conclude with some tentative remarks about the implementation of the system described here. Although a general-purpose first-order logic theorem prover could be used to determine the satisfiability of Sch6nfinkel-Bernays formulae, it should be possible to take advantage of the syntactic restrictions these formulae satisfy to obtain a more efficient implementation. One way in which this might be done is as follows.</Paragraph>
    <Paragraph position="2"> First, the axioms should be expressed in clausal form, i.e. in the form 3xl ... xnV yl ... yn A1 A ... A Am --~ B1 V ... V Bn where the Ai and Bj are atoms. These can be used in a 'forward chaining' inference procedure using 'semi-naive evaluation' (see Genesereth and Nilsson (1987) for details).</Paragraph>
    <Paragraph position="3">  Johnson Features and Formulae For example, the clausal form expansion of axiom (5') for union is 18a. V xyzu union(x,y,z) A in(u,z) ~ in(u,x) V in(u,y) 18b. V xyzu union(x,y,z) A in(u,x) --* in(u,z) 18c. V xyzu union(x,y,z) A in(u,y) ~ in(u,z).</Paragraph>
    <Paragraph position="4"> Second, if efficiency comparable to the standard (purely conjunctive) unification algorithm is to be achieved, it is necessary to efficiently index atoms on their arguments (both from the original constraints and those produced as consequences during the inference process just described). If we were dealing with only purely conjunctive formulae we could use a graph-based representation similar to the one used in the standard attribute-value unification algorithm, but since axioms such as 18a have disjunctive consequents we need a data structure that can represent nonconjunctive formulae, even if all of the linguistic constraints associated with lexical entries and syntactic rules are purely conjunctive. This problem is an instance of the general problem of disjunction, and it seems that some of the techniques proposed in the feature-structure literature to deal with disjunction (e.g. Eisele and D6rre 1988; D6rre and Eisele 1990; Maxwell and Kaplan 1989a, 1989b) can be applied here too.</Paragraph>
  </Section>
class="xml-element"></Paper>