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<Paper uid="E91-1052">
  <Title>An Extension of Earley's Algorithm for S-Attributed Grammars</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4. Finite Partitioning of Attribute Domains
</SectionTitle>
    <Paragraph position="0"> The last remark in the extension of section 3 shows a defect of the Extended Algorithm: It may not terminate in the general case. For the S-attributed case, however, this may happen only if the underlying grammar is ~nfinitely ambiguous or, equivalently, if it has cycles Or derivations of the form A~+A, for some A~ N.</Paragraph>
    <Paragraph position="1"> Consider, for example, the following grammar, which ,measures&amp;quot; the length of each derivation of the sole  Given the input string 'a', the algorithm defines three attributted slate sets:</Paragraph>
    <Paragraph position="3"> Since S1 is infinite, the algorithm does not terminate.</Paragraph>
    <Paragraph position="4"> Cyclic grammars play an important role in most recent linguistic theories, including Government-binding (GB), LexicaI-Functional Grammar (LFG) and GPSG (cf.</Paragraph>
    <Paragraph position="5"> Bcrwick, 1988; Con'ca, 1987b; Kornai and Pullum, 1990). These have in common that they have shifted from rule-based descriptions of language, to declarative orprinciple-based descriptions, in which the role of phrase structure rules or principles is relatively minor.</Paragraph>
    <Paragraph position="6"> Thus, to make the extension of the algorithm useful for natural language applications it becomes necessary to ensure its termination, in spite of cyclic bases.</Paragraph>
    <Paragraph position="7"> - 301 The termination of the Extended Algorithm may be guaranteed while maintaining its full generality, through a finite partition on the attribute domains associated with each cyclic symbol in the grammar. For each such domain dom (a), the partition defines a finite collection of equivalence classes on attribute values. Now, before adding a new state &lt;A--~a'l~, f, ~i&gt; to a state set Si, we test for equivalence (according to the defined partitions) rather than equality to some previously added state; if the new state is equivalent to some other, it is not added. It is easy to show that the number of attributed dotted items in the grammar, and hence the size of the state sets, is now finite. This number is in fact identical to that of Earley's algorithm, except for a constant multiplicative factor, dependent on the grammar and the size of the partitions selected for attribute domains. Since the size of the state sets possible with finite partitioning is now finite, the algorithm always terminates.</Paragraph>
    <Paragraph position="8"> After establishing a correspondence between attribute and unification grammar (UG), we may see that the technique of &amp;quot;restriction&amp;quot; used by Shieber (1985) in his extended algorithm is related to finite partitioning on attribute domains, in fact a particular case which takes advantage of the more structured attribute domains of UG. For attribute grammar, given that the domains involved are more general (e.g., the integers), finite partitioning is the required device.</Paragraph>
  </Section>
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