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<Paper uid="P84-1070">
  <Title>A DISCOVERY PROCEDURE FOR CERTAIN PHONOLOGICAL RULES</Title>
  <Section position="9" start_page="346" end_page="347" type="ackno">
    <SectionTitle>
7. APPENDIX A: DETERMINING A RULE'S CON-
TEXT
</SectionTitle>
    <Paragraph position="0"> In this appendix ! describe an algorithm for calculating the set of rule contexts S c = { C } that satisify equations (2) and (3) repeated below in set notation as (4) and (5). Recall that C b are the contexts in which the alternation did take place, and C a are the contexts in which the alternations did not take place. We want to find (the set, of) contexts that.</Paragraph>
    <Paragraph position="1"> simultaneously match all the Cb, while not matching any C..</Paragraph>
    <Paragraph position="2"> (4) V C~, C C_ C b In this paper 1 adopted strict ordering of all rules because it is one of the more stringent rule ordering hypotheses available.</Paragraph>
    <Paragraph position="3"> e In fact, the sets C a and C b as defined above do not contain quite enough information alone. We must also indicate which segments in these contexts alternate, and what they alternate to. This may form the basis of a very different rule order discovery procedure.</Paragraph>
    <Paragraph position="4"> (5) Vc,. c ; c, We can manipulat.e these into computationally more tractable forms. Starting with (4), we have</Paragraph>
    <Paragraph position="6"> This last equation says thal ever), context thai fulfills the conditions above contains at least one feature that distinguishes it from each C0, and that this feature must be in the intersection of all the C b. If for any C,. C\] - C e=O (the null set of features), then there are no contexts C that simultaneously match all the C b and none of the C,, implying that no rule exists that accounts for the observed ah.ernation.</Paragraph>
    <Paragraph position="7"> We can construct the set S c using this last formula by first, calculating C1, the intersection of all the Cb, and then for each C,, calculating C I : ( C I - Cdeg ), a member of which must be in every 6'. The idea is to keep a set of the minimal C needed to account for the C, so far; if C conl.ains a member of C! we don't need to modify it; if C does not contain a member of C I then we have to add a member of C I to it in order for it to satisfy the equations above. The algorithm below acomplishes this.</Paragraph>
    <Paragraph position="8">  where the subroutine &amp;quot;add&amp;quot; adds a set to S c only if it or its subset is not already present.</Paragraph>
    <Paragraph position="9"> After this algorithm has applied, S c will contain all the minimal different C that satisfy equations (4) and (5) above.</Paragraph>
  </Section>
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