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<Paper uid="C88-1060">
  <Title>An Algorithm for Functional Uncertainty</Title>
  <Section position="4" start_page="2981" end_page="2981" type="intro">
    <SectionTitle>
4. The Smallest Models
</SectionTitle>
    <Paragraph position="0"> The satisfiability of a description in free form is independent of the choice of strings from its uncertainty languages, but of course different string choices result in different satisfying models for the description.</Paragraph>
    <Paragraph position="1"> An infinite number of strings can be chosen from even a very simple functional uncertainty such as (f COMP* SUBJ) : V, and thus there are an infinite nunlber of distinct possible models. This is reminiscent of the infinite nmnber of models for descriptions with no uncertainties at all (just (fsunJ)=v), but in this case the models are systematically related in the natural subsumption ordering on the f~structure lattice.</Paragraph>
    <Paragraph position="2"> There is one smallest structure; the others include the information it contains and thus satisfy the description. But they also include arbitrary amounts of additional information that the description does not call for. This is discussed by/Kaplan and Bresnan 1982/, where the subsumption-minimal structure is defined to be the grammatically relevant one.</Paragraph>
    <Paragraph position="3"> The models corresponding to the choice of different strings from an infinite uncertainty are also systematically related to each other but on an metrle that is orthogonal to the subsumption ordering.</Paragraph>
    <Paragraph position="4"> Again appealing to the Pumping Lemma for regular sets, strings that are longer than the number of states in an uncertainty's minimal-state finite-state accepter include a substring that is accepted by some repeating sequence of transitions. Replicating this substring arbitrarily still yields a string in the uncertainty, so in a certain sense these replications contribute no new grammatically interesting information. Since all the intbrmatim~ is esseutially contained in tile shorter st,rinh~ that has no oeeurreuce of this imrtieular subs(ring, we define this t. be the grammatically relevant representative fin&amp;quot; the whole class. Thus a description with uncertainties has only a finite number of lir~guistically significant models, those that result h'mn the 5nite disjunci:ions that are introduced in converting the description to flee form and fl'om choosing among the finite nmnber of Short strings in the residual uncertainties.</Paragraph>
  </Section>
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