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<Paper uid="P88-1028">
  <Title>Polynomial Learnability and Locality of Formal Grammars</Title>
  <Section position="2" start_page="0" end_page="0" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> Much of the formai modeling of natural language acquisition has been within the classic paradigm of ~identification in the limit from positive examples&amp;quot; proposed by Gold \[7\]. A relatively restricted class of formal languages has been shown to be unleaxnable in this sense, and the problem of learning formal grammars has long been considered intractable. 1 The following two controversiai aspects of this paradigm, however, leave the implications of these negative results to the computational theory of language acquisition inconclusive. First, it places a very high demand on the accuracy of the learning that takes place - the hypothesized language must be exactly equal to the target language for it to be considered &amp;quot;correct&amp;quot;. Second, it places a very permissive demand on the time and amount of data that may be required for the learning - all that is required of the learner is that it converge to the correct language in the limit. 2 Of the many alternative paradigms of learning proposed, the notion of &amp;quot;polynomial learnability ~ recently formulated by Blumer et al. \[6\] is of particular interest because it addresses both of these problems in a unified &amp;quot;Supported by an IBM graduate fellowship. The author gratefully acknowledges his advisor, Scott Weinstein, for his guidance and encouragement throughout this research.  &amp;quot;identification in the limit&amp;quot; that have been proposed to address the first issue, see Osheraon, Stob and Weinstein \[12\]. As for the latter issue, Angluin (\[5\], \[4\]) investigates the feasible learnability of formal languages with the use of powerful oracles such as &amp;quot;MEMBERSHIP&amp;quot; and &amp;quot;EQUIVALENCE&amp;quot;. way. This paradigm relaxes the criterion for learning by ruling a class of languages to be learnable, if each language in the class can be approximated, given only positive and negative examples, a with a desired degree of accuracy and with a desired degree of robustness (probability), but puts a higher demand on the complexity by requiring that the learner converge in time polynomini in these parameters (of accuracy and robustness) as well as the size (complexity) of the language being learned.</Paragraph>
    <Paragraph position="1"> In this paper, we apply the criterion of polynomial learnability to subclasses of formal grammars that are of considerable linguistic interest. Specifically, we present a novel, nontriviai constraint on gra~nmars called &amp;quot;klocality&amp;quot;, which enables context free grammars and indeed a rich class of mildly context sensitive grammars to be feasibly learnable. Importantly the constraint of k-locality is a nontriviai one because each k-locai subclass is an exponential class 4 containing infinitely many infi-Rite languages. To the best of the author's knowledge, ~k-locaiity&amp;quot; is the first nontrivial constraint on grammars, which has been shown to allow a rich cla~s of grammars of considerable linguistic interest to be polynomiaily learnable. We finally mention some recent negative result in this paradigm, and discuss possible implications of its contrast with the learnability of k-locai classes.</Paragraph>
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