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<Paper uid="W06-2917">
  <Title>Learning Auxiliary Fronting with Grammatical Inference</Title>
  <Section position="8" start_page="129" end_page="130" type="evalu">
    <SectionTitle>
4 Experiments
</SectionTitle>
    <Paragraph position="0"> We decided to see whether this algorithm without modification could shed some light on the debate discussed above. The experiments we present here are not intended to be an exhaustive test of the learnability of natural language. The focus is on determining whether learning can proceed in the absence of positive samples, and given only a very weak general purpose bias.</Paragraph>
    <Section position="1" start_page="129" end_page="129" type="sub_section">
      <SectionTitle>
4.1 Implementation
</SectionTitle>
      <Paragraph position="0"> We have implemented the algorithm described above. There are a number of algorithmic issues that were addressed. First, in order to find which pairs of strings are substitutable, the naive approach would be to compare strings pairwise which would be quadratic in the number of sentences. A more efficient approach maintains a hashtable mapping from contexts to congruence classes. Caching hashcodes, and using a union-find algorithm for merging classes allows an algorithm that is effectively linear in the number of sentences.</Paragraph>
      <Paragraph position="1"> In order to handle large data sets with thousands of sentences, it was necessary to modify the algorithm in various ways which slightly altered its formal properties. However for the experiments reported here we used a version which performs the man who is hungry died .</Paragraph>
      <Paragraph position="2"> the man ordered dinner .</Paragraph>
      <Paragraph position="3"> the man died .</Paragraph>
      <Paragraph position="4"> the man is hungry .</Paragraph>
      <Paragraph position="5"> is the man hungry ? the man is ordering dinner .</Paragraph>
      <Paragraph position="6"> is the man who is hungry ordering dinner ? [?]is the man who hungry is ordering dinner ?  above the line were presented to the algorithm during the training phase, and it was tested on examples below the line.</Paragraph>
      <Paragraph position="7"> exactly in line with the mathematical description above.</Paragraph>
    </Section>
    <Section position="2" start_page="129" end_page="130" type="sub_section">
      <SectionTitle>
4.2 Data
</SectionTitle>
      <Paragraph position="0"> For clarity of exposition, we have used extremely small artificial data-sets, consisting only of sentences of types that would indubitably occur in the linguistic experience of a child.</Paragraph>
      <Paragraph position="1"> Our first experiments were intended to determine whether the algorithm could determine the correct form of a polar question when the noun phrase had a relative clause, even when the algorithm was not exposed to any examples of that sort of sentence. We accordingly prepared a small data set shown in Table 1. Above the line is the training data that the algorithm was trained on. It was then tested on all of the sentences, including the ones below the line. By construction the algorithm would generate all sentences it has already seen, so it scores correctly on those. The learned grammar also correctly generated the correct form and did not generate the final form.</Paragraph>
      <Paragraph position="2"> We can see how this happens quite easily since the simple nature of the algorithm allows a straightforward analysis. We can see that in the learned grammar &amp;quot;the man&amp;quot; will be congruent to &amp;quot;the man who is hungry&amp;quot;, since there is a pair of sentences which differ only by this. Similarly, &amp;quot;hungry&amp;quot; will be congruent to &amp;quot;ordering dinner&amp;quot;. Thus the sentence &amp;quot;is the man hungry ?&amp;quot; which is in the language, will be congruent to the correct sentence.</Paragraph>
      <Paragraph position="3"> One of the derivations for this sentence would be: [is the man hungry ?] - [is the man hungry] [?] [is the man] [hungry] [?] - [is] [the man] [hungry] [?] - [is] [the man][who is hungry] [hungry] [?] - null it rains it may rain it may have rained it may be raining it has rained it has been raining it is raining it may have been raining [?]it may have been rained [?]it may been have rain [?]it may have been rain  the line, and testing data below.</Paragraph>
      <Paragraph position="4"> [is] [the man][who is hungry] [ordering dinner] [?]. Our second data set is shown in Table 2, and is a fragment of the English auxiliary system. This has also been claimed to be evidence in favour of nativism. This was discussed in some detail by (Pilato and Berwick, 1985). Again the algorithm correctly learns.</Paragraph>
    </Section>
  </Section>
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