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<Paper uid="W98-0905">
  <Title>Front Back Consistency Overgeneration</Title>
  <Section position="6" start_page="39" end_page="41" type="evalu">
    <SectionTitle>
4 Results
</SectionTitle>
    <Paragraph position="0"> This section summarises the results that have been achieved for complete presentation of data for finite languages (Section 4.1), and incomplete presentation of data for finite (Section 4.3) and infinite languages (Section 4.2).</Paragraph>
    <Paragraph position="1"> The last example (Section 4.3) illustrates how the GA method can be used in conjunction with the formal approach described in Section 2 to automatically discover word-level phonotactic descriptions from raw data sets of phoneme strings.</Paragraph>
    <Section position="1" start_page="39" end_page="40" type="sub_section">
      <SectionTitle>
4.1 German Syllables
</SectionTitle>
      <Paragraph position="0"> Here, the aim was to discover an FSA that accepts a known finite language precisely, so that (following preliminary small-scale tests) the efficiency of the algorithm could be assessed against a medium-sized known language. The data set was generated with the finite-state automaton used by 3usek et al. (1994) to represent the phonotactics of reduced German syllables (shown in Figure 3, double circles indicate final states). This automaton accepts a language of around 11,000 syllables, and because a training set of this size is computationally infeasible, the automaton was divided in half (where the first half covers phonemes preceding the syllable  peak, and the second half covers the remainder), and the corresponding strings were used as separate training sets (the training set for the first half contained 127 strings, that for the second 82).</Paragraph>
      <Paragraph position="1"> Results are summarised in Table 1, and the best automaton that was found is shown in Figure 4. Algorithms based on Hopcroft &amp; Ullman (1979, equivalence proofs pp. 26-27 and 2223, minimisation algorithm p.70) to eliminate e-transitions, determinise and then minimise the original manually constructed automaton showed that the automatically discovered FSAs are the minimal deterministic s equivalents of the two halves of the original automaton, and therefore represent optimal solutions.</Paragraph>
      <Paragraph position="2"> These results show that the GA is able to locate optimal solutions (although it is not guaranteed to find them), and that once an optimum has been found, the population also converges on it, rather than on another, suboptimal solution. null</Paragraph>
    </Section>
    <Section position="2" start_page="40" end_page="41" type="sub_section">
      <SectionTitle>
4.2 Some Non-linguistic Examples
</SectionTitle>
      <Paragraph position="0"> In order to assess the performance of the GA on known infinite languages, and to compare it to another GA-based technique with similar aims, experiments were carried out for four previously investigated learning tasks (Zhou and Grefenstette, 1986). Task 1 was to discover the language (10)', Task 2 was 0&amp;quot;1&amp;quot;0&amp;quot;1&amp;quot;, Task 3 was &amp;quot;all strings with an even number of 0's and an even number of l's&amp;quot;, and Task 4 was &amp;quot;all strings such that the difference between the number of l's and 0's is 3 times n (where n is an integer)&amp;quot;. Zhou &amp; Grefenstette used both SThe 2-dimensional transition matrices that are used as genotypes encode only deterministic FSAs.</Paragraph>
      <Paragraph position="1">  positive and negative examples, and moreover had to &amp;quot;modify the training examples to make the genetic algorithm 'understand' what a perfect concept should be&amp;quot; (p.172). The present approach used positive data only, consisting of all strings up to a certain length L, resulting in datasets of varying size S. L was incremented from 0 until the first value was found for which 5 random runs produced the target automaton in under 200 generations.</Paragraph>
      <Paragraph position="2"> In all four cases, the task was to generalise over the data set, by discovering the recursive nature of the target language from a small sub-set of examples. Results are summarised in Table 2. Zhou &amp; Grefenstette measure the amount of time it takes for a target automaton to be discovered in terms of 'number of trials', but fail to explain exactly what this refers to. It can probably be assumed to refer to the number of generations, as this is the way genetic search performance is usually measured. Given this interpretation, the present method outperformed the earlier approach in all cases. However, the main point is that the method described here discovers the target FSAs without reference to negative examples or manipulation of the data samples. This second set of results shows that</Paragraph>
    </Section>
    <Section position="3" start_page="41" end_page="41" type="sub_section">
      <SectionTitle>
4.3 Russian Data
</SectionTitle>
      <Paragraph position="0"> For the third experiment the words in a data set of 450 bisyllabic feminine Russian nouns were divided into sections in the way described in Section 2.3. This resulted in five learning sets D1-D5 (because the set of final consonants is empty in this training set). For each of the five learning sets, five automata were inferred that precisely generate the learning set. On the basis of these, the degree of generalisation that could be expected given the training sets was estimated, and five automata A1-A5 generalising to between 1.5 and 2.5 times the size of their respective learning sets were then evolved. Finally, two automata C1 and C2 were constructed to encode inter-syllabic constraints, with labels A1-A5 representing the automata that encode intra-syllabic constraints: The resulting phonological grammar hypothesis (the intersection of the two automata encoding inter-syllabic constraints) accepted all words from the original learning set of 450 Russian bisyllabic feminine nouns, generalised to a total set of words of around 10 times this size, and accepted ca. 85% of nouns from a testing set of 200 different such nouns. Generalisation was almost always meaningful in that the greater the similarity between two phonemes (in terms of the phoneme sequences that can follow them), the higher was the hkelihood that they were grouped together on the same transition.</Paragraph>
    </Section>
  </Section>
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