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<Paper uid="P87-1016">
  <Title>ON THE SUCCINCTNESS PROPERTIES OF UNORDERED CONTEXT-FREE GRAMMARS</Title>
  <Section position="5" start_page="113" end_page="114" type="evalu">
    <SectionTitle>
4 Discussion
</SectionTitle>
    <Paragraph position="0"> What do Theorems I and 2 literally mean as far as linguistic descriptions are concerned? First, we notice that the permutation language P,~ really has s counting property: there is exactly one occurrence of each symbol in any string. The same is true if we consider, for fixed m, the strings of length mn in Mn, as n varies.</Paragraph>
    <Paragraph position="1"> Here there must be exactly m occurrences of each symbol in En, in every string. It seems unreasonable to require this counting property as a property of the sublanguage generated by any construction of ordinary language. For example, a list of modifiers, say adjectives, could allow arbitrary repetitions of any of its basic elements, and not insist that there be at most one occurrence of each modifier. So these examples do not have any direct, naturally occurring, linguistic analogues. It is only if we wish to describe permutation-like behavior where the number of occurrences of each symbol is hounded, but with an un- null bounded number of symbols, that we encounter difficulties. null The same observation, however, applies to Barton's NP-cornpleteness result. Exactly the same counting prop-erty is required to make the universal recognition problem intractable. If we do not insist on an n-character alphabet, of course, then the universal recognition problem is only polynomial for ID/LP grammars; and correspondingly, there is a polynomial-size weakly equivalent CFG for each ID/LP grammar. But even with a growing alphabet, it is still possible that direct ID/LP recognition is polynomial on the average. One way to check this possibility empirically would be to examine long utterances (sentences) in actual fragments of free word-order languages, to see whether words are repeated a large number of times in those utterances. If there is a bound, and if all permutations are equally likely, then the above results may have some relevance. It is definitely the case that speculations about the difficulty of processing these languages should be informed by more actual data. However, it is equally true that the conclusions of a theoretical investigation can suggest what data to collect.</Paragraph>
  </Section>
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