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<Paper uid="P87-1016">
  <Title>ON THE SUCCINCTNESS PROPERTIES OF UNORDERED CONTEXT-FREE GRAMMARS</Title>
  <Section position="6" start_page="114" end_page="114" type="concl">
    <SectionTitle>
5 Proof of the IL
</SectionTitle>
    <Paragraph position="0"> Here we repeat the proof of the IL due to Ogden et al.</Paragraph>
    <Paragraph position="1"> It is an excellent example of the combinatory fact known as the Pigeonhole Principle. As we said, we want to encourage more cooperation between theoretical computer science and linguistics, and part of the way to do this is to give a full account of the techniques used in both areas.</Paragraph>
    <Paragraph position="2"> First we restate the lemma.</Paragraph>
    <Paragraph position="3"> Interchange Lemma. Let G be a CFG or ID/LP grammar with fanout r, and with nonterminal alphabet N. Let m and n be any positive natural numbers with r &lt; m &lt;_ n. Let L(n) be the set of length n strings in L(G), and Q(n) be a subset of L(n). Then we can find a k-interchangeable subset .4 of ~(n), such that m/r &lt; k _&lt; m, and such that IAI &gt;_ IQCn)I/(I.'Vl * rib.</Paragraph>
    <Paragraph position="4"> Proof. The proof breaks into two distinct parts: one involving the Pigeonhole Principle, and another involving an argument about paths in derivation trees with fanout r. The two parts are related by the following definition. Fix n, r, and m as in the statement of the IL. A tuple (j, k, B), where j and k are integers between i and n, and where B E N, is said to describe a string z of length n, if (i) there is a (full) derivation tree for z in G, having a subtree whose root is labeled with B, and the subtree exactly covers that portion of z beginning at position j, and having length k; and (ii) k satisfies the inequality stated in the conclusion of the IL. Notice that if one tuple describes every string in a set A, then, since G is context-free, A is k-interchangeable.</Paragraph>
    <Paragraph position="5"> The part of the proof involving derivation trees can now be stated: we claim that every string : in L(G) has at least one tuple describing it. To see that this is true, execute the following algorithm. Let z E L(G). Begin at the root (S) node of a derivation tree for :, and make that the &amp;quot;current node.&amp;quot; At each stage of the algorithm, move the current node down to a daughter node having the longest possible yield length of its dominated subtree, while the yield length of the current node is strictly bigger than m. Let B be the label of the final value of the current node, let j be the position where the yield of the final value of the current node starts, and let k be the length of that yield. By the algorithm, k &lt;_ m. If k &lt; m/r, then since the grammar has fanout r, then the node above the final value of the current node would have yield length less than m, so it would have been the final value of the current node, a contradiction. This establishes the claim.</Paragraph>
    <Paragraph position="6"> Now we give the combinatory part of the proof. Let E and F be finite sets, and let J~ be a binary relation (set of ordered pairs) between E and F. R is said to cover F if every element of F participates in at least one pair of R. Also, we define, for e E E, R(e) = {f \] e R f}.</Paragraph>
    <Paragraph position="7"> One version of the Pigeonhole Principle can be stated as follows.</Paragraph>
    <Paragraph position="8"> Lemma 1 If R covers F, then there is an element e E E such that  a contradiction.</Paragraph>
    <Paragraph position="9"> Now let E be the set of all tuples (j, k, B) where j and k are less than or equal to n, and B E N. Then \]E\[ = iN\[. n 2. Let F = Q(n). Let e R f iff e describes f. By the first part of our proof, R covers F. Thus let e be a tuple given by the conclusion of the Pigeonhole Principle, and let A be R(e). The size of .4 is correct, and since e describes everything in A, then A is k-interchangeable. This completes the proof and the paper.</Paragraph>
  </Section>
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