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<Paper uid="P06-2036">
  <Title>Factoring Synchronous Grammars By Sorting</Title>
  <Section position="7" start_page="444" end_page="444" type="concl">
    <SectionTitle>
5 Conclusion
</SectionTitle>
    <Paragraph position="0"> The algorithm above not only identi es whether a permutation can be factored into a composition of permutations, but also returns the factorization that minimizes the largest rule size, in time O(n log n). The factored SCFG with rules of size at most k can be used to synchronously parse in time O(Nk+4) by dynamic programming with continuous spans in one dimension.</Paragraph>
    <Paragraph position="1"> As mentioned in the introduction, the optimal parsing strategy for SCFG rules with a given permutation may involve dynamic programming states with discontinuous spans in both dimensions. Whether these optimal parsing strategies can be found ef ciently remains an interesting open problem.</Paragraph>
    <Paragraph position="2"> Acknowledgments This work was partially supported by NSF ITR IIS-09325646 and NSF ITR IIS-0428020.</Paragraph>
  </Section>
class="xml-element"></Paper>