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<Paper uid="J99-4004">
  <Title>Semiring Parsing</Title>
  <Section position="8" start_page="601" end_page="603" type="concl">
    <SectionTitle>
8. Conclusion
</SectionTitle>
    <Paragraph position="0"> In this paper, we have given a simple item-based description format that can be used to describe a very wide variety of parsers. These parsers include the CKY algorithm, Earley's algorithm, prefix probability computation, a TAG parsing algorithm, Graham, Harrison, Ruzzo (GHR) parsing, and HMM computations. We have shown that this description format makes it easy to find parsers that compute values in any w-continuous semiring. The same description can be used to find reverse values in commutative w-</Paragraph>
    <Section position="1" start_page="602" end_page="603" type="sub_section">
      <SectionTitle>
Goodman Semiring Parsing
</SectionTitle>
      <Paragraph position="0"> continuous semirings, and in many noncommutative ones as well. This description format can also be used to describe grammar transformations, including transformations to CNF and GNF, which preserve values in any commutative w-continuous semiring.</Paragraph>
      <Paragraph position="1"> While theoretical in nature, this paper is of some practical value. There are three reasons the results of this paper would be used in practice: first, these techniques make computation of the outside values simple and mechanical; second, these techniques make it easy to show that a parser will work in any w-continuous semiring; and third, these techniques isolate computation of infinite sums in a given semiring from the parser specification process.</Paragraph>
      <Paragraph position="2"> Perhaps the most useful application of these results is in finding formulas for outside values. For parsers such as CKY parsers, finding outside formulas is not particularly burdensome, but for complicated parsers such as TAG parsers, GHR parsers, and others, it can require a fair amount of thought to find these equations through conventional reasoning. With these techniques, the formulas can be found in a simple mechanical way.</Paragraph>
      <Paragraph position="3"> The second advantage comes from clarifying the conditions under which a parser can be converted from computing values in the Boolean semiring (a recognizer) to computing values in any w-continuous semiring. We should note that because in the Boolean semiring, infinite summations can be computed trivially and because repeatedly adding a term does not change results, it is not uncommon for parsers that work in the Boolean semiring to require significant modification for other semirings. For parsers like CKY parsers, verifying that the parser will work in any semiring is trivial, but for other parsers the conditions are more complex. With the techniques in this paper, all that is necessary is to show that there is a one-to-one correspondence between item derivations and grammar derivations. Once that has been shown, any w-continuous semiring can be used.</Paragraph>
      <Paragraph position="4"> The third use of this paper is to separate the computation of infinite sums from the main parsing process. Infinite sums can come from several different phenomena, such as loops from productions of the form A --* A; productions involving ~; and left recursion. In traditional procedural specifications, the solution to these difficult problems is intermixed with the parser specification, and makes the parser specific to semirings using the same techniques for solving the summations.</Paragraph>
      <Paragraph position="5"> It is important to notice that the algorithms for solving these infinite summations vary fairly widely, depending on the semiring. On the one hand, Boolean infinite summations are nearly trivial to compute. For other semirings, such as the counting semiring, or derivation forest semiring, more complicated computations are required, including the detection of loops. Finally, for the inside semiring, in most cases only approximate techniques can be used, although in some cases, matrix inversion can be used. Thus, the actual parsing algorithm, if specified procedurally, can vary quite a bit depending on the semiring.</Paragraph>
      <Paragraph position="6"> On the other hand, using our techniques makes infinite sums easier to deal with in two ways. First, these difficult problems are separated out, relegated conceptually to the parser interpreter, where they can be ignored by the constructor of the parsing algorithm. Second, because they are separated out, they can be solved once, rather than again and again. Both of these advantages make it significantly easier to construct parsers. Even in the case where, for efficiency, loops are precomputed offline, as in GHR parsing, the same item-based representation and interpreter can be used.</Paragraph>
      <Paragraph position="7"> In summary, the techniques of this paper will make it easier to compute outside values, easier to construct parsers that work for any w-continuous semiring, and easier  Computational Linguistics Volume 25, Number 4 to compute infinite sums in those semirings. In 1973, Teitelbaum wrote: We have pointed out the relevance of the theory of algebraic power series in noncommuting variables in order to minimize further piecemeal rediscovery (page 199).</Paragraph>
      <Paragraph position="8"> Many of the techniques needed to parse in specific semirings continue to be rediscovered, and outside formulas are derived without observation of the basic formulas given here. We hope this paper will bring about Teitelbaum's wish.</Paragraph>
    </Section>
  </Section>
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