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<Paper uid="C04-1091">
  <Title>An Algorithmic Framework for the Decoding Problem in Statistical Machine Translation</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 Decoding
</SectionTitle>
    <Paragraph position="0"> The decoding problem in SMT is one of finding the most probable translation ^e in the target language of a given source language sentence f in accordance with the Fundamental Equation of SMT (Brown et al., 1993):</Paragraph>
    <Paragraph position="2"> In the remainder of this paper we will refer to the search problem specified by Equation 1 as STRICT DECODING.</Paragraph>
    <Paragraph position="3"> Rewriting the translation model Pr(f|e) assummationtext aPr(f,a|e), where a denotes an alignment between the source sentence and the target sentence, the problem can be restated as:</Paragraph>
    <Paragraph position="5"> Even when the translation model Pr(f|e) is as simple as IBM Model 1 and the language model Pr(e) is a bigram language model, the decoding problem is NP-hard (Knight, 1999).</Paragraph>
    <Paragraph position="6"> Unless P = NP, there is no hope of an efficient algorithm for the decoding problem. Since the Fundamental Equation of SMT does not yield an easy handle to design a solution (exact or even an approximate one) for the problem, most researchers have instead worked on solving the following relatively simpler problem (Germann et al., 2003):</Paragraph>
    <Paragraph position="8"> We call the search problem specified by Equation 3 as RELAXED DECODING.</Paragraph>
    <Paragraph position="9"> Note that RELAXED DECODING relaxes STRICT DECODING to a joint optimization problem. The search in RELAXED DECODING is for a pair (^e,^a). While RELAXED DECODING is simpler than STRICT DECODING, it is also, unfortunately, NP hard for even IBM Model 1 and Bigram language model. Therefore, all practical solutions to RELAXED DECODING have focused on finding suboptimal solutions. The challenge is in devising fast search strategies to find good suboptimal solutions. Table 1 lists the combinatorial optimization problems in the domain of decoding.</Paragraph>
    <Paragraph position="10"> In the remainder of the paper,mandldenote the length of the source language sentence and its translation respectively.</Paragraph>
  </Section>
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