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<Paper uid="C96-2215">
  <Title>Most Probable Tree in Data-Oriented Parsing and Stochastic Tree Grammars. In Proceedings</Title>
  <Section position="5" start_page="1178" end_page="1179" type="concl">
    <SectionTitle>
4 Conclusion and discussion
</SectionTitle>
    <Paragraph position="0"> We conclude that computing the MI)P / MPS / MPP from a sentence / word-graph / word-graph respectively is NP-hard under DOP. Computing the MPS from a word-graph is NP-hard even under SCI,'Gs. Moreover, these results are applicable to STAG as in (Schabes and Waters, 1993).</Paragraph>
    <Paragraph position="1"> The proof of the previous section helps in understanding why computing tt, e MPP in DOP is such a hard problem. The fact that MPS under SCFG is also NP-hard implies that the complexity of the MPPWG, MPS and MPP is due to the definitions of the probabilistic model rather than the complexity of tile syntactic model.</Paragraph>
    <Paragraph position="2"> The main source of NP-completeness is the following common structure of these problems: they all search for an entity that maximizes the sum of the probabilities of processes which depend on that entity. For the MPS problem of SCFGs for example, one searches for the sentence which maximizes the sum of the probabilities of the parses that generate that sentence (i.e. the probability of a parse is also a function of whether it generates the sentence at: hand or not). This is not the  case, for example, when computing the MPD under STSGs (for sentence or even a word-graph), or when computing the MPP under SCFGs (for a sentence or a word-graph).</Paragraph>
    <Paragraph position="3"> The proof in this paper is not a mere theoretical issue. An exponential algorithm can be comparable to a deterministic polynomial algorithm if the grammar-size can be neglected and if the exponential formula is not much worse than the polynomial for realistic sentence lengths. But as soon as the grammar size becomes an important factor (e.g. in DOP), polynomiality becomes a very desirable quality. For example tGI e ~ and IGI n a for n &lt; 7 are comparable but for n = 12 the polynomial is some 94 times faster. If the grammar size is small and the comparison is between 0.001 seconds and 0.1 seconds this might be of no practical importance. But when the grammar size is large and the comparison is between 60 seconds s and 5640 seconds for a sentence of length 12, then things become different.</Paragraph>
    <Paragraph position="4"> To compute the MPP under DOP, one possible solution involves some heuristic that directs the search towards the MPP; a form of this strategy is the Monte-Carlo technique. Another solution might involve assuming Memory-based behavior in directing the search towards the most &amp;quot;suitable&amp;quot; parse according to some heuristic evaluation function that is inferred from the probabilistic model. And a third possible solution is to adjust the probabilities of elementary-trees such that it is not necessary to compute the MPP. The probability of an elementary-tree can be redefined as the sum of the probabilities of all derivations that generate it in the given STSG. This redefinition can be applied by off-line computation and normalization. Then the probability of a parse is redefined as the probability of the MPD that generates it, thereby collapsing the MPP and MPD.</Paragraph>
    <Paragraph position="5"> This method assumes full independence beyond the borders of elementary-trees, which might be an acceptable assumption.</Paragraph>
    <Paragraph position="6"> Finally, it is worth noting that the solutions that we suggested above are merely algorithmic.</Paragraph>
    <Paragraph position="7"> But the ultimate solution to the complexity of probabilistic disambiguation under the current models lies, we believe, only in further incorporation of the crucial elements of the human processing ability into these models.</Paragraph>
    <Paragraph position="8"> SThis is a realistic figure from experiments on the ATIS.</Paragraph>
  </Section>
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