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Empirical Lower Bounds on the Complexity of Translational Equivalence

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Figure 1: (a) Part of a word alignment. (b) Derivation of this word alignment using only binary and nullary productionsrequires one gap per nonterminal, indicated by commas in the production rules.
Figure 2: a) With a parse tree constraining the top sentence,parse tree constraining the bottom sentence, no such align-a hierarchical alignment is possible without gaps
Figure 3: A word alignment that cannot be generated with-out gaps in a manner consistent with both parse trees.
Table 2: Failure rates for hierarchical alignment of bilingualbitexts under word alignment constraints only.
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