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<Paper uid="H05-1101">
  <Title>Proceedings of Human Language Technology Conference and Conference on Empirical Methods in Natural Language Processing (HLT/EMNLP), pages 803-810, Vancouver, October 2005. c(c)2005 Association for Computational Linguistics Some Computational Complexity Results for Synchronous Context-Free Grammars</Title>
  <Section position="2" start_page="0" end_page="803" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> State of the art architectures for machine translation are all based on mathematical models called translation models. Generally speaking, a translation model accounts for all the elementary operations that rule the process of translation between the words and the different word orderings of the source and target languages. Translation models are usually enriched with statistical parameters, to drive the searchtowardthemostlikelytranslation(s). Specialized algorithms are provided for the automatic estimation of these parameters from corpora of translation pairs. Besides the task of natural language translation, statistical translation models are also exploited in other applications, such as word alignment, multilingual document retrieval and automatic dictionary construction.</Paragraph>
    <Paragraph position="1"> The most successful translation models that are found in the literature exploit finite-state machinery.</Paragraph>
    <Paragraph position="2"> The approach started with the so-called IBM models (Brown et al., 1988), implementing a set of elementary operations, such as movement, duplication and translation, that independently act on individual words in the source sentence. These word-to-word models have been later enriched with the introduction of larger units such as phrases; see for instance (Och et al., 1999; Och and Ney, 2002).</Paragraph>
    <Paragraph position="3"> Still, the generative capacity of these models lies within the realm of finite-state machinery (Kumar and Byrne, 2003), so they are unable to handle nested structures and do not provide the expressivity required to process language pairs with very different word orderings.</Paragraph>
    <Paragraph position="4"> Recently, more sophisticated translation models have been proposed, borrowing from the theory of compilers and making use of synchronous rewriting. In synchronous rewriting, two formal grammars are exploited, one describing the source language and the other describing the target language.</Paragraph>
    <Paragraph position="5"> Furthermore, the productions of the two grammars are paired and, in the rewriting process, such pairs are always applied synchronously. Formalisms based on synchronous rewriting have been empowered with the use of statistical parameters, and specialized estimation and translation (decoding) algorithms were newly developed. Among the several proposals, we mention here the models presented in (Wu, 1997; Wu and Wong, 1998), (Alshawi et al., 2000), (Yamada and Knight, 2001), (Gildea, 2003) and (Melamed, 2003).</Paragraph>
    <Paragraph position="6"> In this paper we consider synchronous models based on context-free grammars and probabilistic extensions thereof. This is the most common choice  in statistical translation models that exceed the generative power of finite-state machinery. We focus on two associated computational problems that have been defined in the literature. One is the membership problem, which involves testing whether an input string pair can be generated by the model. The other is the translation problem (also called the decoding problem) which involves the search for a suitable translation of an input string/structure. It has been often informally stated in the literature that the use of structured models results in efficient, polynomial time algorithms for the above problems.</Paragraph>
    <Paragraph position="7"> We show here that sometimes this is not the case.</Paragraph>
    <Paragraph position="8"> The contribution of this paper can be stated as follows: null * we show that the membership problem is NPhard, unless a constant bound is imposed on the length of the productions (Section 3); * we show an exponential time lower bound for the membership problem, in case chart parsing is adopted (Section 3); * we show that translating an input string into the best parse tree in the target language is NPhard, even in case productions are bounded in length (Section 4).</Paragraph>
    <Paragraph position="9"> Investigation of the computational complexity of translation models has started in (Knight, 1999) for word-to-word models. This paper can be seen as the continuation of that line of research.</Paragraph>
  </Section>
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