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<Paper uid="P00-1047">
  <Title>A Polynomial-Time Fragment of Dominance Constraints</Title>
  <Section position="2" start_page="0" end_page="0" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> Dominance constraints are used as partial descriptions of trees in problems throughout computational linguistics. They have been applied to incremental parsing (Marcus et al., 1983), grammar formalisms (Vijay-Shanker, 1992; Rambow et al., 1995; Duchier and Thater, 1999; Perrier, 2000), discourse (Gardent and Webber, 1998), and scope underspeci cation (Muskens, 1995; Egg et al., 1998).</Paragraph>
    <Paragraph position="1"> Logical properties of dominance constraints have been studied e.g. in (Backofen et al., 1995), and computational properties have been addressed in (Rogers and Vijay-Shanker, 1994; Duchier and Gardent, 1999). Here, the two most important operations are satis ability testing { does the constraint describe a tree? { and enumerating solutions, i.e. the described trees. Unfortunately, even the satis ability problem has been shown to be NP-complete (Koller et al., 1998). This has shed doubt on their practical usefulness.</Paragraph>
    <Paragraph position="2"> In this paper, we de ne normal dominance constraints, a natural fragment of dominance constraints whose restrictions should be unproblematic for many applications. We present a graph algorithm that decides satis ability of normal dominance constraints in polynomial time. Then we show how to use this algorithm to enumerate solutions efciently. null An example for an application of normal dominance constraints is scope underspeci cation: Constraints as in Fig. 1 can serve as underspeci ed descriptions of the semantic readings of sentences such as (1), considered as the structural trees of the rst-order representations. The dotted lines signify dominance relations, which require the upper node to be an ancestor of the lower one in any tree that ts the description.</Paragraph>
    <Paragraph position="3">  (1) Some representative of every department in all companies saw a  sample of each product.</Paragraph>
    <Paragraph position="4"> The sentence has 42 readings (Hobbs and Shieber, 1987), and it is easy to imagine how the number of readings grows exponentially (or worse) in the length of the sentence. E cient enumeration of readings from the description is a longstanding problem in scope underspeci cation. Our polynomial algorithm solves this problem. Moreover, the investigation of graph problems that are closely related to normal constraints allows us to prove that many other underspeci cation formalisms { e.g. Minimal Recursion Semantics (Copestake et al., 1997) and Hole Semantics (Bos, 1996) { have NP-hard satis ability problems. Our algorithm can still be used as a preprocessing step for these approaches; in fact, experience shows that it seems to solve all encodings of descriptions in Hole Semantics that actually occur.</Paragraph>
  </Section>
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