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<Paper uid="W06-3606">
  <Title>Practical Markov Logic Containing First-Order Quantifiers with Application to Identity Uncertainty</Title>
  <Section position="2" start_page="0" end_page="41" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> Markov logic networks (MLNs) combine the probabilistic semantics of graphical models with the expressivity of first-order logic to model relational dependencies (Richardson and Domingos, 2004). They provide a method to instantiate Markov networks from a set of constants and first-order formulae.</Paragraph>
    <Paragraph position="1"> While MLNs have the power to specify Markov networks with complex, finely-tuned dependencies, the difficulty of instantiating these networks grows with the complexity of the formulae. In particular, expressions with first-order quantifiers can lead to networks that are large and densely connected, making exact probabilistic inference intractable. Because of this, existing applications of MLNs have not exploited the full richness of expressions available in first-order logic.</Paragraph>
    <Paragraph position="2"> For example, consider the database of researchers described in Richardson and Domingos (2004), where predicates include Professor(person), Student(person), AdvisedBy(person, person), and Published(author, paper). First-order formulae include statements such as &amp;quot;students are not professors&amp;quot; and &amp;quot;each student has at most one advisor.&amp;quot; Consider instead statements such as &amp;quot;all the students of an advisor publish papers with similar words in the title&amp;quot; or &amp;quot;this subset of students belong to the same lab.&amp;quot; To instantiate an MLN with such predicates requires existential and universal quantifiers, resulting in either a densely connected network, or a network with prohibitively many nodes. (In the latter example, it may be necessary to ground the predicate for each element of the power set of students.) However, as discussed in Section 2, there may be cases where these aggregate predicates increase predictive power. For example, in predicting the value of HaveSameAdvisor(ai ...ai+k), it may be useful to know the values of aggregate evidence predicates such as CoauthoredAtLeastTwoPapers(ai ...ai+k), which indicates whether there are at least two papers that some combination of authors from ai ...ai+k have co-authored. Additionally, we can construct predicates such as NumberOfStudents(ai) to model the number of students a researcher is likely to advise simultaneously.</Paragraph>
    <Paragraph position="3"> These aggregate predicates are examples of universal and existentially quantified predicates over observed and unobserved values. To enable these sorts  of predicates while limiting the complexity of the ground Markov network, we present an algorithm that incrementally expands the set of aggregate predicates during the inference procedure. In this paper, we describe a general algorithm for incremental expansion of predicates in MLNs, then present an implementation of the algorithm applied to the problem of identity uncertainty.</Paragraph>
  </Section>
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