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<Paper uid="W04-0902">
  <Title>Solving Logic Puzzles: From Robust Processing to Precise Semantics</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 System Overview
</SectionTitle>
    <Paragraph position="0"> This section explains the languages we use to represent the content of a puzzle. Computing the representations from a text is a complex process with several stages, as shown in Figure 2.</Paragraph>
    <Paragraph position="1"> Most of the stages are independent of the puzzles domain. Section 3 reviews the main challenges in this process, and later sections outline the various processing stages. More details of some of these stages can be found at (Stanford NLP Group, 2004).</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.1 First-Order Logic (FOL)
</SectionTitle>
      <Paragraph position="0"> An obvious way of solving logic puzzles is to use off-the-shelf FOL reasoners, such as theorem provers and model builders. Although most GRE logic puzzles can also be cast as constraint-satisfaction problems (CSPs), FOL representations are more general and more broadly applicable to other domains, and they are closer to the natural language semantics. GRE logic puzzles have finite small domains, so it is practicable to use FOL reasoners.</Paragraph>
      <Paragraph position="1"> The ultimate representation of the content of a puzzle is therefore written in FOL. For example, the representation for the first part of constraint (4) in Figure 1 is: [?]x.room(x) [?]y.sculpture(y)[?]exhibit(y,x). (The treatment of the modal 'must' is explained in SS9.2).</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.2 Semantic Logic (SL)
Representing the meaning of natural language
</SectionTitle>
      <Paragraph position="0"> texts in FOL is not straightforward because human languages employ events, plural entities, modal operations, and complex numeric expressions. We therefore use an intermediate representation, written in Semantic Logic (SL), which is intended to be a general-purpose semantic representation language. SL extends FOL with event and group variables, the modal operators square (necessarily) and (possibly), and Generalized Quantifiers (Barwise and Cooper,  can be [?], [?], at-least(n), etc. To continue the example, the intermediate representation for the constraint is: squareQ([?],x1,room(x1),Q([?]1,x2,sculpture(x2), [?]e.exhibit(e)[?]subj(e,x2)[?]in(e,x1)))</Paragraph>
    </Section>
    <Section position="3" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.3 Non-determinism
</SectionTitle>
      <Paragraph position="0"> Although logic puzzles are carefully designed to reduce ambiguities to ensure that there is exactly one correct answer per question, there are still many ambiguities in the analysis, such as multiple possibilities for syntactic structures, pronominal reference, and quantifier scope. Each module ranks possible output representations; in the event that a later stage reveals an earlier choice to be wrong (it may be inconsistent with the rest of the puzzle, or lead to a non-unique correct answer to a question), the system backtracks and chooses the next-best output representation for the earlier stage.</Paragraph>
    </Section>
  </Section>
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