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<Paper uid="C92-1020">
  <Title>On the Satisfiability of Complex Constraints</Title>
  <Section position="2" start_page="0" end_page="0" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> Most of the more recent formalisms in Computational Linguistics involve complex constraints, i.e. involving either disjunction or and negation of equality constraints \[Kay85, KB82, PS87\].</Paragraph>
    <Paragraph position="1"> From a practical point of view, the main problem arising from the introduction of complex constraints is due to the NP-hardness of checking whether a given set of constraints is satisfiable and, in the affirmative case, of generating tile set of minimal models which satisfy thc constraints.</Paragraph>
    <Paragraph position="2"> Since the NP-hardness of the problem tends to manifest in practical applications in a dra~ matic way there were several proposals of algorithms to minimize this problem \[Kas87, ED88, MK91\].</Paragraph>
    <Paragraph position="3"> However, in our opinion, any practical approach to the problem should rely on applying an exponential satisfaction algorithm as seldom as possible and also, when that cannot be avoided, to reduce as much as possible the size of tile input formulae to which the algorithm is applied.</Paragraph>
    <Paragraph position="4"> A classical way in which one can reduce the size of input formulae is by factoring, i.e. expressing it as a conjunction of smaller formulae which do not have variables in common. As a matter of fact if instead of checking the satisfiability of a formula of length nk we check the satisfiability of n formulae of size k we are gaining an exponential reduction in time. This strategy is also explored in \[MK91\].</Paragraph>
    <Paragraph position="5"> Another way of reducing the size of input formulae which has been used in CLG implementations \[DV89, BDMVg0, DMVglb\], and which can be combined with the previous approaches, is to build a partial model of the input formula and to apply the algorithm to the remaining constraints. Since in many cases this is able to do away with all the constraints this also achieves the goal of using the exponential satisfaction algorithm as seldom as possible. This method can also be seen as a generalization of steps 1 and 2 of the algorithm described in \[Kas87\], while avoiding some of its pitfalls.</Paragraph>
    <Paragraph position="6"> In this paper we start by showing, by translating to the feature logics context the methods for first order terms introduced in \[DMV91a, DMV91b\], how the CLG approach to rewriting constraints while constructing a partial model can greatly reduce the size of the original constraints and thus contribute to reduce the cornputational problem.</Paragraph>
    <Paragraph position="7"> After that will review the notion of factorization for feature logic formulae showing under which conditions it can be done.</Paragraph>
  </Section>
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