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<Paper uid="P06-1018">
  <Title>Sydney, July 2006. c(c)2006 Association for Computational Linguistics Polarized Unification Grammars</Title>
  <Section position="4" start_page="137" end_page="138" type="intro">
    <SectionTitle>
2 Polarities and unification
</SectionTitle>
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    <Section position="1" start_page="137" end_page="137" type="sub_section">
      <SectionTitle>
2.1 Polarized Unification Gramars
</SectionTitle>
      <Paragraph position="0"> Polarized Unification Grammars generate sets of finite structures. A structure is based on objects.</Paragraph>
      <Paragraph position="1"> For instance, for a (directed) graph, objects are nodes and edges. These two types of objects are linked, giving us the proper structure: if X is the set of nodes and U, the set of edges, the graph is defined by two maps p  and p  from U into X which associate an edge with its source and its target.</Paragraph>
      <Paragraph position="2"> Our structures are polarized, that is, objects are associated to polarities. The set P of polarities is provided with an operation noted &amp;quot;.&amp;quot; and called product. The product is commutative and generally associative; (P, . ) is called the system of polarities. A non-empty strict subset N of P contains the neutral polarities. A polarized structure is neutral if all its polarities are neutral. Structures are defined on a colection of objects of various types (syntactic nodes, semantic nodes, syntactic edges ...) and a colection of maps: structural maps linking objects to objects (such as source and target for edges), label maps linking objects to labels and polarity maps linking objects to polarities.</Paragraph>
      <Paragraph position="3"> Structures combine by unification. The unification of two structures A and B gives a new structure A[?]B obtained by &amp;quot;pasting&amp;quot; together these structures and identifying a part of the objects of the first structure with objects of the second structure. When two polarized structures A  A dag is a directed acyclic graph. An n-graph is a graph whose nodes are edges of a (n-1)-graph and a 1-graph is a standard graph.</Paragraph>
      <Paragraph position="4"> and B are unified, the polarity of an object of A[?]B obtained by identifying two objects of A and B is the product of their polarities; if the two objects bear the same map, these maps must be identified and their values, unified. (For instance identifying two edges forces us to identify their sources and targets.) A Polarized Unification Gramar (PUG) is defined by a finite family T of types of objects, a set of maps attached to the objects of each type, a system (P,.) of polarities, a subset N of P of neutral polarities, and a finite subset of elementary polarized structures, whose objects are described by T; one elementary structure is marked as the initial one and must be used exactly once. The structures generated by the grammar are the neutral structures obtained by combining the initial structure and a finite number of elementary structures. In the derivation process, elementary structures combine successively, each new elementary structure combining with at least one object of the previous result; this ensures that the derived structure is continuous. Polarities are only necessary to control the saturation and are not considered when the strong generative capacity of the grammar is estimated. Polarities belong to the declarative part of the grammar, but they rather play a role in the processing of the grammar.</Paragraph>
    </Section>
    <Section position="2" start_page="137" end_page="138" type="sub_section">
      <SectionTitle>
2.2 The system of polarities
In this paper we wil use the system of polarities
</SectionTitle>
      <Paragraph position="0"/>
      <Paragraph position="2"> = absolutely neutral), with N = {#,#}, and a product defined by the folowing array (where [?] represents the imposibility to combine). Note that # is the neutral element of the product. The symbol - can be interpreted as a need and + as the corresponding resource.</Paragraph>
      <Paragraph position="4"> The system {#,#} is used by Nasr (195), while the system {#,#,-,+}, noted {=,-,-,-}, is considered by Bonfante et al. (204), who show advantages of negative and positive polarities for prefiltration in parsing (a set of structures bearing negative and positive polarities can only  be reduced into a neutral structure if the sum of negative polarities of each object type is equal the sum of positive polarities).</Paragraph>
      <Paragraph position="5"> The system (P, . ) we have presented is comutative and associative. Commutativity implies that the combination of two structures is not procedurally oriented (and we can begin a derivation by any elementary structure, provided we use only once the initial structure).</Paragraph>
      <Paragraph position="6"> Asociativity implies that the combination of structures is unordered: if an object B must combine with A and C, there is no precedence order between the combination of A and B and the one of B and C, that is, A[?](B[?]C) = (A[?]B)[?]C. If we leave polarities aside, our formalism is trivially monotonic: the combination of two structures A and B by a PUG gives us a structure A[?]B that contains A and B as substructures. We can add a (partial) order on P in order to make the formalism monotonic.</Paragraph>
      <Paragraph position="7">  Let [?] be this order. In order to give us a monotonic formalism, [?] must verify the folowing monotonicity property: [?]x,y[?]P x.y [?] x. This provides us with the folowing order: # &lt; # &lt; +/- &lt; #. A PUG built with an ordered system of polarities (P, . ,[?]) verifying the monotonicity property is monotonic. Monotonicity implies god computational properties; for instance it allows translating the parsing with PUG into a problem of constraint resolution (Duchier &amp; Thater, 199).</Paragraph>
    </Section>
  </Section>
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