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<Paper uid="C90-2039">
  <Title>Sign NonEmpty Empty V N P ADV</Title>
  <Section position="3" start_page="223" end_page="225" type="intro">
    <SectionTitle>
2. Typed Feature Structures
</SectionTitle>
    <Paragraph position="0"> Ordinary FSs used in unification-based grammar formalisms such as PAT\].{\[Shieher 851 arc classified into two classes, namely, atomic leSs and complex FSs. An atomic FS is represented by an atomic symbol and a complex FS is represented by a set of feature-value pairs.</Paragraph>
    <Paragraph position="1"> Complex FSs are used to partially describe objects by specifying values for certain features or attributes of described objects. Complex FSs can have complex FSs as their feature values and can share certain values among features. For ordinary FSs, unification is defined by using partial ordering based on subsumption relationships. These properties enable flexible descriptions.</Paragraph>
    <Paragraph position="2"> An extension allows complex FSs to have type symbols which define a lattice structure on them, for example, as in \[Pollard and Sag 8&amp;quot;11. The type symbol lattice contains the greatest type symbol Top, which subsumes every type symbol, and the least type symbol Bottom, which is subsumed by every I.ype symbol. An example of a type symbol lattice is shown in Fig. 1.</Paragraph>
    <Paragraph position="3"> An extended complex FS is represented by a type symbol and a set of feature-value pairs. Once complex IeSs are extended as above, an atomic FS can be seen as an extended complex FS whose type symbol has only Top as its greater type symbol and only Bottom as its lesser type symbol and which has an empty set of feature value pairs. Extended complex FSs are called typed feature structures (TFSs). TFSs are denoted by feature-value pair matrices or rooted directed graphs as shown in Fig. 2.</Paragraph>
    <Paragraph position="4"> Among such structures, unification c'm be defined IAP,-Kaci 861 by using the following order; ATFS tl is less than or equal to a TFS t2 if and only if: * the type symbol of tl is less than or equal to the type syn'bol of/2; and * each of the features of t2 exists in t1 and. has as its value a TFS which is not less than its counterpart in tl ; and each of the coreference relationships in t2 is also held in tl.</Paragraph>
    <Paragraph position="5">  greatest lower bound or the meet. A unification example is shown in Fig. 3. In tile directed graph notation, TFS unification corresponds to graph mergi ng. TFSs are very convenient for describing linguistic information in unlfication-based formalisms.</Paragraph>
    <Paragraph position="6"> 3. Wroblewski's Incremental Copy Graph Unifitation</Paragraph>
    <Section position="1" start_page="224" end_page="225" type="sub_section">
      <SectionTitle>
Method and Its Problems
</SectionTitle>
      <Paragraph position="0"> In TFS unification based on Wrobtewski's method, a DG is represented by tile NODE and ARC structures corresponding to a TFS and a feature-value pair respectively, as shown in Fig. 4. The NODE structure has the slots TYPESYMBOL to represent a type symbol, ARCS to represent a set of feature-value pairs, GENERATION to specify the unification process in which the structure has been created, FORWARD, and COPY. When a NODE's GENERATION value is equal to the global value specifying the current unit\]cation process, the structure has been created in the current process or that the structure is currel~l.</Paragraph>
      <Paragraph position="1"> The characteristics which allow nondestructive incremental copy are the NODE's two different slots, FORWARD and COPY, for representing forwarding relationships. A FORWARD slot value represents an eternal relationship while a COPY slot value represents a temporary relationship. When a NODE node1 has a NODE node2 as its FORWARD value, the other contents of tile node1 are ignored and tim contents of node2 are used.</Paragraph>
      <Paragraph position="2"> t{owever, when a NODE has another NODE as its COPY value, the contents of the COPY value are used only when the COPY value is cub:rent. After the process finishes, all COPY slot values are ignored and thus original structures are not destroyed.</Paragraph>
      <Paragraph position="3"> The unification procedure based on this method takes as its input two nodes which are roots of the DGs to be unified. The procedure incrementally copies nodes and ares on the subgraphs of each input 1)G until a node with an empty ARCS value is found.</Paragraph>
      <Paragraph position="4"> The procedure first dereferences both root nodes of the input DGs (i.e., it follows up FORWARD and COPY slot values). If the dereferenee result nodes arc identical, the procedure finishes and returns one of the dereference result nodes.</Paragraph>
      <Paragraph position="5"> Next, the procedure calculates the meet of their type symbol. If the meet is Bottom, which means inconsistency, the procedure finishes and returns Bottom. Otherwise, the procedure obtains the output node with the meet as its TYPESYMBOL. The output node has been created only when neither input node is current; or otherwise the output node is an existing current node.</Paragraph>
      <Paragraph position="6"> Next, the procedure treats arcs. The procedure assumes the existence of two procedures, namely, SharedArcs and ComplementArcs. The SharedArcs procedure takes two lists of arcs as its arguments and gives two lists of arcs each of which contains arcs whose labels exists in both lists with the same arc label order. The ComplementArcs procedure takes two lists of arcs as  In this figure, type symbols are omitted.</Paragraph>
      <Paragraph position="7"> its arguments and gives one list of arcs whose labels are unique to one input list.</Paragraph>
      <Paragraph position="8"> The unification procedure first treats arc pairs obtained by SharedArcs. The procedure applies itself ,'ecursively to each such arc pair values and adds to the output node every arc with the same label as its label and the unification result of their values unless the tmification result is Bottom.</Paragraph>
      <Paragraph position="9"> Next, the procedure treats arcs obtained by ComplementArcs. Each arc value is copied and an arc with the same label and the copied value is added to the output node. For example, consider the case when feature a is first treated at the root nodes of G1 and G2 in Fig. 5. The unification procedure is applied recursively to feature a values of the input nodes. The node specified by the feature path &lt;a&gt; fi'om input graph G1 (Gl/&lt;a&gt;) has an arc with the label c and the corresponding node of input graph G2 does not. The whole subgraph rooted by 6 l/&lt;a c&gt; is then copied. This is because such subgraphs can be modified later. For example, the node Y(G3/&lt;o c g&gt;) will be modified to be the unification result of G 1/&lt;a c g&gt; (or G1/&lt;b d&gt;) and G2/&lt;b d&gt; when the feature path &lt;b d&gt; will be treated.</Paragraph>
    </Section>
    <Section position="2" start_page="225" end_page="225" type="sub_section">
      <SectionTitle>
Incremental Copy Graph Unification
</SectionTitle>
      <Paragraph position="0"/>
      <Paragraph position="2"> The problem with Wroblewski's method is that tile whole result DG is created by using only newly created structures. In the example in Fig. 5, the subgraphs of the result DG surrounded by the dashed rectangle can be shared with subgraphs of input structures G1 and G2, Section 4 proposes a method t.hat avoids this problem, Wroblewski's method first treats arcs with labels that exist in both input nodes and then treats arcs with unique labels. This order is related to the unification failure tendency. Unification fails in treating arcs with common labels more often than in treating arcs with unique labels. Finding a failure can stop further computation as previously described, and thus finding failures first reduces unnecessary computation. This order strategy can be generalized to the EFF and applied to the ordering of arcs with common labels. In Section 5, a method which uses this generalized strategy is proposed.</Paragraph>
    </Section>
  </Section>
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