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<Paper uid="P95-1023">
  <Title>TAL Recognition in O(M(n2)) Time</Title>
  <Section position="3" start_page="0" end_page="166" type="intro">
    <SectionTitle>
2 Tree Adjoining Grammars
</SectionTitle>
    <Paragraph position="0"> A Tree Adjoining Grammar (TAG) consists of a quintuple (N, ~ U {~}, I, A, S), where N is a finite set of nonterminal symbols, is a finite set of terminal symbols disjoint from N, is the empty terminal string not in ~, I is a finite set of labelled initial trees, A is a finite set of auxiliary trees, S E N is the distinguished start symbol The trees in I U A are called elementary trees. All internal nodes of elementary trees are labelled with nonterminal symbols. Also, every initial tree is labelled at the root by the start symbol S and has leaf nodes labelled with symbols from ~3 U {E}. An auxiliary tree has both its root and exactly one leaf (called the foot node ) labelled with the same non-terminal symbol. All other leaf nodes are labelled with symbols in E U {~}, at least one of which has a label strictly in E. An example of a TAG is given in figure 1.</Paragraph>
    <Paragraph position="1"> A tree built from an operation involving two other trees is called a derived tree. The operation involved is called adjunction. Formally, adjunction is an operation which builds a new tree 7, from an auxiliary tree fl and another tree ~ (a is any tree - initial, auxiliary or derived). Let c~ contain an internal node m labelled X and let fl be the auxiliary tree with root node also labelled X. The resulting tree 7, obtained by adjoining fl onto c~ at node m is built as follows (figure 2):  1. The subtree of a rooted at m, call it t, is excised, leaving a copy of m behind.</Paragraph>
    <Paragraph position="2"> 2. The auxiliary tree fl is attached at the copy of m and its root node is identifed with the copy of m.</Paragraph>
    <Paragraph position="3"> 3. The subtree t is attached to the foot node of fl  and the root node of t (i.e. m) is identified with the foot node of ft.</Paragraph>
    <Paragraph position="4"> This definition can be extended to include adjunction constraints at nodes in a tree. The constraints include Selective, Null and Obligatory adjunction constraints. The algorithm we present here can he modified to include constraints.</Paragraph>
    <Paragraph position="5"> For our purpose, we will assume that every internal node in an elementary tree has exactly 2 children. Each node in a tree is represented by a tuple &lt; tree, node index, label &gt;. (For brevity, we will refer to a node with a single variable m whereever there is no confusion) A good introduction to TAGs can be found in (Partee, et al., 1990).</Paragraph>
  </Section>
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