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<Paper uid="J95-4002">
  <Title>Tree Insertion Grammar: A Cubic-Time, Parsable Formalism that Lexicalizes Context-Free Grammar without Changing the Trees Produced Yves Schabes * MERL</Title>
  <Section position="3" start_page="0" end_page="480" type="intro">
    <SectionTitle>
2. Tree Insertion Grammar
</SectionTitle>
    <Paragraph position="0"> Tree insertion grammar (TIG) is a tree generating system that is a restricted variant of tree-adjoining grammar (TAG) (Joshi and Schabes 1992; Schabes 1990). As in TAG, a TIG grammar consists of two sets of trees: initial trees, which are combined by substitution and auxiliary trees, which are combined with each other and the initial trees by adjunction. However, both the auxiliary trees and the adjunction allowed are different than in TAG.</Paragraph>
    <Paragraph position="1"> Definition 6 \[TIG\] A tree insertion grammar (TIG) is a five-tuple (G, NT, L A, S), where ~. is a set of terminal symbols, NT is a set of nonterminal symbols, I is a finite set of finite initial  Schabes and Waters Tree Insertion Grammar trees, A is a finite set of finite auxiliary trees, and S is a distinguished nonterminal symbol. The set I t3 A is referred to as the elementary trees.</Paragraph>
    <Paragraph position="2"> In each initial tree the root and interior--i.e., nonroot, nonleaf--nodes are labeled by nonterminal symbols. The nodes on the frontier are labeled with terminal symbols, nonterminal symbols, or the empty string (e). The nonterminal symbols on the frontier are marked for substitution. By convention, substitutability is indicated in diagrams by using a down arrow (D. The root of at least one elementary initial tree must be labeled S.</Paragraph>
    <Paragraph position="3"> In each auxiliary tree the root and interior nodes are labeled by nonterminal symbols. The nodes on the frontier are labeled with terminal symbols, nonterminal symbols, or the empty string (e). The nonterminal symbols on the frontier of an auxiliary tree are marked for substitution, except that exactly one nonterminal frontier node is marked as the foot. The foot must be labeled with the same label as the root. By convention, the foot of an auxiliary tree is indicated in diagrams by using an asterisk (,). The path from the root of an auxiliary tree to the foot is called the spine. Auxiliary trees in which every nonempty frontier node is to the left of the foot are called left auxiliary trees. Similarly, auxiliary trees in which every nonempty frontier node is to the right of the foot are called right auxiliary trees. Other auxiliary trees are called wrapping auxiliary trees. 2 The root of each elementary tree must have at least one child. Frontier nodes labeled with ~ are referred to as empty. If all the frontier nodes of an initial tree are empty, the tree is referred to as empty. If all the frontier nodes other than the foot of an auxiliary tree are empty, the tree is referred to as empty.</Paragraph>
    <Paragraph position="4"> The operations of substitution and adjunction are discussed in detail below. Substitution replaces a node marked for substitution with an initial tree. Adjunction replaces a node with an auxiliary tree.</Paragraph>
    <Paragraph position="5"> To this point, the definition of a TIG is essentially identical to the definition of a TAG. However, the following differs from the definition of TAG.</Paragraph>
    <Paragraph position="6"> TIG does not allow there to be any elementary wrapping auxiliary trees or elementary empty auxiliary trees. This ensures that every elementary auxiliary tree will be uniquely either a left auxiliary tree or a right auxiliary tree. (Wrapping auxiliary trees are neither. Empty auxiliary trees are both and cause infinite ambiguity.) TIG does not allow a left (right) auxiliary tree to be adjoined on any node that is on the spine of a right (left) auxiliary tree. Further, no adjunction whatever is permitted on a node # that is to the right (left) of the spine of an elementary left (right) auxiliary tree T. Note that for T to be a left (right) auxiliary tree, every frontier node dominated by # must be labeled with ~.</Paragraph>
    <Paragraph position="7"> TIG allows arbitrarily many simultaneous adjunctions on a single node in a manner similar to the alternative TAG derivation defined in Schabes and Shieber (1994). Simultaneous adjunction is specified by two sequences, one of left auxiliary trees and the other of right auxiliary trees that specify the order of the strings corresponding to the trees combined.</Paragraph>
    <Paragraph position="8"> A TIG derivation starts with an initial tree rooted at S. This tree is repeatedly extended using substitution and adjunction. A derivation is complete when every frontier node in the tree(s) derived is labeled with a terminal symbol. By means of adjunction, complete derivations can be extended to bigger complete derivations.</Paragraph>
  </Section>
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