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<Paper uid="P98-2190">
  <Title>Conditions on Consistency of Probabilistic Tree Adjoining Grammars*</Title>
  <Section position="4" start_page="1164" end_page="1164" type="intro">
    <SectionTitle>
2 Notation
</SectionTitle>
    <Paragraph position="0"> In this section we establish some notational conventions and definitions that we use in this paper. Those familiar with the TAG formalism only need to give a cursory glance through this section.</Paragraph>
    <Paragraph position="1"> A probabilistic TAG is represented by (N, E, 2:, A, S, C/) where N, E are, respectively, non-terminal and terminal symbols. 2: U ,4 is a set of trees termed as elementary trees. We take V to be the set of all nodes in all the elementary trees. For each leaf A E V, label(A) is an element from E U {e}, and for each other node A, label(A) is an element from N. S is an element from N which is a distinguished start symbol.</Paragraph>
    <Paragraph position="2"> The root node A of every initial tree which can start a derivation must have label(A) = S.</Paragraph>
    <Paragraph position="3"> 2: axe termed initial trees and ,4 are auxiliary trees which can rewrite a tree node A E V. This rewrite step is called adjunction. C/ is a function which assigns each adjunction with a probability and denotes the set of parameters 1Note that for CFGs it has been shown in (Chaudhari et al., 1983; S~nchez and Bened~, 1997) that inside-outside reestimation can be used to avoid inconsistency. We will show later in the paper that the method used to show consistency in this paper precludes a straightforward extension of that result for TAGs.</Paragraph>
    <Paragraph position="4"> in the model. In practice, TAGs also allow a leaf nodes A such that label(A) is an element from N. Such nodes A are rewritten with initial trees from I using the rewrite step called substitution. Except in one special case, we will not need to treat substitution as being distinct from adjunction.</Paragraph>
    <Paragraph position="5"> For t E 2: U .4, `4(t) are the nodes in tree t that can be modified by adjunction. For label(A) E N we denote Adj(label(A)) as the set of trees that can adjoin at node A E V.</Paragraph>
    <Paragraph position="6"> The adjunction of t into N E V is denoted by N ~-~ t. No adjunction at N E V is denoted by N ~ nil. We assume the following properties hold for every probabilistic TAG G that we  consider: 1. G is lexicalized. There is at least one leaf node a that lexicalizes each elementary tree, i.e. a E E.</Paragraph>
    <Paragraph position="7"> 2. G is proper. For each N E V,</Paragraph>
    <Paragraph position="9"> Adjunction is prohibited on the foot node of every auxiliary tree. This condition is imposed to avoid unnecessary ambiguity and can be easily relaxed.</Paragraph>
    <Paragraph position="10"> There is a distinguished non-lexicalized initial tree T such that each initial tree rooted by a node A with label(A) = S substitutes into T to complete the derivation. This ensures that probabilities assigned to the input string at the start of the derivation are well-formed.</Paragraph>
    <Paragraph position="11"> We use symbols S, A, B,... to range over V, symbols a,b,c,.., to range over E. We use tl,t2,.., to range over I U A and e to denote the empty string. We use Xi to range over all i nodes in the grammar.</Paragraph>
  </Section>
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