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<Paper uid="P98-2192">
  <Title>Restrictions on Tree Adjoining Languages</Title>
  <Section position="4" start_page="0" end_page="1177" type="intro">
    <SectionTitle>
2 Overview
</SectionTitle>
    <Paragraph position="0"> We introduce here the subclass of TAG that we investigate in this paper, and briefly compare it with other proposals in the literature.</Paragraph>
    <Paragraph position="1"> A TAG is a tuple G = (N,~,I,A,S), where N, ~ are the finite sets of nonterminal and terminal symbols, respectively, I, A are the finite  sets of initial and auxiliary trees, respectively, and S E N is the initial symbol. Trees in 112 A are also called elementary trees. The reader is referred to (Joshi, 1985) for the definitions of tree adjunction, tree substitution, and language derived by a TAG.</Paragraph>
    <Paragraph position="2"> The spine of an auxiliary tree is the (unique) path that connects the root and the foot node. An auxiliary tree fl is called a right (left) tree if (i) the leftmost (rightmost, resp.) leaf in ~ is the foot node; and (ii) the spine of fl contains only the root and the foot nodes. An auxiliary tree which is neither left nor right is called a wrapping tree. 1 The TAG restriction we propose is stated as followed: .</Paragraph>
    <Paragraph position="3"> .</Paragraph>
    <Paragraph position="4"> At the spine of each wrapping tree, there is at most one node that can host adjunction of a wrapping tree. This node is called a wrapping node.</Paragraph>
    <Paragraph position="5"> At the spine of each left (right) tree, no wrapping tree can be adjoined and no adjunction constraints on right (left, resp.) auxiliary trees are found.</Paragraph>
    <Paragraph position="6"> The above restriction does not in any way constrain adjunction at nodes that are not in the spine of an auxiliary tree. Similarly, there is no restriction on the adjunction of left or right trees at the spines of wrapping trees.</Paragraph>
    <Paragraph position="7"> Our restriction is fundamentally different from those in (Schabes and Waters, 1993; Schabes and Waters, 1995) and (Rogers, 1994), in that we allow wrapping auxiliary trees to nest inside each other an unbounded number of times, so long as they only adjoin at one place in each others' spines. Rogers, in contrast, restricts the nesting of wrapping auxiliaries to a number of times bounded by the size of the grammar, and Schabes and Waters forbid wrapping auxiliaries altogether, at any node in the grammar.</Paragraph>
    <Paragraph position="8"> We now focus on the recognition problem, and informally discuss the computational advantages that arise in this task when a TAG obeys the above restriction. These ideas are formally developed in the next section. Most of 1The above names are also used in (Schabes and Waters, 1995) for slightly different kinds of trees. the tabular methods for TAG recognition represent subtrees of derived trees, rooted at some node N and having the same span within the input string, by means of items of the form (N,i,p,q,j I. In this notation i, j are positions in the input spanned by N, and p, q are positions spanned by the foot node, in case N belongs to the spine, as we assume in the discussion below.</Paragraph>
    <Paragraph position="9"> i' i p q j j'  The most time expensive step in TAG recognition is the one that deals with adjunction. When we adjoin at N a derived auxiliary tree rooted at some node R, we have to combine together two items (R, i', i, j, j'&gt; and (N, i, p, q, j&gt;. This is shown in Figure 1. This step involves six different indices that could range over any position in the input, and thus has a time cost of O(n~).</Paragraph>
    <Paragraph position="10"> Let us now consider adjunction of wrapping trees, and leave aside left and right trees for the moment. Assume that no adjunction has been performed in the portion of the spine below N. Then none of the trees adjoined below N will simultaneously affect the portions of the tree yield to the left and to the right of the foot node. In this case we can safely split the tree yield and represent item (N,i,p,q, jl by means of two items of a new kind, (Nle~,i,P&gt; and (Wright,q,j&gt;. The adjunction step can now be performed by means of two successive steps. The first step combines (R, i', i, j, j') and (Ntelt, i, p&gt;, producing a new intermediate item I. The second step combines I and (Nright, q, Jl, producing the desired result. In this way the time cost is reduced to O(n5). It is not difficult to see that the above reasoning also applies in cases where no adjunction has been performed at the portion of the spine above N. This suggests that, when pro- null cessing a TAG that obeys the restriction introduced above, we can always 'split' each wrapping tree into four parts at the wrapping node N, since N is the only site in the spine that can host adjunction (see Figure 2(a)). Adjunction of a wrapping tree /3 at N can then be simulated by four steps, executed one after the other. Each step composes the item resulting from the application of the previous step with an item representing one of the four parts of the wrapping tree (see Figure 2(b)).</Paragraph>
    <Paragraph position="11"> We now consider adjunction involving left and right trees, and show that a similar splitting along the spine can be performed. Assume that 7 is a derived auxiliary tree, obtained by adjoining several left and right trees one at the spine of the other. Let x and y be the part of the yield of 7 to the left and right, respectively, of the foot node. From the definition of left and right trees, we have that the nodes in the spine of V have all the same nonterminal label. Also, from condition 2 in the above restriction we have that the left trees adjoined in 7 do not constrain in any way the right trees adjoined in 7. Then the following derivation can always be performed. We adjoin all the left trees, each one at the spine of the other, in such a way that the resulting tree 7te/t has yield x. Similarly, we adjoining all the right trees, one at the spine of the other, in such a way that the yield of the resulting tree &amp;quot;Yright is y. Finally, we adjoin &amp;quot;\[right at the root of 71e/t, obtaining a derived tree having the same yield as 7.</Paragraph>
    <Paragraph position="12"> From the above observations it directly follows that we can always recognize the yield of 7 by independently recognizing 71~/t and 7right. Most important, 71e/t and 7ri~ht can be represented by means of items (Rte/t,i,p) and (Rright,q,j). As before, the adjunction of tree V at some subtree represented by an item I can be recognized by means of two successive steps, one combining I with (Rle~, i,p) at its left, resulting in an intermediate item I t, and the second combining I ~ with (Rright, q, j) at its right, obtaining the desired result.</Paragraph>
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