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<Paper uid="P98-2192">
  <Title>Restrictions on Tree Adjoining Languages</Title>
  <Section position="5" start_page="1177" end_page="1179" type="metho">
    <SectionTitle>
3 Recognition
</SectionTitle>
    <Paragraph position="0"> This section presents the main result of the paper. We provide an algorithm for the recognition of languages generated by the subclass of TAGs introduced in the previous section, and show that the worst case running time is (.9(n5), where n is the length of the input string. To simplify the presentation, we assume the following conditions throughout this section: first, that elementary trees are binary (no more than two children at each node) and no leaf node is labeled by e; and second, that there is always a wrapping node in each wrapping tree, and it differs from the foot and the root node. This is without any loss of generality.</Paragraph>
    <Section position="1" start_page="1177" end_page="1178" type="sub_section">
      <SectionTitle>
3.1 Grammar transformation
</SectionTitle>
      <Paragraph position="0"> Let G = (N, E, I, A) be a TAG obeying the restrictions of Section 2. We first transform A into a new set of auxiliary trees A ~ that will be processed by our method. The root and foot nodes of a tree/3 are denoted R E and FE, respectively.</Paragraph>
      <Paragraph position="1"> The wrapping node (as defined in Section 2) of ~3 is denoted W E.</Paragraph>
      <Paragraph position="2"> Each left (right) tree ~ in A is inserted in A l and is called j3L (j3R). Let 13 be a wrapping tree in A. We split ~ into four auxiliary trees, as informally described in Section 2. Let ~0 be the subtree of fl rooted at W~. We call j3v the tree obtained from/~ by removing every descendant of W~ (and the corresponding arcs). We remove every node to the right (left) of the spine of ~3D and call ~LD (~RD) the resulting tree. Similarly, we remove every node to the right (left) of the spine of ~j and call flnv (~R~\]) the resulting tree. We set F~L D and FER D equal to FE, and set FZL v and FER v equal to W E. Trees ~LU, BRv, ~LD, and ~RD are inserted in A ~ for every wrapping tree/3 in A.</Paragraph>
      <Paragraph position="3">  Each tree in A' inherits at its nodes the adjunction constraints specified in G. In addition, we impose the following constraints: * only trees j3L can be adjoined at the spine of trees ~LD, I~LU; * only trees fir can be adjoined at the spine of trees ~RD, ~RU; * no adjunction can be performed at nodes F~Lu,FZRu.</Paragraph>
    </Section>
    <Section position="2" start_page="1178" end_page="1179" type="sub_section">
      <SectionTitle>
3.2 The algorithm
</SectionTitle>
      <Paragraph position="0"> The algorithm below is a tabular method that works bottom up on derivation trees. Following (Shieber et al., 1995), we specify the algorithm using inference rules. (The specification has been optimized for presentation simplicity, not for computational efficiency.) Symbols N, P, Q denote nodes of trees in A' (including foot and root), c~ denotes initial trees and j3 denotes auxiliary trees. Symbol label(N) is the label of N and children(N) is a string  denoting all children of N from left to right (children(N) is undefined if N is a leaf). We write c~ E Sbst(N) if c~ can be substituted at N. We write f~ E Adj(N) if ~ can be adjoined at N, and nil E Adj(N) if adjunction at N is optional.</Paragraph>
      <Paragraph position="1">  We use two kind of items: * Item &lt;NX,i,j), X E {B,M,T}, denotes a subtree rooted at N and spanning the portion of the input from i to j. Note that two input positions are sufficient, since trees in A ~ always have their foot node at the position of the leftmost or rightmost leaf. We have X -- B if N has not yet been processed for adjunction, X = M if N has been processed only for adjunction of trees f~L, and X = T if N has already been processed for adjunction.</Paragraph>
      <Paragraph position="2"> * Item (~,i,p,q,j) denotes a wrapping tree (in A) with RZ spanning the portion of the input from i to j and with F~ spanning the portion of the input from p to q. In place of ~ we might use symbols \[f~,LD\], \[~, RD\] and \[f~, RU\] to denote the temporary results of recognizing the adjunction of some wrapping tree at W~.</Paragraph>
      <Paragraph position="3"> Algorithm. Let G be a TAG with the restrictions of Section 2, and let A' be the associated set of auxiliary trees defined as in section 3.1. Let aza2...an, n &gt; 1, be an input string. The algorithm accepts the input iff some item (R T, 0, n) can be inferred for some c~ E I. Step 1 This step recognizes subtrees with root N from subtrees with roots in children(N).</Paragraph>
      <Paragraph position="5"> Step 2 This step recognizes the adjunction of wrapping trees at wrapping nodes. We recognize the tree hosting adjunction by composing its four 'chunks', represented by auxiliary trees ~LD, ~RD, ~RU and ~LU in X, around the wrapped tree.</Paragraph>
      <Paragraph position="7"> Due to restrictions on space, we merely claim the correctness of the above algorithm. We now establish its worst case time complexity with respect to the input string length n. We need to consider the maximum number d of input positions appearing in the antecedent of an inference rule. In fact, in the worst case we will have to execute a number of different evaluations of each  inference rule which is proportional to n d, and each evaluation can be carried out in an amount of time independent of n. It is easy to establish that Step 1 can be executed in time O(n 3) and that Step 3 can be executed in time O(n4). Adjunction at wrapping nodes performed at Step 2 is the most expensive operation, requiring an amount of time O(n5). This is also the time complexity of our algorithm.</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="1179" end_page="1180" type="metho">
    <SectionTitle>
4 Linguistic Relevance
</SectionTitle>
    <Paragraph position="0"> In this section we will attempt to show that the restricted formalism presented in Section 2 retains enough generative power to make it useful in the general case.</Paragraph>
    <Section position="1" start_page="1179" end_page="1180" type="sub_section">
      <SectionTitle>
4.1 Athematic and Complement Trees
</SectionTitle>
      <Paragraph position="0"> We begin by introducing the distinction between athematic auxiliary trees and complement auxiliary trees (Kroch, 1989), which are meant to exhaustively characterize the auxiliary trees used in any natural language TAG grammar. 2 An athematic auxiliary tree does not subcategorize for or assign a thematic role to its foot node, so the head of the foot node becomes the head of the phrase at the root. The structure of an athematic auxiliary tree may thus be described as:</Paragraph>
      <Paragraph position="2"> where X n is any projection of category X, y,nax is the maximal projection of Y, and the order of the constituents is variable. 3 A complement auxiliary tree, on the other hand, introduces a lexical head that subcategorizes for the tree's foot node and assigns it a thematic role. The structure of a complement auxiliary tree may be * described as:</Paragraph>
      <Paragraph position="4"> where X rna~ is the maximal projection Of some category X, and y0 is the lexical projection 2The same linguistic distinction is used in the conception of 'modifier' and 'predicative' trees (Schabes and Shieber, 1994), but Schabes and Shieber give the trees special properties in the calculation of derivation structures, which we do not.</Paragraph>
      <Paragraph position="5"> 3The CFG-like notation is taken directly from (Kroch, 1989), where it is used to specify labels at the root and frontier nodes of a tree without placing constraints on the internal structure.</Paragraph>
      <Paragraph position="6"> of some category Y, whose maximal projection dominates X max .</Paragraph>
      <Paragraph position="7"> From this we make the following observations:  1. Because it does not assign a theta role to its foot node, an athematic auxiliary tree may adjoin at any projection of a category, which we take to designate any adjunction site in a host elementary tree.</Paragraph>
      <Paragraph position="8"> 2. Because it does assign a theta role to its foot node, a complement auxiliary tree may only adjoin at a certain 'complement' adjunction site in a host elementary tree, which must at least be a maximal projection of a lexical category.</Paragraph>
      <Paragraph position="9"> 3. The foot node of an athematic auxiliary tree is dominated only by the root, with no intervening nodes, so it falls outside of the maximal projection of the head.</Paragraph>
      <Paragraph position="10"> 4. The foot node of a complement auxiliary  tree is dominated by the maximal projection of the head, which may also dominate other arguments on either side of the foot.</Paragraph>
      <Paragraph position="11"> To this we now add the assumption that each auxiliary tree can have only one complement adjunction site projecting from y0, where y0 is the lexical category that projects yrnax. This is justified in order to prevent projections of y0 from receiving more than one theta role from complement adjuncts, which would violate the underlying theta criterion in Government and Binding Theory (Chomsky, 1981).We also assume that an auxiliary tree can not have complement adjunction sites on its spine projecting from lexical heads other than y0 in order to preserve the minimality of elementary trees (Kroch, 1989; Frank, 1992). Thus there can be no more than one complement adjunction site on the spine of any complement auxiliary tree, and no complement adjunction site on the spine of any athematic auxiliary tree, since the foot node of an athematic tree lies outside of the maximal projection of the head. 4 4It is important to note that, in order to satisfy the theta criterion and minimality, we need only constrain the number of complement adjunctions - not the number of complement adjunction sites - on the spine of an auxiliary tree. Although this would remain within the power of our formalism, we prefer to use constraints expressed in terms of adjunction sites, as we did in Section 2, be- null Based on observations 3 and 4, we can further specify that only complement trees may wrap, because the foot node of an athematic tree lies outside of the maximal projection of the head, below which all of its subcategories must attach. 5 In this manner, we can insure that only one wrapping tree (the complement auxiliary) can adjoin into the spine of a wrapping (complement) auxiliary, and only athematic auxiliaries (which must be left/right trees) can adjoin elsewhere, fulfilling our TAG restriction in Section 2.</Paragraph>
    </Section>
    <Section position="2" start_page="1180" end_page="1180" type="sub_section">
      <SectionTitle>
4.2 Possible Extensions
</SectionTitle>
      <Paragraph position="0"> We may want to weaken our definition to include wrapping athematic auxiliaries, in order to account for modifiers with raised heads or complements as in Figure 3: &amp;quot;They so revered him that they built a statue in his honor.&amp;quot; This can be done within the above algorithm as long as the athematic trees do not wrap productively (that is as long as they cannot be adjoined one at the spine of the other) by splitting the athematic auxiliary tree down the spine and treating the two fragments as tree-local multicomponents, which can be simulated with non-recursive features (Hockey and Srinivas, 1993).</Paragraph>
      <Paragraph position="1">  Since the added features are non-recursive, this extension would not alter the (9(n 5) result reported in Section 3.</Paragraph>
    </Section>
    <Section position="3" start_page="1180" end_page="1180" type="sub_section">
      <SectionTitle>
4.3 Comparison of Coverage
</SectionTitle>
      <Paragraph position="0"> In contrast to the formalisms of Schabes and Waters (Schabes and Waters, 1993; Schabes and Waters, 1995), our restriction allows wrapping complement auxiliaries as in Figure 4 (Schabes and Waters, 1995). Although it is difficult to find examples in English which are excluded by cause it provides a restriction on elementary trees, rather than on derivations.</Paragraph>
      <Paragraph position="1"> 5Except in the case of raising, discussed below.</Paragraph>
      <Paragraph position="2"> Rogers' regular form restriction (Rogers, 1994), we can cite verb-raised complement auxiliary trees in Dutch as in Figure 5 (Kroch and Santorini, 1991). Trees with this structure may adjoin into each others' internal spine nodes an unbounded number of times, in violation of Rogers' definition of regular form adjunction, but within our criteria of wrapping adjunction at only one node on the spine.</Paragraph>
    </Section>
  </Section>
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