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<Paper uid="P95-1021">
  <Title>D-Tree Grammars</Title>
  <Section position="4" start_page="152" end_page="154" type="metho">
    <SectionTitle>
2 Definition of D-Tree Grammars
</SectionTitle>
    <Paragraph position="0"> A d-tree is a tree with two types of edges: domination edges (d-edges) and immediate domination edges (i-edges). D-edges and i-edges express domination and immediate domination relations between nodes. These relations are never rescinded when d-trees are composed. Thus, nodes separated by an i-edge will remain in a mother-daughter relationship throughout the derivation, whereas nodes separated by an d-edge can be equated or have a path of any length inserted between them during a derivation.</Paragraph>
    <Paragraph position="1"> D-edges and i-edges are not distributed arbitrarily in d-trees. For each internal node, either all of its daughters are linked by i-edges or it has a single daughter that is linked to it by a d-edge. Each node is labelled with a terminal symbol, a nonterminal symbol or the empty string. A d-tree containing n d-edges can be decomposed into n + 1 components containing only i-edges.</Paragraph>
    <Paragraph position="2"> D-trees can be composed using two operations: subsertion and sister-adjunction. When a d-tree a is subserted into another d-tree/3, a component of a is substituted at a frontier nonterminal node (a substitution node) of/3 and all components of a that are above the substituted component are inserted into d-edges above the substituted node or placed above the root node. For example, consider the d-trees a and /3 shown in Figure 3. Note that components are shown as triangles. In the composed d-tree 7 the component a(5) is substituted at a substitution node in /3. The components, a(1), a(2), and a(4) of a above a(5) drift up the path in/3 which runs from the substitution node. These components are then inserted into d-edges in/3 or above the root of/3. In general, when a component c~(i) of some d-tree a is inserted into a d-edge between nodes ~/1 and r/2 two new d-edges are created, the first of which relates r/t and the root node of a(i), and the second of which relates the frontier</Paragraph>
    <Paragraph position="4"> the substituted node to drift arbitrarily far up the d-tree and distribute themselves within domination edges, or above the root, in any way that is compatible with the domination relationships present in the substituted d-tree. DTG provide a mechanism called subsertion-insertlon constraints to control what can appear within d-edges (see below).</Paragraph>
    <Paragraph position="5"> The second composition operation involving d-trees is called sister-adjunction. When a d-tree a is sister-adjoined at a node y in a d-tree fl the composed d-tree 7 results from the addition to /~ of a as a new leftmost or rightmost sub-d-tree below 7/. Note that sister-adjunction involves the addition of exactly one new immediate domination edge and that severM sister-adjunctions can occur at the same node. Sister-adjoining constraints specify where d-trees can be sister-adjoined and whether they will be right- or left-sister-adjoined (see below).</Paragraph>
    <Paragraph position="6"> A DTG is a four tuple G = (VN, VT, S, D) where VN and VT are the usual nonterminal and terminal alphabets, S E V~ is a distinguished nonterminal and D is a finite set of elementary d-trees. A DTG is said to be lexicalized if each d-tree in the grammar has at least one terminal node. The elementary d-trees of a grammar G have two additionM annotations: subsertion-insertion constraints and sister-adjoining constraints* These will be described below, but first we define simultaneously DTG derivations and subsertion-adjoining trees (SAtrees), which are partial derivation structures that can be interpreted as representing dependency information, the importance of which was stressed in the introduction 5.</Paragraph>
    <Paragraph position="7"> Consider a DTG G = (VN, VT,S, D). In defining SA-trees, we assume some naming convention for the elementary d-trees in D and some consistent ordering on the components and nodes of elementary d-trees in D. For each i, we define the set of d-trees TI(G) whose derivations are captured by SA-trees of height i or less. Let To(G) be the set D of elementary d-trees of G. Mark all of the components of each d-tree in To(G) as being substitutable 6. Only components marked as substitutable can be substituted in a subsertion operation. The SA-tree for ~ E To(G) consists of a single node labelled by the elementary d-tree name for a.</Paragraph>
    <Paragraph position="8"> For i &gt; 0 let ~(G) be the union of the set ~-I(G) with the set of all d-trees 7 that can be produced as follows. Let a E D and let 7 be the result of subserting or sister-adjoining the d-trees 71,- *., 7k into a where 71, * -., 7k are all in Ti- I(G), with the subsertions taking place at different substitution nodes in as the footnote. Only substitutable components of 71,..-, 3'k can be substituted in these subsertions. Only the new components of 7 that came from a are marked as substitutable in 7. Let Vl,..., ~'k be the SA-trees for 71,..-,7k, respectively. The SA-tree r for 7 has root labelled by the name for a and k sub-trees rt,. *., rk. The edge from the root of r to the root of the subtree ri is labelled by li (1 &lt; i &lt; k) defined as follows. Suppose that 71 was subserted into a and the root of r/is labelled by the name of some c~ ~ E D. Only components of a ~ will have been marked as substitutable in 7/- Thus, in this subsertion some component cJ(j) will have been substituted at a node in a with address n. In this case, the label l~ is the pair (j, n). Alternatively, 7i will have S I)ue to space limitations, in the following definitions we are forced to be somewhat imprecise when we identify a node in a derived d-tree with the node in the elementary d-trees (elementary nodes) from which it was derived. This is often done in TAG literature, and hopefully it will be clear what is intended.</Paragraph>
    <Paragraph position="9"> eWe will discuss the notion of substitutability further in the next section. It is used to ensure the $A-tree is a tree. That is, an elementary structure cannot be subserted into more than one structure since this would be counter to our motivations for using subsertion for complementation.</Paragraph>
    <Paragraph position="10">  been d-sister-adjoined at some node with address n in a, in which case li will be the pair (d, n) where d e { left, right }.</Paragraph>
    <Paragraph position="11"> The tree set T(G) generated by G.is defined as the set of trees 7 such that: 7' E T/(G) for some i 0; 7 ~ is rooted with the nonterminal S; the frontier of 7' is a string in V~ ; and 7 results from the removal of all d-edges from 7'. A d-edge is removed by merging the nodes at either end of the edge as long as they are labelled by the same symbol. The string language L(G) associated with G is the set of terminal strings appearing on the frontier of trees in T(G).</Paragraph>
    <Paragraph position="12"> We have given a reasonably precise definition of SA-trees since they play such an important role in the motivation for this work. We now describe informally a structure that can be used to encode a DTG derivation. A derivation graph for 7 E T(G) results from the addition of insertion edges to a SA-tree r for 7. The location in 7 of an inserted elementary component a(i) can be unambiguously determined by identifying the source of the node (say the node with address n in the elementary d-tree a') with which the root of this occurrence of a(i) is merged with when d-edges are removed. The insertion edge will relate the two (not necessarily distinct) nodes corresponding to appropriate occurrences of a and a' and will be labelled by the pair (i, n).</Paragraph>
    <Paragraph position="13"> Each d-edge in elementary d-trees has an associated subsertion-insertion constraint (SIC). A SIC is a finite set of elementary node addresses (ENAs). An I=NA ~} specifies some elementary d-tree a E D, a component of a and the address of a node within that component of a. If a ENA y/is in the SIC associated with a d-edge between 7z and r/2 in an elementary d-tree a then ~/cannot appear properly within the path that appears from T/t to T/2 in the derived  Each node of elementary d-trees has an associated sister-adjunction constraint (SAC). A SAC is a finite set of pairs, each pair identifying a direction (left or right) and an elementary d-tree. A SAC gives a complete specification of what can be sister-adjoined at a node. If a node ~/is associated with a SAC containing a pair (d, a) then the d-tree a can be d-sister-adjoined at r/. By definition of sisteradjunction, all substitution nodes and all nodes at the top of d-edges can be assumed to have SACs that are the empty-set. This prevents sister-adjunction at these nodes.</Paragraph>
    <Paragraph position="14"> In this section we have defined &amp;quot;raw&amp;quot; DTG. In a more refined version of the formalism we would associate (a single) finite-valued feature structure with each node 7. It is a matter of further research to determine to what extent SICs and SACs can be stated globally for a grammar, rather than being attached  structures.</Paragraph>
    <Paragraph position="15"> to d-edges/nodes s. See the next section for a brief discussion of linguistic principles from which a grammar's SICs could be derived.</Paragraph>
  </Section>
  <Section position="5" start_page="154" end_page="157" type="metho">
    <SectionTitle>
3 Linguistic Examples
</SectionTitle>
    <Paragraph position="0"> In this section, we show how an account for the data introduced in Section 1 can be given with DTG.</Paragraph>
    <Section position="1" start_page="154" end_page="156" type="sub_section">
      <SectionTitle>
3.1 Getting Dependencies Right: English
</SectionTitle>
      <Paragraph position="0"> In Figure 4, we give a DTG that generates sentence (1). Every d-tree is a projection from a lexical anchor. The label of the maximal projection is, we assume, determined by the morphology of the anchor. For example, if the anchor is a finite verb, it will project to S, indicating that an overt syntactic (&amp;quot;surface&amp;quot;) subject is required for agreement with it (and perhaps case-assignment). Furthermore, a finite verb may optionally also project to S' (as in the d-tree shown for claims), indicating that a wh-moved or topicalized element is required. The finite verb seems also projects to S, even though it does not itself provide a functional subject. In the case of the to adore tree, the situation is the inverse: the functional subject requires a finite verb Sin this context, it might be beneficiM to consider the expression of a feature-based lexicalist theory such as HPSG in DTG, similar to the compilation of HPSG to TAG (Kasper et al., 1995).</Paragraph>
      <Paragraph position="1">  to agree with, which is signaled by the fact that its component's root and frontier nodes are labelled S and VP, respectively, but the verb itself is not finite and therefore only projects to VP\[-fin\]. Therefore, the subject will have to raise out of its clause for agreement and case assignment. The direct object of to adore has wh-moved out of the projection of the verb (we include a trace for the sake of clarity).</Paragraph>
      <Paragraph position="2">  We add SlCs to ensure that the projections are respected by components of other d-trees that may be inserted during a derivation. A SIC is associated with the d-edge between VP and S node in the seems d-tree to ensure that no node labelled S ~ can be inserted within it - i.e., it can not be filled by with a wh-moved element. In contrast, since both the subject and the object of to adore have been moved out of the projection of the verb, the path to these arguments do not carry any SIC at all 9.</Paragraph>
      <Paragraph position="3"> We now discuss a possible derivation. We start out with the most deeply embedded clause, the adores clause. Before subserting its nominal arguments, we sister-adjoin the two adjectival trees to the tree for hotdogs. This is handled by a SAC associated with the N' node that allows all trees rooted in AdjP to be left sister-adjoined. We then subsert this structure and the subject into the to adore d-tree. We subsert the resulting structure into the seems clause by substituting its maximal projection node, labelled VP\[fin: -\], at the VP\[fin: -\] frontier node of seems, and by inserting the subject into the d-edge of the seems tree. Now, only the S node of the seems tree (which is its maximal projection) is substitutable. Finally, we subsert this derived struc9We enforce island effects for wh-movement by using a \[+extract\] feature on substitution nodes. This corresponds roughly to the analysis in TAG, where islandhood is (to a large extent) enforced by designating a particular node as the foot node (Kroch &amp; Joshi, 1986).</Paragraph>
      <Paragraph position="4"> ture into the claims d-tree by substituting the S node of seems at the S complement node of claims, and by inserting the object of adores (which has not yet been used in the derivation) in the d-edge of the claims d-tree above its S node. The derived tree is shown in Figure 5. The SA-tree for this derivation corresponds to the dependency tree given previously in Figure 2.</Paragraph>
      <Paragraph position="5"> Note that this is the only possible derivation involving these three d-trees, modulo order of operations. To see this, consider the following putative alternate derivation. We first subsert the to adore d-tree into the seems tree as above, by substituting the anchor component at the substitution node of seems. We insert the subject component of fo adore above the anchor component of seems. We then subsert this derived structure into the claims tree by substituting the root of the subject component of to adore at the S node of claims and by inserting the S node of the seems d-tree as well as the object component of the to adore d-tree in the S'/S d-edge of the claims d-tree. This last operation is shown in Figure 6. The resulting phrase structure tree would be the same as in the previously discussed derivation, but the derivation structure is linguistically meaningless, since to adore world have been subserted into both seems and claims. However, this derivation is ruled out by the restriction that only substitutable components can be substituted: the subject component of the adore d-tree is not substitutable after subsertion into the seems d-tree, and therefore it cannot be substituted into the claims d-tree.</Paragraph>
      <Paragraph position="6">  In the above discussion, substitutability played a  central role in ruling out the derivation. We observe in passing that the SIC associated to the d-edge in the seems d-tree also rules out this derivation. The derivation requires that the S node of seems be inserted into the SI/S d-edge of claims. However, we would have to stretch the edge over two components which are both ruled out by the SIC, since they violate the projection from seems to its S node. Thus, the derivation is excluded by the independently motivated Sits, which enforce the notion of projection. This raises the possibility that, in grammars that express certain linguistic principles, substitutability is not needed for ruling out derivations of this nature.</Paragraph>
      <Paragraph position="7"> We intend to examine this issue in future work.</Paragraph>
    </Section>
    <Section position="2" start_page="156" end_page="157" type="sub_section">
      <SectionTitle>
3.2 Getting Word Order Right: Kashmiri
</SectionTitle>
      <Paragraph position="0"> for sentence (2b). We use the node label VP throughout and use features such as top (for topic) to differentiate different levels of projection. Observe that in both trees an argument has been fronted. Again, we will use the SlCs to enforce the projection from a lexical anchor to its maximal projection. Since the direct object of kor has wh-moved out of its clause, the d-edge connecting it to the maximal projection of its verb has no SIC. The d-edge connecting the maximal projection of baasaan to the Aux component, however, has a SIC that allows only VP\[wh: +, top: -\] nodes to be inserted.</Paragraph>
      <Paragraph position="1">  The derivation proceeds as follows. We first subsert the embedded clause tree into the matrix clause tree. After that, we subsert the nominal arguments and function words. The derived structure is shown in Figure 8. The associated SA-tree is the desired, semantically motivated, dependency structure: the embedded clause depends on the matrix clause.</Paragraph>
      <Paragraph position="2"> In this section, we have discussed examples where the elementary objects have been obtained by projecting from lexical items. In these cases, we overcome both the problems with TAG considered in Section 1. The SlCs considered here enforce the same notion of projection that was used in obtaining the elementary structures. This method of arriving at SlCs not only generalizes for the English and Kashmiri examples but also appears to apply to the case of long-distance scrambling and topicalization in German.</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="157" end_page="157" type="metho">
    <SectionTitle>
4 Recognition
</SectionTitle>
    <Paragraph position="0"> It is straightforward to &amp;quot;.~lapt the polynomial-time El&lt;Y-style recognition algorithm for a lexicalized UVG-DI. of Rarnbow (1994b) for DTG. The entries in this array recording derivations of substrings of input contain a set of elementary nodes along with a multi-set of components that must be in~rted above during bottom-up recognition. These components are added or removed at substitution and insertion.</Paragraph>
    <Paragraph position="1"> The algorithm simulates traversal of a derived tree; checking for SICS and SACs can be done easily. Becanse of lexicalization, the size of these multi-sets is polynomially bounded, from which the polynomial time and space complexity of the algorithm follows.</Paragraph>
    <Paragraph position="2"> For practical purposes, especially for lexicalized grammars, it is preferable to incorporate some element of prediction. We are developing a polynomial-time Earley style parsing algorithm. The parser returns a parse forest encoding all parses for an input string. The performance of this parser is sensitive to the grammar and input. Indeed it appears that for grammars that lexicalize CFG and for English grammar (where the structures are similar to the I_TAG developed at University of Pennsylvania (XTAG Research Group, 1995)) we obtain cubic-time complexity. null</Paragraph>
  </Section>
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