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<Paper uid="C86-1048">
  <Title>Tree Adjoining and Head Wrapping~</Title>
  <Section position="3" start_page="204" end_page="206" type="metho">
    <SectionTitle>
3. Equivalence of MttG's and TAG,'s
</SectionTitle>
    <Paragraph position="0"> We will now straw that the class MIlL is equal to the class TAL. The complete proofs for the results presented here are given in \[4\].</Paragraph>
    <Paragraph position="1"> 3.1.. Inclusion of TAL in MIlL Based on the earlier observation concerning the similarity between the wrapping and adjoining operations, we shall now present a scheme for transforming a given TAG G -(VtC/, VT, S, I, A) to an equivalent MHG G' - (V\]~, VT, S, P). In this section, we have generalized the concatentation operations of MHG's to be of the form Cj for j _&gt; 1. It is obvious that these operations can always be simulated using just C1 and C2.</Paragraph>
    <Paragraph position="2"> We shall first describe the algorithm convert informally. If r\] is a node of some elementary tree '7, applying convert to ~1 returns a sequence of productions in the MHG formalism capturing the structure of the subtree of q rooted at ~. The wrapping operation is used to simulate the effect of adjunction; the concatenation operations Ci eoncatentate the strings derivable from the daughters of a node. The choice of i depends on which child is the ancestor of the foot node. The exact structure'of a tree can be captured by using nonterminals that are named by the addresses of nodes of elementary trees rather than the nonterminMs labelling tim nodes.</Paragraph>
    <Paragraph position="3"> The main idea of our scheme is as follows. Let (fl, i) be the address of a node in an auxiliary tree fl, and '7 belongs to P((fl, i)) with a frontier WlXtO 2, We have a nonterminal corresponding to this node (denoted by \[fl, i\]) which derives the split string watw~.. In particular, when (fl, i) is the root of fl (i.e., i =: e), then the nonterminal \[fl, c\] should derive the split strings WlTW2 whenever there is a tree in P(fl) with frontier WlXW 2. That is, the split point appears in a position corresponding to the foot node.</Paragraph>
    <Paragraph position="4"> Thus, the wrapping operation W can be used to simulate the effect of adjoining in the following manner. If ('7, i) is a node at which fl is adjoinable, we have a production corresponding to adjunction of fl at ('7, i).</Paragraph>
    <Paragraph position="5"> b,i\] -, w(\[fl,4, b, il) where \['7, i1 derives strings derivable from the children of ('7, i). We also have the rule \['7, i\] -~ \['7, i\] for the case when no adjunction takes place at ('7, i). Since \['7, i\] is supposed to derive strings derivable by the concatenation of the frontiers of subtrees dominated by the children of ('7, i), we have the production, \[% i\]&amp;quot; -* Cj(\[%i. 1\] .... , \[%i .j\] ..... \['7, i. k\]) where @,i. 1&gt;,..., ('7, i. j&gt;,..., ('7, i. k&gt; correspond to the h children of ('7,i) and where the jth child is the ancestor of the foot node. The operation Cj is used so that the split point appears in the same position as the foot node. By convention, we let j be I when (%i) is not the ancestor of the foot node.</Paragraph>
    <Paragraph position="6"> We are now in a position to define the conversion process. The algorithm is as fi)llows: for each initial tree a, let S -&gt; \[a,e\] G P.</Paragraph>
    <Paragraph position="7"> for each elementary tree % call convert((% e)) where the procedure convert is as defined below.</Paragraph>
    <Paragraph position="8"> define convert (('7, i)); case 1: (%i) is a leaf node if ('7, i) has label a Ci VT U {A} then  for each/3 in SA constraint of ('7, i) if ('7, i} does not have OA constraint then step 5: add \[3,i\] -, \[%i\] if ('7, i) is ancestor of foot node then step 6: add \['7, i\] -, Cj(\['7, i. 1\] ..... \[V,i. k\]) where j~h child dominates foot node; if ('7, i) is not ancestor of foot node then step 7: add \['7,i\[-, C1(\['7, i. 1\],... ,\['7, i' kl) for 1 &lt;j &lt; k do eonvert((%i.j)).</Paragraph>
    <Paragraph position="9"> We prove the inclusion of L(G) in L(G'), by induction oll the height of the trees derived from all subtrees of elementary trees, where the inductive hypothesis states: For all elementary trees &amp;quot;7, and addresses i in q, if there is a tree &amp;quot;7' in P(('7, i}) of height less than k, and the frontier of &amp;quot;71= WlXW2 or wlw2, then \['7, i\]--~ wl~w2.</Paragraph>
    <Paragraph position="10"> It will be easy to simw the inclusion of L(G) in L(G') by induction, considering steps 4, 5, 6 and 7. The base cases correspond to steps 1, 2 and 3.</Paragraph>
    <Paragraph position="11">  We show the inclusion in the other direction by induction on the length of derivation of split strings in G'. The induction hypothesis is given by: ir \[-~,i\] _~ ~,~Tw.,. in k steps, then there is a &amp;quot;~' c p((~,i)) such that the frontier ofq' is wlXw2 or wlw2, depending on whether the foot node of % labelled by X if it exists, is a descendant of &lt;'~,i&gt; or not.</Paragraph>
    <Paragraph position="12"> 3.2. Inclusion of MHL in TAL When we convert a TAG into a MHG, each elementary tree generates a set of productions. The sets generated by any two distinct elementary trees are disjoint and, furthermore, have a constrained form encoding the hierarchical structure of the tree. The task of converting a MHG to a TAG cannot simply involve the inversion of this construction since it is not in general possible to find groupings of productions in a MHG that have the required structure.</Paragraph>
    <Paragraph position="13"> The approach used to convert MHG's to TAG's is based on satisfying the following requirement: for each derivation in the MHG there must be a derived tree in the TAG for the same string, in which the foot is positioned at the split point: i.e., X ~ wHw2 in MHG if and only if there is a derived auxiliary tree &amp;quot;~ having no OA constraints, with root labelled X and frontier wlXw2, Suppose we had derived trees corresponding to derivations for B and C (as shown in the center of figure 3.1) that satisfied the above requirement.</Paragraph>
    <Paragraph position="14"> We can capture the effect of each MHG production directly by associating exactly one elementary tree with each production. For example, figure 3.1 illustrates trees associated with the productions A --~ CI(B, C) (on the left) and A -~ W(B,C) (on the right). We position the foot node in the elementary trees to ensure that the split point and foot node appear at the same position. When the tree corresponding to wrapping is used the string derived from B is wrapped around the string derived from C. The foot of the resulting tree will appear immediately under the foot of the derived tree for C.</Paragraph>
    <Paragraph position="16"> The TAG that we produce could be viewed as simulating rewriting of nonterminMs. Each rewriting corresponds to one use of the adjoining operation. The NA constraints at the root and the foot node of each auxiliary tree ensure that each occurence of a nontermin~l is rewritten only once. The OA constraints are used to ensure that every nonterminal introduced is rewritten.</Paragraph>
    <Paragraph position="17"> We now present the complete construction. Without loss of generality, we will assume a normal form that uses productions of the following form:  A-~ f(B,C) or A-~ ?a or A-~a$ where A,B,C e VN, a C VT tO {A} and f e {CI,C2, W}. The conversion proceeds as follows:  The set I of&amp;quot; initial trees consists of the single tree a: ~X: t5 0,I E A We prove that L(G) C_ L(G') by induction on the length of the MHG derivation. We show that if X =% wlTw= then there is a derived auxiliary tree having no OA constraints, with root labelled X and frontier wlXw2.</Paragraph>
    <Paragraph position="18"> We prove that L(G') C L(G) by induction on the height of the &amp;quot;rAG derivation tree. We show that if q E P (fl), has no OA constraints, and has frontier wlXw2 then X =% wlrw2. While straightforward, the proof given above does not capture the linguistic motivation underlying TAG's. The auxiliary trees directly reflect the use of the concatenation and the wrapping operations. It is also interesting to note that a consequence of the equivalence of MHG's and TAG's and the construction used in proving the inclusion of MHL in TAL is that we have the following normal form for TAG's. For any TAG there is an equivalent TAG with exactly one initial tree and auxiliary trees which are of five possible forms shown above.</Paragraph>
    <Paragraph position="19"> C) E P or A ---+ C2(B,C) 6 P then</Paragraph>
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