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<Paper uid="C86-1048">
  <Title>Tree Adjoining and Head Wrapping~</Title>
  <Section position="2" start_page="0" end_page="204" type="intro">
    <SectionTitle>
1. Introduction
</SectionTitle>
    <Paragraph position="0"> This paper discusses the relationstfip between Tree Adjoining Grammars (TAG's) and :Head Grammars (HG's). TAG's and HG's, introduced to capture certain structural properties of natural languages, were developed independently.</Paragraph>
    <Paragraph position="1"> TAG's deal with a set of elementary trees which are composed by means of an operation called adjoining. HG's are like Context-free Grammars, except for the fact that besides concatenation of strings, string wrapping operations are permitted. TAG's were first introduced in 1975 by Joshi, Levy and Wakahashi \[3\]. Joshi \[2\] investigated some formal and linguistic properties of TAG's with local constraints.</Paragraph>
    <Paragraph position="2"> The formulation of local constraints was then modified and formal properties were investigated by Vijay-Shanker and Joshi \[9\]. The linguistic properties were studied in detail by Kroeh and Joshi \[5\]. HG's were first introduced by Pollard \[6\] in 1983 and their formal properties were investigated by Roach \[7\]. It was observed that the two systems seemed to possess similar generative power and since they also appear to have the same closure properties \[7,9\] as well as similar parsing algorithn~ \[6,9\] a significant amount of indirect evidence existed to suggest that they were formally equivalent. In the present paper, we will attempt to provide a characterization of the formal relationship between HG's and TAG's. In \[10\] we consider various linguistic aspects of the relationship: in particular what might be referred to as the strong equivalence of the two formalisms.</Paragraph>
    <Paragraph position="3"> Vijay-Shanker and Joshi \[9\] provided a brief description of the intuition behind the inclusion of Tree Adjoining Langages (TAL) in the class of languages generated by a variant of HG's called Modified Head Grammars (MHG's). In the present paper, we give a proof of this result as well as a proof for the inclusion of Modified Head Languages (MHL) in TAL: hence we show that MHG's and TAG's are equivalent. This result is presented in section 3. In section 2, we discuss the relationship between MHG's and HG's.</Paragraph>
    <Paragraph position="4"> J&amp;quot; This work was partially supported by NSF grants MCS- null where Vlv and V T are finite sets of nonterminals and terminals respectively, S is a distinguished nonterminal, I is a finite set of initial trees and A is a finite set of auxiliary trees.</Paragraph>
    <Paragraph position="5"> Initial trees and auxiliary trees have the following form: /\</Paragraph>
    <Paragraph position="7"> Except for one node, which is called the foot node, all the nodes in the frontier of an auxiliary tree are labelled by terminal symbols. The foot node is labelled by the same nonterminal syrnbol as the root. All initial aud auxilia:3r trees, referred to as elementa:-y trees, have a height of at least one.</Paragraph>
    <Paragraph position="8"> We now define the adjoining operation. Let ~ be some node labelled X in a tree % Let fl be an auxiliary tree with root and foot labelled by X. The tree obtained by adjoilfing fl at r/ is given in tigurc 1.2.</Paragraph>
    <Paragraph position="10"> is inserted in its place and the excised subtree is inserted at the foot of/~.</Paragraph>
    <Paragraph position="11"> As defined above it is possible to adjoin any auxiliary tree at a node as long as the label of the node was the same as that of the root and the foot of the auxiliary tree. How-. ever, in general, adjoining will be constrained ss follows. Associated with each node is a selective adjoining (SA) constraint specifying that subset of the auxiliary tree which can be adjoined at this node. Trees can only be included in the SA constraint associated with a particular node if their root and foot are labelled with the same nonterminal that labels the node .There are two special cases: (a) the subset specified by SA is the entire set of auxiliary trees~ in this  case the entire set need not be explicitly listed; (b) the sub-set specified is empty i.e., no adjoining is possible. We call this the null adjoining (NA) constraint. A node may be associated with a a so-called obligatory adjoining (OA) constraint which can be used to ensure that an adjunction is obligatorily performed at a node.</Paragraph>
    <Paragraph position="12"> Example 1.1 We now present an example TAG G, which illustrates the notation used to specify the constraints associated with a node. There is one initial tree a and two auxiliary trees fll and f12: 0(: S o~ : A&amp;quot; figure 1.3 Having introduced SA and OA constraints, we must extend the definition of the adjunction operation. Suppose we adjoin an auxiliary tree fl at a node y of a tree ~/producing the tree ~'. For those nodes in ~' that do not correspond to nodes of fl, the constraints remain the same as those in % The remaining nodes in &amp;quot;7 ~ have the same constraints as those for the corresponding nodes of ft. For example, consider a sample derivations in the grammar G as given below in figure 1.4. We use an * to indicate the node at which adjunetion is performed.</Paragraph>
    <Paragraph position="14"> We will now present an alternative (yet equivalent) definition of the adjoining operation. So far, our definition allowed us to adjoin only with auxiliary trees, and allowed adjunetion only into sentential trees. This can be generalized to allow adjunctions of any tree derived from an auxiliary tree into any derived ti'ee. Consider the derivation given in figure 1.4. Given this generalization of adjunetion, we can also derive the same tree q2 by composing trees in the following sequence.</Paragraph>
    <Paragraph position="15"> The derived auxiliary tree &amp;quot;-/~ can be obtained by adjoining f12 in fl~. ~/~ can then be adjoined in a to give q~. Notice that trees derived from an auxiliary tree fl, will always have the property that their root and foot are labelled with the same nonterminal as those of ft. Viewing a derivation in this manner considerably simplifies several proofs of formal properties of 'rAG's \[9\].</Paragraph>
    <Paragraph position="17"> We use the notation P(3) to denote the set of trees derived from the elementary tree 3 using 0 or more adjunctions.</Paragraph>
    <Paragraph position="18"> The tree set T(G) of a TAG C is T(G) = U,~I P(a).</Paragraph>
    <Paragraph position="19"> The string language L(G) generated by a TAG G is given by L(G) = { w I &amp;quot; is the frontier of some &amp;quot;y in T(G) } Now we can see that tile language Lt, generated by the example grammar G, is L1 =: { a'~gbnfcnh \]n &gt; 0 } It is useful to further generalize the notion of a derived tree to include trees derived from subSrees of elementary trees. If is a node in some elementary tree, then P(~) represents the set of trees derived from the subtree rooted at y. Nodes are represented using an extension of the tree addressing scheme of Gorn \[11. Each node in an elementary tree is given a unique name in the following manner: the pMr (% e} denotes the root of ~,; if (% i} is a node in % then (3, i. j) represents the jth daughter of this node.</Paragraph>
    <Paragraph position="20"> 1.2. tlead Grammars Before giving the formal definition of I:lead Grammars, the notion of a beaded string will be described. A headed string is a string of symbols containing one distinguished symbol referred to as the head of tile string. Formally, this can be represented as a pair consisting of a string w and an integer that indicates the position of the \]lead in the string. In this paper, we use one of two notations to denote this string: when we wish to explicitly mention the head we use the representation wlSw2 where wlaw2 =- w; alternatively, we can simply denote the headed string by ~'. This allows us to denote the headed empty string as ~.</Paragraph>
    <Paragraph position="21">  Definition 2. A Itead Grammar, G, is given by a 4-tupk; (VN,VT, S,P). Productions in P are of the form: A -~ f(al,...,a~) or A -~ al where A C VN, a~ either belongs  to Vlv or is a headed string.</Paragraph>
    <Paragraph position="22"> f C \[.J~ 1{ LCi, LLi, LRi, RCi, RLi, RRi } We now define the operations LCi, LLi, and LRi for i C {1, 2}. Definitions of the other operations can be found in \[6\] and are not given here, since Roach \[7\] has shown that there is a normal form for Head Grammars which only uses these operations.</Paragraph>
    <Paragraph position="24"> Both Pollard \[6\] and Roach \[7\] define these operations as partial functions. Pollard's definition of headed strings includes the headed empty string (A). However, mathematically, A does not have the same status as other headed strings: for example, LCI(A,~) is undefined. In general, the term fi(~-T,...,~~,... ,~--~) is undefined when w-T = A.</Paragraph>
    <Paragraph position="25"> This nonuniformity has led to difficulties in proving certain formal results about Head Grammars \[7\], and has caused problems in showing the equivalence of MHG's and HG's (see section 2).</Paragraph>
    <Paragraph position="26"> The language generated by a HG G is defined as follows:  We find it convenient to consider a formalism that closely resembles HG's: referred to as Modified Head Grammars (MHG's). Instead of headed strings, MHG's have split strings. A split string has a distinguished position between two strings in V~, about which it may be split. We will denote a split string as Wl~W2 where wlw2 C V T. Notice that we can represent the split empty string as ATA , though this will be denoted A whenever the context makes it obvious that we are referring to a split string. In MHG's, there are 3 operations on split strings: W, C1, and C2, defined as follows: W(wl~w2, ul~u2) = wlu,Tu2w2</Paragraph>
    <Paragraph position="28"> The operations C1 and C2 correspond to the operations LC1 and LC2 in ttG's. The operation W has been defined such that the split point of its second argument becomes the split point of the string resulting from application of the operation (like the HG operations LL2 and Lit2).</Paragraph>
    <Paragraph position="29"> Since the split point is not a symbol but a position between strings, separate operations corresponding to LL2 and LR2 are not needed. In addition, unlike HG's, which distinguish the two wrapping operations LL1 and LL2, W suffices as a substitute for both of these operations. Suppose  Y ~ WlTW2 and Z ~ u~yu2 and we want X to derive W~lU~U2W 2. This can be achieved with the following two productions: Z f~ ~ el(A, Z) and X -+ W(Y, Z1~).</Paragraph>
    <Paragraph position="30"> Example 1.3 We now give a MHG generating L~.</Paragraph>
    <Paragraph position="32"> We will defer the discussion of both the formal and linguistic relationship between HG~s and MHG's until section 2. It is worth noting at this point that the definition of MHG's given here coincides with the definition of HG's given in  Rounds \[8\]. As we shall see in section 2, these formalisms are very closely related.</Paragraph>
    <Paragraph position="33"> 1.4. Tree Adjunetion and Wrapping Before showing the formal equivalence of MHG's and TAG's, it is instructive to consider the relationship between the wrapping operation W of MHG's and the adjoining operation of TAG's. Suppose that we have the production p X --~ W(Y, Z) in a MHG G, and that we have two derivations from the nonterminals Y and Z deriving the headed strings wltw2 and VlTV2 respectively. Given the production p, we can derive the split string wlvllv2w2 from X.</Paragraph>
    <Paragraph position="34"> Suppose there is a derived auxiliary tree &amp;quot;7 corresponddeg ing to the above derivation of wl~w2, from Y where the foot node appears at the split point, as shown in figure 1.6 below. Also assume that there is a node ~/dominating a sub-tree that corresponds to a derivation of vl~v2 from Z where, as before~ we assume that the foot node appears at the split point. Consider the tree resulting from the adjunction of *7 at the node ~?, also shown in figure 1.6. The resulting tree can be thought of as corresponding to the derivation of the split string wlvl)v2w2 from X.</Paragraph>
    <Paragraph position="35"> w, t ~va ~oot figure 1.6 This example illustrates the basic intuition behind the constructions involved in the following proofs showing the equivalence of MHG's and TAG's.</Paragraph>
    <Paragraph position="36"> 2. Head Grammars and Modified Head Grammars In this section, we shall discuss the relationship between MHG's and HG's. First we present the outline of a construction showing that for evelT HG G there is an equivalent MHG G'. We then briefly discuss the linguistic relationship between MHG's and HG%.</Paragraph>
    <Paragraph position="37"> Suppose X ~ wl-hw2. This headed string can be split in two ways: into the substrings wt and hw2; or wlh and w2. This depends on whether X is used in a left or right wrapping operation. Since in MHG's we can only split a string in one place, we use two nonterminals, X ~ and X r deriving wth;w2 and Wl~hW 2 respectively. Thus, for example, the production Z ---+ W(X~,Y) can be used in place of Z ~ LL2(X,Y). A further complication arises when a headed string is split first to the right of its head and then the resulting string is split to the left of the same head. The problem is resolved by introducing nonterminals X $h, that derive split strings of the form wl~w2 whenever X derives wl-hw2 in the HG. We can reintroduce the missing head with the following productions: X ~ -~ W(X Th, ht) and X r ---+ W(X Th, Th) Complete details of this proof are given in \[4\].</Paragraph>
    <Paragraph position="38"> We are unable to prove the inclusion of MHL's in HL's.</Paragraph>
    <Paragraph position="39"> The problems faced when attempting to find such a proof are a result of the operations in HG's not being total functions. For example, CI(A,W) is defined in MItG's, whereas LCI(~,~) is undefined in the HG's framework. We have not found any way of getting around this technical problem in a systematic manner. All TAG's considered by the authors so far have an equivalent HG. We feel that the problem of the empty headed string in the HG formalism does not result from an important difference between the formalisms.</Paragraph>
    <Paragraph position="40"> In the following discussion, we propose that MHG's can be given a linguistic interpretation if we retain the notion of a head terminal in a split string. The split point should be viewed a~ determining the position of the head. As far as the authors are aware, Ilead Grammars for natural languages use only one kind of wrapping operation: either only the left wrapping operations LLi, or only the right wrapping operations LRi. Thus, any headed string can be split on only one 'fide of the head. For example, if wl-hw2 is a headed string, and only the left wrapping operations were used, then the headed string can only be split as wlh and w~.. For any HG using only left wrapping operations there exists an equivalent MHG such that split strings will have their split points in~nediately to the right of the actual head. However, obviously not every MHG (:an be given a linguistic interpretation in this way.</Paragraph>
  </Section>
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