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<?xml version="1.0" standalone="yes"?> <Paper uid="H86-1020"> <Title>SOME COMPUTATIONAL PROPERTIES OF TREE ADJOINING GRAMMARS*</Title> <Section position="4" start_page="0" end_page="215" type="metho"> <SectionTitle> 2. TREE ADJOINING GRAMMARS--TAG's </SectionTitle> <Paragraph position="0"> We now introduce tree adjoining grammars (TAG's). TAG's are more powerful than CFG's, both weakly and strongly. ! TAG's were first introduced in \[Joshi, Levy, and Takahashi,1O7$\] and \[Joshi,1983\]. We include their description in this section to make the paper self-contained.</Paragraph> <Paragraph position="1"> We can define a tree ~ grammar as follows. A tree adjoining grammar G is'-~pair (I,A) where \] ~ a set of initial trees, and A is a set of auxiliary trees.</Paragraph> <Paragraph position="2"> A tree o is an initial tree if it is of the form</Paragraph> <Paragraph position="4"> That L% the root node of a is labelled S and the frontier nodes are all terminal symbols. The internal nodes are aU non-terminaL.</Paragraph> <Paragraph position="5"> A tree ~ is an a~xiliary tree if it is of the form</Paragraph> <Paragraph position="7"> That is, the root node of ~0 m labelled with a non-termb~al X and the frontier nodes are all labelled with terminals symbols except one which is labelled X. The node labelled by X on the frontier will be called the foot node of ~. The frontiers of initial trees belong to L ~, whereas the frontiers of the auxiliary trees belong to ~ N L~ + O ~+ N L'.</Paragraph> <Paragraph position="8"> We will now define a composition operation called adjoininL (or adjunetion) which composes an auxiliary tree fl with a tree &quot;I. Let '7 be a tree with a node n labelled X and let B be an auxiliary tree with the root labelled with the same symbol X. (Note that must have, by definition, a node (and only one) labelled X on the frontier.) IGr~ramm GI tad G2 are weakly equivalent if the string language of GI. L(GI) ~ the strbqE italr,~e4e of G2, L(G2). GI tad G2 m strongly equlvsleut they are weakly eqolwJent tad for each w In L(GI) ~ t~G2), both GI sad G2 mlga the 8Lme structural description to w. A Ip'ffimmu G iz weakbr adequate for a (string) hmgeqe L, if L(G) -- L. G iJ strongly adequate for L if L(G) -- L tad for each w in L, G as~slgne ta eappropriatea structural description to w. The notion of strung adequacy is undoubtedly not preclsC/ beesasC/ it depends on the notion of appropriate structural descriptions Adjoining can now be defined as follows, if p is adjoined to &quot;I st the node n then the resulting tree &quot;ft' is as shown in Fig. 2.1 below.</Paragraph> <Paragraph position="10"> The tree t dominated by X in &quot;~ is excised, ~ is inserted at&quot; the node n in &quot;1 and the tree t is attached to the foot node (labelled X) of ,C/. i.e., ~ is inserted or adjoined to the node n in 7 pushing t do*swards. Note that adjoining is not a substitution operation.</Paragraph> <Paragraph position="11"> We will now define T(G): The set of all truce derived in G starting from initial trees in I. This set will be called the tree net of G.</Paragraph> <Paragraph position="12"> L(G): The set of all terminal strings which appear in the frontier of the trees in T(G). This set will be called the string I.xngeage (or language) of G. If L is the string language of a TAG G thee we say that L is a Tree-Adjoining Language ITAL\]. The relationship between TAG's , context-free grammars, and the corresponding string languages can be summarized as follows (\[Joehi, Levy, and Takahashi, 1975\], \[aoehi, *~SSl).</Paragraph> <Paragraph position="13"> Theorem 2.1: For every context-free grammar, G', there is an equivalent TAG, G, both weakly and strongly.</Paragraph> <Paragraph position="14"> Theorem 2.2: For ever,/ TAG, G, we have the following situations: *. I,(G) is context-free and there is a context4ree grammar G' that is strongly (end therefore weakly) equivalent to G.</Paragraph> <Paragraph position="15"> b. L(G) is context-free and there is no context4ree grammar G' that is equivalent to G. Of course, there must be a context-free grammar that is weakly equivalent to G.</Paragraph> <Paragraph position="16"> C/. L(G) is strictly context-sensitive. Obviously in this case, there is no context-free grammar that is weakly equivalent to G.</Paragraph> <Paragraph position="17"> Parts (a) and (e) of Theorem 2.2 appear in (\[Joehi, Levy, and Tskahashi, 1075\]). Part (b) is implicit in that paper, but it is imro~taut to state it explicitly u we have done here because of its linguistic significance. Example 2.1 illustrates part (a). We will now illustrate parts (b) and (c).</Paragraph> <Paragraph position="18"> Example 2.2: Let G -- (I,A) where</Paragraph> <Paragraph position="20"> adjoined at S as indicated in 70- adjoined at T as indicated in 7=.</Paragraph> <Paragraph position="21"> Clearly, L(G), tlie string language of G is L= (**eh=/n > o) which is a context-free language. Thus, there must exist a context,.. free grammar, G', which is at least weakly equivalent to G. It can be shown however that there is no context4ree grammar G' which is strongly equivalent to G, i.e., T(G) -- T(G'). This follows from the fact that the set T(G) (the tree set of G) is non-recognizable, Le., there is no finite state bottom-up tree automaton that can recognize precisely T(G). Thus a TAG may generate a context-free language, yet assign structural descriptions to the strings that cannot be assigned by any context-free grammar.</Paragraph> <Paragraph position="22"> Example 2.3: Let G , (LA) where</Paragraph> <Paragraph position="24"> The precise definition of L(G) is as follows: L(G) - LI ffi {w * e u / u > o, w is a string of a's and b', such that (1) the n*mber of *% ~ the number of b's -- *i and (2) for any initial substring of w, the n*mber of a's .~ the number of b's. } L! is a strictly context-sensitive language (i.e., * context-sensitive language that is not context-free). This can be shown as follows. Intersecting L with the regular language a* b* e c* results in the language L, = { na bS e ca / n ~.. o } ~ffi LI I'l a&quot; b&quot; e c * L 2 is well-known strictly co*text-sensitive language. The result of intersecting a context-free language with a regular language is always a context-free language; hence, L i is *ot * context-free language. It is thus a strictly context-sensitive language. Example 2.3 thus illustrates part (c) of Theorem 2.2.</Paragraph> <Paragraph position="25"> TAG's have more power than CFG's. However, the extra power is quite limited. The language L! has equal number of a's, b's and c's; however, the a's and b's are mixed in * certain way. The language L~ is similar to Li, except that a's come before all b's. TAG's as defined so far are not powerful enough to generate L=. This can be seen as follows. Clearly, for any TAG for L2, each initial tree must contain equal *amber of *'% b's and e's (including zero), and each auxiliary tree must also contain equal number of a'n, b's and c's. Further in each case the a's meet precede the b's. Then it is easy to see from the grammar of Example 2.3, that it will not be po~ible to avoid getting the a's and b's mixed. However, L~ can be generated by a TAG with local constraints (see Sectio* 2.1) The sotailed copy language.</Paragraph> <Paragraph position="26"> Lffi {wew/w~ {a,b}&quot; ) also cannot be generated by * TAG, however, again, with local constraints. It is thus clear that TAG's can generate more than context-free languages. It can be shown that TAG's cannot generate all context-sensitive languages \[Joehi ,1984\].</Paragraph> <Paragraph position="27"> Although TAG's are more powerful than CFG's, this extra power is highly constrained and appace*tly it is just the right kind for characterizing certain structural description. TAG's share almost all the formal properties of CFG's (more precisely, the correspo*ding classes of la*guages), as we shall see in sectio* 4 of this paper and \[Vijay-Shankar and Joshi01985\]. I* addition,the string languages of TAG's can also be parsed in polynomial time, in particular in O(ne). The parsing algorithm is described in detail in section 3.</Paragraph> <Paragraph position="28"> 2.1. TAG's with Local Constraints on Adjoining The adjoining operation as defined in Sectio* 2.1 is &quot;contextfree'. An a*xiliary tree, say,</Paragraph> <Paragraph position="30"> is adjoinable to * tree t at * *ode, say, *, if the label of that node is X. Adjoining does *or depend on the context (tree context) around the node n. I* this sense, adjoining is co*text-free.</Paragraph> <Paragraph position="31"> In \[Joshi ,1983\]. local constraints o* adjoining similar to those investigated by \[Joshi and Levy ,1977\] were considered.These are a generalization of the context-sensitive constraints studied by \[Peters and Ritchie .1069\]. it was soon recognized, however, that the full power of these co*straints was never fully utilized, both in the linguistic context as well as in the &quot;formal languages&quot; of TAG's. The so-called proper analysis contexts and domination contexts (as defined i* \[Joshi and Levy ,1977l) as used in \[Joshi .10831 always turned out to be such that the context elements were always in a specific elementary tree i.e.. they were further localized by being in the same elementary tree. Based on this observation and a suggestio* in \[Joshi, Levy and Takahashi ,1975\], we will describe a new way of introducing local C/o*strainta. This approach *ot only captures the insight stated above, but it is truly in the spirit of TAG's. The earlier approach was not so, although it was certainly adequate for the investigatio* in \[Joshi ,1983\]. A precise characterization of that approach still remains an ope* problem.</Paragraph> <Paragraph position="32"> G ~ (I,A) be a TAG with local constraints if for each elementary tree t 6 I U A, and for each node, n, in t, we specify the set fl of auxiliary trees that can be adjoined *t the node n. Note that if there is no constraint then *H auxiliary trees are adjoinablC/ at n (of course, only those whose root has the same label as the label of the node n). Thee, in general, ~ is a subnct of the set of all the auxiliary trees adjoinable at n.</Paragraph> <Paragraph position="33"> We will adopt the following conventions.</Paragraph> <Paragraph position="34"> 1. Since, by definition, no auxiliary trees are adjoinable to a node labelled by a terminal symbol, *o co*straint ha8 to be stated for node labelled by a terminal.</Paragraph> <Paragraph position="35"> 2. If there is no constraint, i.e., all auxiliary trees (with the appropriate root label) are adjoinable *t a node, say, n, then we will not state this explicitly.</Paragraph> <Paragraph position="36"> 3. If no auxiliary trees are adjoinabie at a *ode n, then we will write the constraint as (~b), where C/b de*ores the null set.</Paragraph> <Paragraph position="37"> 4. We will als,~ allow for the po~ihility that for a node at least one adjoining is obligatory, of course, from the set of all possible auxiliary trees adjoinable at that node.</Paragraph> <Paragraph position="38"> Hence, a'TAG with local constraints is defined as follows. G = (1, A) is a TAG with local constraints if for each node, n. in each tree t, be specify one (and only one) of the f'ollowing constraints.</Paragraph> <Paragraph position="39"> 1. Selective Adjoining ~SA:) Only a specified subset of the set of all auxiliary trees are adjoinable at n. SA is written as (C), where C is a subset of the set of all auxiliary trees adjoinable at n.</Paragraph> <Paragraph position="40"> If C equals the set of all auxiliary trees adjoinable at n, then we do not explicitly state this at the node u.</Paragraph> <Paragraph position="41"> 2. Null Adjoinin~ INA:) No attxiliary tree is adjoinable at the node N. NA will be writte* a8 (~).</Paragraph> <Paragraph position="42"> 3. Obligating Adjoining IOA: ) At least one (oat of all the auxiliary trees adjoinable at n) must be adjoined at *.</Paragraph> <Paragraph position="43"> OA is writte* as (OA), or as O(C) where C i~ * subset of the set of all auxiliary trees adjoinable at n.</Paragraph> <Paragraph position="45"> In 01 no auxiliary trees can be adjoined to the root node. Only ~1 is ~ljoinable to the left S node at depth 1 and only /9 s is adjoinable to the right S node at depth 1. In ~1 only Pi is ad\]oinable at the root node and no auxiliary trees are adjoinable at the \[(~,~.</Paragraph> <Paragraph position="46"> node. Similarly for PS&quot; We must now modify our definition of adjoining to take care of the local constraints, given a tree &quot;1 with a node, say, n, labelled A and given an auxiliary tree, say, ~, with the root node labelled A, we define adjoining as follows. # is adjoinable to &quot;1 at the node n if ~ E #, where B is the constraint associated with the node n in &quot;1. The result of adjoining p to 7 will be as defined in earlier, except that the constraint C associated with n will be replaced by C', the constraint associated with the root node ore and by C deg, the constraint associated with the foot node of ~. Thus. given</Paragraph> <Paragraph position="48"> We also adopt the convention that any derived tree with a node which has an OA constraint associated with it will not be included in the tree set associated with a TAG, G. The string language L of G is then defined as the set of all terminal strings of all trees derived in G (starting with initial trees) which have no OA constraints left-in them.</Paragraph> <Paragraph position="50"> There are no constraints in a t. In ~ no auxiliary trees are adjoJnable st the root node sad the foot node and for the center S node there are no constraints.</Paragraph> <Paragraph position="51"> Starting with a I and adjoining ,8 to a ! at the root node we It is easy to see that G generates the string language L= {a&quot;b~ec&quot;/n >o} Other languages such ~ L'f{a u' In >I}, L&quot; ---- {a u: I n _> I} also cannot be generated by TAG's. This is because the strings of a TAt grow linearly (for a detailed definite of the property called * contact growth&quot; property, see \[Joshi ,198.3J.</Paragraph> <Paragraph position="52"> For those familiar with IJoshi, 1983\], it is worth pointing out that the SA constraint is only abbreviating, i.e., it does not affect the power of TAG's. The NA and OA constraints however do affect the power of TAG's. This way of looking at local constraints has only greatly simplified their statement, but it has also allowed us to capture the insight that the 'locality' of the constraint is statable in terms of the elementary trees themselves! l.I. Simple Linguistic Examples We now give a couple of lingnistle examples. Readers may refer to \[Kroch and Joshi, 1985J for details.</Paragraph> <Paragraph position="54"> The girl vho net Blll 18 * senior 2. Starting with the initial tree 3`1 ~ a2 and adjoining ~2 at the'indicated node in a 2 we obtain 3`2-</Paragraph> <Paragraph position="56"/> <Paragraph position="58"> if try John persuaded Bill to invite Mary Note that the initial tree a 2 is not * matrix sentence. In order for it to become * matrix sentence, it must undergo an adjunction at its root node, for example, by the auxiliary tree ~it as show* above. Thus, for o 2 we will specify a local constraint O(~t) for the root * node, indicating that o= requires for it to undergo tn adjunct\on at the root node by an auxiliary tree ~2- In * fuller grammar there will be, of course, some alternatives in the scope of O().</Paragraph> </Section> <Section position="5" start_page="215" end_page="217" type="metho"> <SectionTitle> 3. PARSING TREE-ADJOINING LANGUAGES 3,1, Definitions </SectionTitle> <Paragraph position="0"> We will give * few additional definitions. These are not necessary for defining derivations in * TAG as defined in section 2.</Paragraph> <Paragraph position="1"> However, they are introduced to help explain the parsing algorithm and the proofs for some of the closure properties of TAL's.</Paragraph> <Paragraph position="2"> DEFINITION 3.1 Let %3`' be two trees.We say 3` I--- 3`' if i* 3` we adjoin an auxiliary tree to obtain 3`'.</Paragraph> <Paragraph position="3"> I--* is the reflexive,transitive closure of \[--.</Paragraph> <Paragraph position="4"> DEFINITION 3.2 7' is called * derived tree if 3` \],--&quot; 3`' for some elementary tree % We then say &quot;7' 6 D(3`).</Paragraph> <Paragraph position="5"> The frontier of any derived tree &quot;I belongs to either L ~ N E + U LE t- N E deg if 3'6 D($) for some auxiliary tree ~0, or to E* if 3` 6 D(o) for some initial tree C/x. Note if 3` 6 D(c~) for some initial tree C/x, then 3` is also * sentential tree.</Paragraph> <Paragraph position="6"> If ~ is an auxiliary tree, 3` 6 D(~) and the frontier of 3` is w s X w 2 (X is * nonterminal,Wl,W 2 6 L ~') the* the leaf node having this non-terminal symbol X at the frontier is called the foot of 3`.</Paragraph> <Paragraph position="7"> Sometimes we will be loosely using the phrase &quot;adjoining with a derived tree&quot; ,7 6 D(~) for some auxiliary tree ~8. What we mean is that suppose we adjoin ,8 at some node and then adjoin within ~8 and so on, we can derive the desired derived tree 6 D(~) which uses the same adjoining sequence and use~this resulting tree to &quot;adjoin&quot; at the original node.</Paragraph> <Paragraph position="8"> 3.~. The Parsing Algorithm The algorithm, we present here to parse Tree-Adjoining Languages (TALe), is a modification of the CYK algorithm (which is described in detail in \[Aho and Ullman,1073\]), which *sea a dynamic programming technique to parse CFL's. For the sake of making our description of the parsing algorithm simpler, we shall present the algorithm for parsing without considering local constraints. We will later show how to handle local constraints.</Paragraph> <Paragraph position="9"> We shall a~ume that any node in the elementary trees in the grammar hal *tmost two children. This assumption ca* be made without **y loss of generality, bee*use it can be easily shown that for any TAG G there is *n equivalent TAG G ! such that **y node in any elementary tree in G l has utmost two children. A similar assumption is made in CYK algorithm. We use the terms ancestor and descendant, throughout the paper as * transitive and reflexive relation, for example, the foot *ode may be called the ancestor of the foot node.</Paragraph> <Paragraph position="10"> The algorithm works as follows. Let al...a, n be the input to be &quot;parsed. We use * four.dimensional array A; each element of the array contains * subset of the nodes of derived trees. We nay * node X of * derived tree 3` belongs to A\[i~,k01J if X dominates * nab-tree of 3` whose frontier is given by either ai+i...a j Y nk+l...a u (where the foot node of &quot;7 is labelled by V) or ai+v..a u (i.e., j ~ k. This corresponds to the case When -f is * sentential tree). The indices (iJ,k,I) refer to the positions between the input symbols and range over 0 through n. If i -- 5 say, then it refers to the gap between at and a s.</Paragraph> <Paragraph position="11"> Initially, we fill A\[i,i+l,i+l,i+l\] with those nodes in the frontier of the elementary trees whose label is the same as the input ti+ 1 for 0 < i < n*l. The foot nodes of auxiliary trees will belong to all Aii,i,jj I. such that i -- j.</Paragraph> <Paragraph position="12"> We are now in n position to fill in *11 the elements of the array A. There are five cases to be considered.</Paragraph> <Paragraph position="13"> Case 1. We know that if a node X in a derived tree is the ancestor of the foot node, and node Y is its right sibling, such that X E Ali,j,k,l\] and Y E All,m,m,n\], then their parent, sayt Z should belong to Alij.k,n l, see Fig 3.1a.</Paragraph> <Paragraph position="14"> Case 2. U the right sibling Y is the ancestor of the foot node such that it belongs to A\[I,m,n,p\] and its left sibling X belongs to A\[i,jj,I\], then we know that the parent Z of X and Y belongs to A\[i,m,n,p\], see Fig 3.1b Case 3. If neither X nor its right sibling Y are the ancestors of the foot node ( or there is no foot node) then if X E Ali,j,j,l\] and Y 6 All,re,*,el then their parent Z belongs to A\[i,j,j,n\].</Paragraph> <Paragraph position="15"> Case 4. If a node Z has only one child X, and if X E Alij,k,I\], then obviously Z E A\[i,j,k,l\]. ~&quot; Case 5. U a node X E Ali,j,k,I\], and the root Y of * derived tree &quot;/having the same label as that of X, belong,s to Alm,i,l,n\], then adjoining ? at X makes the resulting node to be in Almj,k,n\], see Fig</Paragraph> <Paragraph position="17"> Although we have stated that the elements of the array contain * subset of the nodes of derived trees, what really goes in there ate the addressee of nodes in the elementary trees. Thus the the size of any set is bounded by * constant, determined by the grammar. It is hoped that the presentation of the algorithm below will make it clear why we do m.</Paragraph> <Paragraph position="18"> a.a. The allgorlthm The compkteMgorithmk given below Step 1 For i=0 to n-I step 1 do Step 2 put all nodes in the frontier of elementary true v hose lnbel i8 *t*t in a\[i.i*l,i*l.l*l\].</Paragraph> <Paragraph position="19"> Step 3 For i=O to n-I step I do Step 4 for J=l to n-I step 1 do Step 6 put foot nodes of all nuxilinry trees In ancestor of the foot node. The parent is put in A\[Q,k,I\] if the left sibling is in A\[i,j,k,m\] and the right sibling is in A\[m,p,p,l\], where k _ m < I, m ~ p, p < I. Therefore Case I is written as For n=k to 1-I step I do for p= n to 1,step I do if there is a left sibling in A\[i,J.k.n\] and the right sibling in A\[a.p.p.1\] satisfying appropriate restrictions then put their parent in A\[i,J.k.1\].</Paragraph> <Paragraph position="20"> (b) Case 2 corresponds to the ease where the right sibling is the ancestor of the foot node. If the left sibling is in A\[i,m,m,p\] and the right sibling is in A\[p,j,k,I\], i < m < p and p < j, then we put their parent in A\[i,j,k,I\]. This may be written as For n=i to J-! step I do For p=a*l to \] step i do for all left siblin~ in A\[i.n,m,p\] ud right siblings in A\[p.J.k.1\] satisfying appropriate rentrictionn put thei= parents in A\[i.J.k.1\].</Paragraph> <Paragraph position="21"> (C/) Case 3 corresponds to the case where *either ehildre* *re ancestors of the foot *ode. If the left sibling E A\[i,j,j,m\] and the right</Paragraph> <Paragraph position="23"> |), This may be written as for * = J to 1-1 step I do for p = J to 1 step i do for all left 81blLngs In ACL,|,J.a\] and right slblings in A\[nop.p.l\] 8atisfylng the appropriate restrictions pet their parent in A\[ioJ,Jdegl\]. (e) Case 5 corresponds to adjoining. If X is * node in A\[m~,k,p\] and Y is the root of a auxiliary tree with same symbol as that of X, such thatYisiuA\[i,m,p,I\]((i < m < p < Iori < m < p < l) and(m < j < k_ porm_~ j < k < p)). Thls may be written as for * = ~. to |step t do for p = * to I step I do if t node X 6 A\[n,J.k,p\] tad the root of auxiliary tree Is in A\[i,a.pol\] then put X in A\[l.J.k.1\] Case 4 corresponds to the case where a node Y has only one child X If X E A\[i,j,k,I\] then put Y in A\[i,j,k,I I. Repeat Case 4 again if Y has no siblings.</Paragraph> <Section position="1" start_page="217" end_page="217" type="sub_section"> <SectionTitle> 3.4. Complexity of the Algorithm </SectionTitle> <Paragraph position="0"> It is obvious that steps 10 through 15 (cases ~-e) are completed in O(e:~), because the different cases have at most two nested for loop statements, the iterating variables taking values in the range 0 through n. They are repeated atmost O(n 4) times, because of the four loop statements in steps 6 through 9. The initialization phase (steps 1 through 5) has a time complexity of O(n + n 2) = O(n2).</Paragraph> <Paragraph position="1"> Step 15 is completed in O(n). Therefore, the time complexity of the parsing algorithm is O(nS).</Paragraph> </Section> <Section position="2" start_page="217" end_page="217" type="sub_section"> <SectionTitle> 3.5. Correctness of the Algorithm </SectionTitle> <Paragraph position="0"> The main issue in proving the algorithm correct, is to show that while computing the contents of an element of the array A, we must have already determined the contents of other elements of the array needed to correctly complete this entry. We can show this inductively by considering each case individually. We give an informal argument below.</Paragraph> <Paragraph position="1"> Case l: We need to know the contents of A{i,j,k,m\], A\[m,p,p,l l where m < I, i < m, when we ate trying to compute the contents of A\[i,j,k,l\]. Since I is the variable itererated in the outermost loop (step 6), we nan assume {by induction hypothesis) that for all m < I and for all p,q,r, the co*teats of AIp,q,r,m \] are already computed. Hence, the contents of A\[i,j,k,m\] are known. Similarly, for all m > i. and for all p,q, and r _ I, A\[m,p,q,r i would have been computed. Thus, A\[m,p,p,! ! would also have bee* computed.</Paragraph> <Paragraph position="2"> A\[ij,i,I\], we *end to know the *odes in A\[i,i,i,p\] and A\[p,q,q,I\]. Note i > i or j < I. Hence, we know that the co*teats of A\[i,j,j,p\] and A\[p,q,q,I\] would have been compared already.</Paragraph> <Paragraph position="3"> Case 5: The co*tents of A\[i,m,p,I\] and A\[m,j,k,p\] mesa be know* in order to compute A\[i,j,k,ll, where ( i < m < p < I or i m_p_<l)and(m <_j <k<porm<j_<k_<p). Since either m > i or p < I, contents of A\[m,j0k,p\] will be known.</Paragraph> <Paragraph position="4"> Similarly, since either m < j or k < p, the contents of A\[i,m,p,l\] would have been computed.</Paragraph> </Section> <Section position="3" start_page="217" end_page="217" type="sub_section"> <SectionTitle> 3.6. Pining with Local Coustrslnt6 </SectionTitle> <Paragraph position="0"> So far,we have ~*med that the give* grammar has *o local constraints, if the grammar has local constraints, it is easy to modify the above algorithm to take care of them. Note that in Case 5, if an adjuectio* occurs at a node X, we add X again to the element of the array we are computing. This seems to be in contrast with our definition of how to associate local constraints with the nodes in a sentential tree. We should have added the root of the auxiliary tree instead to the element of the array being competed, since so far as the local constraints are concerned,this *ode decides the local constraints at this node in the derived tree. However, this scheme cannot be adopted in our algorithm for obvious reasons. We let pairs of the form {X,C) belong to elements of the array, where X is as before and C represents the local constraints to be associated with this node.</Paragraph> <Paragraph position="1"> We then alter the algorithm as follows. If (X,Ct) refers to * node at which we attempt to adjoin with *n auxiliary tree {whose root is denoted by (Y,Ca)). then adjunctioa would determined by C t.</Paragraph> <Paragraph position="2"> If adjunction is allowed, then we can add (X,C2) in the corresponding element of the array. In cases 1 through 4, we do not attempt to add a new element if any one of the children has a* obligatory constraint.</Paragraph> <Paragraph position="3"> Once it has been determined that the given string belongs to the language, we can find the parse in a way similar to the scheme adopted in CYK algorithm.To make this process simpler and more efficient, we can use pointers from the new element added to the elements which caused it to be put there. For example, consider Case 1 of the algorithm (step 10 ). if we add a node Z to A\[i,j,k,I\], because of the presence of its children X and Y in A\[i,j,k,m\] and A\[m,p,p,I\] respectively, then we add pointers from this node Z in A\[i,j,k,I\] to the nodes X, Y in Ali,j,k,m\] and A\[m,p,p,I\]. Once this has been done, the parse can be found by traversing the tree formed by these pointers.</Paragraph> <Paragraph position="4"> A parser based on the techniques described above is currently being implemented and will be reported at time of presentation.</Paragraph> </Section> </Section> <Section position="6" start_page="217" end_page="220" type="metho"> <SectionTitle> 4. CLOSURE PROPERTIES OF TAG's </SectionTitle> <Paragraph position="0"> In this section, we present some closure results for TALe. We now informally sketch the proofs for the closure properties.</Paragraph> <Paragraph position="1"> Interested readers may refer to \[Vijay-Shankar and Joshi,19851 fort the complete proofs.</Paragraph> <Section position="1" start_page="217" end_page="220" type="sub_section"> <SectionTitle> 4.1. Closure under Union </SectionTitle> <Paragraph position="0"> Let G 1 and G 2 be two TAGs generating L! and ~ respectively.</Paragraph> <Paragraph position="1"> We can construct a TAG G such that L(G)~L! tJ L2.</Paragraph> <Paragraph position="2"> Let G 1 = ( ! !, A v N v S ), and G 2 ---- ( 12 , A 2, N 2, S ).</Paragraph> <Paragraph position="3"> Without loss of generality, we may Lssume that the N! f'l N 2 ~ #.</Paragraph> <Paragraph position="5"> Let x 6 L l UI, 2 . Then x 6 L! or x 6 L2. If x 6 Ll, thee it must be possible to generate the string x in G , since I 1 , A! ate in G. Hence x E L(G). Si~nilarly if x E ~ , we can show that x E L(G).</Paragraph> <Paragraph position="6"> Hence L 1 LIL 2 ~ L(G). If x E L(G), then x is derived using either only I l,A Ioronly 12, A 2sinceN! f'IN2~ ~. Hence, x6L! orx6 l..~z. Thus, L(G) C_ L I V L2. Therefore, L(G} = L, O L=z.</Paragraph> <Paragraph position="7"> 4.S. Closure under Coneatenntton Let G, --(lt.At,Nt.St), G s -- (la,As.Ns.Sa) be two TAGs generating LI, 1,2 respectively, such that N 1 I&quot;1 N2 ,m at. We can construct * TAG G == (I, A, N, S) such that L(G)== L t . L a. We chooeeSsucbthatSisnotinN n UNa. We let N == N t U N2U {S), A ffi= A i U A 2. For all t I E ! l, tz E 1 2, we add tlz to !, as shown in Fig 4.2.1. Therefore, I ffi= ( t12 \[ t I E It, ta E lz), where the nodes in the subtrees t I and t z of the tree t12 have the same C/oustrxints associnted with them as in the original grammars G s ned G s. It is eMy to show that L(G) ~ L 1 . L2. once we note that there are no auxiliary trees in G rooted with the symbol S, and that N 1 13 N z == as.</Paragraph> <Paragraph position="9"> Let G 1 ~ (Ii.Ai.NI.Si) be a TAG generating L 1. We can show that we can construct a TAG G such that L(G) = Ls'. Let S be a symbol ant in Ni, and let N == N t U (S). We let the set I of initial trees of G be (te}, where t e is the tree shown in Fig 4.3a. The set of auxiliary trees A is dermed M A= (tsx/t IEIt}UA t.</Paragraph> <Paragraph position="10"> The tree teA is as shown in Fig 4.3b, with the constraints on the root of each ttA being the null adjoining constraint, no constraints on the foot, and the constraints on the nodes of the subtreee t I of the trees tlA being the same as those for the corresponding nodes in the initial tree t I of G t.</Paragraph> <Paragraph position="11"> To see why L(G) .- Lt&quot; , consider x (~ L(G). Obviously, the tree derived (whose frontier is given by x ) must be of the form shown in Fig 4.3C/, where each t i' is a eeutential tree in Gl,such t i' E D(ti), for an initial tree t i in G I. Thus, L(G) _ Lt'.</Paragraph> <Paragraph position="12"> On the other hand, if x E Lt', then x ~ wt...wn, w i 6 L l for 1 i _~ n. Let each w i thee be the frontier of the eenteutial tree t i' of G t such that t i' E D(tl) , t i E ! t. Obviously, we can derive the tree T, using the initial tree re. and have a sequence of adjoining'operations using the auxiliary trees tiA for I < i _< n. From T we can obviously obtain the tree T' the same as give* by Fig 4.3C/, using only the * ~xiliary trees in A t . The frontier of T' is obviously ws...w n. Hence, x G L(G). Therefore, L t. G L(G). Thus L(G) -- L,'.</Paragraph> <Paragraph position="13"> Let L T be a TAL and L R be a regular language. Let G be a TAG generating L T and M = (Q , E , 6, q0 , QF) be a fruits state automaton recognizing L R. We can construct a grammar G and will 8how that L(GI) -- L T N L R.</Paragraph> <Paragraph position="14"> Let a be an elementary tree in G. We shall negotiate with each node a quadruple (ql,q2,q~,q4) where ql,q2,qa,q4 E Q. Let (ql,qa,qs,q4) be associated with a node X in a. Let us assume that a is an auxiliary tree, and that X is an ancestor of tbe foot node of n, ud hence, the ancestor of the foot node of any derived tree -/iu D(a). Let Y be the label of the root and foot nodes of a. If the frontier of '7 ('r in D(a)) is w I w 2 Y w s w4, and the frontier of the subtree of 7 rooted at Z, which corresponds to the node X in a is w z Y wt. The idea of associating (ql,q2.qs,q4) with X is that it must be the ease that 6&quot;(ql, w2) = q2, and 6&quot;(q~, ws) ffi q4- When &quot;t becomes a part of the sentential tree 7' whose frontier is given by u w I w z v w s w 4 w, then it must be the case that 6&quot;(q2, v) == qs. Following this reasoning, we must make q2 ~ qa, if Z is not the ancestor of the foot node of % or if &quot;7 is in D(a) for some initial tree a in G.</Paragraph> <Paragraph position="15"> We have assumed here. as in the case of the parsing algorithm prcsented earlier, that any node in any elementary tree has atmoet two children.</Paragraph> <Paragraph position="16"> From G we can obtain G s as follows. For each initial tree a, ar~ociate with the root the quadruple (q0, q, q, qt) where qo is the initial state of the finite state automaton M, nnd qf E QF- For each auxiliary tree 0 of G, a~5ociate with the root the quadruple (qt,q2,q.q,q4), where q,ql,q2,q~,q4 are some variables which will later be given values from Q. Let X be some *ode in some elementary tree a. Let (qt,q2,q3,q4) be associated with X. Then, we have to consider the follo'~ing caacs.</Paragraph> <Paragraph position="17"> Case 1: X has two children Y and Z. The left child Y is the ancestor of the foot node of a. Then associate with Y the quadruple ( P, q2, q3, q ), and ( q, r, r, s ) with Z, and associate with X the constraint that only those trees whose root has the quadruple ( ql, P, e, q4 ), among those which were allowed in the original grammar, &quot; may be adjoined at this node. If ql ~ p, or q4 ~ u , then the constraint as6ociated with X must be made obligatory. If in the original grammar X had an obligatory constraint aasocinted with it then we retain the obligatory constraint regarding of the relationship between ql and p, and q4 and s. If the constraint a~mciated with X is a null adjoining constraint, we sumociate ( qt, el,q, qa, q ), and ( C/b r, r. q4 ) with Y and Z respectively, and associate the null adjoining constraint with X. If the label of Z is ~, where * E E, then we choose s and q such that 6 ( q, a ) ~ s. In the null adjoining constraint ease, q is chosen such that 6 ( q, a ) ~ q4.</Paragraph> <Paragraph position="18"> child } be the ancestor of the the foot node the tree a. Then we shall associate (p,q,q,r), (r,qs,q3,s) with Y and Z. The associated constraint with X shall be that only those trees among those which were allowed in the orignal grammar may be adjoined provided their root has the quadruple (ql,p,s,qt) associated with it. If q, ~ p or q4 ~ r then we make the constraint obligatory. If the original grammar had obligatory constraint we will retain the obligatory constraint. Null constraint in the original grammar will force us to use null constraint and not consider the cases where it is not the case that ql == P and q4 -- s. If the label of Y is a terminal 'a' then we choose r such that oe'(p,n) ~ r. If the constraint at X is n null adjoining constraint, then * o~(ql,a) = r.</Paragraph> <Paragraph position="19"> Case 3: This corresponds to the case where neither the left child Y nor the right child Z of the node X is the ancestor of the foot node of o or if ~ is a initial tree. Then q2 ~ q~ ffi q- We will associate with Y and Z the quadruples (p,r,r,q) and (q,s,s,t) reap. The constraints are assigned as before , in this case it is dictated by the quadruple (ql,p,t,q4). If it is not the case that qt ~ P and ql ~ t, then it becomes an OA constraint. The OA and NA constraints at X are treated similar to the previous cases, and so is the case if either Y or Z is labelled by a terminal symbol.</Paragraph> <Paragraph position="20"> Case 4: If (qt,q2,~,q4) is associated with a node X, which has only one child Y, then we can deal with the various cases as follows. We will associate with Y the quadruple (p,q20q~,s) and the constraint that root of the tree which can be adjoined at X should have the quadruple (ql,P,e,q4) associated with it among the trees which were allowed in the original grammar, if it is to be adjoined at X. The cases where the original grammar had null or obligatory &quot;constraint associated with this *ode or Y is labelled with a terminal symbol, are treated similar to how we dealt with them ia the previous cases.</Paragraph> <Paragraph position="21"> Once this has bee* done, let ql,&quot;',qm be the independent variables for this elementary tree a, then we produce as many copies of a so that ql,&quot;',qm take all possible values from Q. The only difference among the various copies of a so produced will be constraints associated with the *odes in the trees. Repeat the process for all the elementary trees in G !. Once this has bee* done and each tree given * unique name we can write the constraints in terms of these *ames. We will now show why L(GI) =ffi L T f3 L R.</Paragraph> <Paragraph position="22"> Let w E L(GI). Theu there is s seque*ce of adjoining operatio*s starting with an initial tree a to derive w. Obviously, w 6 LT, also since corresponding to eseh tree used in deriving w, there is . correspo*ding tree ia G, which differs only in the C/onstrai*ts associated with its *odes. Note, however, that the C/o*strai*ts associated with the *odes in trees in G t are just * restriction of the correspo*ding o*es in G, or an obHgatoiT C/o*straint where there was * o*e in G. Now, if we can assume ( by inductin* hypothesis ) that if ~fter n adjoining operatio*s we can derive &quot;f 6 D(~x'), then there is a correspo*ding tree &quot;T 6 D(a) iu G, which has the same tree structure as ~/' but differing o*ly in the constraints associated with the corresponding *odes, then if we adjoin at some node in &quot;~' to obtain &quot;h', we can adjoin in &quot;~ to obtain &quot;h (corresponding to gl')-Therefore, if w can be derived in Gi, then it can dcfmitely be derived inG.</Paragraph> <Paragraph position="23"> If we can also show th~ L(Gi) C L a. then we can co*clods that L(GI) C L T N Lit. We can use induetio* to prove this. The induction hypothesis is that if all derived trees obtained after k < n adjoining operations have the property P then so will th* derived trees after * .4- I adjoiniugs where P is defi*ed as, the tree p to which X belo*gs labelled Y as a desce*dant such that w s Y w s is the frontier of the subtree of # rooted at X, then if (qs,q2,~,q4) had been associated with X, 6'(qvwl) ,~ qz and ~(q3,w2) ~ q4, a*d if w is the frontier of the subtree under the foot * ode of # in '7 is then ~(qs,w) ~= q~. If X is not the ancestor of the foot *ode of # then the subtree of # below is of the form wlw s.</Paragraph> <Paragraph position="24"> Suppose X has associated with it (qt,q,q,q2) then ~(ql,wa) -~ q, ~*(q,w2) ffi q2&quot; Actually what we mean by an adjoining operation is *of necessarily just o*e adjoining operatio* but the minimum number so that no obligatory co*straints are associated with any *odes in the derived trees. Similarly, the base case *teed ant cousider o*ly elementary trees, but the smallest (in terms of the *umber of adjoining operatin*s) tree starting with eleme*tary trees which has * o obligatory coustrai*t associated with any of its *odes. The base case ca* be see* easily co*sidering the way the grammar was built (it can be shown formally by induction ou the height of the tree) The inductive step is obvious. Note that the derived tree we are going to use for adjoining will have the property P, and so will the tree at which we adjoin; the former because of the way we designed the grammar and a~ig*ed constraints, and the latter because of induction hypothesis. Thus so will the new derived tree. Once we have proved this, all we have to do to show that L(Gx) C L R is to consider tho6e derived trees which are se*tential trees and observe that the roots of these trees obey property P.</Paragraph> <Paragraph position="25"> Now. if n string x E L T 13 L R, we ca* show that x E L(G). To do that, we make use of the foUowing claim.</Paragraph> <Paragraph position="26"> Let ~ be an auxiliary tree in G with root labelled Y= and let &quot;y 6 D(~). We claim that there is a 8' in G I with the same structure as 8, such that there is a &quot;;' in D(bet~0)' ) where &quot;I' has the same structure as 7, such that there is *o OA coustraint in '7'. Let X be a *ode in fit which was used in deriving -;. Then there is a *ode X' in 7' such that X' belongs to the auxillixry tree #l' (with the same structure as ~|. There are several cams to co*sider Case I: X. is the ancestor of the foot node of 81, such that the fro*tier of the subtree of ,81 rooted at X is wsYw 4 and the frontier of the subtree of 7 rooted at X is w,wlZwsw4. Let ~(ql,ws) ~ q, ~(q,wl) -~- q2, ~(qS,w2) = r, and ~(r,w4) ~ q4. Then X' will have (ql,q,r,q4) associated with it, and there will be *o OA co*straint in '7'* Case 2: X is the ancestor of the foot *ode of ~l, and the frontier of the subtree of fll rooted at X is wsYw 4. Let the frontier of the aubtree of 'T rooted at X is wawlwsw 4. Then we claim that X' in -;' will have associated with it the quadruple (ql,q,r,q4), if G*(qt,wa) q, f(q,wl) = p0 6&quot;(p,wz} = r, and ~(r,w4} = q4-Case 3: Let the froutier of the subtree of ~i (and also &quot;7) rooted at X is wlw 2. Let f(q,wl) = p, 6*(p,ws) = r. The* X' will have associated with it the quadruple (q,p,p,r).</Paragraph> <Paragraph position="27"> We shall prove our claim by inductio* o* the *umber of adjoining operations used to derive &quot;I. The base case (where &quot;1 == 0) is obvious from the way the grammar G 1 was built. We shall *ow assume that for all derived trees % which have bee* derived from p using k or less adjoining operatio*s, have the property as required in our claim. Let ~ be a derived tree in p after k adju*ctio*s. By our inductive hypothesis we may ~asume the existence of the corresponding derived tree &quot;y' E D(~') derived in G I. Let X be a uode in 7 as shown ia Fig. 4.4.1. Then the *ode X* in &quot;y' eorrespo*di*g to X will have associated with it the quadruple (ql',q2S,q~l',q4&quot;)- Note we are aseumin~ here that the left child Y' of X' is the ancestor of the foot node of D'- The quedruples (qt',q~',qa',P) and (P,Pt*Pt,q4&quot;) will be associated with Y' and Z' (by the induction hypothesis). Let ~t be derived from ~ by edjoining Pt at X as in Fig. 4.4.2. We have to slaw the existence of It' in GI such that the root of this auxili~f tt~ has asmeinted with K the quedruple (q,qt',q4O,r). The existence 0( the tree follows from induction hypothesis (k m 0). We have also got to show that there exkts '71' with the mane structure as q' but one that allows It' to be adjoined at the required noC/le. But this should be so, since from the way we obtained the trees in GI, there will exist &quot;/1&quot; such that X t' has the quadruple (q,q:t',qs',r) and the constraints st X 1' are dictated by the quadruple (q,qt',q4eJ'), but such that the two children of X t' will have the same quedruple as in 1'. We san now adjoin It' in 7t deg to obtain &quot;Yl'- It can be shown that lt' has the required property to establish our claim.</Paragraph> <Paragraph position="29"> FiKure 4.4.1. Fi~re 4.4.2 Firstly, any node below the foot of PI' in 7t' will satisfy our requirement~ as they are the same as the corresponding nodes in &quot;/l'-Since ~t' satisfy* the requirement, it is simple to observe that the sods* in 01' will, even after the edjuuction of I1' in &quot;el'&quot; Howcver, because the quadruple associated with X l' are different, the quadruples of the nodes above X i' must reflect this change. It is easy to chock the existence of an auxiliary tree such that the nodes above X l' satisfy the requirements as stated above. It can also be argued an the by*is of the design of grammar GI, that there exists trees which allow this new auxiliary tree to be adjoined at the appropriate pi~ce.</Paragraph> <Paragraph position="30"> This then allows us to conclude that there exist * derived tree for e~h derived tree bebngin to D(0) as in our el~timo The next step is to extend our claim to take into amount all derived trees (Le., including the centennial truest This can be done in * manner similar to our treatment of derived trees belonging to D(~) for some auxiliary tree I as above. Of course, we have to consider only the C/~-~e where the finite state automaton starts from tlie initial state q0, and teaches some fmal state deg4 on the ihput which is the frontier of tome sentential tree in (3. This, then allows us to conclude that L T f3 Ln c_ L(C,)..nose, C(G,} -- C r n t~.</Paragraph> <Paragraph position="31"> ?</Paragraph> </Section> </Section> <Section position="7" start_page="220" end_page="222" type="metho"> <SectionTitle> 5. HEAD GRA.MMARS AND TAG's </SectionTitle> <Paragraph position="0"> In this section, we attempt to show that Heed Grtmmar* (JIG) are remarktbly similar to Tree Adjoining Grammars. it appears that the bask: intuition behind the two systems is more ~ lea the same.</Paragraph> <Paragraph position="1"> Head Grammars were introduced in \[Pollard,10841, .but we follow the notations used in \[Roach,1084\]. It has been observed that TAG's ud HG's share s lot of common formal properties such as ahnoet identical cloture results, similar pummping leman.</Paragraph> <Paragraph position="2"> Consider the bask operation in llead Grammars - the Heed Wrapping operation. A derivation from * non-terminal produces * pair (i,*t...ai...sa) (* more convenient representation for this pair is * l...ailai/l...~ ).* The arrow denotes the head of the string, which in turn determines where the string is split up when wrapping operation takes pl~e. For example, consider X->LI~(A,B), and let A=t*WhlX and B=t'Uglv.Thcn we say, X=t'whUglVX'We shall define some functions used in the HG formalism, which we need here. If A derives in 0 or more steps the heeded string whx~ and B derives u~v, then</Paragraph> <Paragraph position="4"> X derives vhx~v 4, Nov consider hov* derLvttion Lu TAGs proceeds Let ~ be In auxilliary tree and let a be * eentential tree as in Fig 5.1. Adjoining ~ st the root of the sub-tree &quot;r gives us the seutential tree in File 5.1. We can, now see how the string whx has &quot;wrapped around&quot; the sub-tree i.e,tbe string ugv. This seems to suggest that there is something similiar in the role played by the foot in *n auxilliary tree and the head in a Head Grammar how the adjoining operations and head-wrapping operations operate on strings. We could say that if X is the root of an auxillizry tree ~ ted * l...al X ~i+l...an is the frontier of a derived tree &quot;1 6 D(~), then the derivation of &quot;/would correspond to * derivation from a non-terminal X to the string at...a i lai+t...tu in HG and the nee of &quot;f in some sentential tree would correspond to how the strings st... a i and ai+t...* a are used in deriving a string in IlL.</Paragraph> <Paragraph position="5"> Based on this observation, we attempt to show the close relationship of TAL% and llL's. It k more convinient for us to think of the headed string (i,a,.:.aa) as the string al...a a with the head pointing in between the symbok a I and el+ , rather than at the symbol t 1. The defmition of the derivation oporatom can be extended in, straightforward manner to take this into aeeount. However, we can acheive the same eHeet by considering the definitions of the operators LL,LC,etc. Pollard suggest~ that cases such u LL2~,~ ) be left undefined. We shall assume that if ~&quot; mwty then L L~,k) -andLC,(X~) ~ kx.</Paragraph> <Paragraph position="6"> We, then ;ay tha~t if G is x Head Grammar, then w I -- whx belongs to L(G) if and only if S derives the headed string w~or whkx.</Paragraph> <Paragraph position="7"> With this new definition, we shall show, without giving the pro~* f, that the ci-,ss of TAL'e is contained in the clan of HL's. by systematically converting any TAG G to n HG G'. We shall assume, without loss of generality, that the constraints expressed at the nodes of elementary trees of G are 1) Nothing can be adjoined at a node {NA).</Paragraph> <Paragraph position="8"> 2) Any appropriate tree (symbols at the node and root of the auxilliary tree must match) can be adjoined {AA), or 3) Adjoining at the node is obligatory {OA).</Paragraph> <Paragraph position="9"> It is easy to show that these constraints are enough, and that selective adjoining can be expressed in terms of these and additional non-terminals. We know give a procedural description of obtaining an equivalent Head Grammar from a Tree-Adjoining Grammar. The procedure works as follows. It is n recursive p~rocedure {Convert to HG) which takes in two parameters, the first representing the node on which it is being applied and the second the label appearing on the left-hand side of the HG productions for this node. If X is a nonterminal, for each auxiliary tree #.whose root has the label X, we obtain a sequence of productions such that the first one has X on the left-hand side. Using these productions, we can derive the string w|Xw z where a derived tree in D(~) has a frontier wlYw =. If Y is a#node with with label X in rome tree where adjoining is allowed, we introduce the productions Y' -> LL2(X,N') (so that s derived tree with root Iabel X nay wrap around the string derived free the 8ubtree below this node} N' -> LCi(A , ..... Aj) {asstming that there are J children of this node and the t tit child in the ancestor of the foot node. By calling the procedure recursively for all the J children of Y with Ak.k ranging from I through J, ve can derive frou N' the frontier of the subtree below Y} Y' -) N' ( this i8 to handle the case where no adJunctton takes place at Y} If G k a TAGthen we do the following Repeat for every Initial tree Convert to HG(root,S') {S' viii be the start symbol of the new Head Grammar).</Paragraph> <Paragraph position="10"> Repeat for etch Auxillinry tree Convert td~_HG(root,rootsyabol) where Convert to HG(node,ntue) Is defined u follows if node is an internal node then cue 1 If the constraint *t the node t8 AA add productioan Sym->LL=(node nyabol,|').</Paragraph> <Paragraph position="11"> r->LCt(AI*. .... Ai'o .... Aj*) SYm->LCt(AI'. .... At'. .... tj') where N'.AI'.~'o...A J' are new non-teraintl uymbolu.A t ..... Aj correspond to the |children of the node and i=i if foot node is not * descendant of node else =1 such that the 1 ~ child of node to ancestor of foot node0J=nuaber of children of node for k=l to J step I do Convert to Hf(k t& child of node,Ak').</Paragraph> <Paragraph position="12"> Cue 2 The constraint at the node in NA.</Paragraph> <Paragraph position="13"> Same as Case 1 except don't add the productions Sya->LLl(node nyabol.g').</Paragraph> <Paragraph position="14"> N'->LCi(At'. .... Aj').</Paragraph> <Paragraph position="15"> Case 3 The constraint at the node i80A.</Paragraph> <Paragraph position="16"> State as Case I except that we don't add Syn->LCi(AI',...Aj') else if the node has t terainai syabol a.</Paragraph> <Paragraph position="17"> then add the production Sya ->~ e'lse {it i8 a foot node } if the constraint at the foot node is AA then add the productions - --Sya ->LL2(node eysbolok)/k if the constraint is 0A then add only the production Sya ->LL2(node syabol~) if the constraint is NA add the production Sym.->X We shall now give an example of converting a TAG G to a HG. G contains a single initial tree a, and a single auxiliary tree as in Fig. 5.2.</Paragraph> <Paragraph position="18"> Applying the procedure Convert_to_HG to this grammar we obtain the HG whose productions are gives by- null be verified that this grma~ gennratsn exactly It can L(6).</Paragraph> <Paragraph position="19"> It is worth emphaaising that the main point of this exercise was to show the similarities between Head Grammars and Tree Adjoining Grammars. We have shown how a HG G' (using our extended definitions) can be obtained in a systematic fashion from a TAG G. It is our belief that the extension of the definition may not necessary. Yet, this conversion process should help us understand the similarities between the two formalisms.</Paragraph> </Section> class="xml-element"></Paper>