| { |
| "paper_id": "C86-1048", |
| "header": { |
| "generated_with": "S2ORC 1.0.0", |
| "date_generated": "2023-01-19T13:14:19.721519Z" |
| }, |
| "title": "Tree Adjoining and Head Wrapping~", |
| "authors": [ |
| { |
| "first": "Vijay-Shanker", |
| "middle": [], |
| "last": "David", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "University of Pennsylvania Philadelphia", |
| "location": { |
| "postCode": "19104", |
| "region": "PA" |
| } |
| }, |
| "email": "" |
| }, |
| { |
| "first": "J", |
| "middle": [], |
| "last": "Weir", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "University of Pennsylvania Philadelphia", |
| "location": { |
| "postCode": "19104", |
| "region": "PA" |
| } |
| }, |
| "email": "" |
| }, |
| { |
| "first": "Aravind", |
| "middle": [ |
| "K" |
| ], |
| "last": "Joshi", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "University of Pennsylvania Philadelphia", |
| "location": { |
| "postCode": "19104", |
| "region": "PA" |
| } |
| }, |
| "email": "" |
| } |
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| "identifiers": {}, |
| "abstract": "In this paper we discuss the formal relationship between the classes of languages generated by Tree Adjoining Grammars and Head Grammars. In particular, we show that Head Languages are included in Tree Adjoining Languages and that Tree Adjoining Grammars are equivalent to a modification of Head Grammars called Modified Head Grammars. The inclusion of MHL in HL, and thus the equivalence of HG's and TAG's, in the most general case remains to be established.", |
| "pdf_parse": { |
| "paper_id": "C86-1048", |
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| "abstract": [ |
| { |
| "text": "In this paper we discuss the formal relationship between the classes of languages generated by Tree Adjoining Grammars and Head Grammars. In particular, we show that Head Languages are included in Tree Adjoining Languages and that Tree Adjoining Grammars are equivalent to a modification of Head Grammars called Modified Head Grammars. The inclusion of MHL in HL, and thus the equivalence of HG's and TAG's, in the most general case remains to be established.", |
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| "section": "Abstract", |
| "sec_num": null |
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| "body_text": [ |
| { |
| "text": "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. 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. 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.", |
| "cite_spans": [ |
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| "start": 533, |
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| "text": "[3]", |
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| "text": "9]", |
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| "text": "[10]", |
| "ref_id": "BIBREF10" |
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| "section": "Introduction", |
| "sec_num": "1." |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [ |
| { |
| "start": 24, |
| "end": 27, |
| "text": "[9]", |
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| "section": "Introduction", |
| "sec_num": "1." |
| }, |
| { |
| "text": "J\" This work was partially supported by NSF grants MCS-82-19116-CER, MCS-82-07294 and DC1~-84-10413. We are grateful bo Tony Kroch and Carl Pollard for valuable discus~ sions.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Introduction", |
| "sec_num": "1." |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "'.\u00a3ree Adjoining Grammars Definition 1. A TAG is a 54uple G -= (VN,V.r,S,I,A)", |
| "sec_num": "1.1." |
| }, |
| { |
| "text": "Initial trees and auxiliary trees have the following form:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "'.\u00a3ree Adjoining Grammars Definition 1. A TAG is a 54uple G -= (VN,V.r,S,I,A)", |
| "sec_num": "1.1." |
| }, |
| { |
| "text": "/\\ ~ V. w~wz ~Vx \"{ x~ V~ figure 1.l", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "'.\u00a3ree Adjoining Grammars Definition 1. A TAG is a 54uple G -= (VN,V.r,S,I,A)", |
| "sec_num": "1.1." |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "'.\u00a3ree Adjoining Grammars Definition 1. A TAG is a 54uple G -= (VN,V.r,S,I,A)", |
| "sec_num": "1.1." |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "'.\u00a3ree Adjoining Grammars Definition 1. A TAG is a 54uple G -= (VN,V.r,S,I,A)", |
| "sec_num": "1.1." |
| }, |
| { |
| "text": "LL V vu 'v,/f \"\"/Z figure 1.;~ V", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "'.\u00a3ree Adjoining Grammars Definition 1. A TAG is a 54uple G -= (VN,V.r,S,I,A)", |
| "sec_num": "1.1." |
| }, |
| { |
| "text": "The subtree under ~ is excised from % the auxiliary tree [~ is inserted in its place and the excised subtree is inserted at the foot of/~. 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 subset 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. 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~ :", |
| "cite_spans": [], |
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| "eq_spans": [], |
| "section": "'.\u00a3ree Adjoining Grammars Definition 1. A TAG is a 54uple G -= (VN,V.r,S,I,A)", |
| "sec_num": "1.1." |
| }, |
| { |
| "text": "A\"", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "'.\u00a3ree Adjoining Grammars Definition 1. A TAG is a 54uple G -= (VN,V.r,S,I,A)", |
| "sec_num": "1.1." |
| }, |
| { |
| "text": "figure 1.3", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "'.\u00a3ree Adjoining Grammars Definition 1. A TAG is a 54uple G -= (VN,V.r,S,I,A)", |
| "sec_num": "1.1." |
| }, |
| { |
| "text": "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 \"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. 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. The derived auxiliary tree \"-/~ 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] .", |
| "cite_spans": [ |
| { |
| "start": 1425, |
| "end": 1428, |
| "text": "[9]", |
| "ref_id": "BIBREF9" |
| } |
| ], |
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| "eq_spans": [], |
| "section": "'.\u00a3ree Adjoining Grammars Definition 1. A TAG is a 54uple G -= (VN,V.r,S,I,A)", |
| "sec_num": "1.1." |
| }, |
| { |
| "text": "O~ S~ 5~ Sr~ figure 1.5 We use the notation P(3) to denote the set of trees derived from the elementary tree 3 using 0 or more adjunctions.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "/\\", |
| "sec_num": null |
| }, |
| { |
| "text": "The tree set T(G) of a TAG C is T(G) = U,~I P(a).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "/\\", |
| "sec_num": null |
| }, |
| { |
| "text": "Now we can see that tile language Lt, generated by the example grammar G, is L1 =: { a'~gbnfcnh ]n > 0 }", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "The string language L(G) generated by a TAG G is given by L(G) = { w I \" is the frontier of some \"y in T(G) }", |
| "sec_num": null |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "The string language L(G) generated by a TAG G is given by L(G) = { w I \" is the frontier of some \"y in T(G) }", |
| "sec_num": null |
| }, |
| { |
| "text": "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 ~.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "tlead Grammars", |
| "sec_num": "1.2." |
| }, |
| { |
| "text": "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. The language generated by a HG G is defined as follows:", |
| "cite_spans": [ |
| { |
| "start": 115, |
| "end": 118, |
| "text": "[6]", |
| "ref_id": "BIBREF6" |
| } |
| ], |
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| "eq_spans": [], |
| "section": "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. f C [.J~ 1{ LCi, LLi, LRi, RCi, RLi, RRi }", |
| "sec_num": null |
| }, |
| { |
| "text": "L(G) ={wlS~:~} Example 1.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "LC](Ul~U2", |
| "sec_num": null |
| }, |
| { |
| "text": "2 We now present a sample Head Grammar for the language L1 = { a\"gb'~fcr*h J n > 0 }", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "LC](Ul~U2", |
| "sec_num": null |
| }, |
| { |
| "text": "Sa -~ nL2(S2,bc)", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "S ~ LL2(S,,f), S ---+ LL2(S2,-f), S~ -.~ LC2(-~, Sa), $2 --+ \"~h, S 3 ---+ Ln2(S~,bc),", |
| "sec_num": null |
| }, |
| { |
| "text": "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:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Modified Head Grammars", |
| "sec_num": "1.3." |
| }, |
| { |
| "text": "W(wl~w2, ul~u2) = wlu,Tu2w2 C1(WllW2, ul~u2) = WllW2ulu2 C2(WllW2,Ul~U2) = wlw2ul~u2", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Modified Head Grammars", |
| "sec_num": "1.3." |
| }, |
| { |
| "text": "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).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Modified Head Grammars", |
| "sec_num": "1.3." |
| }, |
| { |
| "text": "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", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Modified Head Grammars", |
| "sec_num": "1.3." |
| }, |
| { |
| "text": "suffices as a substitute for both of these operations. Suppose", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Modified Head Grammars", |
| "sec_num": "1.3." |
| }, |
| { |
| "text": "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~). 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.", |
| "cite_spans": [ |
| { |
| "start": 392, |
| "end": 395, |
| "text": "[8]", |
| "ref_id": "BIBREF8" |
| } |
| ], |
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| "eq_spans": [], |
| "section": "Y ~", |
| "sec_num": null |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Tree Adjunetion and Wrapping", |
| "sec_num": "1.4." |
| }, |
| { |
| "text": "Suppose there is a derived auxiliary tree \"7 correspond\u00b0 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 subtree 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. This example illustrates the basic intuition behind the constructions involved in the following proofs showing the equivalence of MHG's and TAG's.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Tree Adjunetion and Wrapping", |
| "sec_num": "1.4." |
| }, |
| { |
| "text": "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%.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Head Grammars and Modified Head Grammars", |
| "sec_num": "2." |
| }, |
| { |
| "text": "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", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Head Grammars and Modified Head Grammars", |
| "sec_num": "2." |
| }, |
| { |
| "text": "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:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Z ---+ W(X~,Y) can be used in place of Z ~ LL2(X,Y). A", |
| "sec_num": null |
| }, |
| { |
| "text": "X ~ -~ W(X Th, ht) and X r ---+ W(X Th, Th)", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Z ---+ W(X~,Y) can be used in place of Z ~ LL2(X,Y). A", |
| "sec_num": null |
| }, |
| { |
| "text": "Complete details of this proof are given in [4] .", |
| "cite_spans": [ |
| { |
| "start": 44, |
| "end": 47, |
| "text": "[4]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Z ---+ W(X~,Y) can be used in place of Z ~ LL2(X,Y). A", |
| "sec_num": null |
| }, |
| { |
| "text": "We are unable to prove the inclusion of MHL's in HL's. 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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Z ---+ W(X~,Y) can be used in place of Z ~ LL2(X,Y). A", |
| "sec_num": null |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Z ---+ W(X~,Y) can be used in place of Z ~ LL2(X,Y). A", |
| "sec_num": null |
| }, |
| { |
| "text": "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] .", |
| "cite_spans": [ |
| { |
| "start": 129, |
| "end": 132, |
| "text": "[4]", |
| "ref_id": null |
| } |
| ], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Equivalence of MttG's and TAG,'s", |
| "sec_num": "3." |
| }, |
| { |
| "text": "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 --", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "3.1.. Inclusion of TAL in MIlL", |
| "sec_num": null |
| }, |
| { |
| "text": "In this section, we have generalized the concatentation operations of MHG's to be of the form Cj for j _> 1. It is obvious that these operations can always be simulated using just C1 and C2.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "(Vt\u00a2, VT, S, I, A) to an equivalent MHG G' -(V]~, VT, S, P).", |
| "sec_num": null |
| }, |
| { |
| "text": "We shall first describe the algorithm convert informally.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "(Vt\u00a2, VT, S, I, A) to an equivalent MHG G' -(V]~, VT, S, P).", |
| "sec_num": null |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "(Vt\u00a2, VT, S, I, A) to an equivalent MHG G' -(V]~, VT, S, P).", |
| "sec_num": null |
| }, |
| { |
| "text": "The main idea of our scheme is as follows. Let That is, the split point appears in a position corresponding to the foot node. 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). 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,", |
| "cite_spans": [], |
| "ref_spans": [ |
| { |
| "start": 230, |
| "end": 240, |
| "text": "If ('7, i)", |
| "ref_id": null |
| }, |
| { |
| "start": 355, |
| "end": 369, |
| "text": "([fl,4, b, il)", |
| "ref_id": "FIGREF6" |
| } |
| ], |
| "eq_spans": [], |
| "section": "(Vt\u00a2, VT, S, I, A) to an equivalent MHG G' -(V]~, VT, S, P).", |
| "sec_num": null |
| }, |
| { |
| "text": "[% i]\" -* Cj([%i. 1] .... , [%i .j] ..... ['7, i. k])", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "(Vt\u00a2, VT, S, I, A) to an equivalent MHG G' -(V]~, VT, S, P).", |
| "sec_num": null |
| }, |
| { |
| "text": "where @,i. 1>,..., ('7, i. j>,..., ('7, i. k> 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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "(Vt\u00a2, VT, S, I, A) to an equivalent MHG G' -(V]~, VT, S, P).", |
| "sec_num": null |
| }, |
| { |
| "text": "We are now in a position to define the conversion process. The algorithm is as fi)llows:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "(Vt\u00a2, VT, S, I, A) to an equivalent MHG G' -(V]~, VT, S, P).", |
| "sec_num": null |
| }, |
| { |
| "text": "for each initial tree a, let S -> [a,e] G P.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "(Vt\u00a2, VT, S, I, A) to an equivalent MHG G' -(V]~, VT, S, P).", |
| "sec_num": null |
| }, |
| { |
| "text": "for each elementary tree % call convert((% e))", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "(Vt\u00a2, VT, S, I, A) to an equivalent MHG G' -(V]~, VT, S, P).", |
| "sec_num": null |
| }, |
| { |
| "text": "where the procedure convert is as defined below.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "(Vt\u00a2, VT, S, I, A) to an equivalent MHG G' -(V]~, VT, S, P).", |
| "sec_num": null |
| }, |
| { |
| "text": "define convert (('7, i) (['7, i. 1] .", |
| "cite_spans": [], |
| "ref_spans": [ |
| { |
| "start": 15, |
| "end": 23, |
| "text": "(('7, i)", |
| "ref_id": null |
| }, |
| { |
| "start": 24, |
| "end": 35, |
| "text": "(['7, i. 1]", |
| "ref_id": null |
| } |
| ], |
| "eq_spans": [], |
| "section": "(Vt\u00a2, VT, S, I, A) to an equivalent MHG G' -(V]~, VT, S, P).", |
| "sec_num": null |
| }, |
| { |
| "text": "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 <j < k do eonvert((%i.j)).", |
| "cite_spans": [], |
| "ref_spans": [ |
| { |
| "start": 90, |
| "end": 129, |
| "text": "add ['7,i[-, C1(['7, i. 1],... ,['7, i'", |
| "ref_id": null |
| } |
| ], |
| "eq_spans": [], |
| "section": ".... [V,i. k])", |
| "sec_num": null |
| }, |
| { |
| "text": "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:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": ".... [V,i. k])", |
| "sec_num": null |
| }, |
| { |
| "text": "For all elementary trees \"7, and addresses i in q, if there is a tree \"7' in P(('7, i}) of height less than k, and the frontier of \"71= WlXW2 or wlw2, then ['7, i]--~ wl~w2.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": ".... [V,i. k])", |
| "sec_num": null |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": ".... [V,i. k])", |
| "sec_num": null |
| }, |
| { |
| "text": "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 \"~' 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 <'~,i> or not.", |
| "cite_spans": [], |
| "ref_spans": [ |
| { |
| "start": 198, |
| "end": 211, |
| "text": "\"~' c p((~,i)", |
| "ref_id": null |
| } |
| ], |
| "eq_spans": [], |
| "section": ".... [V,i. k])", |
| "sec_num": null |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Inclusion of MHL in TAL", |
| "sec_num": "3.2." |
| }, |
| { |
| "text": "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 \"~ 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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Inclusion of MHL in TAL", |
| "sec_num": "3.2." |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Inclusion of MHL in TAL", |
| "sec_num": "3.2." |
| }, |
| { |
| "text": "I figure 3.1", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "C oa", |
| "sec_num": null |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "C oa", |
| "sec_num": null |
| }, |
| { |
| "text": "We now present the complete construction. Without loss of generality, we will assume a normal form that uses productions of the following form:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "C oa", |
| "sec_num": null |
| }, |
| { |
| "text": "or A-~ ?a or A-~a$ where A,B,C e VN, a C VT tO {A} and f e {CI,C2, W}.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "A-~ f(B,C)", |
| "sec_num": null |
| }, |
| { |
| "text": "The conversion proceeds as follows:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "A-~ f(B,C)", |
| "sec_num": null |
| }, |
| { |
| "text": "1. IfA-~ Ta 6PorA--+ a t EPthenincludefll orfl2 in A respectively.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "A-~ f(B,C)", |
| "sec_num": null |
| }, |
| { |
| "text": "A The set I of\" initial trees consists of the single tree a:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "#':", |
| "sec_num": null |
| }, |
| { |
| "text": "~X: t5 0,I E A", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "#':", |
| "sec_num": null |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "#':", |
| "sec_num": null |
| }, |
| { |
| "text": "We prove that L(G') C L(G) by induction on the height of the \"rAG derivation tree. We show that if q E P (fl), has no OA constraints, and has frontier wlXw2 then X =% wlrw2.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "#':", |
| "sec_num": null |
| }, |
| { |
| "text": "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.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "#':", |
| "sec_num": null |
| }, |
| { |
| "text": "In this paper we have shown the equivalence of TAL and MHL. Since we have also established the inclusion of HL in MHL we have shown that HLs are included in TALs. The inclusion of MHL in HL, and thus the equivalence of HG's and TAG's, in the most general case remains to be established We briefly discuss the relationship between MHG's and HG's and argue that it is close, both linguistically and formally. ", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Conclusion", |
| "sec_num": "4." |
| } |
| ], |
| "back_matter": [], |
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| "FIGREF0": { |
| "type_str": "figure", |
| "text": "figure 1.4", |
| "num": null, |
| "uris": null |
| }, |
| "FIGREF1": { |
| "type_str": "figure", |
| "text": ",V) = ulafu2v, LC2(~,vlh-~v2) :: uvla2v2, LLI(ulSTlu2,~) = ul5~lvu~., LL2(ul~u2,V-) = ulal~'u2, Lm(u,~;~,v) -~.~u~, LR2(u~.~,v) = ~lVa~u2 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.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).", |
| "num": null, |
| "uris": null |
| }, |
| "FIGREF2": { |
| "type_str": "figure", |
| "text": "We now give a MHG generating L~.,5' --~ W(S,, It), S --~ W(S2, f%), S1 \"--+ C2(a~,Sa), S2 ~ g~h, s~ -~ w(s~,Nd, ss -, w(s~,Nd", |
| "num": null, |
| "uris": null |
| }, |
| "FIGREF3": { |
| "type_str": "figure", |
| "text": "figure 1.6", |
| "num": null, |
| "uris": null |
| }, |
| "FIGREF4": { |
| "type_str": "figure", |
| "text": "(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.", |
| "num": null, |
| "uris": null |
| }, |
| "FIGREF5": { |
| "type_str": "figure", |
| "text": "If A -~ W(B,C) ~ P then include fls.", |
| "num": null, |
| "uris": null |
| }, |
| "FIGREF6": { |
| "type_str": "figure", |
| "text": "1 provides a summary of these results.", |
| "num": null, |
| "uris": null |
| } |
| } |
| } |
| } |
