Datasets:

WINGNUS
/

ACL-OCL

	{
	"paper_id": "H86-1020",
	"header": {
	"generated_with": "S2ORC 1.0.0",
	"date_generated": "2023-01-19T03:35:13.485730Z"
	},
	"title": "SOME COMPUTATIONAL PROPERTIES OF TREE ADJOINING GRAMMARS*",
	"authors": [
	{
	"first": "Ar~vlnd",
	"middle": [
	"K"
	],
	"last": "Joshl",
	"suffix": "",
	"affiliation": {
	"laboratory": "",
	"institution": "University of Pennsylvania Philadelphia",
	"location": {
	"postCode": "PA 191C4"
	}
	},
	"email": ""
	}
	],
	"year": "",
	"venue": null,
	"identifiers": {},
	"abstract": "Tree Adjoining Grammar (TAG) is a formalism for natural language grammars. Some of the basic notions of TAG's were introduced in [Joshi,Levy, and Takahashi 19751 and by [Joshi, 1083]. A detailed investigation of the linguistic relevance of TAG's has been carried out in [Kroch and Joshi,1985]. In this paper, we will describe some new results for TAG's, especially in the following areas: (I) parsing complexity of TAG's, (2) some closure results for TAG's, and (3) the relationship to Head grammars.",
	"pdf_parse": {
	"paper_id": "H86-1020",
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	"abstract": [
	{
	"text": "Tree Adjoining Grammar (TAG) is a formalism for natural language grammars. Some of the basic notions of TAG's were introduced in [Joshi,Levy, and Takahashi 19751 and by [Joshi, 1083]. A detailed investigation of the linguistic relevance of TAG's has been carried out in [Kroch and Joshi,1985]. In this paper, we will describe some new results for TAG's, especially in the following areas: (I) parsing complexity of TAG's, (2) some closure results for TAG's, and (3) the relationship to Head grammars.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Abstract",
	"sec_num": null
	}
	],
	"body_text": [
	{
	"text": "Investigation of constrained grammatical systems from the point of view of their linguistic adequacy and thei\u00a2 computational tractability has been a major concern of computational linguists for the last several years.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "INTRODUCTION",
	"sec_num": "1."
	},
	{
	"text": "Generalized Phrase Structure grammars (GPSG), Lexical Functional grammars\" (LFG), Phrase Linking grammars (PLG), and Tree Adjoining grammars (TAG) are some key examples of grammatical systems that have been and still continue to be investigated along these lines.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "INTRODUCTION",
	"sec_num": "1."
	},
	{
	"text": "Some of the basic notions of TAG% were introduced in [Joshi, Levy, and Takahashi,1975] and [Jo6hi, lQ83] . Some preliminary investigations of the linguistic relevance and some computational properties were also carried out in [Jo6hi,1983] . More recently, a detailed investigation of the linguistic relevance of TAG's were carried out by [Kroch and Joshi,1985] .",
	"cite_spans": [
	{
	"start": 53,
	"end": 86,
	"text": "[Joshi, Levy, and Takahashi,1975]",
	"ref_id": null
	},
	{
	"start": 91,
	"end": 98,
	"text": "[Jo6hi,",
	"ref_id": null
	},
	{
	"start": 99,
	"end": 104,
	"text": "lQ83]",
	"ref_id": null
	},
	{
	"start": 226,
	"end": 238,
	"text": "[Jo6hi,1983]",
	"ref_id": null
	},
	{
	"start": 338,
	"end": 360,
	"text": "[Kroch and Joshi,1985]",
	"ref_id": null
	}
	],
	"ref_spans": [],
	"eq_spans": [],
	"section": "INTRODUCTION",
	"sec_num": "1."
	},
	{
	"text": "In this paper, we will describe some new results for TAG's, especially in the following areas: (1) parsing complexity of TAG's, (2) some closure results for TAG's, and (3) the relationship to Head grammars. These topics will be covered in Sections 3, 4, and S respectively. In section 2, we will give an introduction to TAG's. In section 6, we will state some properties not discussed here. A detailed exposition of these results is given in [Vijay-Shankar and Joshi,1985] . \"This work wu partially supported by NSF Grants MCS-821011e-CER. MCS-82~7204. We want to thuk Carl P'oUard, Kelly Roach, David SeaM, tad David Weir. We have benefited enormously by valuable dbcussion8 with them.",
	"cite_spans": [
	{
	"start": 128,
	"end": 131,
	"text": "(2)",
	"ref_id": "BIBREF1"
	},
	{
	"start": 461,
	"end": 472,
	"text": "Joshi,1985]",
	"ref_id": null
	}
	],
	"ref_spans": [],
	"eq_spans": [],
	"section": "INTRODUCTION",
	"sec_num": "1."
	},
	{
	"text": "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.",
	"cite_spans": [
	{
	"start": 183,
	"end": 195,
	"text": "[Joshi,1983]",
	"ref_id": null
	}
	],
	"ref_spans": [],
	"eq_spans": [],
	"section": "TREE ADJOINING GRAMMARS--TAG's",
	"sec_num": "2."
	},
	{
	"text": "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. 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'.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "TREE ADJOINING GRAMMARS--TAG's",
	"sec_num": "2."
	},
	{
	"text": "We will now define a composition operation called adjoininL (or adjunetion) which composes an auxiliary tree fl with a tree \"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.)",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "TREE ADJOINING GRAMMARS--TAG's",
	"sec_num": "2."
	},
	{
	"text": "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 precls\u00a2 beesas\u00a2 it depends on the notion of appropriate structural descriptions Adjoining can now be defined as follows, if p is adjoined to \"I st the node n then the resulting tree \"ft' is as shown in Fig. 2 ",
	"cite_spans": [],
	"ref_spans": [
	{
	"start": 705,
	"end": 711,
	"text": "Fig. 2",
	"ref_id": "FIGREF3"
	}
	],
	"eq_spans": [],
	"section": "TREE ADJOINING GRAMMARS--TAG's",
	"sec_num": "2."
	},
	{
	"text": "I x \\ t --i \\-- I \\ --x--p IX i \\+--t Fisure ~",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "TREE ADJOINING GRAMMARS--TAG's",
	"sec_num": "2."
	},
	{
	"text": "The tree t dominated by X in \"~ is excised, ~ is inserted at\" the node n in \"1 and the tree t is attached to the foot node (labelled X) of ,\u00a2. i.e., ~ is inserted or adjoined to the node n in 7 pushing t do*swards. Note that adjoining is not a substitution operation.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "TREE ADJOINING GRAMMARS--TAG's",
	"sec_num": "2."
	},
	{
	"text": "We will now define",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "TREE ADJOINING GRAMMARS--TAG's",
	"sec_num": "2."
	},
	{
	"text": "The set of all truce derived in G starting from initial trees in I. This set will be called the tree net of G.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "T(G):",
	"sec_num": null
	},
	{
	"text": "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). Theorem 2.1: For every context-free grammar, G', there is an equivalent TAG, G, both weakly and strongly. Theorem 2.2: For ever,/ TAG, G, we have the following situations:",
	"cite_spans": [
	{
	"start": 366,
	"end": 401,
	"text": "([Joehi, Levy, and Takahashi, 1975]",
	"ref_id": null
	}
	],
	"ref_spans": [],
	"eq_spans": [],
	"section": "L(G):",
	"sec_num": null
	},
	{
	"text": "*. I,(G) is context-free and there is a context4ree grammar G' that is strongly (end therefore weakly) equivalent to G.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "L(G):",
	"sec_num": null
	},
	{
	"text": "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. \u00a2. L(G) is strictly context-sensitive. Obviously in this case, there is no context-free grammar that is weakly equivalent to G.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "L(G):",
	"sec_num": null
	},
	{
	"text": "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). 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. The precise definition of L(G) is as follows: In [Joshi ,1983] . local constraints o\u2022 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\u2022straints was never fully utilized, both in the linguistic context as well as in the \"formal languages\" of TAG's. The so-called proper analysis contexts and domination contexts (as defined i\u2022 [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.",
	"cite_spans": [
	{
	"start": 983,
	"end": 996,
	"text": "[Joshi ,1983]",
	"ref_id": null
	},
	{
	"start": 1063,
	"end": 1085,
	"text": "[Joshi and Levy ,1977]",
	"ref_id": "BIBREF2"
	},
	{
	"start": 1461,
	"end": 1484,
	"text": "[Joshi and Levy ,1977l)",
	"ref_id": null
	},
	{
	"start": 1496,
	"end": 1504,
	"text": "[Joshi .",
	"ref_id": null
	}
	],
	"ref_spans": [],
	"eq_spans": [],
	"section": "L(G):",
	"sec_num": null
	},
	{
	"text": "L(G) \u2022-LI \u2022ffi {w \u2022 e u / u > o,",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "L(G):",
	"sec_num": null
	},
	{
	"text": "Based on this observation and a suggestio\u2022 in [Joshi, Levy and Takahashi ,1975 ], we will describe a new way of introducing local \u00a2o\u2022strainta. This approach \u2022ot 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\u2022 in [Joshi ,1983] .",
	"cite_spans": [
	{
	"start": 46,
	"end": 78,
	"text": "[Joshi, Levy and Takahashi ,1975",
	"ref_id": null
	},
	{
	"start": 334,
	"end": 347,
	"text": "[Joshi ,1983]",
	"ref_id": null
	}
	],
	"ref_spans": [],
	"eq_spans": [],
	"section": "L(G):",
	"sec_num": null
	},
	{
	"text": "A precise characterization of that approach still remains an ope\u2022 problem. 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 \u2022t the node n. Note that if there is no constraint then \u2022H auxiliary trees are adjoinabl\u00a2 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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "L(G):",
	"sec_num": null
	},
	{
	"text": "We will adopt the following conventions.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "L(G):",
	"sec_num": null
	},
	{
	"text": "1. Since, by definition, no auxiliary trees are adjoinable to a node labelled by a terminal symbol, \u2022o co\u2022straint ha8 to be stated for node labelled by a terminal.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "L(G):",
	"sec_num": null
	},
	{
	"text": "If there is no constraint, i.e., all auxiliary trees (with the appropriate root label) are adjoinable \u2022t a node, say, n, then we will not state this explicitly.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "2.",
	"sec_num": null
	},
	{
	"text": "3. If no auxiliary trees are adjoinabie at a \u2022ode n, then we will write the constraint as (~b), where \u00a2b de\u2022ores the null set.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "2.",
	"sec_num": null
	},
	{
	"text": "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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "2.",
	"sec_num": null
	},
	{
	"text": "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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "2.",
	"sec_num": null
	},
	{
	"text": "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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "2.",
	"sec_num": null
	},
	{
	"text": "If C equals the set of all auxiliary trees adjoinable at n, then we do not explicitly state this at the node u. ",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "2.",
	"sec_num": null
	},
	{
	"text": "(#=) /\\ /\\ / \\ / \\ = s (4,) (\u00a2) s b",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Null Adjoinin~",
	"sec_num": "2."
	},
	{
	"text": "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 [(~,~. node. Similarly for PS\"",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Null Adjoinin~",
	"sec_num": "2."
	},
	{
	"text": "We must now modify our definition of adjoining to take care of the local constraints, given a tree \"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 \"1 at the node n if ~ E #, where B is the constraint associated with the node n in \"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 \u00b0, the constraint associated with the foot node of ~. Thus. given 7= p=",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Null Adjoinin~",
	"sec_num": "2."
	},
	{
	"text": "s A (c') / \\ node n I \\ / ^ (c) / \\ I1\\ I \\ ll \\\\ 1 \\ II \\\\ I \\ . . . . . . . . . . . . . . . . It .....",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Null Adjoinin~",
	"sec_num": "2."
	},
	{
	"text": "The resultant tree \"/' is \".",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(c l)",
	"sec_num": null
	},
	{
	"text": "-/' = S /\\ / \\ / \\ / A (c') / /\\ \\ ---/ \\--- / \\ / x (c') / /\\ \\ ---/ \\--- / \\",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(c l)",
	"sec_num": null
	},
	{
	"text": "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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(c l)",
	"sec_num": null
	},
	{
	"text": "Example 2.5: where",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(c l)",
	"sec_num": null
	},
	{
	"text": "Z : 0 = Let G = (I,A) be a TAG with local constraints A: ~= s (~) /I II a S /1\\ I1\\ b I c s (~)",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(c l)",
	"sec_num": null
	},
	{
	"text": "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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(c l)",
	"sec_num": null
	},
	{
	"text": "Starting with a I and adjoining ,8 to a ! at the root node we obtain",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(c l)",
	"sec_num": null
	},
	{
	"text": "3' = s (4O /I /I t S /1\\ /1\\ b I \u00a2 S (\u00a2) I @",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(c l)",
	"sec_num": null
	},
	{
	"text": "Adjoining ~ to the center S node (the only node at which adjunction can be made} we have",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(c l)",
	"sec_num": null
	},
	{
	"text": "s (~) II II n/s\" (~) / II ~ //I 'aS ~ ,-- P / I1\\ / II\\ I b I c t L. ..... II\\ b l c s (~) I U",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(c l)",
	"sec_num": null
	},
	{
	"text": "It is easy to see that G generates the string language L= {a\"b~ec\"/n >o} Other languages such ~ L'f{a u' In >I}, L\" ----{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 \u2022 contact growth\" property, see [Joshi , 198 .3J.",
	"cite_spans": [
	{
	"start": 302,
	"end": 310,
	"text": "[Joshi ,",
	"ref_id": null
	},
	{
	"start": 311,
	"end": 314,
	"text": "198",
	"ref_id": null
	}
	],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(c l)",
	"sec_num": null
	},
	{
	"text": "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!",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(c l)",
	"sec_num": null
	},
	{
	"text": "We now give a couple of lingnistle examples. Readers may refer to [Kroch and Joshi, 1985J for details.",
	"cite_spans": [
	{
	"start": 66,
	"end": 89,
	"text": "[Kroch and Joshi, 1985J",
	"ref_id": null
	}
	],
	"ref_spans": [],
	"eq_spans": [],
	"section": "l.I. Simple Linguistic Examples",
	"sec_num": null
	},
	{
	"text": "1. Starting with \"II m a I which is an initial tree and then adjoining Pl (with appropriate lexical insertions) at the indicated node in a I. we obtain '/s- ",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "l.I. Simple Linguistic Examples",
	"sec_num": null
	},
	{
	"text": "71 = ~tl = $ /\\ Nps /\\ I\\ DET Ii V",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "l.I. Simple Linguistic Examples",
	"sec_num": null
	},
	{
	"text": "We will give \u2022 few additional definitions. These are not necessary for defining derivations in \u2022 TAG as defined in section 2. However, they are introduced to help explain the parsing algorithm and the proofs for some of the closure properties of TAL's. DEFINITION 3.1 Let %3`' be two trees.We say 3` I---3`' if i\u2022 3` we adjoin an auxiliary tree to obtain 3`'. I--* is the reflexive,transitive closure of [--. DEFINITION 3.2 7' is called \u2022 derived tree if 3` ],--\" 3`' for some elementary tree % We then say \"7' 6 D(3`).",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "PARSING TREE-ADJOINING LANGUAGES 3,1, Definitions",
	"sec_num": "3."
	},
	{
	"text": "The frontier of any derived tree \"I belongs to either L ~ N E + U LE t-N E \u00b0 if 3'6 D($) for some auxiliary tree ~0, or to E* if 3` 6 D(o) for some initial tree \u00a2x. Note if 3` 6 D(c~) for some initial tree \u00a2x, then 3` is also \u2022 sentential tree.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "PARSING TREE-ADJOINING LANGUAGES 3,1, Definitions",
	"sec_num": "3."
	},
	{
	"text": "If ~ is an auxiliary tree, 3` 6 D(~) and the frontier of 3` is w s X w 2 (X is \u2022 nonterminal,Wl,W 2 6 L ~') the\u2022 the leaf node having this non-terminal symbol X at the frontier is called the foot of 3`. Sometimes we will be loosely using the phrase \"adjoining with a derived tree\" ,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 \"adjoin\" at the original node.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "PARSING TREE-ADJOINING LANGUAGES 3,1, Definitions",
	"sec_num": "3."
	},
	{
	"text": "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 \u2022sea 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.",
	"cite_spans": [
	{
	"start": 145,
	"end": 166,
	"text": "[Aho and Ullman,1073]",
	"ref_id": null
	}
	],
	"ref_spans": [],
	"eq_spans": [],
	"section": "3.~. The Parsing Algorithm",
	"sec_num": null
	},
	{
	"text": "We shall a~ume that any node in the elementary trees in the grammar hal \u2022tmost two children. This assumption ca\u2022 be made without \u2022\u2022y loss of generality, bee\u2022use it can be easily shown that for any TAG G there is \u2022n equivalent TAG G ! such that \u2022\u2022y 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 \u2022 transitive and reflexive relation, for example, the foot \u2022ode may be called the ancestor of the foot node.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "3.~. The Parsing Algorithm",
	"sec_num": null
	},
	{
	"text": "The algorithm works as follows. Let al...a, n be the input to be \"parsed. We use \u2022 four.dimensional array A; each element of the array contains \u2022 subset of the nodes of derived trees. We nay \u2022 node X of \u2022 derived tree 3` belongs to A[i~,k01J if X dominates \u2022 nab-tree of 3` whose frontier is given by either ai+i...a j Y nk+l...a u (where the foot node of \"7 is labelled by V) or ai+v..a u (i.e., j ~ k. This corresponds to the case When -f is \u2022 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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "3.~. The Parsing Algorithm",
	"sec_num": null
	},
	{
	"text": "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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "3.~. The Parsing Algorithm",
	"sec_num": null
	},
	{
	"text": "We are now in n position to fill in \u202211 the elements of the array A. There are five cases to be considered. 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. ~\"",
	"cite_spans": [],
	"ref_spans": [
	{
	"start": 330,
	"end": 336,
	"text": "Fig 3.",
	"ref_id": null
	}
	],
	"eq_spans": [],
	"section": "3.~. The Parsing Algorithm",
	"sec_num": null
	},
	{
	"text": "Case 5. U a node X E Ali,j,k,I], and the root Y of \u2022 derived tree \"/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 3. 1c.",
	"cite_spans": [],
	"ref_spans": [
	{
	"start": 199,
	"end": 206,
	"text": "Fig 3.",
	"ref_id": null
	}
	],
	"eq_spans": [],
	"section": "3.~. The Parsing Algorithm",
	"sec_num": null
	},
	{
	"text": "T /\\ /\\ / \\ / \\ / \\ / \\ / \\ / \\ / z' \\ / \\ / /\\ \\ / \\ / I \\ \\ / \\ / / \\ \\ / \\ / V' y. \\ .......... X ........ I /\\ I\\ \\ I\\ / / \\ / \\ \\ I / \\ I i / \\/ \\ \\ n / \\ n ......... x ................. / \\ ...... / \\ t iI i i i .........",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "(t) X' (c)",
	"sec_num": null
	},
	{
	"text": "i Jk l n n t",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "$--:",
	"sec_num": null
	},
	{
	"text": "I I I (b) X\" /\\ i j k 1 I \\ / \\ / \\ I Z' \\ / /\\ \\ / / \\ \\ / / \\ \\ / v' T' \\ / /\\ /\\ \\ / / \\ I \\ \\ / / \\1 \\ \\ ................. X t ........",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "$--:",
	"sec_num": null
	},
	{
	"text": "Fil~ure 3.__.~",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "I I I II I i J 1 nn p",
	"sec_num": null
	},
	{
	"text": "Although we have stated that the elements of the array contain \u2022 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 \u2022 constant, determined by the grammar. It is hoped that the presentation of the algorithm below will make it clear why we do m.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "I I I II I i J 1 nn p",
	"sec_num": null
	},
	{
	"text": "The compkteMgorithmk given below",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 1 For i=0 to n-I step 1 do",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 2 put all nodes in the frontier of elementary true v hose lnbel i8 \u2022tt in a[i.il,il.ll].",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 3 For i=O to n-I step I do",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 4",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "for J=l to n-I step 1 do",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 6 put foot nodes of all nuxilinry trees In A[l.t.J.J]",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 6 For 1=0 to n step 1 do",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 7",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "For i=l to 0 step -I do ~Step 8 For J=i to I step \| do",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 9 For k=l to J step -1 do",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 10 do Cue 1",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 11",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "do Cane 2",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 12 do Case 8",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 13 do Case 6",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 14 do Case 4",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "Step 18 Accept if root of sons initial tree E A[O.J.J,n], 0<j<n",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "where, ",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "a.a. The allgorlthm",
	"sec_num": null
	},
	{
	"text": "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).",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Complexity of the Algorithm",
	"sec_num": "3.4."
	},
	{
	"text": "Step 15 is completed in O(n). Therefore, the time complexity of the parsing algorithm is O(nS).",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Complexity of the Algorithm",
	"sec_num": "3.4."
	},
	{
	"text": "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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Correctness of the Algorithm",
	"sec_num": "3.5."
	},
	{
	"text": "We give an informal argument below. ",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Correctness of the Algorithm",
	"sec_num": "3.5."
	},
	{
	"text": "So far,we have ~\u2022med that the give\u2022 grammar has \u2022o 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\u2022 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 \u2022ode 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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Pining with Local Coustrslnt6",
	"sec_num": "3.6."
	},
	{
	"text": "We then alter the algorithm as follows. If (X,Ct) refers to \u2022 node at which we attempt to adjoin with \u2022n auxiliary tree {whose root is denoted by (Y,Ca)). then adjunctioa would determined by C t. 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\u2022 obligatory constraint.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Pining with Local Coustrslnt6",
	"sec_num": "3.6."
	},
	{
	"text": "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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Pining with Local Coustrslnt6",
	"sec_num": "3.6."
	},
	{
	"text": "A parser based on the techniques described above is currently being implemented and will be reported at time of presentation.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Pining with Local Coustrslnt6",
	"sec_num": "3.6."
	},
	{
	"text": "In this section, we present some closure results for TALe. We now informally sketch the proofs for the closure properties. Interested readers may refer to [Vijay-Shankar and Joshi,19851 fort the complete proofs.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "CLOSURE PROPERTIES OF TAG's",
	"sec_num": "4."
	},
	{
	"text": "Let G 1 and G 2 be two TAGs generating L! and ~ respectively. We can construct a TAG G such that L(G)~L! tJ L2. ",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Closure under Union",
	"sec_num": "4.1."
	},
	{
	"text": "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\"1 N2 ,m at. We can construct \u2022 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 \u00a2oustrxints 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. 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) \u2022= 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.",
	"cite_spans": [],
	"ref_spans": [
	{
	"start": 324,
	"end": 330,
	"text": "Fig 4.",
	"ref_id": "FIGREF4"
	},
	{
	"start": 921,
	"end": 927,
	"text": "Fig 4.",
	"ref_id": "FIGREF4"
	}
	],
	"eq_spans": [],
	"section": "4.S. Closure under Coneatenntton",
	"sec_num": null
	},
	{
	"text": "t,= / \\ ~= / \\ / \\ I \\ I \\ / \\ t12 = S I\\ / \\ I \\ / \\ st I \\ / \\ / 4~ 1 \\ / ta \\",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "4.S. Closure under Coneatenntton",
	"sec_num": null
	},
	{
	"text": "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.",
	"cite_spans": [],
	"ref_spans": [
	{
	"start": 28,
	"end": 34,
	"text": "Fig 4.",
	"ref_id": "FIGREF4"
	}
	],
	"eq_spans": [],
	"section": "4.S. Closure under Coneatenntton",
	"sec_num": null
	},
	{
	"text": "To see why L(G) .-Lt\" , consider x (~ L(G). Obviously, the tree derived (whose frontier is given by x ) must be of the form shown in Fig 4.3\u00a2 , 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'.",
	"cite_spans": [],
	"ref_spans": [
	{
	"start": 133,
	"end": 141,
	"text": "Fig 4.3\u00a2",
	"ref_id": "FIGREF4"
	}
	],
	"eq_spans": [],
	"section": "4.S. Closure under Coneatenntton",
	"sec_num": null
	},
	{
	"text": "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\u2022 by Fig 4. 3\u00a2, using only the \u2022 ~xiliary trees in A t . The frontier of T' is obviously ws...w n. Hence, x ..__.33",
	"cite_spans": [],
	"ref_spans": [
	{
	"start": 396,
	"end": 402,
	"text": "Fig 4.",
	"ref_id": "FIGREF4"
	}
	],
	"eq_spans": [],
	"section": "4.S. Closure under Coneatenntton",
	"sec_num": null
	},
	{
	"text": "G L(G). Therefore, L t. G L(G). Thus L(G) --L,'.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "4.S. Closure under Coneatenntton",
	"sec_num": null
	},
	{
	"text": "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",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	},
	{
	"text": "(GI) --L T N L R.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	},
	{
	"text": "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\"(ql, w2) = q2, and 6\"(q~, ws) \u2022ffi q4-When \"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\"(q2, v) == qs. Following this reasoning, we must make q2 ~ qa, if Z is not the ancestor of the foot node of % or if \"7 is in D(a) for some initial tree a in G.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	},
	{
	"text": "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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	},
	{
	"text": "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 \u2022ode in some elementary tree a. Let (qt,q2,q3,q4) be associated with X. Then, we have to consider the follo'~ing caacs. 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, \" 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 ( \u00a2b r, r. q4 ) with Y and Z respectively, and associate the null adjoining constraint with X. If the label of Z is ~, where \u2022 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.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	},
	{
	"text": "Case 2: This corresponds to the ease where a node X has two children Y and Z, with (ql,qs,qs,q4) ~mocinted at X. Let Z ( the right 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 \u2022 o~(ql,a) = r.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	},
	{
	"text": "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~ \u2022ffi 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.",
	"cite_spans": [],
	"ref_spans": [
	{
	"start": 351,
	"end": 361,
	"text": "(ql,p,t,q4",
	"ref_id": "FIGREF4"
	}
	],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	},
	{
	"text": "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 \"constraint associated with this \u2022ode or Y is labelled with a terminal symbol, are treated similar to how we dealt with them ia the previous cases.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	},
	{
	"text": "Once this has bee\u2022 done, let ql,\"',qm be the independent variables for this elementary tree a, then we produce as many copies of a so that ql,\"',qm take all possible values from Q. The only difference among the various copies of a so produced will be constraints associated with the \u2022odes in the trees. Repeat the process for all the elementary trees in G !. Once this has bee\u2022 done and each tree given \u2022 unique name we can write the constraints in terms of these \u2022ames. We will now show why L(GI) =ffi L T f3 L R.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	},
	{
	"text": "Let w E L(GI). Theu there is s seque\u2022ce of adjoining operatio\u2022s 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\u2022ding tree ia G, which differs only in the \u00a2onstrai\u2022ts associated with its \u2022odes. Note, however, that the \u00a2o\u2022strai\u2022ts associated with the \u2022odes in trees in G t are just * restriction of the correspo\u2022ding o\u2022es in G, or an obHgatoiT \u00a2o\u2022straint where there was \u2022 o\u2022e in G. Now, if we can assume ( by inductin\u2022 hypothesis ) that if ~fter n adjoining operatio\u2022s we can derive \"f 6 D(~x'), then there is a correspo\u2022ding tree \"T 6 D(a) iu G, which has the same tree structure as ~/' but differing o\u2022ly in the constraints associated with the corresponding \u2022odes, then if we adjoin at some node in \"~' to obtain \"h', we can adjoin in \"~ to obtain \"h (corresponding to gl')-Therefore, if w can be derived in Gi, then it can dcfmitely be derived inG.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	},
	{
	"text": "If we can also show th~ L(Gi) C L a. then we can co\u2022clods that L(GI) C L T N Lit. We can use induetio\u2022 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\u2022 derived trees after \u2022 .4-I adjoiniugs where P is defi\u2022ed as, Property P: If any node X in a derived tree '3' has the foot-node of the tree p to which X belo\u2022gs labelled Y as a desce\u2022dant 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\u2022d if w is the frontier of the subtree under the foot \u2022 ode of # in '7 is then ~(qs,w) ~= q~. If X is not the ancestor of the foot \u2022ode of # then the subtree of # below is of the form wlw s. Suppose X has associated with it (qt,q,q,q2) then ~(ql,wa) -~ q, ~*(q,w2) ffi q2\" 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\u2022ed 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\u2022tential trees and observe that the roots of these trees obey property P. Now. if n string x E L T 13 L R, we ca\u2022 show that x E L(G). To do that, we make use of the foUowing claim.",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	},
	{
	"text": "Let ~ be an auxiliary tree in G with root labelled \u00a5 and let \"y 6 D(~). We claim that there is a 8' in G I with the same structure as 8, such that there is a \";' in D(bet~0)' ) where \"I' has the same structure as 7, such that there is \u2022o OA coustraint in '7'. Let X be a \u2022ode in fit which was used in deriving -;. Then there is a \u2022ode X' in 7' such that X' belongs to the auxillixry tree #l' (with the same structure as ~\|. There are several cams to co\u2022sider -Case I: X. is the ancestor of the foot node of 81, such that the fro\u2022tier 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 \u2022o OA co\u2022straint in '7'* Case 2: X is the ancestor of the foot \u2022ode 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\"(p,wz} = r, and ~(r,w4} = q4-Case 3: Let the froutier of the subtree of ~i (and also \"7) rooted at X is wlw 2. Let f(q,wl) = p, 6(p,ws) = r. The\u2022 X' will have associated with it the quadruple (q,p,p,r).",
	"cite_spans": [],
	"ref_spans": [],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	},
	{
	"text": "We shall prove our claim by inductio\u2022 o\u2022 the \u2022umber of adjoining operations used to derive \"I. The base case (where \"1 == 0) is obvious from the way the grammar G 1 was built. We shall \u2022ow assume that for all derived trees % which have bee\u2022 derived from p using k or less adjoining operatio\u2022s, have the property as required in our claim. Let ~ be a derived tree in p after k adju\u2022ctio\u2022s. By our inductive hypothesis we may ~asume the existence of the corresponding derived tree \"y' E D(~') derived in G I. Let X be a uode in 7 as shown ia Fig. 4.4.1 . Then the \u2022ode X* in \"y' eorrespo\u2022di\u2022g to X will have associated with it the quadruple (ql',q2S,q~l',q4\")-Note we are aseumin~ here that the left child Y' of X' is the ancestor of the Applying the procedure Convert_to_HG to this grammar we obtain the HG whose productions are gives by- 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. 2) senillnsurtty and Parikh-boundednus8.",
	"cite_spans": [],
	"ref_spans": [
	{
	"start": 539,
	"end": 549,
	"text": "Fig. 4.4.1",
	"ref_id": "FIGREF4"
	}
	],
	"eq_spans": [],
	"section": "Clolure under Intersection with Regulu Languages",
	"sec_num": "4.4."
	}
	],
	"back_matter": [
	{
	"text": "foot node of D'-The quedruples (qt',q~',qa',P) and (P,Pt*Pt,q4\") 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 no\u00a2le. But this should be so, since from the way we obtained the trees in GI, there will exist \"/1\" 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 \u00b0 to obtain \"Yl'-It can be shown that lt' has the required property to establish our claim. Firstly, any node below the foot of PI' in 7t' will satisfy our requirement~ as they are the same as the corresponding nodes in \"/l'-Since ~t' satisfy\u2022 the requirement, it is simple to observe that the sods\u2022 in 01' will, even after the edjuuction of I1' in \"el'\" 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\u2022is of the design of grammar GI, that there exists trees which allow this new auxiliary tree to be adjoined at the appropriate pi~ce. This then allows us to conclude that there exist \u2022 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 \u2022 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 \u00a2~-~e where the finite state automaton starts from tlie initial state q0, and teaches some fmal state \u00b04 on the ihput which is the frontier of tome sentential tree in (3. This, then allows us to conclude that L T f3Ln c_ L(C,)..nose, C(G,} --C r n t~. ?",
	"cite_spans": [],
	"ref_spans": [
	{
	"start": 182,
	"end": 188,
	"text": "Fig. 4",
	"ref_id": null
	}
	],
	"eq_spans": [],
	"section": "annex",
	"sec_num": null
	},
	{
	"text": "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. 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. 1. Adjoining ~ st the root of the sub-tree \"r gives us the seutential tree in File 5.1. We can, now see how the string whx has \"wrapped around\" 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 \u2022n 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 \u2022 l...al X ~i+l...an is the frontier of a derived tree \"1 6 D(~), then the derivation of \"/would correspond to \u2022 derivation from a non-terminal X to the string at...a i lai+t...tu in HG and the nee of \"f in some sentential tree would correspond to how the strings st... a i and ai+t...\u2022 a are used in deriving a string in IlL.",
	"cite_spans": [
	{
	"start": 281,
	"end": 293,
	"text": "[Roach,1084]",
	"ref_id": null
	}
	],
	"ref_spans": [],
	"eq_spans": [],
	"section": "HEAD GRA.MMARS AND TAG's",
	"sec_num": "5."
	},
	{
	"text": ".~_l_~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 ~\" mwty then L L~,k) --andLC,(X~) ~ kx.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.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).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).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} Same as Case 1 except don't add the productions Sya->LLl(node nyabol.g').N'->LCi(At'. .... Aj').Case 3 The constraint at the node i80A.State as Case I except that we don't add Syn->LCi(AI',...Aj') else if the node has t terainai syabol a.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.->XWe 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 ",
	"cite_spans": [],
	"ref_spans": [
	{
	"start": 2910,
	"end": 2913,
	"text": "Fig",
	"ref_id": null
	}
	],
	"eq_spans": [],
	"section": "S /\\ I \\ I z \\ / I-\\ \\ / ,/---\\,<_.l~__~ /_3",
	"sec_num": null
	}
	],
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	},
	"ref_entries": {
	"FIGREF0": {
	"type_str": "figure",
	"uris": null,
	"text": "tree o is an initial tree if it is of the form CZ~ s L% the root node of a is labelled S and the frontier nodes are all terminal symbols. The internal nodes are aU non-terminaL. A tree ~ is an a~xiliary tree if it is of the form",
	"num": null
	},
	"FIGREF1": {
	"type_str": "figure",
	"uris": null,
	"text": "Let G --(I,A) where with pa adjoined at S as indicated in 70-adjoined at T as indicated in 7=.",
	"num": null
	},
	"FIGREF2": {
	"type_str": "figure",
	"uris": null,
	"text": "Let G \u2022, (LA) where",
	"num": null
	},
	"FIGREF3": {
	"type_str": "figure",
	"uris": null,
	"text": "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], seeFig 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,\u2022,el then their parent Z belongs to A[i,j,j,n].",
	"num": null
	},
	"FIGREF4": {
	"type_str": "figure",
	"uris": null,
	"text": "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].",
	"num": null
	},
	"FIGREF5": {
	"type_str": "figure",
	"uris": null,
	"text": "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\u2022teats 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\u2022 computed. Case 2: By \u2022 similar reasoning, the contents of A[i,m,m,p] and Alp,i,k,l] are known since p < i and p > i. Case 3: When we are trying to compute the contents of some A[ij,i,I], we \u2022end to know the \u2022odes in A[i,i,i,p] and A[p,q,q,I]. Note i > i or j < I. Hence, we know that the co\u2022teats of A[i,j,j,p] and A[p,q,q,I] would have been compared already.",
	"num": null
	},
	"FIGREF6": {
	"type_str": "figure",
	"uris": null,
	"text": "The co\u2022tents of A[i,m,p,I] and A[m,j,k,p] mesa be know\u2022 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. Similarly, since either m < j or k < p, the contents of A[i,m,p,l] would have been computed.",
	"num": null
	},
	"FIGREF7": {
	"type_str": "figure",
	"uris": null,
	"text": "( ! !, A v N v S ), and G 2 ----( 12 , A 2, N 2, S ). Without loss of generality, we may Lssume that the N! f'l N 2 ~ #. LetG ~ (I IU 12, AtUA 2,N, t.JN 2, S ). We claim that L(G) -L I UL2 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). 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.",
	"num": null
	},
	"FIGREF8": {
	"type_str": "figure",
	"uris": null,
	"text": "Clo,ure under Kleene St.m.",
	"num": null
	},
	"FIGREF9": {
	"type_str": "figure",
	"uris": null,
	"text": "Figure 4 ..__.33",
	"num": null
	},
	"FIGREF10": {
	"type_str": "figure",
	"uris": null,
	"text": "\u2022 Ltq(S,b~c) or t' ->l,lq(S,l~c) be verified that this grma~ gennratsn exactly It can L(6).",
	"num": null
	},
	"FIGREF11": {
	"type_str": "figure",
	"uris": null,
	"text": "PROPERTIES OFTAG's ~. Additional formal properties of TAG's have been discussed in \|Vijay-Shtakar and Joshi,1985]. Some of them are listed below il Pumping lemma for TAG's TAL's are closed under substitution and homomorphlsms TAL's ate not closed under the following operations a) intersection with TALs b) intersection with CFL8 \u00a2) conplsntatation Some other properties that have been considered in [Vijay* Shankar tad Joehi,1985] are as follows 1) closure under the following propertieu a) inverse hmsoaorphimt b) gem napplng8",
	"num": null
	},
	"TABREF1": {
	"html": null,
	"text": "w is a string of a's and b', such that (1) the n\u2022mber of \u2022% ~ the number of b's --\u2022i and (2) for any initial substring of w, the n\u2022mber of a's .~ the number of b's. } L! is a strictly context-sensitive language (i.e., \u2022 contextsensitive language that is not context-free).",
	"content": "<table><tr><td/><td>This can be shown as</td></tr><tr><td colspan=\"2\">follows. Intersecting L with the regular language a* b* e c* results in</td></tr><tr><td>the language</td><td/></tr><tr><td colspan=\"2\">L, = { na bS e ca / n ~.. o } ~ffi LI I'l a\" b\" e c *</td></tr><tr><td colspan=\"2\">L 2 is well-known strictly co\u2022text-sensitive language. The result</td></tr><tr><td colspan=\"2\">of intersecting a context-free language with a regular language is</td></tr><tr><td colspan=\"2\">always a context-free language; hence, L i is \u2022ot \u2022 context-free</td></tr><tr><td colspan=\"2\">language. It is thus a strictly context-sensitive language. Example</td></tr><tr><td colspan=\"2\">2.3 thus illustrates part (c) of Theorem 2.2.</td></tr><tr><td colspan=\"2\">TAG's have more power than CFG's. However, the extra</td></tr><tr><td colspan=\"2\">power is quite limited. The language L! has equal number of a's, b's</td></tr><tr><td colspan=\"2\">and c's; however, the a's and b's are mixed in \u2022 certain way. The</td></tr><tr><td colspan=\"2\">language L~ is similar to Li, except that a's come before all b's.</td></tr><tr><td colspan=\"2\">TAG's as defined so far are not powerful enough to generate L=.</td></tr><tr><td colspan=\"2\">This can be seen as follows. Clearly, for any TAG for L2, each</td></tr><tr><td colspan=\"2\">initial tree must contain equal \u2022amber of \u2022'% b's and e's (including</td></tr><tr><td colspan=\"2\">zero), and each auxiliary tree must also contain equal number of a'n,</td></tr><tr><td colspan=\"2\">b's and c's. Further in each case the a's meet precede the b's. Then</td></tr><tr><td colspan=\"2\">it is easy to see from the grammar of Example 2.3, that it will not be</td></tr><tr><td colspan=\"2\">po~ible to avoid getting the a's and b's mixed. However, L~ can be</td></tr><tr><td colspan=\"2\">generated by a TAG with local constraints (see Sectio\u2022 2.1) The so-</td></tr><tr><td>tailed copy language.</td><td/></tr><tr><td colspan=\"2\">also cannot be generated by \u2022 TAG, however, again, with local</td></tr><tr><td colspan=\"2\">constraints. It is thus clear that TAG's can generate more than</td></tr><tr><td colspan=\"2\">context-free languages. It can be shown that TAG's cannot generate</td></tr><tr><td colspan=\"2\">all context-sensitive languages [Joehi ,1984].</td></tr><tr><td colspan=\"2\">Although TAG's are more powerful than CFG's, this extra</td></tr><tr><td colspan=\"2\">power is highly constrained and appace\u2022tly it is just the right kind</td></tr><tr><td colspan=\"2\">for characterizing certain structural description. TAG's share almost</td></tr><tr><td colspan=\"2\">all the formal properties of CFG's (more precisely, the correspo\u2022ding</td></tr><tr><td colspan=\"2\">classes of la\u2022guages), as we shall see in sectio\u2022 4 of this paper and</td></tr><tr><td colspan=\"2\">[Vijay-Shankar and Joshi01985]. I\u2022 addition,the string languages of</td></tr><tr><td colspan=\"2\">TAG's can also be parsed in polynomial time, in particular in O(ne).</td></tr><tr><td colspan=\"2\">The parsing algorithm is described in detail in section 3.</td></tr><tr><td colspan=\"2\">2.1. TAG's with Local Constraints on Adjoining</td></tr><tr><td colspan=\"2\">The adjoining operation as defined in Sectio\u2022 2.1 is \"context-</td></tr><tr><td>free'. An a\u2022xiliary tree, say,</td><td/></tr><tr><td>X</td><td/></tr><tr><td>/\\</td><td/></tr><tr><td>/</td><td>\\</td></tr><tr><td>/</td><td>\\</td></tr><tr><td colspan=\"2\">---X---</td></tr><tr><td colspan=\"2\">is adjoinable to \u2022 tree t at \u2022 \u2022ode, say, \u2022, if the label of that</td></tr><tr><td colspan=\"2\">node is X. Adjoining does \u2022or depend on the context (tree context)</td></tr><tr><td colspan=\"2\">around the node n. I\u2022 this sense, adjoining is co\u2022text-free.</td></tr></table>",
	"type_str": "table",
	"num": null
	},
	"TABREF3": {
	"html": null,
	"text": "Note that the initial tree a 2 is not \u2022 matrix sentence. In order for it to become \u2022 matrix sentence, it must undergo an adjunction at its root node, for example, by the auxiliary tree ~it as show\u2022 above. Thus, for o 2 we will specify a local constraint O(~t) for the root \u2022 node, indicating that o= requires for it to undergo tn adjunct\\on at the root node by an auxiliary tree ~2-In \u2022 fuller grammar there will be, of course, some alternatives in the scope of O().",
	"content": "<table><tr><td/><td/><td/><td/><td/><td>PI z</td></tr><tr><td/><td/><td/><td/><td/><td>lip /\\</td></tr><tr><td/><td/><td/><td/><td/><td>liP $ /\\</td></tr><tr><td/><td/><td colspan=\"2\">MP</td><td/></tr><tr><td>I</td><td>I</td><td colspan=\"2\">I I\\</td><td/><td>/\\</td></tr><tr><td>I</td><td>I</td><td colspan=\"2\">I I\\</td><td/><td>Ifl ~ Yp</td></tr><tr><td colspan=\"6\">the glrl I DET N is I I</td><td>I \u2022</td><td>/\\ \u00a5 liP</td></tr><tr><td/><td/><td colspan=\"4\">t senior</td><td>I</td><td>I</td></tr><tr><td/><td/><td/><td/><td/><td>net II</td></tr><tr><td colspan=\"6\">the girl t8 a senior</td><td>I</td></tr><tr><td/><td/><td/><td/><td/><td>Bill</td></tr><tr><td>\"~2 =</td><td/><td/><td/><td/></tr><tr><td/><td/><td/><td/><td/><td>S</td></tr><tr><td/><td/><td/><td/><td/><td>I \\</td></tr><tr><td/><td/><td/><td/><td colspan=\"2\">,-d</td><td>I</td><td>\\ \\</td></tr><tr><td/><td/><td/><td colspan=\"3\">tNP ~ %/\\ ~</td><td>VP /\\</td></tr><tr><td/><td/><td/><td/><td/><td>\\\\ v</td><td>NP</td></tr><tr><td/><td/><td colspan=\"4\">DgT/\\N~ S~ I / \\~i8 DET N /\\</td></tr><tr><td/><td/><td>I</td><td/><td/><td>law</td><td>s\\</td><td>I I</td></tr><tr><td/><td colspan=\"5\">the girl~ ~T/ \\ X\\ \u2022 senior</td></tr><tr><td/><td/><td/><td/><td/><td>VP \\</td></tr><tr><td/><td/><td/><td/><td/><td>I I</td><td>/\\</td><td>\\</td></tr><tr><td/><td/><td/><td/><td/><td>le</td><td>V m'\\</td></tr><tr><td/><td/><td/><td/><td/><td>net I ~ ~pt</td></tr><tr><td/><td/><td/><td/><td/><td>\\</td><td>N</td><td>t</td></tr><tr><td/><td/><td/><td/><td/><td>\\</td><td>I</td><td>I</td></tr><tr><td/><td/><td/><td/><td/><td>Blll</td></tr><tr><td/><td colspan=\"5\">The girl vho net Blll 18 \u2022 senior</td></tr><tr><td colspan=\"6\">2. Starting with the initial tree 3`1 ~ a2 and adjoining ~2 at</td></tr><tr><td colspan=\"6\">the'indicated node in a 2 we obtain 3`2-</td></tr><tr><td>3`1 = el2 =</td><td/><td/><td/><td/></tr><tr><td/><td colspan=\"2\">* S</td><td/><td/><td>0(82)</td><td>S</td></tr><tr><td/><td/><td>/\\</td><td/><td/><td>/\\</td></tr><tr><td/><td colspan=\"4\">NP VP</td><td>NP VP</td></tr><tr><td/><td>l</td><td colspan=\"2\">IX</td><td/><td>I</td><td>II\\</td></tr><tr><td/><td colspan=\"5\">PRO To vP</td><td>H</td><td>I I \\</td></tr><tr><td/><td/><td/><td/><td colspan=\"2\">I\\</td><td>I v ~</td><td>S (~)</td></tr><tr><td/><td/><td/><td colspan=\"3\">V lip</td><td>John J \\</td></tr><tr><td/><td/><td/><td colspan=\"3\">I I</td><td>I \\</td></tr><tr><td/><td colspan=\"3\">invite</td><td/><td>Ii</td><td>persuaded N</td></tr><tr><td/><td/><td/><td/><td/><td>I</td><td>I</td></tr><tr><td/><td/><td/><td/><td/><td>Mary</td><td>Bill</td></tr><tr><td/><td colspan=\"5\">PRO to invite Mary</td><td>John persuaded Bill S</td></tr><tr><td>3`2 =</td><td/><td/><td/><td/></tr><tr><td/><td/><td/><td/><td/><td>4 S N / /\\ ,,</td></tr><tr><td/><td/><td/><td>/</td><td colspan=\"2\">/tIP</td><td>VP ~</td></tr><tr><td/><td/><td>/</td><td/><td/><td>I</td><td>II\\ \"~</td><td>~</td><td>.8 2</td></tr><tr><td/><td/><td>1</td><td/><td/><td>N II\\</td><td>\\</td></tr><tr><td/><td/><td colspan=\"4\">I John I</td></tr><tr><td/><td/><td>I</td><td/><td/><td>I</td></tr><tr><td/><td/><td colspan=\"4\">I persuaded I~ I /\\</td></tr><tr><td/><td/><td>t</td><td/><td/><td>I; IrovP</td></tr><tr><td/><td/><td>\\</td><td colspan=\"2\">%</td><td>..</td><td>I tPa0 I\\ Bill~ V NP</td></tr><tr><td/><td/><td/><td/><td/><td>....</td><td>I</td><td>J</td></tr><tr><td/><td/><td/><td/><td/><td>invite N</td></tr><tr><td/><td/><td/><td/><td/><td>I</td></tr><tr><td/><td/><td/><td/><td/><td>if try</td></tr><tr><td/><td/><td colspan=\"4\">John persuaded Bill to invite Mary</td></tr></table>",
	"type_str": "table",
	"num": null
	},
	"TABREF4": {
	"html": null,
	"text": "Case 3 corresponds to the case where \u2022either ehildre\u2022 \u2022re ancestors of the foot \u2022ode. If the left sibling E A[i,j,j,m] and the right \u2022 sibling E A[m,p,p,l I then we ca\u2022 pet the parent in A[i,j,j,l] if it is the \u00a2g~ulthat(i<j <_ mori_<j <m) and(m < p< Iorm < p < \|), This may be written as",
	"content": "<table><tr><td>(\u00a2) for \u2022 = J to 1-1 step I do</td></tr><tr><td>for p = J to 1 step i do</td></tr><tr><td>for all left 81blLngs In ACL,\|,J.a] and</td></tr><tr><td>right slblings in A[nop.p.l] 8atisfylng the appropriate</td></tr><tr><td>restrictions pet their parent in A[ioJ,J\u00b0l].</td></tr><tr><td>(e) Case 5 corresponds to adjoining. If X is \u2022 node in A[m~,k,p] and</td></tr><tr><td>Y is the root of a auxiliary tree with same symbol as that of X, such</td></tr><tr><td>thatYisiuA[i,m,p,I]((i < m < p < Iori < m < p < l) and(m</td></tr><tr><td>< j < k_ porm_~ j < k < p)). Thls may be written as</td></tr><tr><td>for \u2022 = ~. to \| step t do</td></tr><tr><td>for p = \u2022 to I step I do</td></tr><tr><td>if t node X 6 A[n,J.k,p] tad the root of</td></tr><tr><td>auxiliary tree Is in A[i,a.pol] then put X in A[l.J.k.1]</td></tr><tr><td>Case 4 corresponds to the case where a node Y has only one child X</td></tr><tr><td>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</td></tr><tr><td>no siblings.</td></tr><tr><td>Came 1 corresponds to situation where the left sibling is the</td></tr><tr><td>ancestor of the foot node. The parent is put in A[Q,k,I] if the left</td></tr><tr><td>sibling is in A[i,j,k,m] and the right sibling is in A[m,p,p,l], where k</td></tr><tr><td>_ m < I, m ~ p, p < I. Therefore Case I is written as</td></tr><tr><td>For n=k to 1-I step I do</td></tr><tr><td>for p= n to 1,step I do</td></tr><tr><td>if there is a left sibling in A[i,J.k.n] and the</td></tr><tr><td>right sibling in A[a.p.p.1] satisfying appropriate</td></tr><tr><td>restrictions then put their parent</td></tr><tr><td>in A[i,J.k.1].</td></tr><tr><td>(b) Case 2 corresponds to the ease where the right sibling is the</td></tr><tr><td>ancestor of the foot node. If the left sibling is in A[i,m,m,p] and the</td></tr><tr><td>right sibling is in A[p,j,k,I], i < m < p and p < j, then we put their</td></tr><tr><td>parent in A[i,j,k,I]. This may be written as</td></tr><tr><td>For n=i to J-! step I do</td></tr><tr><td>For p=a*l to ] step i do</td></tr><tr><td>for all left siblin~ in A[i.n,m,p] ud right</td></tr><tr><td>siblings</td></tr><tr><td>in A[p.J.k.1] satisfying appropriate rentrictionn put</td></tr><tr><td>thei= parents</td></tr><tr><td>in A[i.J.k.1].</td></tr></table>",
	"type_str": "table",
	"num": null
	},
	"TABREF5": {
	"html": null,
	"text": "Actually what we mean by an adjoining operation is \u2022of necessarily just o\u2022e adjoining operatio\u2022 but the minimum number so that no obligatory co\u2022straints are associated with any \u2022odes in the derived trees. Similarly, the base case \u2022teed ant cousider o\u2022ly elementary trees, but the smallest (in terms of the \u2022umber of adjoining operatin\u2022s) tree starting with eleme\u2022tary trees which has \u2022 o obligatory coustrai\u2022t associated with any of its \u2022odes. The base case ca\u2022 be see\u2022 easily co\u2022sidering",
	"content": "<table/>",
	"type_str": "table",
	"num": null
	}
	}
	}
	}