Base_JSON/prefixJ/json/J92/J92-4004.json

{
    "paper_id": "J92-4004",
    "header": {
        "generated_with": "S2ORC 1.0.0",
        "date_generated": "2023-01-19T02:44:48.258309Z"
    },
    "title": "Using Descriptions of Trees in a Tree Adjoining Grammar",
    "authors": [
        {
            "first": "K",
            "middle": [],
            "last": "Vijay-Shanker",
            "suffix": "",
            "affiliation": {
                "laboratory": "",
                "institution": "University of Delaware",
                "location": {}
            },
            "email": ""
        }
    ],
    "year": "",
    "venue": null,
    "identifiers": {},
    "abstract": "This paper describes a new interpretation of Tree Adjoining Grammars (TAG) that allows the embedding of TAG in the unification framework in a manner consistent with the declarative approach taken in this framework. In the new interpretation we present in this paper, the objects manipulated by a TAG are considered to be descriptions of trees. This is in contrast to the traditional view that in a TAG the composition operations of adjoining and substitution combine trees. Borrowing ideas from Description Theory, we propose quasi-trees as a means to represent partial descriptions of trees. Using quasi-trees, we are able to justify the definition of feature structurebased Tree Adjoining Grammars (FTAG) that was first given in Vijay-Shanker (1987) and Vijay-Shanker and Joshi (1988). In the definition of the FTAG formalism given here, we argue that a grammar manipulates descriptions of trees (i.e., quasi-trees); whereas the structures derived by a grammar are trees that are obtained by taking the minimal readings of such descriptions. We then build on and refine the earlier version of FTAG, give examples that illustrate the usefulness of embedding TAG in the unification framework, and present a logical formulation (and its associated semantics) of FTA G that shows the separation between descriptions of well-formed structures and the actual structures that are derived, a theme that is central to this work. Finally, we discuss some questions that are raised by our new interpretation of the TAG formalism: questions dealing with the nature and definition of the adjoining operation (in contrast to substitution), its relation to multi-component adjoining, and the distinctions between auxiliary and initial structures.",
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        "abstract": [
            {
                "text": "This paper describes a new interpretation of Tree Adjoining Grammars (TAG) that allows the embedding of TAG in the unification framework in a manner consistent with the declarative approach taken in this framework. In the new interpretation we present in this paper, the objects manipulated by a TAG are considered to be descriptions of trees. This is in contrast to the traditional view that in a TAG the composition operations of adjoining and substitution combine trees. Borrowing ideas from Description Theory, we propose quasi-trees as a means to represent partial descriptions of trees. Using quasi-trees, we are able to justify the definition of feature structurebased Tree Adjoining Grammars (FTAG) that was first given in Vijay-Shanker (1987) and Vijay-Shanker and Joshi (1988). In the definition of the FTAG formalism given here, we argue that a grammar manipulates descriptions of trees (i.e., quasi-trees); whereas the structures derived by a grammar are trees that are obtained by taking the minimal readings of such descriptions. We then build on and refine the earlier version of FTAG, give examples that illustrate the usefulness of embedding TAG in the unification framework, and present a logical formulation (and its associated semantics) of FTA G that shows the separation between descriptions of well-formed structures and the actual structures that are derived, a theme that is central to this work. Finally, we discuss some questions that are raised by our new interpretation of the TAG formalism: questions dealing with the nature and definition of the adjoining operation (in contrast to substitution), its relation to multi-component adjoining, and the distinctions between auxiliary and initial structures.",
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                "section": "Abstract",
                "sec_num": null
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            {
                "text": "A number of grammatical systems and linguistic theories, such as Functional Unification Grammars (FUGs), Lexical Functional Grammars (LFGs), Generalized Phrase Structure Grammars (GPSGs), and Head-driven Phrase Structure Grammars (HPSGs), are said to take the unification-based approach to grammars. A common aspect shared by these grammars or theories is that they are based on specifying constraints that define well-formed structures. This work discusses viewing Tree Adjoining Grammars (TAG) in such a manner and embedding it in a unification-based framework.",
                "cite_spans": [],
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                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "Tree Adjoining Grammars (TAG) were first introduced by Joshi, Levy, and Takahashi (1975) . A preliminary study of this formalism, from the point of view of its formal properties and linguistic applicability, was carried out by Joshi (1985) . A detailed study of the linguistic relevance of TAG was done by Kroch and Joshi (1985) . Abeille et al. (1990) discuss a fairly substantial grammar for English using the lexicalized approach to TAG that was originally proposed by Schabes, Abeille, and Joshi (1988) .",
                "cite_spans": [
                    {
                        "start": 55,
                        "end": 88,
                        "text": "Joshi, Levy, and Takahashi (1975)",
                        "ref_id": "BIBREF10"
                    },
                    {
                        "start": 227,
                        "end": 239,
                        "text": "Joshi (1985)",
                        "ref_id": "BIBREF7"
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                    {
                        "start": 306,
                        "end": 328,
                        "text": "Kroch and Joshi (1985)",
                        "ref_id": null
                    },
                    {
                        "start": 331,
                        "end": 352,
                        "text": "Abeille et al. (1990)",
                        "ref_id": "BIBREF0"
                    },
                    {
                        "start": 472,
                        "end": 506,
                        "text": "Schabes, Abeille, and Joshi (1988)",
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                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "\u2022 Department of Computer and Information Sciences, University of Delaware, Newark, DE 19716. (~) 1992 Association for Computational Linguistics TAG is defined as a tree rewriting system. In the definition given traditionally, a TAG is defined by a finite set of trees and an operation called adjoining to compose trees. One of the basic intuitions underlying the use of the TAG formalism is that these trees provide a large enough structure that most cooccurrence restrictions (dependencies) can be stated over (localized within) these trees. Predicate-argument, wh-dependencies, and filler-gap dependencies are examples of dependencies that can be localized in a TAG.",
                "cite_spans": [
                    {
                        "start": 65,
                        "end": 92,
                        "text": "Delaware, Newark, DE 19716.",
                        "ref_id": null
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                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "Our aim is to view a TAG as defining constraints on well-formed structures (according to the linguistic intuitions being instantiated in the grammar). In this paper, we argue that if we chose to interpret the objects manipulated by a TAG as trees (as is done currently) then it is not possible to embed TAG in a unification framework in a straightforward manner. We show that this follows from the fact that the adjoining operation on trees is such that it does not preserve the structural relationships that have been specified in the structures being combined. We argue that instead we should view the objects manipulated (to be distinguished from derived) by a TAG as (partial) descriptions of trees. In particular, these descriptions include the partial specification of domination, as in description theory or D-theory (Marcus, Hindle, and Fleck 1983) , in addition to the specification of immediate domination. We argue that this is a well-motivated interpretation that is consistent with certain assumptions made in the lexicalized approach to TAG. We introduce quasi-trees as a means to structurally depict partial specifications of trees. Using this interpretation, we show that the resulting structure obtained after adjoining preserves the structural relationships described in the structures being composed.",
                "cite_spans": [
                    {
                        "start": 824,
                        "end": 856,
                        "text": "(Marcus, Hindle, and Fleck 1983)",
                        "ref_id": "BIBREF15"
                    }
                ],
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                "eq_spans": [],
                "section": "Introduction",
                "sec_num": "1."
            },
            {
                "text": "For the sake of contrasting the two definitions, we start by giving the currently used definition of TAG. In Section 2, we show that this definition is not consistent with the assumptions made in the unification framework. We propose a novel way of interpreting the basic objects of a TAG, borrowing ideas from description theory (D-theory). By means of an example, we introduce the notion of quasi-trees. We then show how TAG can be embedded in a unification-based framework. This interpretation of the objects manipulated by a TAG grammar as quasi-trees not only leads to our current definition of FTAG, but also explains the earlier definition (Vijay-Shanker 1987; Vijay-Shanker and Joshi 1988) . In Section 3, we give examples to show why the introduction of feature structures and unification adds to the descriptive capabilities of TAG. In particular, we focus on the implementation of the so-called adjoining constraints (that determine locally which structures can be used for adjoining and whether adjunction is mandatory). We will show that not only can adjoining constraints be specified in a linguistically more appealing manner now, but also that in several cases redundant specifications of structural descriptions can be avoided.",
                "cite_spans": [
                    {
                        "start": 647,
                        "end": 667,
                        "text": "(Vijay-Shanker 1987;",
                        "ref_id": "BIBREF21"
                    },
                    {
                        "start": 668,
                        "end": 697,
                        "text": "Vijay-Shanker and Joshi 1988)",
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                ],
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                "section": "Outline of the Paper",
                "sec_num": "1.1"
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            {
                "text": "In Section 5, we consider some possible implications of the new interpretation of the formalism proposed here. One particular question that arises is whether the operations of adjoining and even multi-component adjoining (as used in Multi-component Tree Adjoining Grammar) can be considered to be the same as the substitution operation where the characteristics of the adjoining and multi-component adjoining operations can be derived from the fundamental (linguistic) assumptions that concern the make-up of elementary objects of a grammar. Questions related to this issue, such as whether a distinction between initial and auxiliary structures (the two types of basic structures used in a TAG) needs to be made, are also raised. Further work along the lines suggested in this section depends on investigation of certain linguistic issues involved in the use of the TAG formalism that is beyond the scope of this work. A1- though we provide no definitive answers to these questions, these topics are raised in this paper because they are brought out by the new interpretation of the TAG formalism that we propose. In Section 4, we propose a logical formulation of FTAG grammars (along the lines of the logical formulation of Functional Unification Grammars given by Rounds and Manaster-Ramer [1987] ) and then show how the denotation of a b-TAG grammar can be obtained. The logical formulation is given, in part, to show the separation between the descriptions of well-formed structures (as specified in a FTAG grammar) and the models that satisfy these descriptions.",
                "cite_spans": [
                    {
                        "start": 1267,
                        "end": 1299,
                        "text": "Rounds and Manaster-Ramer [1987]",
                        "ref_id": "BIBREF18"
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                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Outline of the Paper",
                "sec_num": "1.1"
            },
            {
                "text": "We would like to note that the work presented in this paper concerns a formalism and not linguistic issues. A deliberate attempt has been made to only discuss the TAG formalism in general terms rather than focusing on linguistic issues. By doing so, our intent is to pay closer attention to the formalism itself and uncover the aspects of the definition of TAG that are stipulations and those that fall out as a corollary of a formalism that tries to localize dependencies. The use of linguistic examples in this paper by no means indicates the suitability of any linguistic theories. The only assumption that we make is that a grammar will attempt to localize dependencies to the extent possible.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Outline of the Paper",
                "sec_num": "1.1"
            },
            {
                "text": "A Tree Adjoining Grammar (TAG) as defined traditionally is said to be specified by a finite set of elementary trees. Unlike the string rewriting formalisms that write recursion into the rules that generate the phrase structure, a TAG factors recursion and dependencies into a finite set of elementary trees. The elementary trees in a TAG correspond to minimal linguistic structures that localize the dependencies such as subcategorization, and filler-gap. There are two kinds of elementary trees: initial trees and auxiliary trees. Originally, initial trees (e.g., c~1 and OL 2 in Figure 1 ) were defined to correspond to minimal sentential structures. Therefore, the root of an initial tree was required to be labeled by the symbol S. With the advent of lexicalized TAG and the use of the substitution operation, this assumption is no longer made (see c~3).",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 581,
                        "end": 589,
                        "text": "Figure 1",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Introduction to Tree Adjoining Grammars",
                "sec_num": "1.2"
            },
            {
                "text": "Auxiliary trees (ill, t2 in Figure 2 ) are usually defined to correspond to minimal recursive constructions. Thus, if the root of an auxiliary tree is labeled by a nonterminal symbol, A, then there is a distinguished node, called the foot node, in the frontier of this tree that is also labeled by A. The foot nodes of auxiliary trees, fll and /32, are indicated with an asterisk.",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 28,
                        "end": 36,
                        "text": "Figure 2",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Introduction to Tree Adjoining Grammars",
                "sec_num": "1.2"
            },
            {
                "text": "The adjoining operation is used to compose trees. An auxiliary tree, whose root and foot node are labeled A, can be adjoined at a node that is also labeled A. Adjoining may be described as follows: the subtree below the node of adjunction is excised; the auxiliary tree is inserted in its place; and the excised subtree is substituted at the foot node of the inserted auxiliary tree. Figure 3 shows the result of adjoining fll at the VP node in OL 1 (to yield 3'I) as well as the adjunction of t32 in O~ 2 to yield ~2. The latter example illustrates a key feature of TAG, i.e., localization of dependencies. The tree oL 2 indicates the topicalization of the object, localizing the filler-gap dependency. Notice that although the dependent nodes (the two nodes labeled NPi) are stretched apart in 3`2, the adjoining operation does not alter any dependency present in the original trees being composed.",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 384,
                        "end": 392,
                        "text": "Figure 3",
                        "ref_id": null
                    }
                ],
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                "section": "Introduction to Tree Adjoining Grammars",
                "sec_num": "1.2"
            },
            {
                "text": "1.2.1 Lexicalized Approach to TAG and Substitution. In the traditional approach to TAG, adjunction was the only operation used to compose trees. In the lexicalized approach to TAG as proposed by Schabes, Abeille, and Joshi (1988) , the substitution operation is also used. In this approach, elementary trees are associated with lexical items. These lexical items (indicated by ~) are said to be the anchors of the trees. These trees define the arguments required by the anchor. Figure 1 shows two initial trees ~1 and c~2 whose anchors are transitive verbs. The two trees specify the arguments required by the anchor (a transitive verb) and describe the structure for the simple declarative form and for the case where the object is topicalized. Note in both these trees, the argument (subject and object NP) nodes are not elaborated any further. This elaboration is done instead by substituting other initial trees at these nodes. The tree 3'3 ( Figure 3 ) is the result of substituting o~ 3 at the subject NP node in el. In a lexicalized TAG, frontier nodes labeled by nonterminals (with the exception of foot nodes) are marked for substitution (specified by ~).",
                "cite_spans": [
                    {
                        "start": 195,
                        "end": 229,
                        "text": "Schabes, Abeille, and Joshi (1988)",
                        "ref_id": "BIBREF20"
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                ],
                "ref_spans": [
                    {
                        "start": 478,
                        "end": 486,
                        "text": "Figure 1",
                        "ref_id": null
                    },
                    {
                        "start": 947,
                        "end": 955,
                        "text": "Figure 3",
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                ],
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                "section": "Introduction to Tree Adjoining Grammars",
                "sec_num": "1.2"
            },
            {
                "text": "1.2.2 Adjoining Constraints. So far, the only restriction we have placed on the set of auxiliary trees that can be adjoined at a node is that the label of that node must be the same as the label of the root (and the foot) node of the auxiliary tree. However, often it becomes necessary to allow only a subset of such auxiliary trees to be adjoined at 71 :",
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                "section": "Introduction to Tree Adjoining Grammars",
                "sec_num": "1.2"
            },
            {
                "text": "S NP~. VP VP adv /k v NP$ 72 : S NPi$ S NP$ VP A v S NP,I, vP 73 : s v NPi NP VP \u2022 Det n v NP$ Figure 3",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Introduction to Tree Adjoining Grammars",
                "sec_num": "1.2"
            },
            {
                "text": "Some examples of adjoining and substitution.",
                "cite_spans": [],
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                "section": "Introduction to Tree Adjoining Grammars",
                "sec_num": "1.2"
            },
            {
                "text": "a node. In a TAG, associated with each node is a list of auxiliary trees that can be adjoined at that node. This specification of a set of auxiliary trees with each node is called the Selective Adjoining (SA) constraints of the nodes. A node is said to have a Null Adjoining (NA) constraint if no auxiliary tree is allowed to be adjoined at that node. An NA constraint is specified by associating an empty set with a node. In current TAG literature NA constraints are therefore said to be a special case of SA constraints. In addition, for some nodes it is necessary to insist that adjunction is mandatory at a node. In such a case, we say that the node has an Obligatory Adjoining (OA) constraint.",
                "cite_spans": [],
                "ref_spans": [],
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                "section": "Introduction to Tree Adjoining Grammars",
                "sec_num": "1.2"
            },
            {
                "text": "A more detailed description of TAG, the use of adjoining constraints, their propagation during derivation, and their usefulness in providing linguistic analyses may be found in Kroch and Joshi (1985) . At this point we would like to note that by the specification of such adjoining constraints are stipulations of the adjunction possibil-ities at that node. On the other hand, we will see that: in the version of FTAG we define here, decisions such as the choice of auxiliary trees that can be adjoined at a node or whether adjunction is mandatory at a node follows from the assertions (stated in terms of feature structures) about the linguistic features of individual nodes, rather than being specific to the adjoining operation. In fact, in this paper, we would like to highlight this issue while addressing the usefulness of this \"unification-based approach\" to TAG.",
                "cite_spans": [
                    {
                        "start": 177,
                        "end": 199,
                        "text": "Kroch and Joshi (1985)",
                        "ref_id": null
                    }
                ],
                "ref_spans": [],
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                "section": "Introduction to Tree Adjoining Grammars",
                "sec_num": "1.2"
            },
            {
                "text": "In the unification-based approach to grammars, the rules of a grammar are viewed as constraints that define well-formedness. At any point during derivation, the structures built reflect the information known at (or the constraints specified up to) that point. Further derivation leads to more constraints being specified. We begin this section by illustrating why the traditional definition of TAG is incompatible with this aspect of the unification-based approach to grammars.",
                "cite_spans": [],
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                "eq_spans": [],
                "section": "A Unification-Based Approach to Tree Adjoining Grammars (FTAG)",
                "sec_num": "2."
            },
            {
                "text": "Given al (Figure 1 ), we can state that there is a relationship between the S node and the v node that is fixed by the fact that we have stated that al is a tree. For instance, one of the assertions we can make is that (since we consider al as a tree) following two immediate domination (ID) links from the S node leads us to the v node. Now consider the tree \"yl ( Figure 3 ) obtained by adjoining at the VP node (of al) that lies along the path from the S node to the v node. In \"Yl, although the S and v node are still present, the v node is no longer the grandchild (two ID links) of the root node. This example illustrates that, in general, the adjoining operation on trees nullifies certain assertions that can be made about the component trees (that are composed). The reason that the traditional definition of TAG is not compatible with the unification approach is that it defines that the grammar manipulates (composes) fully specified structures (trees in this case) rather than partially specified structures. The composition operation of adjoining creates a new structure that does not maintain all of the properties that held in the original (fully specified) structures of which it is composed.",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 9,
                        "end": 18,
                        "text": "(Figure 1",
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                    {
                        "start": 366,
                        "end": 374,
                        "text": "Figure 3",
                        "ref_id": null
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                ],
                "eq_spans": [],
                "section": "Adjoining of Trees",
                "sec_num": "2.1"
            },
            {
                "text": "In the rest of the paper we will discuss an alternate definition of TAG and argue that our proposed definition is more compatible with the unification approach. Unlike the traditional definition of TAG, we do not consider the objects manipulated by a grammar to be trees. Rather, we will say that although the elementary objects do specify tree structures, they do so only in a partial fashion.",
                "cite_spans": [],
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                "section": "Adjoining of Trees",
                "sec_num": "2.1"
            },
            {
                "text": "We start by examining the nature of objects that are manipulated by a TAG. The only assumption we will make about these objects is that the elementary objects of the grammar give a sufficiently enlarged domain of locality that allows localization of statements of dependencies such as subcategorization, and filler-gap.",
                "cite_spans": [],
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                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
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                "text": "2.2.1 Quasi-Trees. Let us reconsider al (shown in Figure 1 ), which is assumed to be one of the tree structures associated with a transitive verb. Let us consider which information captured in this tree is important for asserting the cooccurrence dependencies involved. We must represent the obligatory arguments required by a transitive verb. If we look at the relationship between the obligatory arguments and the anchor captured by the tree al, we notice that the sentence structure is formed by combining the subject NP node with a VP node. This information is often captured by a rule (1): S --* NP VP. Also, notice that a VP that captures the combination of the lexical anchor with the other obligatory arguments must be formed. In the case of a transitive verb, such information can be captured by a rule (2): VP ---* vNP. Thus we can see that the essential information captured by c~l includes the simultaneous use of the two rules and can be described by stating the relationships between the six entities (three for each rule) involved in the rules.",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 50,
                        "end": 58,
                        "text": "Figure 1",
                        "ref_id": null
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                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
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            {
                "text": "These relationships can be stated by means of some assertions about the individual entities. At this point it is useful to use some names (identifiers) to refer to these entities) Let these names be Sl, npl, vpl (for the three symbols in rule (1)) and vp2, v2, np2 (for the three symbols in rule (2)). The assertions given below (that can be captured by the structural representation, ~4, given in Figure 4 ) can be stated to minimally describe a structure anchored by a transitive verb.",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 398,
                        "end": 406,
                        "text": "Figure 4",
                        "ref_id": "FIGREF7"
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                ],
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                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
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            {
                "text": "1. The label of the entity referred by Sl is S. It immediately dominates the entities referred to by npl and vpl. npl and vpl correspond to the occurrences of NP and VP in the right-hand side of rule (1) and hence the immediate domination.",
                "cite_spans": [],
                "ref_spans": [],
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                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
            },
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                "text": "2. npl refers to the subject of v2 and is labeled by the symbol NP. It is one of the obligatory arguments required by the anchor.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
            },
            {
                "text": "3. vpl is labeled by VP and is used to indicate the combination with the subject (i.e., npl) to yield a sentence.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
            },
            {
                "text": "4. vp2 (also labeled VP), corresponds to the occurrence of VP in the left-hand side of rule (2). It immediately dominates v2 and np2. vp2 is used to indicate the result of the combination of the transitive verb with the obligatory object argument (given by np2).",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
            },
            {
                "text": "5. Since the combination of the anchor with the subcategorized arguments (given by npl and np2) will yield a sentence, the sl dominates v2 by a path of length at least two. Furthermore, the nodes named vpl and vp2 lie on the path from the v2 node to the Sl node. Since vpl must dominate v2, we can conclude that the node named vpl must dominate the node named vp2 (indicated by a dashed link in c~6) and thus, in turn the v2 node. Immediate domination, on the other hand, is represented in the usual fashion.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
            },
            {
                "text": "Here we define the domination relation to be reflexive (i.e., a node dominates itself) in addition to being transitive and antisymmetric. Therefore, we are not stating that the nodes named as vpl and vp2 are necessarily different. Notice that the above assertions have been made independent of TAG or the commitment to use trees for the elementary objects. In TAG, given the decision to use trees, a (minimal) tree that satisfies these assertions will be used. It is due to this minimality requirement that the nodes named as vpl and vp2 are assumed to be the same.",
                "cite_spans": [],
                "ref_spans": [],
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                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
            },
            {
                "text": "On the other hand, the only decision we have committed to is to use structures large enough to localize subcategorization. In this case, we have given some assertions that describe the structure for simple declarative sentences anchored by a transitive verb. Although compatible (though different) assertions have been made about the 1 We adopt this practice of naming nodes following D-theory. This choice to incorporate ideas from D-theory arose from an observation made by S. M. Shieber. The domination relations.",
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                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
            },
            {
                "text": "nodes referred by vpl and vp2, (from these assertions) we cannot conclude whether these nodes are different or are the same node. In fact, this is the reason that structures such as \")/1 (which represents the case where the two are different) in Figure 3 as well as O~1 (where vpl and vp2 both refer to the same node), given in Figure 1 , can both be derived. The structure given by c~4 (with the dashed link indicating possible separation) partially describes the phrase structure tree for both cases. Since vpl and vp2 can both refer to the same node, to avoid confusion, henceforth we will call them quasi-nodes. Thus a node such as the VP node in ~1 ( Figure 1 ) is represented by a pair of quasinodes in ~4. We will refer to these quasi-nodes as the top (for example, vpl) quasi-node and the bottom (vp2) quasi-node. Structures such as ~4 will be called quasi-trees to indicate that they are not trees but (partial) descriptions of trees.",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 246,
                        "end": 254,
                        "text": "Figure 3",
                        "ref_id": null
                    },
                    {
                        "start": 328,
                        "end": 336,
                        "text": "Figure 1",
                        "ref_id": null
                    },
                    {
                        "start": 656,
                        "end": 664,
                        "text": "Figure 1",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
            },
            {
                "text": "A second example that also motivates the proposed interpretation of TAG where the elementary objects are taken to be descriptions of trees (quasi-trees with domination and immediate domination links) rather than trees involves the tree structure in the case of topicalization. The topicalization of the object of a transitive verb can be described by the quasi-tree ~5 (in Figure 4) , which is the counterpart of ~2 (Figure 1 ) used traditionally in TAG. If the elementary structures in a TAG are supposed to depict the localization of dependencies such as those arising due to subcategorization and movement, then we claim structures like ~5 are indeed the appropriate structures to consider. For instance, no treatment of topicalization can justify the identification of the nodes referred to by s2 and s3. Thus, a pair of quasi-nodes is appropriate for their representation. As in the case of vpl and vp2 quasi-nodes in ~4, one can only claim that s2 quasi-node dominates s3 quasi-node (again, by domination, we also allow for the possibility that s2 and s3 can refer to the same node). It may be interesting to contrast this lack of information in ~5 (whether or not they refer to the same node) with the use of functional uncertainty in LFG (Kaplan and Maxwell 1988) to account for long-distance dependency.",
                "cite_spans": [
                    {
                        "start": 1246,
                        "end": 1271,
                        "text": "(Kaplan and Maxwell 1988)",
                        "ref_id": "BIBREF11"
                    }
                ],
                "ref_spans": [
                    {
                        "start": 373,
                        "end": 382,
                        "text": "Figure 4)",
                        "ref_id": "FIGREF7"
                    },
                    {
                        "start": 416,
                        "end": 425,
                        "text": "(Figure 1",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
            },
            {
                "text": "In order to consistently maintain the distinction between descriptions of trees with trees, while discussing the proposed interpretation of TAG we will use the terms ",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A New Interpretation of TAG Objects",
                "sec_num": "2.2"
            },
            {
                "text": "Associating feature structures with quasi-nodes.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 5",
                "sec_num": null
            },
            {
                "text": "initial quasi-tree, auxiliary quasi-tree, quasi-root, and quasi-foot in place of initial tree, auxiliary tree, root node, and foot node, respectively.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 5",
                "sec_num": null
            },
            {
                "text": "Let us now consider a4 given in Figure 4 and the pair of VP quasi-nodes. In the version of FTAG formalism we define here, the feature structure that we associate with quasinodes simply reflects the assertions that we make about them. For instance, suppose a constraint VP.head.subj.agr = NP.head.agr was used in conjunction with the rule S --* NP VP; and the constraint VP.head = v.head was used with the rule VP --* v NP. These two rules (and associated constraint equations) when used together produce ~6, shown in Figure 5 . Notice that the feature structure associated with a top quasi-node can be considered as constraints on it (and hence a constraint on the nature of tree that is rooted at this quasi-node) that are made on the basis of its ancestors and siblings. Similarly, the feature structure associated with a bottom quasi-node reflects the nature of tree that is rooted at this quasi-node (that is its descendants).",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 32,
                        "end": 40,
                        "text": "Figure 4",
                        "ref_id": "FIGREF7"
                    },
                    {
                        "start": 517,
                        "end": 525,
                        "text": "Figure 5",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Associating Feature Structures with Quasi-Nodes",
                "sec_num": "2.3"
            },
            {
                "text": "Instead of explicitly using a pair of quasi-nodes and drawing the domination (dashed) link between them, we can also depict it in a more traditional manner found in TAG literature (see ol 7 in Figure 5 ). In such a case a node, such as the VP node in c~7, will have two feature structures (the ones associated with the two quasi-nodes) associated with it. This matches the previous definition of feature structure-based Tree Adjoining Grammars where these two feature structures were called the top and bottom feature structures associated with a node. In fact, this correspondence was independently observed by Henderson (1990) and was used in the translation of an FTAG to a Structure Unification Grammar. When convenient, we will use \"a node with two associated feature structures\" instead of \"a pair of quasi-nodes (with one feature structure associated with each quasi-node).\"",
                "cite_spans": [
                    {
                        "start": 612,
                        "end": 628,
                        "text": "Henderson (1990)",
                        "ref_id": "BIBREF4"
                    }
                ],
                "ref_spans": [
                    {
                        "start": 193,
                        "end": 201,
                        "text": "Figure 5",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Associating Feature Structures with Quasi-Nodes",
                "sec_num": "2.3"
            },
            {
                "text": "If the objects manipulated by a TAG are considered as quasi-trees, a natural question arises when one considers what would be a node in a tree as a pair of quasi-nodes. For our current purposes, this aspect is not relevant. For instance, the auxiliary quasitrees, f13, f14, fls,/36 in Figure 6 , are equally acceptable (well-formed structures) in the formalism. No matter which one is used, for an auxiliary quasi-tree, we have to state the quasi-root node and the quasi-foot node. As shown in Figure 6 , for the auxiliary quasi-trees, f13~ f14~ f15~ f16, they are given by the pairs of names vp3, vp4; vps, vp6; vp7~ vps; and vp9, vplo, respectively. ",
                "cite_spans": [
                    {
                        "start": 593,
                        "end": 622,
                        "text": "vp3, vp4; vps, vp6; vp7~ vps;",
                        "ref_id": null
                    }
                ],
                "ref_spans": [
                    {
                        "start": 285,
                        "end": 293,
                        "text": "Figure 6",
                        "ref_id": null
                    },
                    {
                        "start": 494,
                        "end": 502,
                        "text": "Figure 6",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Associating Feature Structures with Quasi-Nodes",
                "sec_num": "2.3"
            },
            {
                "text": "We will now define the adjoining operation on quasi-trees and see that (unlike previously) this operation has the property that in the resulting structure all the structural relations specified in the objects being composed are preserved. We will see that this, in turn, allows for a straightforward embedding of TAG in the unification-based framework. Recall that by considering a pair of quasi-nodes we allow for possible separation. We now define the operation of adjoining as the operation that achieves this separation. Consider a quasi-tree, as shown in Figure 7 , with a pair of top and bottom quasi-nodes referred by the names, say ~t and ~/b respectively. In this figure, we have deliberately chosen to indicate that feature structures (ft and fb) as labeling quasinodes (~/t and ~/b respectively). Consider a auxiliary quasi-tree, fl, with the quasi-root node and quasi-foot node referred by the names root~ and foot~ as shown in Figure 7 .",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 560,
                        "end": 568,
                        "text": "Figure 7",
                        "ref_id": null
                    },
                    {
                        "start": 940,
                        "end": 948,
                        "text": "Figure 7",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "The Adjoining Operation",
                "sec_num": "4"
            },
            {
                "text": "Adjoining fl at the pair of quasi-nodes 0/t, ~b) in ~ is now defined by stating that the domination relation between the pair of quasi-nodes is specified to be the same domination relation that exists between the quasi-root node of fl and the quasi-foot node of ft. Thus, the pair of quasi-nodes (referred by the names ~t and ~b) get separated when the quasi-nodes referred by ?lt (~b) and root~ (foot/~) are identified. Adjunction is thus defined as a pair of simultaneous substitutions. 2 Note that the adjoining operation is defined only if the point of adjunction is specified as a pair of quasi-nodes. We stated that the feature structure associated with a quasi-node is an encoding of the assertions (in terms of feature-value pairs) that are made about it. Let ft, fb, froot, and ffoot be the feature structures that satisfy the constraints stated about the quasi-nodes referred by rh, 71b, rootfl, and footfl respectively. Then, because adjunction is defined as identifying the quasi-nodes referred by 7/t and root~ (as well as those referred by ~/b and foot~), ft and froot (fb and ffoot) satisfy the constraints expressed about the same quasi-node. Thus they have to be unified (as shown in Figure 8 ). Note that we have motivated ft as a reflection on the constraints about the tree below the corresponding c' 3~;",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 1201,
                        "end": 1209,
                        "text": "Figure 8",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "The Adjoining Operation",
                "sec_num": "4"
            },
            {
                "text": "fl:",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "The Adjoining Operation",
                "sec_num": "4"
            },
            {
                "text": "Ffroo % ; I I !",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "The Adjoining Operation",
                "sec_num": "4"
            },
            {
                "text": "The adjoining operation in FTAG.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 7",
                "sec_num": null
            },
            {
                "text": "\"/4 S NP~, VP ve Ts VP adv VP v NP,I. S /X NP$ VP VP adv A v NP$",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 7",
                "sec_num": null
            },
            {
                "text": "Examples of the adjoining operation.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 8",
                "sec_num": null
            },
            {
                "text": "quasi-node, one that possibly arises due to the relationship of this quasi-node with its ancestors and siblings in % The feature structure froot also reflects the nature of the tree below the quasi-root node. Since these two quasi-nodes are now required to be the same, the unification of ft with froot gives a feature structure that reflects all constraints when the quasi-nodes are identified. Figure 8 shows the result of adjoining at the paired VP quasi-nodes in c~4 by the auxiliary quasi-trees t3 (resulting in q/4) and t6 (75)\"",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 396,
                        "end": 404,
                        "text": "Figure 8",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Figure 8",
                "sec_num": null
            },
            {
                "text": "The substitution operation used in TAG is the same as that used in context-free grammars (CFGs), where one considers a CFG as a tree-rewriting formalism rather than a string-rewriting formalism. In this case, given two trees, the substitution operation can be defined as the tree obtained by identifying the root node of one tree with the target E foot-1",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "The Substitution Operation",
                "sec_num": "2.5"
            },
            {
                "text": "The substitution operation.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 9",
                "sec_num": null
            },
            {
                "text": "(of the substitution operation) node appearing in the frontier of the other tree. Due to this identification, the feature structures associated with the two nodes in question are unified.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 9",
                "sec_num": null
            },
            {
                "text": "A similar definition will be used to define the substitution operation here. Let refer to a quasi-node in the frontier (see Figure 9 ). The substitution of \"~ at the quasinode ~ is defined as the quasi-tree obtained by identifying the quasi-nodes ~ and the quasi-root of % Thus the feature structures associated with these quasi-nodes get unified as shown in Figure 9 .",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 124,
                        "end": 132,
                        "text": "Figure 9",
                        "ref_id": null
                    },
                    {
                        "start": 359,
                        "end": 367,
                        "text": "Figure 9",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Figure 9",
                "sec_num": null
            },
            {
                "text": "We can make the following observations at this stage. The dashed link between a pair of quasi-nodes indicates that it is possible for the two to be the same. However it is possible to insist that such a pair of quasi-nodes are distinct. This is possible, by stating incompatible assertions about them. On the other hand, as was noted by Marcus, Hindle, and Fleck (1983) , without explicitly stating so, we cannot make assertions about a pair of quasi-nodes that will indicate that they are the same. These observations will be further elaborated in Section 3 to capture obligatory adjoining (OA) constraints.",
                "cite_spans": [
                    {
                        "start": 337,
                        "end": 369,
                        "text": "Marcus, Hindle, and Fleck (1983)",
                        "ref_id": "BIBREF15"
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Some Observations",
                "sec_num": "2.6"
            },
            {
                "text": "Another point that can be noted is that the adjoining operation and its use of auxiliary trees can itself be motivated from the definition of quasi-trees. Notice that we have introduced the concept of quasi-trees simply from the motivation of considering structures with enlarged domains of locality in order to localize dependencies such as subcategorization and filler-gap. In defining quasi-trees we stated that pairs of quasinode can be separated (i.e., they need not be the same node). If a pair of associated quasi-nodes are to be separated by the use of a composition operation, it is easy to see that it can only be done by an operation like adjunction, and the kind of structure that can fit between them must have the general form of an auxiliary tree. Of course, with the use of the new notation, the insistence that the root and foot nodes (of auxiliary trees) be labeled by the same nonterminal symbol (as well as for the target of adjunction) is only a stipulation (and not required by the formalism). Let us consider the labeling of nodes (quasi-nodes) by atomic symbols (such as S, NP). In contrasting the traditional definition of TAG with the definition given here, suppose we make a correspondence between a node in a tree (using the traditional definition) with a pair of quasi-nodes in a quasi-tree. It must be the case that such a pair of quasi-nodes are labeled by the same atomic symbol (since they correspond to a single node according to the traditional definition of TAG). Proceeding with the assumption that pairs of quasi-nodes are labeled by the same symbol we note from the definition of adjoining given in Section 2.4, it follows that for any auxiliary tree to be adjoined at this node, the quasi-root of this auxiliary tree and the quasi-foot must also be labeled the same. Thus the above-mentioned stipulation is a statement that recursion is factored out of elementary trees. In fact, as we will see, if instead of nonterminal symbols we consider category structures (as specified in GPSG) as labeling nodes then almost all pairs of quasi-nodes in the trees we will consider here will be labeled differently (by compatible or incompatible categories). In fact, it is the relationship between the two labels that will determine the subset of auxiliary trees that can be adjoined at a node. Further discussion on this matter can be found in Section 3.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Some Observations",
                "sec_num": "2.6"
            },
            {
                "text": "In the definition here, since we do not start by assuming that trees are composed, there is no need to make such an assumption that a pair of quasi-nodes separated by the domination (dashed) link must be labeled the same, unless if it follows from some linguistic principle/intuition being expressed using the TAG formalism. At this point we would like to note that enforcing such a stipulation has significant consequences on the definition of the formalism. Some of these consequences are noted in Section 5, where we contrast multi-component adjoining with adjoining.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Some Observations",
                "sec_num": "2.6"
            },
            {
                "text": "We have stated that an FTAG grammar manipulates (partial) descriptions of trees (i.e., quasi-trees). We will now state that a grammar derives trees (with nodes labeled by feature structures).",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Objects Derived by a Grammar",
                "sec_num": "2.7"
            },
            {
                "text": "The composition operations of adjoining and substitution compose quasi-trees to build more complex (and more specific) quasi-trees. Each quasi-tree obtained during the derivation process specifies a set of trees. The set of trees derived can be obtained by taking the circumscriptive reading of the domination relation indicated in the quasi-trees obtained. The domination link between a pair of quasi-nodes represents the situation that they may or may not refer to the same object. In the absence of further information (for instance at the end of the derivation process) we shall consider that the pair of quasi-nodes refer to the same (single) node. Thus, given a quasi-tree, its minimal reading leads to the derived tree that is obtained by explicitly equating the related top and bottom quasi-nodes for each pair of quasi-nodes (since by the domination relation specified here any quasi-node dominates itself). Thus, in a derived tree (such as c~2, in Figure 1 , obtained by taking the circumscriptive reading of the domination relationship specified by the quasi-tree c~5 given in Figure 4 ) only one feature structure is associated with each node.",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 958,
                        "end": 966,
                        "text": "Figure 1",
                        "ref_id": null
                    },
                    {
                        "start": 1088,
                        "end": 1096,
                        "text": "Figure 4",
                        "ref_id": "FIGREF7"
                    }
                ],
                "eq_spans": [],
                "section": "Objects Derived by a Grammar",
                "sec_num": "2.7"
            },
            {
                "text": "The discussion given above justifies the unification (or coindexing) of the top and bottom feature structures of a node at the end of the derivation process as specified in the previous definition of FTAG. Of course, due to the associativity of the unification operation, the coindexing of the top and bottom feature structures for all nodes does not have to be delayed until the end of the derivation process. Such unifications for a node can be done whenever one decides that there will be no more adjunctions at that particular node.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Objects Derived by a Grammar",
                "sec_num": "2.7"
            },
            {
                "text": "In the traditional definition of TAGs, a derived tree cannot have nodes with OA constraints, even though intermediate trees can have nodes with OA constraints. This requirement on derived trees is analogous to the use of ANY in FUG. In our current definition a tree is derived (in the above-mentioned manner) only if the corresponding quasi-tree has compatible feature structures associated with each pair of quasi-nodes. If this were not the case, i.e., some pair of quasi-nodes had incompatible feature structures associated with them, then taking the circumscriptive reading of the domination relation will not be possible. Such quasi-trees do describe a set of trees, but the one obtained by equating the pairs of top and bottom quasi-nodes is not one of them. Obvi-ously this should be the case, since incompatible assertions about a pair of quasi-nodes indicates that they do refer to different nodes (and hence specify OA constraints).",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Objects Derived by a Grammar",
                "sec_num": "2.7"
            },
            {
                "text": "A question arises whether (as in standard CFG-based unification grammars) one could associate just one (rather than two) feature structure per node, i.e., whether it is necessary to consider pairs of quasi-nodes. In fact, Harbusch (1990) defined such a treatment of TAG where only one feature structure is associated with each node.",
                "cite_spans": [
                    {
                        "start": 222,
                        "end": 237,
                        "text": "Harbusch (1990)",
                        "ref_id": "BIBREF3"
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Using One (Rather than Two) Feature Structure",
                "sec_num": "2.8"
            },
            {
                "text": "One could argue that it may be inefficient (for instance, when implementing the formalism as defined here) to start with the pairs of quasi-nodes and then try to merge them eventually when possible. Strategies to improve processing may be considered particularly if we believe that, on an average, a relatively small proportion of potential sites will become actual targets of adjunctions during a derivation of a sentence. Then (to improve performance) we could specify that by defimlt the associated pair of top and bottom quasi-nodes are to be identified. That is, we will not consider a node as a pair of quasi-nodes unless there is reason to believe it is necessary (if adjunction has to be performed). So we can even state that there is just one feature structure per node, which has to be the one obtained by unifying the feature structures associated with the top and bottom quasi-nodes. Now if adjunction takes place at a node in some tree that has been derived, then the \"unification\" that has been performed has to be undone to recover the top (relating it with its ancestors and siblings) and bottom (based on the structure it dominates) feature structures. This undoing can be quite complex, especially if the pair of quasi-nodes in question is a part of a derived object rather than an elementary structure specified by the grammar. The above description essentially captures the definition of the formalism presented by Harbusch (1990) .",
                "cite_spans": [
                    {
                        "start": 1435,
                        "end": 1450,
                        "text": "Harbusch (1990)",
                        "ref_id": "BIBREF3"
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Using One (Rather than Two) Feature Structure",
                "sec_num": "2.8"
            },
            {
                "text": "Another point can be made about the scheme presented above. Consider a node whose top and bottom feature structures are incompatible and hence nonunifiable. If we were to insist that only one feature structure were to be associated with every node then we can only unify the compatible parts of the top and bottom feature structures and somehow (perhaps with the use of a device like ANY) retain (effectively) the OA constraint machinery.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Using One (Rather than Two) Feature Structure",
                "sec_num": "2.8"
            },
            {
                "text": "In the traditional definition of a TAG, the adjoining possibilities at a node is determined by the association of adjoining constraints with each node. In this section we consider how such constraints may be captured by the use of feature structures and then contrast the two methods of determining the adjoining possibilities. Since we attempt to contrast the adjoining possibilities, in this section we will make correspondences between nodes in trees (used in the traditional definition of TAG) with pairs of quasi-nodes that are linked by the domination (dashed) link. That is, we talk of such pairs of quasi-nodes as the target of adjunction. Also, if we have a pair of quasi-nodes given by ~/1 and r/2 where ?]1 quasi-node dominates ~/2, we will say that the 72 is the bottom quasi-node paired with ~/1 or that ~/1 quasi-node is the top quasi-node paired with 7/2.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Feature Structures and Adjoining Constraints",
                "sec_num": "3."
            },
            {
                "text": "In the definition of TAG, given in Section 1.2, it was stated that if a node has an OA constraint, then adjoining is mandatory at that node. In terms of quasi-nodes this means that the corresponding pair of quasi-nodes must be separated. Therefore, the use of an OA constraint at a node may be interpreted as stating that the related pair Figure 10 \"OA\" constraints.",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 339,
                        "end": 348,
                        "text": "Figure 10",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "3.10A Constraint",
                "sec_num": null
            },
            {
                "text": "of quasi-nodes are indeed distinct, i.e., there must be some feature that distinguishes them. Hence the linguistic basis for making the claim that the node has an OA constraiht must be stated in such a way that the feature structures on the two quasi-nodes are incompatible. As an example, consider c~8 given in Figure 10 . The feature structure of the quasi-root of c~8 has a value of + for the tense attribute to specify that any tree rooted at this quasi-node must satisfy the constraint that it describes a tensed sentence. On the other hand, the feature structure of the paired bottom quasi-node has a value of -for the tense attribute since it only reflects the descendants. Since these two feature structures are incompatible, this pair of quasi-nodes has an \"OA constraint\" (since it is not possible to stop the derivation process and identify the top with the bottom quasi-node). However, % that results from the adjoining of/37 does not have any pair of quasi-nodes with an \"OA constraint.\"",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 312,
                        "end": 321,
                        "text": "Figure 10",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "3.10A Constraint",
                "sec_num": null
            },
            {
                "text": "Recall that an SA constraint of a node lists a subset of auxiliary trees that can be adjoined at this node. The definition of adjunction used here is stated in terms of a pair of substitutions (and thus adjunction involves two unifications). In terms of quasi-trees, we allow the \"SA\" constraints to be determined as a consequence of the unifications required by identifications of quasi-nodes. If an auxiliary quasi-tree cannot be adjoined at a pair of quasi-nodes, then it must be the case that there is an incompatibility among the relevant pairs of feature structures that we unify when we attempt adjunction. When we attempt adjunction the feature structure of the top quasi-node (in the pair where adjunction is attempted) and the feature structure of the quasi-root (of the auxiliary quasi-tree) are unified, as are the feature structure associated with the bottom quasi-node (in the pair where adjunction is attempted) and the feature structure of the quasi-foot (of the quasi-tree being adjoined). If at least one of these unifications fails then adjunction is not possible. Consider f18 given in Figure 11 . This quasi-tree cannot be adjoined at the pair (s], s2) in c~8 ( Figure 10 ) but can be adjoined at the pair (sl, s2) of ag. On the other hand, we saw that f17 can be adjoined at the pair (sl, s2) of a8. Thus we can say that the pair (sl, s2) of a8 has an \"SA constraint\" that includes/37 but not f18.",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 1106,
                        "end": 1115,
                        "text": "Figure 11",
                        "ref_id": null
                    },
                    {
                        "start": 1183,
                        "end": 1192,
                        "text": "Figure 10",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "SA Constraints",
                "sec_num": "3.2"
            },
            {
                "text": "Recall that a node with an NA constraint cannot be the target of an adjunction. Traditionally, this is specified by stating that the set of auxiliary trees that can be adjoined at such a node is the empty set. For this reason, it is often stated that NA constraints are special form of SA constraints. There are two possible ways of interpreting \"NA constraints\" in the quasi-tree framework. Firstly, a pair of quasi-nodes with an \"NA constraint\" may be interpreted as a stipulation that insists that no quasi-tree can be adjoined at this pair; a statement made regardless of the nature of the auxiliary quasi-trees in the grammar. This may for instance be made if we wish to allow only certain derivation sequences. One could argue that the reason for insisting that foot nodes of complement 3 auxiliary trees have NA constraints, as is the case in most TAG accounts, is to avoid certain derivation sequences (Kroch and Joshi 1985) .",
                "cite_spans": [
                    {
                        "start": 910,
                        "end": 932,
                        "text": "(Kroch and Joshi 1985)",
                        "ref_id": null
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "NA Constraints",
                "sec_num": "3.3"
            },
            {
                "text": "On the other hand, we may also interpret the association of \"NA constraint\" with a pair of quasi-nodes as a statement that none of the auxiliary quasi-trees in the grammar matches the requirements of the type of auxiliary quasi-trees that can be adjoined at this pair (as determined by the associated feature structures). Unlike the previous case, adjunction is not barred per se. Instead, attempting to adjoin at such a pair will never yield well-formed structures. This is because of the nature of such a pair and of the auxiliary quasi-trees in the given grammar. In the TAG formalism, both these interpretations are captured by the same operational mechanism.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "NA Constraints",
                "sec_num": "3.3"
            },
            {
                "text": "The first kind of NA constraint is easily stated. According to this interpretation, for each pair of quasi-nodes with an \"NA constraint,\" the two quasi-nodes are indeed the same node (since we are stating that there is no possible separation). Since the two quasi-nodes are to be identified, the feature structure associated with the resulting quasi-node must reflect both the relationship of the quasi-node with its ancestor (which we assume stands for the top feature structure) as well as its relationship with its descendants (the bottom feature structure).",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "NA Constraints",
                "sec_num": "3.3"
            },
            {
                "text": "Earlier we had stated that the target of an adjunction operation must be a pair of quasi-nodes that have not been identified (i.e., merged). Suppose that a pair of quasi-nodes (71,72) were merged. Let the quasi-root and quasi-foot of some auxiliary quasi-trees fl be given by r and f. Adjoining fl at the pair given by 71 and ~2 (after they have been identified) will result in the identification of 71 with r and 72 with f and thus r with f. If we stipulate that in all auxiliary quasi-trees, the quasi-root and quasi-foot do not refer to the same node (i.e., the quasi-root properly dominates the quasi-foot), then no adjunction can occur at a pair of quasi-nodes that have been identified. Thus the identification of a pair of quasi-nodes captures \"NA constraints\" of the first kind.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "NA Constraints",
                "sec_num": "3.3"
            },
            {
                "text": "As far as the second kind of \"NA constraints\" is concerned, we note that it is only a specific case of \"SA constraints.\" Therefore, given a pair of quasi-nodes, if the associated feature structures are such that no auxiliary quasi-tree can be adjoined at this pair then it has an \"NA constraint\" (of the second kind). However, because of the nature of feature structures (in that they capture only partial information), it is hard to detect if a pair of quasi-nodes has such an \"NA constraint.\" In Section 3.4, we will consider such an example.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "NA Constraints",
                "sec_num": "3.3"
            },
            {
                "text": "In the TAG formalism, selective adjoining constraints are specified by enumeration, and hence are stipulations stating which trees can be adjoined at a node. Hence, specifying adjoining constraints in such a way is not a linguistically appealing solution. Obviously, such stipulations are needed because the information content of the labels of nodes in a TAG is often insufficient to determine the trees that can be adjoined at various nodes. In the case of FFAG, labeling of quasi-nodes by symbols such as NP, S is only a part of information contained in the feature structures associated with them. We associate with a pair of quasi-nodes feature structures that describe the features of the top and bottom quasi-nodes. The fact that only appropriate quasi-trees get adjoined is a corollary of the fact that only those consistent with these declarations are acceptable. Additionally, in a FTAG, \"adjoining constraints\" can be dynamically instantiated and are not pre-specified as in a TAG.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Comparing the Implementation of Adjoining Constraints",
                "sec_num": "3.4"
            },
            {
                "text": "We will now point out some differences between the implementation of adjoining constraints in TAG and b-TAG that arise because of different methods adopted in adjoining constraint specification. Of course, if the constraints are prespecified as in TAG, then little work has to be done (say by a parser) to verify whether an auxiliary tree can be adjoined at a node during the derivation process. This is not the case in FTAG, because of dynamic instantiation of \"constraints\" in b-TAG. For example, instead of f17 (Figure 10 ), suppose we consider/39 shown in Figure 12 . The result of adjoining f19 at the pair (S1~$2) of O~ 9 is ')'7. There is a pair of quasi-nodes, ($3~$4) , in \"~7 with values of -and + for the tense attribute (thus giving rise to \"OA constraints\"). ",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 514,
                        "end": 524,
                        "text": "(Figure 10",
                        "ref_id": null
                    },
                    {
                        "start": 560,
                        "end": 569,
                        "text": "Figure 12",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Comparing the Implementation of Adjoining Constraints",
                "sec_num": "3.4"
            },
            {
                "text": "Comparison of adjunction constraints--Example 1.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 12",
                "sec_num": null
            },
            {
                "text": "In a TAG grammar, the SA constraints at the root of tree corresponding to a9 would be given to disallow this adjunction. In the case of FTAG, as shown in Figure 12 , this adjunction is allowed, because the associated unifications did not fail. Now suppose (as one might expect) the auxiliary quasi-trees in the grammar were such that none of them had their quasi-root with a feature structure compatible with tense: -and quasi-foot with a feature structure compatible with tense: +. In this case, although the adjunction of/39 was permitted, no tree can ever be derived from the result of adjunction. In fact, until we try all possible adjunctions at the node ~ in 77, we cannot realize that adjunction of f19 at the root of a9 can result in a final acceptable tree. Thus, the pair (s3, s4) has an NA constraint of the second kind. Now we will consider an example where specification of constraints in TAG suffers in comparison with the implementation of \"constraints\" in FTAG. Consider the following well-formed sentences",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 154,
                        "end": 163,
                        "text": "Figure 12",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Figure 12",
                "sec_num": null
            },
            {
                "text": "(1) Who did John see?",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 12",
                "sec_num": null
            },
            {
                "text": "(2) Who did Peter think John saw?",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 12",
                "sec_num": null
            },
            {
                "text": "(3) I wonder who John saw. Comparison of adjunction constraints--Example 2.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 12",
                "sec_num": null
            },
            {
                "text": "(7) I wonder who did John see?",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 12",
                "sec_num": null
            },
            {
                "text": "(8) Who Peter thought John saw.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 12",
                "sec_num": null
            },
            {
                "text": "We will first consider a TAG account (in traditional style). The trees (without considering adjoining constraints) given in Figure 13 have been suggested in literature to account for the well-formed sentences above. We have drawn these trees accounting for substitution at the NP nodes. From the well-formedness of (1) and ill-formedness of (6) it follows that the node 7/of eel 0 must have an OA constraint with fl12 in its SA constraint. On the other hand, from the well-formedness of (3) and ill-formedness of (6) it follows that the root of ~10 must have an OA constraint with fit0 in its SA constraint. However, the requirement of an OA constraint on these two nodes in ~10 is mutually exclusive. Because of this, a TAG grammar that accounts for the sentences above must have two trees, that have exactly the same tree structure but only differ in the adjoining constraints attached at the nodes. Now, from the well-formedness of (5), which can be derived by adjoining/311 at the root of c~11, we can conclude that there need not be an OA constraint on the root of /311. However, suppose we adjoin flu at the node ~ in ~10 such that the frontier matches with (8). From the ill-formedness of (8) and the well-formedness of (2) we realize that there must be an OA constraint on the root of fin with/312 in its OA constraint. Thus, again we will need two trees (corresponding to fl11), with identical tree structure but differing in the adjoining constraints.",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 124,
                        "end": 133,
                        "text": "Figure 13",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Figure 12",
                "sec_num": null
            },
            {
                "text": "We will see that such replication of tree structure is not necessary. Now consider the FTAG fragment (inspired by similar treatment in Abeille [1991] ) given in Figure 14 Comparison of adjunction constraints--Example 2.",
                "cite_spans": [
                    {
                        "start": 135,
                        "end": 149,
                        "text": "Abeille [1991]",
                        "ref_id": "BIBREF1"
                    }
                ],
                "ref_spans": [
                    {
                        "start": 161,
                        "end": 170,
                        "text": "Figure 14",
                        "ref_id": "FIGREF7"
                    }
                ],
                "eq_spans": [],
                "section": "Figure 12",
                "sec_num": null
            },
            {
                "text": "of quasi-nodes labeled S will obtain an \"OA constraint\" and vice versa as required (hence (6) can not be derived). In fact, if fl13 were adjoined at the root of Oq2 (and thus showing (3) is well formed) then it will no longer be possible to derive (7). Likewise, by adjoining ills at the pair of s3 and s4 quasi-nodes in a~2, we can derive (1) but will no longer be able to derive (7). Proceeding in this manner we can show the well-formedness of (1)-(5) and the illformedness of (6)-(8). Thus we have shown that if appropriate assertions can be stated about the individual nodes then a more succinct grammar can be given: one that does not require replication of tree structures, due to the fact that adjoining constraints are not pre-specified as in a TAG.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 12",
                "sec_num": null
            },
            {
                "text": "A central theme in our definition of FTAG has been the view that the objects manipulated by a grammar are descriptions of trees (rather than trees). This separation of descriptions of trees from the trees (models) derived has been crucial in embedding TAG in the unification framework. The question of which language to use to describe trees (together with its semantics) arises. We have used quasi-trees (as the descriptions themselves) in order to focus on TAG, and have not introduced some general formal framework for describing trees. The discussion below does not constitute a suggestion about how such general descriptions may be given, but is one way to specify an FTAG that will be convenient for our purposes here.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A Logical Formulation",
                "sec_num": "4."
            },
            {
                "text": "In this section, we describe a logical formulation of the unification-based approach to TAGs. The purpose of providing a logical formulation of FTAG is so that we can find the denotation of an FTAG grammar (the set of structures generated) as well as contrast it with context-free grammar-based unification grammars. To define the denotation of an FTAG grammar, we will first describe how an FTAG grammar can be represented. This representation uses the logical formulation of feature structures as given by Kasper and Rounds (1986) and Johnson (1988) and is similar in approach to the logical formulation of Functional Unification Grammar (FUG) given by Rounds and Manaster-Ramer (1987) .",
                "cite_spans": [
                    {
                        "start": 508,
                        "end": 532,
                        "text": "Kasper and Rounds (1986)",
                        "ref_id": "BIBREF12"
                    },
                    {
                        "start": 537,
                        "end": 551,
                        "text": "Johnson (1988)",
                        "ref_id": "BIBREF5"
                    },
                    {
                        "start": 655,
                        "end": 687,
                        "text": "Rounds and Manaster-Ramer (1987)",
                        "ref_id": "BIBREF18"
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A Logical Formulation",
                "sec_num": "4."
            },
            {
                "text": "In the framework of Rounds and Manaster-Ramer (1987) , an FUG (or any contextfree grammar with associated unification equations as in, say PATR-II) can be represented by means of a set of equations, using the formulae of Kasper-Rounds to represent feature structures. For example, a context-free grammar rule S --* NP VP can be represented as s ::= CAT : S A 1 : np A 2 : vp. Here s, np, and vp are type variables. The attributes 1 and 2 are used to indicate the first and second children respectively. Using standard techniques to derive fixed points from a set of recursive rules, the denotation of type variables are obtained. The denotation of the type variables gives the set of structures derived from the corresponding nonterminals. Now suppose we wish to express reentrancy in feature structures by using variables; it is clear that we have to use individual variables and not type variables. As in Johnson (1988) , we use individual variables and equalities to express reentrancy. The syntax we adopt to describe attribute-value structures is as follows. Firstly, the set of terms is defined as",
                "cite_spans": [
                    {
                        "start": 20,
                        "end": 52,
                        "text": "Rounds and Manaster-Ramer (1987)",
                        "ref_id": "BIBREF18"
                    },
                    {
                        "start": 907,
                        "end": 921,
                        "text": "Johnson (1988)",
                        "ref_id": "BIBREF5"
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A Logical Formulation",
                "sec_num": "4."
            },
            {
                "text": "t::= a x l(tl)",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A Logical Formulation",
                "sec_num": "4."
            },
            {
                "text": "where a is an atomic value where x is an individual variable where 1 is a label (or attribute) and tl is a term.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A Logical Formulation",
                "sec_num": "4."
            },
            {
                "text": "The set of formulae is defined as 0::= tl ~ t2 01 A 02",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A Logical Formulation",
                "sec_num": "4."
            },
            {
                "text": "0~ v 02",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A Logical Formulation",
                "sec_num": "4."
            },
            {
                "text": "where tl,t2 are terms where 01,02 are formulae where 01~ 02 are formulae.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A Logical Formulation",
                "sec_num": "4."
            },
            {
                "text": "For example, (l(x) = y) A (h(x) = z) A (g(y) = z) A (z = a) describes (among others) the following feature structure.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "A Logical Formulation",
                "sec_num": "4."
            },
            {
                "text": "Note that individual variables (that stand for individual feature structures) are being used to capture reentrancy, whereas typed variables play a role analogous to the role of nonterminals in grammars (such as CFGs) and stand for a set of feature structures. For the purpose of describing an FTAG, we need individual variables to specify reentrancy (as well as to refer to quasi-nodes) and \"typed\" variables to denote the set of structures derived from elementary quasi-trees. To distinguish between these two kinds of variables, in our framework, we will use monadic predicate instead of typed variables.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": ": g : [-i-~ ] h : [~a",
                "sec_num": "1"
            },
            {
                "text": "Firstly, we note that quasi-initial trees are analogous to nonterminals in CFGs. Thus, as indicated above, quasi-initial trees will be represented by monadic predicates. If a is a quasi-initial tree, then we will use a predicate symbol ~ to represent this quasi-tree. If a structure ,,4 is derivable in the grammar starting from a then we would like to have .4 to belong to the set denoted by ~. For example, any structure described by Ot14 can be assumed to satisfy the requirements on the variable x in",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "cat(x) ~ S A Dom(x,y) A cat(y) ~ S A l(y) ~ z A count(y) ~ zero A cat(z) ~, c.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "This description is intended to not only describe the features of nodes, but also the structure of the subtrees rooted at each node (with attributes 1,2,... used to specify the first, second, ... child of a node). In the formula given above, x represents the quasi-root node. Therefore, we will define ~14 by",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "~14(X) K===~ cat(x) ~ S A Dom(x,y) A cat(y) ~ S A l(y) ~ z A count(y) ,~ zero A cat(z) ~ c.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "In this case, Dom(x, y) is used to indicate that the quasi-root (x) dominates the associated bottom quasi-node (given by y). Now if we view the definition of 0t14 independent of the rest of the grammar, then Dom(x,y) represents domination in any arbitrary manner. However, the rest of the grammar specifies the constraints on the domination relation by defining the actual possibilities for the domination. This is because a pair of quasi-nodes (say as given by x and y in c~14) is intended to mean that either they are the same objects or are different nodes that are related by proper domination. In our definition, the separation can take place only by adjunction. So given a grammar, we can specify that the domination relationship is actually defined by",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "Dom(x,y) 4==~ x ~ y V fll(x~y) V ... V -'fin(X~y)",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "where {ill,..., fin} are the quasi-auxiliary trees in the grammar. Here we assume that fl captures the (domination) relationship between its quasi-root and quasi-foot nodes of the quasi-auxiliary tree ft. Since the actual definition of the domination between a pair of quasi-nodes is determined by the quasi-trees of the grammar, it is appropriate to consider fixed-point semantics to define the denotation of a grammar.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "Before we discuss the fixed point we will complete our discussion about how we can specify a grammar. Let us define another monadic predicate Inittree by",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "Inittree(x) 4=~ ~l(x) V ... V ~m(X)",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "where {al,... ,OLm} is the set of initial trees. If we further wish to stipulate that a structure is derived in a FTAG if it is derived from some quasi-initial tree and is rooted in S we can define",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "Grammar(x) 4=~ Inittree(x) A cat(x) = S.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "Note that for a quasi-node (referred to as x) where substitution can take place, we can specify Inittree(x) to specify the substitution.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "We will now illustrate the representation of an FTAG grammar, shown pictorially in Figure 15 . This grammar contains c~14 and fl16. Apart from the cat information, the only other attribute used in the feature structures are count (counts the number of adjoining operations used in deriving a tree), one (used in counting), and attributes 1,2, 3 (which are used for specifying the children of a node).",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 83,
                        "end": 92,
                        "text": "Figure 15",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "To compare our representational scheme for FTAG with that for FUG given by Rounds and Manaster-Ramer (1987) , we have used predicate symbols instead of type variables. The use of monadic predicates alone is sufficient to represent FUG (or actually a CFG-based unification grammar) since only \"substitution\" is used. Binary Example: An FTAG grammar and its representation.",
                "cite_spans": [
                    {
                        "start": 75,
                        "end": 107,
                        "text": "Rounds and Manaster-Ramer (1987)",
                        "ref_id": "BIBREF18"
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "predicates are used to capture adjunction (which is defined as a pair of substitutions) in FTAG.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Expressing an FTAG",
                "sec_num": "4.1"
            },
            {
                "text": "As mentioned before, the set of terms is defined recursively as",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Fixed-Point Semantics (Denotation of an FTAG Grammar)",
                "sec_num": "4.2"
            },
            {
                "text": "t::= a x l(h)",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Fixed-Point Semantics (Denotation of an FTAG Grammar)",
                "sec_num": "4.2"
            },
            {
                "text": "where a is an atomic value where x is an individual variable where I is a label and tl is a term.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Fixed-Point Semantics (Denotation of an FTAG Grammar)",
                "sec_num": "4.2"
            },
            {
                "text": "However the set of formulae is now defined by \u00a2::= tl ~ t2 P(tl,...,tn)",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Fixed-Point Semantics (Denotation of an FTAG Grammar)",
                "sec_num": "4.2"
            },
            {
                "text": "\u00a21 A \u00a22 41 V \u00a22",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Fixed-Point Semantics (Denotation of an FTAG Grammar)",
                "sec_num": "4.2"
            },
            {
                "text": "where tl, t2 are terms where tl,. \u2022., tn are terms and P is a n-ary predicate symbol where \u00a21, \u00a22 are formulae where \u00a21, \u00a22 are formulae.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Fixed-Point Semantics (Denotation of an FTAG Grammar)",
                "sec_num": "4.2"
            },
            {
                "text": "From the discussion given in the previous section any FTAG can be stated as Pl(tl,1,..., tin,l) \": ~. \" 01",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Fixed-Point Semantics (Denotation of an FTAG Grammar)",
                "sec_num": "4.2"
            },
            {
                "text": "Pn (h,n,..., tm,n) ~ \u00a2~ where \u00a21,..., Cn are formulae and t1,1,.. \u2022 tm,1, tl,n \u2022 .., tm,n are terms such that for 1 _< i,j < n, if i \u00a2 j then the symbol Pi ~ Pj. Of course for describing an FFAG, monadic and binary predicates are enough. The structures that terms denote are the finite state automata (actually equivalence classes containing such automata; for details, we refer to Moshier [1988] for a discussion about these structures) as defined by Kasper and Rounds (1986) and used in defining the satisfiability of formulae in their logic. We can give a fixed point semantics of a grammar in the standard way.",
                "cite_spans": [
                    {
                        "start": 3,
                        "end": 18,
                        "text": "(h,n,..., tm,n)",
                        "ref_id": null
                    },
                    {
                        "start": 66,
                        "end": 89,
                        "text": "\u2022 tm,1, tl,n \u2022 .., tm,n",
                        "ref_id": null
                    },
                    {
                        "start": 382,
                        "end": 396,
                        "text": "Moshier [1988]",
                        "ref_id": "BIBREF16"
                    },
                    {
                        "start": 452,
                        "end": 476,
                        "text": "Kasper and Rounds (1986)",
                        "ref_id": "BIBREF12"
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Fixed-Point Semantics (Denotation of an FTAG Grammar)",
                "sec_num": "4.2"
            },
            {
                "text": "Let p be a function that maps each variable to an automaton. We define a Value function as a partial function that returns the denotation of a term (an automaton) relative to an environment (mapping variables to automata)\u2022 Let p be an environment function and I be an interpretation mapping predicate symbols to their denotations, i.e\u2022, if P is a n-ary predicate symbol then I maps P to some set of n-tuples of automata\u2022 Given an interpretation function I and an environment p we define ~ in the following way.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Definition",
                "sec_num": null
            },
            {
                "text": "(I, p) b ~1 A ~2 iff (I, p) ~ ~1 and (I, p) ~ $2 (I,P) V ~bl V ~b2 iff (I,p) ~ ~1 or (I,p) ~ ~b2 (I, p) ~ tl ~ t2 iff Valuep(tl) and Valuep(t2) are defined and Valuep(tl) = Valuep(t2) (I, p) ~ P (h,..., tn) iff Valuep(ti) is defined (1 < i < n) and (Valuep(tl),..., Valuep(t,) ) E I(P).",
                "cite_spans": [
                    {
                        "start": 118,
                        "end": 128,
                        "text": "Valuep(tl)",
                        "ref_id": null
                    },
                    {
                        "start": 195,
                        "end": 206,
                        "text": "(h,..., tn)",
                        "ref_id": null
                    },
                    {
                        "start": 249,
                        "end": 276,
                        "text": "(Valuep(tl),..., Valuep(t,)",
                        "ref_id": null
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Definition",
                "sec_num": null
            },
            {
                "text": "We now define a transformation function mapping interpretations in the following way. For some m ~ 1, let Pi (ti,l,..., ti,ni) 4==~ dpi (1 < i < m) be the grammar specification. We define the transformation function, To, such that given an interpretation,/, Tc returns an interpretation TG (I) given by",
                "cite_spans": [
                    {
                        "start": 109,
                        "end": 126,
                        "text": "(ti,l,..., ti,ni)",
                        "ref_id": null
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Definition",
                "sec_num": null
            },
            {
                "text": "For all substitutions, p, where Valuep(ti,j) is defined for 1 <_ j <_ ni, (Valuen(tia),..., Valuep(ti,nl) ) E TG(I)(Pi) iff (I,p) ~ \u00a25i.",
                "cite_spans": [
                    {
                        "start": 67,
                        "end": 105,
                        "text": "<_ ni, (Valuen(tia),..., Valuep(ti,nl)",
                        "ref_id": null
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Definition",
                "sec_num": null
            },
            {
                "text": "We use the ordering relationship, f-, as defined by Rounds and Kasper (1986) i.e., `41 ___ `42 iff there is a homomorphism mapping the states of Jt I to the states of -/~2 that preserves the transition and output functions. We extend this ordering relation to an ordering on n-tuples and state that (.41, ... , .An) u (131, . .. ~13nl iff for 1 < i < n `4/ __13/. Among pairs of sets of n-tuples of automata, say 191,/92, we use the same ordering as that used by Rounds and Manaster-Ramer (1987) and state that/91 G /92 iff/91 G /92. The least element among the sets of n-tuples of automata is the empty set. The ordering among interpretation functions is defined as h G /2 iff for all predicate symbols P, h(P) f-/2(P), i.e., h(P) c_ I2(P). If/1 _/2, then for all environments, p, and formulae, q~, if (11, p) ~ ~ then (/2, P) ~ q~.",
                "cite_spans": [
                    {
                        "start": 52,
                        "end": 76,
                        "text": "Rounds and Kasper (1986)",
                        "ref_id": "BIBREF12"
                    },
                    {
                        "start": 299,
                        "end": 304,
                        "text": "(.41,",
                        "ref_id": null
                    },
                    {
                        "start": 305,
                        "end": 310,
                        "text": "... ,",
                        "ref_id": null
                    },
                    {
                        "start": 311,
                        "end": 323,
                        "text": ".An) u (131,",
                        "ref_id": null
                    },
                    {
                        "start": 324,
                        "end": 325,
                        "text": ".",
                        "ref_id": null
                    },
                    {
                        "start": 463,
                        "end": 495,
                        "text": "Rounds and Manaster-Ramer (1987)",
                        "ref_id": "BIBREF18"
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Ordering relations",
                "sec_num": null
            },
            {
                "text": "This can be easily shown by using induction on the structure of the formula ~. Let/1 _/2. We have to show for all P that Tc(I1)(P) C Tc(I2)(P). Let P(h,..., tn) ~ \u00a23 be a part of the grammar specification and let (`41,...,. An) E TG(I1)(P). Thus, for any environment p such that (h, p) ~ ~b and for 1 < i < n we have Valuep(ti) = `4i. By the above lemma, we also have I2,p ~ 4 and hence (`41,... ,An) C Tc(I2)(P). Thus, T~(I1)(P) C TG(I2)(P) and To(h) _E To(/2).",
                "cite_spans": [
                    {
                        "start": 213,
                        "end": 223,
                        "text": "(`41,...,.",
                        "ref_id": null
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Ordering relations",
                "sec_num": null
            },
            {
                "text": "We will call an interpretation, L finite if for all predicate symbols, P, I(P) is a finite set. ~b then there is a finite This can be shown by a straightforward induction on the structure of ~b, and by constructing I0 in the obvious manner.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Ordering relations",
                "sec_num": null
            },
            {
                "text": "Consider the derivation of:",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Ordering relations",
                "sec_num": null
            },
            {
                "text": "(1) Which picture did you buy a copy of?",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Ordering relations",
                "sec_num": null
            },
            {
                "text": "(2) Which picture did you buy a photograph of a copy of?",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Ordering relations",
                "sec_num": null
            },
            {
                "text": "This form of long-distance dependency cannot be localized in a TAG if we wish to localize the predicate-argument dependencies as well (for details, see Kroch [1987] ). On the other hand, an analysis has been given using a version of multi-component adjoining. Multi-Component Tree Adjoining Grammar (MCTAG) differs from (the traditional definition of) TAG in that the elementary objects of the grammar are sets of trees rather than trees, and multi-component adjoining involves the composition of these elementary sets of trees 4 (rather than elementary trees). See Joshi (1987) for more details on Multi-Component Tree Adjoining Grammars (MCTAG). The multi-component sets, given in Figure 16 , may be used to give an account for sentences (1) and (2). Obtained by adjoining the two components of fl. 17 in C~lS, % can be used for sentence (1) ( Figure 17) .",
                "cite_spans": [
                    {
                        "start": 152,
                        "end": 164,
                        "text": "Kroch [1987]",
                        "ref_id": "BIBREF14"
                    },
                    {
                        "start": 566,
                        "end": 578,
                        "text": "Joshi (1987)",
                        "ref_id": "BIBREF6"
                    }
                ],
                "ref_spans": [
                    {
                        "start": 683,
                        "end": 692,
                        "text": "Figure 16",
                        "ref_id": null
                    },
                    {
                        "start": 846,
                        "end": 856,
                        "text": "Figure 17)",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Ordering relations",
                "sec_num": null
            },
            {
                "text": "The need for introducing multi-component sets and multi-component adjoining (in this case, at least) arises because of the decision in traditional TAGs to compose trees (rather than descriptions of trees, i.e., quasi-trees). In particular, the domination relations allow us to give partial descriptions of trees such as oL16 (in Figure 18 ) that captures the same information as in the multi-component set fl17 (in Figure 16 ). Note that OL16 can be described by using the same principles that relate (X13 and O~12 (see Figure 14) . If, for a moment, we consider c~15 to be an auxiliary quasi-tree (rather than an initial quasi-tree) and use it for \"adjoining\" (treating the n2 quasi-node as the quasi-foot) then we obtain the same structure as \"Y8 ( Figure 17) .",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 329,
                        "end": 338,
                        "text": "Figure 18",
                        "ref_id": null
                    },
                    {
                        "start": 415,
                        "end": 424,
                        "text": "Figure 16",
                        "ref_id": null
                    },
                    {
                        "start": 520,
                        "end": 530,
                        "text": "Figure 14)",
                        "ref_id": "FIGREF7"
                    },
                    {
                        "start": 751,
                        "end": 761,
                        "text": "Figure 17)",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Ordering relations",
                "sec_num": null
            },
            {
                "text": "Two issues can be raised with respect to this example. Firstly we can question whether such uses of multi-component adjoining (and where the foot node of one component dominates the root of the other components in the eventual structure 5) can be considered to be adjoining in the quasi-tree framework; and secondly whether these operations can be thought of as essentially the substitution operation when viewed in this framework (that uses quasi-trees rather than trees). However, c~15 would normally be called an initial quasi-tree, and we would have considered substitution at the 1\"/2 quasi-node rather than treating C~lS as an auxiliary quasi-tree and the n2 quasi-node as the quasi-foot. Nevertheless, this \"adjunction\" of oL15 seems to be really playing the role of substitution (with a sub-quasi tree though).",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Ordering relations",
                "sec_num": null
            },
            {
                "text": "Addressing the first issue, in the case of the multi-component adjoining example used here, we believe the need for multi-component adjoining arises from the fact that objects being composed were defined to be trees. Even in the previous version of FTAG, it was assumed that the objects being composed were trees despite the fact that two feature structures were associated with each node. These top and bottom feature structures associated with a node were supposed to account for a view of that node from two different perspectives (from the top and from below). However, 4 There are three different definitions of multi-component adjoining that have been proposed. The version considered here is the simplest kind: one where a set of trees are simultaneously adjoined into a single tree. This version leads to a system weakly equivalent to TAG. The other definitions include the case where sets of trees are adjoined simultaneously into nodes in trees that belong to another set and finally where a set of auxiliary trees are adjoined simultaneously without any restriction on the adjoining sites, 5 Although not a part of the definition of multi-component adjoining, in all analyses we are aware of, it is the case that the foot node of one component dominates the root of the other component in the eventual structure. ",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Ordering relations",
                "sec_num": null
            },
            {
                "text": "A multi-component tree adjoining grammar.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 16",
                "sec_num": null
            },
            {
                "text": "because one was dealing with a single node, it was taken for granted that the two feature structures associated with the single node would assign the node the same label (S, NP,...), no matter which perspective (viewing a node from above or from below) one took. That is, we could not consider the possibility of a node whose top and bottom parts were labeled by S and NP. Therefore, instead of using one quasi-tree (c~16), a multi-component set, fl~7, composed of two trees is used. Assuming the possibility of stating domination between quasi-nodes with different labels (as in c~16), we can similarly extend the definition of \"auxiliary quasi-trees\" to allow for the quasi-root and quasi-foot nodes being labeled differently. This is the assumption we made when we \"adjoined\" the quasi-tree c~lS to capture the effect of multi-component adjoining. Assuming that c~15 is an auxiliary structure points out the similarities between multi-component adjoining and adjoining. However, it is more natural to assume it is an initial quasi-tree and use substitution at the object NP node (rather than call c~15 an auxiliary quasi-tree and n2 quasi-node, without any justification, a quasi-foot).",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 16",
                "sec_num": null
            },
            {
                "text": "A similar situation arises when we consider the so-called complement auxiliary trees (see Kroch [1987] ). fl19, an elementary quasi-tree anchored by a verb such as \"think,\" would be defined to be a complement auxiliary quasi-tree because the quasi-foot is present due to the subcategorization requirements of the anchor. In general, in the lexicalized approach to TAG, it is assumed that such an argument node is expanded by substitution. This is consistent with Figure 19 where we could call fl19 an initial quasi-tree and substitute a17 at the supposed quasi-foot (sz) to derive a structure for Peter thinks John saw Mary. However, a quagi-tree such as fl19 must be treated as an auxiliary structure in order that we could use it for adjoining so that it can be adjoined in a18 (see Figure 18 ) at the pair (s3, s4} to derive a structure for who did Peter think John saw.",
                "cite_spans": [
                    {
                        "start": 90,
                        "end": 102,
                        "text": "Kroch [1987]",
                        "ref_id": "BIBREF14"
                    }
                ],
                "ref_spans": [
                    {
                        "start": 463,
                        "end": 472,
                        "text": "Figure 19",
                        "ref_id": null
                    },
                    {
                        "start": 785,
                        "end": 794,
                        "text": "Figure 18",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Figure 16",
                "sec_num": null
            },
            {
                "text": "The question about the basis of deciding when one should call an elementary structure auxiliary or initial remains. It is hard to justify that s2 quasi-node of fl19 is the quasi-foot on the basis of factoring of recursion (the original reason for introducing auxiliary structures). However, while developing a grammar, the s2 node in fl19 is not expanded further because we wish to factor recursion, but because it is required by the subcategorization of the anchor and such nodes are expanded as a result of a derivation step. Among the quasi-nodes that appear in the frontier of flw, the s2 quasinode is called the quasi-foot because extraction cannot occur from a tree that can appear below the subject NP quasi-node, whereas it can in the case of s2 quasi-node. However, on this basis, one could also call a15 an auxiliary quasi-tree and state that the n2 quasi-node is the quasi-foot.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 16",
                "sec_num": null
            },
            {
                "text": "Structures such as fl19 and alS (of Figure 20) raise the question of whether there is an essential difference between initial and (complement) auxiliary quasi-trees, and whether adjoining is only a special form of substitution. It appears that in the case of the two examples above, we came to the situation of calling certain structures auxiliary (~15 : ",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 36,
                        "end": 46,
                        "text": "Figure 20)",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Figure 16",
                "sec_num": null
            },
            {
                "text": "Initial or auxiliary?",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 19",
                "sec_num": null
            },
            {
                "text": "structures solely for the purpose of using the adjoining operation. If we wish to claim that there is no essential difference between initial and auxiliary structures (at least of the complement auxiliary tree variety), then we must account for the apparent difference between substitution and adjoining operations. We argue now that it may not be necessary to make this distinction if we take a closer look at the adjoining and substitution operations. Recall that the substitution operation was defined by the identification of two quasi-nodes. So far this has been illustrated by identifying a quasi-node that appears in the frontier of a quasi-tree with the quasi-root of another quasi-tree. However, now consider fl19 (see Figure 21 ) and \"substitution\" at s2 by the subtree rooted at s4 (i.e.,",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 728,
                        "end": 737,
                        "text": "Figure 21",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Figure 19",
                "sec_num": null
            },
            {
                "text": "0~18 \u2022 S | S S NPI S ,dP---~ ill9 : I ] i z g , i . S I~ ~ NP VP A I A John Peter v NP i v S ~ s., f J I saw \u2022 think",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 19",
                "sec_num": null
            },
            {
                "text": "Adjoining versus substitution. Adjoining seen as substitution.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 20",
                "sec_num": null
            },
            {
                "text": ":",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "0~20",
                "sec_num": null
            },
            {
                "text": "al]~OL21 : V.. \"71o : X C~22 : X / X B ,f,, b 2 .... C,~c 1",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "0~20",
                "sec_num": null
            },
            {
                "text": "Adjoining.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Figure 22",
                "sec_num": null
            },
            {
                "text": "identify the nodes referred by s2 and s4). If we insist that the resulting structure must describe a tree, then we must have either sl dominate s3 or s3 dominate s~. Now suppose there are some fundamental linguistic principles (perhaps those principles that govern the makeup of elementary structures and hence also the characteristics of the domination link between paired quasi-nodes) that determine that it is the case that s 3 must dominate sl and not vice versa. In this case we obtain 79 (as shown in Figure 21 ), a structure obtained by \"adjoining\" fl19. In fact, that s3 must dominate Sl must be derivable from any reasonable linguistic theory that is used to produce the elementary structures concerned (for otherwise a wrong sequence of words would be predicted)\u2022 One possible explanation of why s3 dominates Sl could be given by importing a device like the functional uncertainty machinery (Kaplan and Maxwell 1988) used in LFG\u2022 The treatment used in LFG, when imported here, would suggest that zero or more structures of the form given by fl19 would fit in the gap specified by the domination link between s3 and s4. Thus when the identification of s2 and s4 takes place, s3 must dominate sl and again zero or more structures of the form of fl19 could fit between s3 and sl now (see Joshi and Vijay-Shanker [1989] ) for a discussion of the treatment of long-distance dependency in TAG and LFG). Another way to explain the domination of s3 over sl could be done by using the notions of maximal government domains discussed by Kroch (1989) and using it now to define the characteristics of the domination links such as that between s3 and s4. Note that once the nature of the adjoining operation has been derived, one can pre-compile out the linguistic principles and machinery used to express it. Thus even if one uses, say, the functional uncertainty machinery or maximal government domains, these additional devices (used during the developmental stages) of the grammar need not be used again during the derivation process once we have derived the adjoining operation\u2022 This is analogous to the situation with elementary structures. Some linguistic theory will be involved in defining the elementary structures of a TAG. However, once the grammar has been developed, these principles are no longer directly involved during the derivation phase\u2022 This is because the principles have been pre-compiled into the elementary structures built\u2022 Figure 22 describes the general situation that may be used to contrast substitution, adjoining and multi-component adjoining\u2022 As usual, the identification of the bl and b2 quasi-nodes defines the substitution of the O~21 at the bl quasi-node of c~20. Now suppose instead of considering a (quasi) root such as the one named b2 we consider a pair of hence also determining the nature of domination links). Thus, this raises the question that although adjoining is used in defining the TAG formalism, could it too (like the elementary structures) be precompiled from some more fundamental principles?",
                "cite_spans": [
                    {
                        "start": 901,
                        "end": 926,
                        "text": "(Kaplan and Maxwell 1988)",
                        "ref_id": "BIBREF11"
                    },
                    {
                        "start": 1295,
                        "end": 1325,
                        "text": "Joshi and Vijay-Shanker [1989]",
                        "ref_id": "BIBREF8"
                    },
                    {
                        "start": 1537,
                        "end": 1549,
                        "text": "Kroch (1989)",
                        "ref_id": "BIBREF13"
                    }
                ],
                "ref_spans": [
                    {
                        "start": 507,
                        "end": 516,
                        "text": "Figure 21",
                        "ref_id": null
                    },
                    {
                        "start": 2449,
                        "end": 2458,
                        "text": "Figure 22",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Figure 22",
                "sec_num": null
            },
            {
                "text": "In this section we show that the new interpretation of the TAG formalism allows the possibility of representing a grammar in a more compact fashion. This is illustrated by means of an example. The structure named \"/12 ( Figure 24 ) pictorially represents the normal (or default) tree structure that can be associated with any verb, whereas 3'13 will be used specifically in the case of a simple transitive verb. The default structure associated with a simple transitive verb can be obtained by considering the description illustrated pictorially by 3'13 and inheriting the description (3'12) that is common for all verbs. Now since the Vl and v2 nodes have to be identified, we have the following.",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 220,
                        "end": 229,
                        "text": "Figure 24",
                        "ref_id": "FIGREF7"
                    }
                ],
                "eq_spans": [],
                "section": "Describing the Elementary Objects of a Grammar",
                "sec_num": "5.2"
            },
            {
                "text": "\u2022 The domination link between vpl and Vl quasi-nodes indicates a path length greater than or equal to 0. However, in this case since the labels of these quasi-nodes are different, they cannot refer to the same node. Thus, in this case we have a path length that is greater than 0.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Describing the Elementary Objects of a Grammar",
                "sec_num": "5.2"
            },
            {
                "text": "\u2022 vp2 quasi-node immediately dominates the v2 quasi-node (i.e., path length=l).",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Describing the Elementary Objects of a Grammar",
                "sec_num": "5.2"
            },
            {
                "text": "\u2022 Since Vl and v2 quasi-nodes are identified and since we insist on a tree structure, we have vpl and vp2 quasi-nodes in the domination relation. In fact vpl quasi-node must dominate vp2 quasi-node in the resulting structure by a path of length 0 or more (from the two observations above).",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Describing the Elementary Objects of a Grammar",
                "sec_num": "5.2"
            },
            {
                "text": "Thus we get the structure given by c~2a as desired. Rogers and Vijay-Shanker (1992) describe a proof system that can be used to perform the type of reasoning involved in constructing the structure o~23 as described above.",
                "cite_spans": [
                    {
                        "start": 52,
                        "end": 83,
                        "text": "Rogers and Vijay-Shanker (1992)",
                        "ref_id": "BIBREF17"
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Describing the Elementary Objects of a Grammar",
                "sec_num": "5.2"
            },
            {
                "text": "In the manner described above we can build the default structure for every subcategorization frame. Such structures will be specified in any lexicalized TAG; the difference (in the envisaged specification method) is that we no longer precompile out all possibilities (thus repeating the structure ~'12 in all structures associated with every type of verb). To complete the description of the rest of the elementary quasi-trees one would have to use transformations, meta-rules, or lexical rules to specify the structures for passivization, wh-movement, topicalization, etc. Work along this direction is being carried out (Vijay-Shanker and Schabes 1992).",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Describing the Elementary Objects of a Grammar",
                "sec_num": "5.2"
            },
            {
                "text": "In this paper, we have embedded TAG in the unification framework in a manner consistent with the constraint-based approach used in this framework. Starting from first principles and taking the localization of dependencies within the elementary structures of a TAG grammar as the only basic principle, we have argued that the objects manipulated by such a grammar are not trees but descriptions of well-formed syntactic structures. From D-Theory, we have adopted the use of domination relation and use of identifiers to refer to nodes while describing such structures. Quasi-trees were introduced to depict pictorially partial descriptions of trees. The pairing of quasi-nodes (with domination link between them) was then used to explain the association of two feature structures with individual nodes in previous definition of Feature structurebased Tree Adjoining Grammars (FTAG). In fact, we also show that the formalism defined in Harbusch (1990) (where only one feature structure is associated with every node) turns out to be similar to the use of FTAG with an additional decision to merge every pair of quasi-nodes by default. We argue that such defaults lead to nonmonotonic behavior.",
                "cite_spans": [
                    {
                        "start": 934,
                        "end": 949,
                        "text": "Harbusch (1990)",
                        "ref_id": "BIBREF3"
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Conclusions",
                "sec_num": "6."
            },
            {
                "text": "One can now view FTAG as a generalization of TAG in that arbitrary categories (as used in GPSG) can label nodes, instead of just atomic symbols (nonterminals) as in TAG. In fact, by not insisting that a pair of quasi-nodes be labeled by the same category in FTAG, as was done in TAG, we argue that the \"adjoining constraints\" follow from the definition of adjunction and the labeling of quasi-nodes, thus making unnecessary the stipulations of SA and OA constraints. In addition, contrary to the assumptions made in current literature on TAG, we show that there are two possible interpretations of NA constraints, only one of which is a special case of SA constraint. We note that as the information associated with quasi-nodes grows during derivation, \"adjoining constraints\" get instantiated dynamically in an FTAG. We make use of this property in order to give examples to show how FTAG can give more succinct descriptions than TAG.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Conclusions",
                "sec_num": "6."
            },
            {
                "text": "We have given a logical formulation of FTAG. This builds on a similar treatment of FUG given by Rounds and Manaster-Ramer. We view this logical formulation as a description of those trees and associated feature structures that are built by CFG-based unification grammars. Unlike a CFG-based formalism that allows only for substitution operation, for an FTAG one has to depict adjunction in addition to substitution. Our treatment captures both these cases. We end by giving a presentation of the semantics that can be used to give the denotation of a grammar, i.e., in our case, the structures derived by a grammar.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Conclusions",
                "sec_num": "6."
            },
            {
                "text": "We have emphasized throughout the paper that we are only interested in the definition of the FlAG formalism. In particular, we have not been concerned with linguistic analyses. However, we have raised a few questions about the formalism that we believe can only be answered on linguistic grounds. In the context of the new interpretation, some of these include whether the linguistic uses of multi-component adjoining can be simulated as the adjoining operation; whether there is an essential need to divide the elementary structures of the grammar as initial and auxiliary structures; and whether the adjoining operation itself can be defined as a substitution operation, the apparent differences between these operations being derived on the basis of some more fundamental linguistic principles used in the design of the elementary structures of the grammar. Even if the answer is in the affirmative, we believe there is considerable advantage to be gained by deriving this operation in order that we can manipulate directly the elementary structures that localize various forms of the dependencies. As observed earlier, with the derivation of this operation (like the derivation of the elementary structures of the grammar), we can disregard (i.e., not reason with) the principles used (to derive them) during the derivation of more complex structures. Finally we have also shown that the new interpretation of the TAG formalism proposed here allows for the possibility of a more compact representation of a TAG grammar.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Conclusions",
                "sec_num": "6."
            },
            {
                "text": "In this way, we have defined adjoining to make it compatible with the traditional definition of adjoining. Recall that in Section 1.2 adjoining in TAG was defined as substituting the auxiliary tree at the node of adjunction with the excised subtree being substituted at the foot node. In Section 5, we look more closely at the definition of adjoining, in particular whether the definition can be made to follow from linguistic principles or stipulations, i.e., whether adjoining is a derived operation and not a fundamental one.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            },
            {
                "text": "A complement tree (for example, the tree f12 inFigure 2) is one where the foot node corresponds to One of the arguments required by the anchor of the tree.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "",
                "sec_num": null
            }
        ],
        "back_matter": [
            {
                "text": "This work was partially supported by NSF grant IRI-9016591. I am extremely grateful to A. AbeiUe, A. K. Joshi, A. Kroch, K. E McCoy, Y. Schabes, S. M. Shieber, and D. J. Weir. Their suggestions and comments at various stages have played a substantial role in the development of this work. I am thankful to the reviewers for many useful suggestions. Many of the figures in this paper have been drawn by XTAG (Schabes and Paroubek 1992) , a workbench for Tree-Adjoining Grammars. I would like to thank Yves Schabes for making this available to me.",
                "cite_spans": [
                    {
                        "start": 407,
                        "end": 434,
                        "text": "(Schabes and Paroubek 1992)",
                        "ref_id": "BIBREF19"
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Acknowledgments",
                "sec_num": null
            },
            {
                "text": "This can be easily established using Lemma 4.1 and Lemma 4.2.Since Tc is continuous, the least fixed point of T6 can be obtained aswhere I\u00b1 is the least interpretation function and is given by I_L(P) the empty set for all predicate symbols P. Let Ic be the fixed point of TG. Then the set of structures derived by a grammar G is given by It(Grammar) , where Grammar is the distinguished predicate symbol as defined earlier.",
                "cite_spans": [
                    {
                        "start": 338,
                        "end": 349,
                        "text": "It(Grammar)",
                        "ref_id": null
                    }
                ],
                "ref_spans": [],
                "eq_spans": [],
                "section": "annex",
                "sec_num": null
            },
            {
                "text": "The logical formulation of FTAG given above is similar to the formulation of FUG and the associated semantics given by Rounds and Manaster-Ramer. This logical formulation of FUG essentially captures CFG-based unification grammars where substitution (and associated unifications) is the operation used for composition. This can be seen from their semantic treatment where type variables are repeatedly substituted for. Rather than using type variables for \"nonterminals,\" in our formulation predicate symbols represent the nonterminals. Although \"substitution\" at frontier nodes can be effectively captured by Rounds-Manaster-Ramer calculus, we found it less cumbersome to express adjunction operation and FTAG in the above DCG-like style.The domination relation and adjunction operation are easily captured by using binary predicates and their substitutions. Despite these syntactic differences, the presentation of the semantics is essentially the same traditional fixed-point semantics. Not only do we capture the substitution operation, as was done in the Rounds-Manaster-Ramer calculus, but we are also able to contrast FUG (and CFG-based unification grammars) with FTAG by capturing adjoining as a pair of substitutions.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Some Remarks",
                "sec_num": "4.3"
            },
            {
                "text": "So far we have concerned ourselves with an interpretation of TAG that is compatible with the constraint-based approach to grammars. We will now briefly discuss some possible implications that this new interpretation may have on design or development of TAG grammars. The point of this section is simply to raise certain possibilities and questions. Providing definitive answers and solutions involves exploring linguistic issues that are beyond the scope of this work.",
                "cite_spans": [],
                "ref_spans": [],
                "eq_spans": [],
                "section": "Some Consequences of the New Interpretation",
                "sec_num": "5."
            },
            {
                "text": "We defined the adjoining operation as an operation that fits a structure in the gap between a pair of associated quasi-nodes. Although the nature of the adjoining operation itself has not been examined in much detail in this paper (apart from defining it in terms of quasi-nodes in a manner such that it is similar to the traditional definition), questions that arise from this work are: how different is the adjoining operation from the more commonly used substitution operation; and whether the definition of adjoining itself (as stated here) follows from some more fundamental linguistic assumptions. To motivate our arguments, we start by considering an example using the so-called multi-component adjoining. Multi-component adjoining.quasi-nodes, such as Cl and b3, that are interior quasi-nodes. Now suppose we unify the b~ and b3 quasi-nodes. Since we will assume that the resulting structure must be a description of a tree, we must have the al quasi-node dominate Cl quasi-node or vice versa. If the cl quasi-node dominates al (as in \"710), we have a structure that appears like the one obtained by adjoining. Suppose there is some principle that predicts this situation to occur when substitution takes place; then we can conclude that adjoining is not a fundamental operation in itself but rather a derived operation. Trying to capture the above-mentioned principle would involve specifying the characteristics of the domination link between pairs of quasi-nodes such as that specified by Cl and b3 and the makeup of elementary structures of a grammar. Let us now consider the other case. Suppose we substitute at the bl node with the quasi-tree rooted by b3; there is no reason to assume that cl must dominate al.Consider the case when al dominates cl. In this case, the structure ~20 must be spliced into two ({~0 and 0\u00a3~) as indicated in Figure 23 . There are several possibilities. First, c~0 may appear above all of 0\u00a322 as indicated by \"711. This appears to correspond to the version of multi-component adjoining where different components of a set ({{~0, ~}) are adjoined simultaneously into another multi-component set, ({0\u00a3~2, ~})\" Other possibilities include 0\u00a320 and O~2 splintered into some number of pieces (depending on the domination links found in them) and interleaved in a more complex fashion.To summarize, when we substitute at bl by identifying it with a quasi-root of another structure, we have the standard substitution. On the other hand, when we substitute at bl by identifying bl and b3, if cl dominates al then the resulting structure appears to be the one formed after adjunction. When al dominates Cl the situation seems to be comparable with that of multi-component adjoining, where (~20 and 0\u00a322 are multi-component sets made up of 0\u00a320,' c~20\" and 0~22 ,' 0\u00a322 ,'' respectively. Such multicomponent adjoining has been used previously in providing linguistic analyses. Since both cases occur (al dominates cl or vice versa), we believe it only further justifies our claim that in situations where we consider substitutions as above, whether we have cl dominating al (adjoining) or not (multi-component adjoining) depends on the linguistic principles being instantiated during the development of elementary structures (and",
                "cite_spans": [],
                "ref_spans": [
                    {
                        "start": 1852,
                        "end": 1861,
                        "text": "Figure 23",
                        "ref_id": null
                    }
                ],
                "eq_spans": [],
                "section": "Adjoining, Multi-Component Adjoining, and Substitution",
                "sec_num": "5.1"
            }
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        "ref_entries": {
            "FIGREF0": {
                "type_str": "figure",
                "num": null,
                "text": "Figure 1 Initial trees.",
                "uris": null
            },
            "FIGREF1": {
                "type_str": "figure",
                "num": null,
                "text": "Figure 2 Auxiliary trees.",
                "uris": null
            },
            "FIGREF2": {
                "type_str": "figure",
                "num": null,
                "text": "Figure 4",
                "uris": null
            },
            "FIGREF4": {
                "type_str": "figure",
                "num": null,
                "text": "Figure 6 Some possible auxiliary quasi-trees.",
                "uris": null
            },
            "FIGREF5": {
                "type_str": "figure",
                "num": null,
                "text": "Figure 11 \"SA\" constraints.",
                "uris": null
            },
            "FIGREF7": {
                "type_str": "figure",
                "num": null,
                "text": "I wonder who Peter thought John saw.(5) Peter thought John saw Mary. and the following, which are not well-formed sentences.(6) Who John saw?",
                "uris": null
            },
            "FIGREF8": {
                "type_str": "figure",
                "num": null,
                "text": ". If the feature structures of sl and s2 quasi-nodes of c~12 are unified then the other pair",
                "uris": null
            },
            "FIGREF9": {
                "type_str": "figure",
                "num": null,
                "text": "x) ,~ S A Dom(;G y) A cat(y) ~. SA l(y) ~ z A count(y) ,~ zero A cat(z) ~ c x) ~. s A Dora(x. z) A cat(~) ~ s ^ on,(coast(z)) ~ co~,,t(z.2 ADorn(z2, y) A cat(z2) ~ S A cat(z3) ~ b A cat(y) ~ S Aco~nt(z2) ~ count(y)Figure 15",
                "uris": null
            },
            "FIGREF11": {
                "type_str": "figure",
                "num": null,
                "text": "The transformation function is monotonic.",
                "uris": null
            },
            "FIGREF12": {
                "type_str": "figure",
                "num": null,
                "text": "For all environments, p, and interpretations, L if (I, p) interpretation I0 such that I0 u I and (I, p) ~ ~b.",
                "uris": null
            },
            "FIGREF13": {
                "type_str": "figure",
                "num": null,
                "text": "Figure 17 Derivations in MCTAG.",
                "uris": null
            },
            "FIGREF14": {
                "type_str": "figure",
                "num": null,
                "text": "Figure 18 Multi-component adjoining seen as adjoining.",
                "uris": null
            },
            "FIGREF15": {
                "type_str": "figure",
                "num": null,
                "text": "Figure 21",
                "uris": null
            },
            "FIGREF16": {
                "type_str": "figure",
                "num": null,
                "text": "Figure 24 Representation of an elementary structure.",
                "uris": null
            }
        }
    }
}