Using Descriptions of Trees in a Tree 
Adjoining Grammar 
K. Vijay-Shanker* 
University of Delaware 
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 structure- 
based 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. 
1. Introduction 
A number of grammatical systems and linguistic theories, such as Functional Unifi- 
cation 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 de- 
fine well-formed structures. This work discusses viewing Tree Adjoining Grammars 
(TAG) in such a manner and embedding it in a unification-based framework. 
Tree Adjoining Grammars (TAG) were first introduced by Joshi, Levy, and Taka- 
hashi (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 de- 
tailed 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 lexi- 
calized approach to TAG that was originally proposed by Schabes, Abeille, and Joshi 
(1988). 
• Department of Computer and Information Sciences, University of Delaware, Newark, DE 19716. 
(~) 1992 Association for Computational Linguistics 
Computational Linguistics Volume 18, Number 4 
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 com- 
pose 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. 
Our aim is to view a TAG as defining constraints on well-formed structures (ac- 
cording 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 adjoin- 
ing 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 spec- 
ification 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 de- 
pict 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. 
1.1 Outline of the Paper 
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 interpret- 
ing 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. 
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 oper- 
ations 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 oper- 
ation where the characteristics of the adjoining and multi-component adjoining oper- 
ations 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- 
482 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
O~ 1 : 
0~2 : S 
NP,~ s 
S NP$ VP 
~ 0~3: NP 
NP$ ~VP v~ NPil 
v, NP,~ , det$ nO 
Figure 1 
Initial trees. 
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. 
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. 
1.2 Introduction to Tree Adjoining Grammars 
A Tree Adjoining Grammar (TAG) as defined traditionally is said to be specified by a fi- 
nite 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 depen- 
dencies into a finite set of elementary trees. The elementary trees in a TAG correspond 
to minimal linguistic structures that localize the dependencies such as subcategoriza- 
tion, 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). 
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 
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Computational Linguistics Volume 18, Number 4 
f12 : S 
w 
v A 
VP* adv  vo s* 
Figure 2 
Auxiliary trees. 
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. 
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. 
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 ~). 
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 
484 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
71 : 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 
• Det n v NP$ 
Figure 3 
Some examples of adjoining and substitution. 
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. 
A more detailed description of TAG, the use of adjoining constraints, their prop- 
agation 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- 
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Computational Linguistics Volume 18, Number 4 
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. 
2. A Unification-Based Approach to Tree Adjoining Grammars (FTAG) 
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. 
2.1 Adjoining of Trees 
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. 
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. 
2.2 A New Interpretation of TAG Objects 
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. 
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 depen- 
dencies 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 
486 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
(1): S --* NP VP. Also, notice that a VP that captures the combination of the lexical an- 
chor 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. 
These relationships can be stated by means of some assertions about the indi- 
vidual 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. 
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. 
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. 
3. vpl is labeled by VP and is used to indicate the combination with the 
subject (i.e., npl) to yield a sentence. 
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). 
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. 
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. 
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. 
487 
Computational Linguistics Volume 18, Number 4 
C~ 4 
nPl----'~ 
jh 
s 
NP$ VP ~ v~ 
vv "~-v h 
vo NP,k ~ rip 2 
S /\ 
NPi4. S 
S 
NP$ VP 
VP 
vO NPi 
Figure 4 
The domination relations. 
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 quasi- 
nodes 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. 
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 domina- 
tion 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 de- 
pict 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. 
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 
488 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
$ 
E~' 6 : 
++ \['.,,=">Cl ,',\[,....,= \[...h,= \[.,;<,>| 
I I 
VP \[head: <2> \[1 
S 
O/,/ : 
\[h,.: <I~t 
• + \[h=a= <I> \[l ~* 
Figure 5 
Associating feature structures with quasi-nodes. 
initial quasi-tree, auxiliary quasi-tree, quasi-root, and quasi-foot in place of initial 
tree, auxiliary tree, root node, and foot node, respectively. 
2.3 Associating Feature Structures with Quasi-Nodes 
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 quasi- 
nodes 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). 
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)." 
If the objects manipulated by a TAG are considered as quasi-trees, a natural ques- 
tion 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 quasi- 
trees, 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. 
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Computational Linguistics Volume 18, Number 4 
f13 " VP 
7 ,% 
vP 
VP advO 
flS" VP //'\ 
VP7 VP advO 
f14: VP 7 
vP 
VP* advO 
,% 
VP* ~ VP* 
,a,4 
Figure 6 
Some possible auxiliary quasi-trees. 
f16 : vp '~'"vt'9 
VP* advO 
~o 
2.4 The Adjoining Operation 
We will now define the adjoining operation on quasi-trees and see that (unlike pre- 
viously) this operation has the property that in the resulting structure all the struc- 
tural 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 quasi- 
nodes (~/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. 
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 domi- 
nation 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 
2 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. 
490 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
c'3~; 
fl: Ffroo  
% 
; 
I 
I ! 
Dfoot u fl~ 
Figure 7 
The adjoining operation in FTAG. 
"/4 S 
NP~, VP 
ve Ts 
VP adv 
VP 
v NP,I. 
S /X 
NP$ VP 
VP adv 
A 
v NP$ 
Figure 8 
Examples of the adjoining operation. 
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)" 
2.5 The Substitution Operation 
The substitution operation used in TAG is the same as that used in context-free gram- 
mars (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 
491 
Computational Linguistics Volume 18, Number 4 
E foot-1 
Figure 9 
The substitution operation. 
(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. 
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 quasi- 
node ~ 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. 
2.6 Some Observations 
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. 
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 quasi- 
node 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, 
492 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
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. 
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. 
2.7 Objects Derived by a Grammar 
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). 
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. 
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. 
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 struc- 
tures 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- 
493 
Computational Linguistics Volume 18, Number 4 
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). 
2.8 Using One (Rather than Two) Feature Structure 
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 neces- 
sary 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. 
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). 
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. 
3. Feature Structures and Adjoining Constraints 
In the traditional definition of a TAG, the adjoining possibilities at a node is deter- 
mined 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 correspon- 
dences 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. 
3.10A Constraint 
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 
494 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
37 s \[\] 
S\[tense. q S \[tense ." +\]. 
5/I S\[tense: -\] NP* VP 
VP 
NP VP /~~ 
PRO v S \[tense : <1> -\] 
tO V \] 
I ! I ' 
tries S \[tense : <1~\] win 
Sltense : + 1 
I 
S\[,e.se: *\] 
NP$ VP 
i i , 
vp 
v $ \[tense : <1> .\] 
I ! i 
tries S \[tense: <1>\] 
NP VP 
PRO 
to v 
I 
win 
Figure 10 
"OA" constraints. 
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 con- 
straiht 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 sen- 
tence. 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." 
3.2 SA Constraints 
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 
495 
Computational Linguistics Volume 18, Number 4 
C~ 0 " 
J 
S\[t,ns,: q 
i i 
i 
t S\[,eos,: q 
NP~. VP 
VP 
v NP$ 
met 
rid" 
S\[tense: +\] 
i 
s \[t,,,~ : +\] 
/N 
NP$ VP 
VP 
,, s \[re.s, : <~> +\] 
I ' i i ! 
thinks S \[tense : <1>\] 
Figure 11 
"SA" constraints. 
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. 
3.3 NA Constraints 
Recall that a node with an NA constraint cannot be the target of an adjunction. Tradi- 
tionally, 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). 
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 
3 A complement tree (for example, the tree f12 in Figure 2) is one where the foot node corresponds to One 
of the arguments required by the anchor of the tree. 
496 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
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. 
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). 
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. 
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. 
3.4 Comparing the Implementation of Adjoining Constraints 
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. 
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"). 
497 
Computational Linguistics Volume 18, Number 4 
99 : 
"/r : ! 
NP.I. vP 
S\[,e°,e. +3 S\[,e,,,e-÷\] "I"I h J 
vP ,,i ~9 : :, ~ 
S\[,e.,° : ÷\] , .i--" ~2 s\[,,,,. 
+\] • ~I, ..... 1 "i"'s3 /% 
I , 
tri. S \[tease : .~ "~" s 4 NPL VP 
t NP,I. VP /~ 
i i / \ 
VP 1 Np* vP 
vP 
v S \[,ense : "l /~ 
v NP,I. I v NP,~ 
i I I 
tries S \[\] met ta,t 
Figure 12 
Comparison of adjunction constraints--Example 1. 
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 suf- 
fers in comparison with the implementation of "constraints" in FTAG. Consider the 
following well-formed sentences 
(1) Who did John see? 
(2) Who did Peter think John saw? 
(3) I wonder who John saw. 
(4) 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? 
498 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
(~10 " 
fllo : 
S 
NPI S 
who NP VP 
A 
John v NPI 
$11W e 
S 
fl11: 
NiP Vl' 
I v S* I 
wonder 
Cgll : 
$ 
NP VP 
Peter v $* 
J think 
$ 
NP ~ 
John • 
taw Mary 
ill2 " s A 
StCt S e 
J 
did 
Figure 13 
Comparison of adjunction constraints--Example 2. 
(7) I wonder who did John see? 
(8) Who Peter thought John saw. 
We will first consider a TAG account (in traditional style). The trees (without consid- 
ering 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, 
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Computational Linguistics Volume 18, Number 4 
again we will need two trees (corresponding to fl11), with identical tree structure but 
differing in the adjoining constraints. 
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. 
If the feature structures of sl and s2 quasi-nodes of c~12 are unified then the other pair 
813 : 
~12 : 
s \[main : +\] 
I i 
I 
Nv, S\[m,:<l>\] 
I 
who 
S\[in,. "1 
NP ~ 'A 
John 
v NP i 
I 1 
SaW • 
s En,-'. • ÷\] 
I i 
I I 
s\[,,,. -I 
NP ~ 
\[wh: + 
81a : 
wonder 
Cg13 : S \[tu,la: ~\] 
I 
I~ Irp 'A 
J~ 
I I saw Mary 
s\[..,° : ,\] 
I s 6.-." q 
IA 
think 
815 : s\[\] 
I 
u. s 0 .... \] I 
did 
Figure 14 
Comparison of adjunction constraints--Example 2. 
500 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
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 ill- 
formedness 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. 
4. A Logical Formulation 
A central theme in our definition of FTAG has been the view that the objects manip- 
ulated 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. 
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). 
In the framework of Rounds and Manaster-Ramer (1987), an FUG (or any context- 
free grammar with associated unification equations as in, say PATR-II) can be rep- 
resented 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 vari- 
ables; 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 
t::= a 
x 
l(tl) 
where a is an atomic value 
where x is an individual variable 
where 1 is a label (or attribute) and tl is a term. 
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Computational Linguistics Volume 18, Number 4 
The set of formulae is defined as 
0::= tl ~ t2 
01 A 02 
0~ v 02 
where tl,t2 are terms 
where 01,02 are formulae 
where 01~ 02 are formulae. 
For example, (l(x) = y) A (h(x) = z) A (g(y) = z) A (z = a) describes (among others) 
the following feature structure. 
1 : g : \[-i-~ \] 
h : \[~a 
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. 
4.1 Expressing an FTAG 
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 
cat(x) ~ S A Dom(x,y) A cat(y) ~ S A l(y) ~ z A count(y) ~ zero A cat(z) ~, c. 
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 
~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. 
In this case, Dom(x, y) is used to indicate that the quasi-root (x) dominates the associ- 
ated 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 
Dom(x,y) 4==~ x ~ y V fll(x~y) V ... V -'fin(X~y) 
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 
502 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
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. 
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 
Inittree(x) 4=~ ~l(x) V ... V ~m(X) 
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 
Grammar(x) 4=~ Inittree(x) A cat(x) = S. 
Note that for a quasi-node (referred to as x) where substitution can take place, we can 
specify Inittree(x) to specify the substitution. 
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). 
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 ac- 
tually a CFG-based unification grammar) since only "substitution" is used. Binary 
Inittree(x) 
Dora(z, y) 
?G(x, y) 
s\[\] .~.-.- x 
I 
C~14 : S \[count: \[o .... I> \[~ ~ z s\[\] 
y....~ s Dnnt, o\] . s \[con.,,,~,\] b 
, \ 
, ~ c S\[~ou.t,,,~d % ~-..y 
Inittree(x) A cat(x) ~ S 
4==~ cat(x) ,~ S A Dom(;G y) A cat(y) ~. SA 
l(y) ~ z A count(y) ,~ zero A cat(z) ~ c 
~,~(x, y) v (x ~ y) 
~. c.t(x) ~. s A Dora(x. z) A cat(~) ~ s ^ on,(coast(z)) ~ co~,,t(z.2 
^I(~) ~ z, ^ 20) ~ z~ A 3(2) ~ :~ ^ c~t(z,) ~. a 
ADorn(z2, y) A cat(z2) ~ S A cat(z3) ~ b A cat(y) ~ S 
Aco~nt(z2) ~ count(y) 
Figure 15 
Example: An FTAG grammar and its representation. 
503 
Computational Linguistics Volume 18, Number 4 
predicates are used to capture adjunction (which is defined as a pair of substitutions) 
in FTAG. 
4.2 Fixed-Point Semantics (Denotation of an FTAG Grammar) 
As mentioned before, the set of terms is defined recursively as 
t::= a 
x 
l(h) 
where a is an atomic value 
where x is an individual variable 
where I is a label and tl is a term. 
However the set of formulae is now defined by 
¢::= tl ~ t2 
P(tl,...,tn) 
¢1 A ¢2 
41 V ¢2 
where tl, t2 are terms 
where tl,. •., tn are terms and P is a n-ary predicate symbol 
where ¢1, ¢2 are formulae 
where ¢1, ¢2 are formulae. 
From the discussion given in the previous section any FTAG can be stated as 
Pl(tl,1,..., tin,l) ": ~." 01 
Pn(h,n,..., tm,n) ~ ¢~ 
where ¢1,..., Cn are formulae and t1,1,.. • tm,1, tl,n • .., tm,n are terms such that for 1 _< 
i,j < n, if i ¢ 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 equiva- 
lence 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. 
Definition 
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)• 
• Valuep(x) = p(x) where x is a variable• 
• Valuep(a) = .G where a is an atom, where Aa is the atomic structure that 
corresponds to the atom a. 
• Valuep(l(t)) -- fit~l, if fit/l is defined, where 1 is an attribute, t is a term 
and Valuep(t) = fit. If Valuep(t) is not defined or Valuep(t) = fit but fit/l is 
not defined then Valuep(l(t)) is not defined. 
Let p be an environment function and I be an interpretation mapping predicate 
symbols to their denotations, i.e•, if P is a n-ary predicate symbol then I maps P to some 
set of n-tuples of automata• Given an interpretation function I and an environment p 
we define ~ in the following way. 
504 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
Definition 
(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). 
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 specifica- 
tion. We define the transformation function, To, such that given an interpretation,/, 
Tc returns an interpretation TG (I) given by 
Definition 
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) ~ ¢5i. 
Ordering relations 
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). 
Lemma 4.1. 
If/1 _/2, then for all environments, p, and formulae, q~, if (11, p) ~ ~ then (/2, P) ~ q~. 
This can be easily shown by using induction on the structure of the formula ~. 
Theorem 4.1. 
The transformation function is monotonic. 
Let/1 _/2. We have to show for all P that Tc(I1)(P) C Tc(I2)(P). Let P(h,..., tn) ~ ¢3 
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). 
We will call an interpretation, L finite if for all predicate symbols, P, I(P) is a finite 
set. 
Lemma 4.2. 
For all environments, p, and interpretations, L if (I, p) 
interpretation I0 such that I0 u I and (I, p) ~ ~b. 
~b then there is a finite 
This can be shown by a straightforward induction on the structure of ~b, and by con- 
structing I0 in the obvious manner. 
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Computational Linguistics Volume 18, Number 4 
Theorem 4.2. 
The transformation function is continuous. 
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 as 
U Ti (I.a_) 
i~o 
where I± 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. 
4.3 Some Remarks 
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 for- 
mulation of FUG essentially captures CFG-based unification grammars where substi- 
tution (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 predi- 
cate 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. 
5. Some Consequences of the New Interpretation 
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. 
5.1 Adjoining, Multi-Component Adjoining, and Substitution 
We defined the adjoining operation as an operation that fits a structure in the gap be- 
tween 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 adjoin- 
ing 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. 
506 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
Consider the derivation of: 
(1) Which picture did you buy a copy of? 
(2) Which picture did you buy a photograph of a copy of? 
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 tra- 
ditional 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). 
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). 
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). 
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. 
507 
Computational Linguistics Volume 18, Number 4 
/ 18 : 
~17 : 
S ~ 
NPI $ 
which painting 
NP 
DET N' 
a N PP 
A 
copy p NP(, 
of 
NP 
Cgl5 : S 
DET N' 
a N PP 
did N'P VP I A IA 
photograph p NP i you v NIP 
I II 
of buy • 
Figure 16 
A multi-component tree adjoining grammar. 
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). 
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 
508 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
7s : S ~ 
NFI S 
which painting aux $ 
did NP VP 
you v NP 
buy DET I 
a 
N ~ 
N PP 
copy p ~i t 
~ • 
Figure 17 
Derivations in MCTAG. 
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. 
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 quasi- 
node 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. 
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 
509 
Computational Linguistics Volume 18, Number 4 
(~15 : S 
aux S 
did NP I 
you 
~16 " 
vP A 
i NP ~ 112 
buy 
S ~ 
NP~ S 
which painting NP 
DET 1 
a 
~t 
N PP IA 
copy p NPI 
of e 
Figure 18 
Multi-component adjoining seen as adjoining. 
C~17 : 
S S 
l /~19 : g 
, I 
S S 
NP VP NP VP 
John Peter v NP v S 
I I I 
saw Mary think 
Figure 19 
Initial or auxiliary? 
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., 
510 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
0~18 • 
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 • think 
Figure 20 
Adjoining versus substitution. 
G¢18 " 
S 
I 
S 
NPi S 
; ill9 : 
who S 
NP VP 
John 
v NPi 
I 1 
saw • 
79 : 
NP VP 
Peter v 5 
I 
think 
s 
s 
NPi S 
f i t 
who 5 
NP VP 
Peter v S 
think 
NP VP 
John 
v NPi 
I I 
S&W e 
• ,c--s 2 
Figure 21 
Adjoining seen as substitution. 
511 
Computational Linguistics Volume 18, Number 4 
0~20 : al\]~OL21 : 
V.. 
"71o : X 
C~22 : X / X 
B ,f,, b 2 .... C,~c 1 , Z~ al 
Figure 22 
Adjoining. 
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)• 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• 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 discus- 
sion 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 max- 
imal 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• 
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• This is because the principles have been pre-compiled into the 
elementary structures built• 
Figure 22 describes the general situation that may be used to contrast substitution, 
adjoining and multi-component adjoining• 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 
512 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
")11 : A G'2 
J 
x 
d2 c i 
, 
i 
E B 
Figure 23 
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£~) as indicated in Figure 23. There are several possibilities. First, 
c~0 may appear above all of 0£22 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£~2, ~})" Other pos- 
sibilities include 0£20 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£22 
are multi-component sets made up of 0£20,' c~20" and 0~22 ,' 0£22 ,'' respectively. Such multi- 
component 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 linguis- 
tic principles being instantiated during the development of elementary structures (and 
513 
Computational Linguistics Volume 18, Number 4 
")'12 " $ 
A 
NP VP 
"/13 " 
J 
Figure 24 
Representation of an elementary structure. 
C~23 • 
S 
A 
NP VP 
VP VP 
A A 
v NP v NP 
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? 
5.2 Describing the Elementary Objects of a Grammar 
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. 
• 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. 
• vp2 quasi-node immediately dominates the v2 quasi-node (i.e., path 
length=l). 
• 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). 
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. 
In the manner described above we can build the default structure for every sub- 
categorization frame. Such structures will be specified in any lexicalized TAG; the 
difference (in the envisaged specification method) is that we no longer precompile out 
514 
K. Vijay-Shanker Using Descriptions of Trees in a Tree Adjoining Grammar 
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). 
6. Conclusions 
In this paper, we have embedded TAG in the unification framework in a manner con- 
sistent with the constraint-based approach used in this framework. Starting from first 
principles and taking the localization of dependencies within the elementary struc- 
tures 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 syntac- 
tic 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 structure- 
based Tree Adjoining Grammars (FTAG). In fact, we also show that the formalism 
defined in Harbusch (1990) (where only one feature structure is associated with ev- 
ery 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. 
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. 
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. 
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 essen- 
tial 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 
515 
Computational Linguistics Volume 18, Number 4 
operation, the apparent differences between these operations being derived on the 
basis of some more fundamental linguistic principles used in the design of the ele- 
mentary 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 com- 
plex 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. 
Acknowledgments 
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. 
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