Feature Structures Based Tree Adjoining Grammars 1 
K. Vijay-Shanker 
Department of Computer and Information Sciences 
University of Delaware 
Newark, DE 19711 
U.S.A 
A. K. Joshi 
Del)artment of Computer and Information Science 
University of Pennsylvania 
Philadelphia, PA 19104 
U.S.A 
Abstract We have embedded Tree Adjoining Grammars (TAG) in a fea- 
ture structure based unification system. The resulting system, Feature 
Structure based Tree Adjoining Grammars (FTAG), captures the princi- 
ple of factoring dependencies and recursion, fundamental to TAG's. Wc 
show that FTAG has an enhanced descriptive capacity compared to TAG 
formalisnr. We consider some restricted versions of this system and some 
possible linguistic stipulations that can be made. We briefly describe a 
calculus to represent the structures used by this system, extending on 
the work of Rounds, and Kasper \[Rounds et al. 1986, Kasper et al. 1986\] 
involving the logical formulation of feature structures. 
S 
S Np / ~'~ VP 
DET N V 
I I I 
the man met 
U I U 2 U 3 
Figure 1: Initial Trees 
NNNp / 
DET 
I 
~c 
\ 
N 
I 
woman 
1 Introduction 
Tree Adjoining Grammars (TAG) were first introduced by Joshi, Levy, 
and Takalmshi \[Joshi et al. 1975\]. The first study of this system, from 
the point of view of its formal properties and linguistic applicability, was 
carried out by Joshi in \[Joshi 1985\]. TAG's have been used in providing 
linguistic analyses; a detailed study of the linguistic relevance was done 
by Kroch and Joshi in \[Kroch et al. 1985\]. 
In this paper, we show lmw TAG's can be embedded in a feature struc- 
ture based framework. Feature structure based Tree Adjoining Grammars 
(FTAG) are introduced in Section 2, and is f611owed by a comparsion of 
the descriptive capacity of FTAG and TAG. A restricted version of FTAG 
is proposed and some possible linguistic stipulations are considered. In 
Section 3, we introduce a calculus, which is an extension of the logical 
calculus of Rounds and Kasper \[Rounds et al. 1986, Kasper et al. 1986\] 
allowing A-abstraction and application, in order to describe the structures 
used in FTAG's. Finally, in Section 4, we summarize the work presented 
in this paper. 
1.1 Introduction to Tree Adjoining Grammars 
Tree Adjoining Grammars (TAG), unlike other grammatical systems used 
in computational linguistics, is a tree rewriting system. Unlike the string 
rewriting formalisms which writes recursion into the rules that generate 
the phrase structure, a TAG factors reeursion and dependencies into a 
finite set of elementary trees. The elementary trees in a TAG correspond 
to minimal linguistic structures that localize the dependencies such as 
agreement, subcategorization, and filler-gap. There are two kinds of el- 
enrentary trees: the initial trees and auxiliary trees. The initial trees 
roughly (Figure 1) correspond to simple sentences. Thus, the root of an 
initial trce is labelled by the symbol S. They are required to have a 
frontier made up of terminals. 
The auxiliary trees (Figure 2) correspond roughly to minimal recur- 
sive constructions. Thus, if the root of an auxiliary tree is labelled by a 
nonterminal symbol, X, then there is a node (called the foot node) in the 
frontier of this tree which is labelled by X. The rest of the nodes in the 
frontier are labelled by terminal symbols. 
1This work was partially supported by NSF grants MCS-82-19116-CER, DCR-84- 
10413,ARO ffrant DAA29-84-9-~027, and DARPA grant N0014-85-K0018 
714 
X 
vl ~v2 
foot node 
NP 
/ ~s 
who I I 
e V 
Figure 2: Auxiliary Trees 
We will now define the operation of adjunction. Let 7 be a tree with 
a node labelled by X. Let fl be an auxiliary tree, whose root and foot 
node are also labelled by X. Then, adjoining/3 at the node labelled by 
X in 7 will result in tbe tree illustrated in Figure 3. In Figure 3, we also 
S 
u 2 
NP 
wu/ DL:rr N 
I l 1 Nf ~ man who j I 
e V 
$ 
vP 
V/" ~NP 
I /N 
Dl~r N 
I I ~e Woman 
Figure 3: The operation of adjoinfng 
show tl~e result of adjoining the auxiliary tree fll at the subject NP node 
of the initial tree al. 
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 the node must be the 
same as the label of tile root (and the foot) node of the auxiliary tree. 
Fm'ther restriction on this set of auxiliary trees is done by enumerating 
with each node the subset of anxiliary trees which can be adjoined at that 
node. This specification of a set of auxiliary trees, which can be adjoined 
at a node, is called the Selective Adjoining (SA) constraints. In tim case 
where we specify the empty set, we say that the node has a Nail Adjoining 
(NA) constraint:~. It is possible to insist that adjunction is mandatory at 
a node. In such a case, wc say that the node has an Obligatory Adjoining 
(OA) constraint. 
A more detailed description of TAG's and their linguistic relevance 
may be found in \[Kroeh et al. 1985\]. 
1.2 Feature Structure Based Grammatical Systems 
Several different approaches to natural language granunars have devel- 
oped the notion of feature structures to describe linguistic objects. In 
order to capture certain linguistic phenomena such as agreement, subcat- 
egorization, cte., a number of. recent grammatical systems have added, 
on top of a CFG skclcton, a feature based informatioual element. Ex- 
ample or" sncb systems (see \[Shieber 1985a\]) include Generalized Phrase 
Structure Grammars (GPSG), Lexical functional Grammars (LFG), and 
tIead-driven Phrase Structure Grammars (IIPSG). A feature structure 
(as given below) is essentially a set of attribute-value pairs where values 
may be atomic ~*ymbols or another feature structure. 
cat : S 
cat 
\[ : \[\] agr 
cat : 
2 : agr : 
subject 
53 "1 
\[\] 
\[\] 
Tim notation of the co-indexing box (\[\] in this example) is used to ex- 
press the f;~ct that the values of two subfeatures are the stone. Feature 
structures with co-indexing boxes have also been called reentrant feature 
structures in the literature. 
We can define a partial ordering, E, on a set of feature structures 
using tbe notion of subsnmption (carries less in/ormalion or is more gen- 
eral). Unification of two feat,re structures (if it is defined) corresponds 
to the feature ~;tructure that has all the information contained in the 
original two feal;nre structures and nothing more. We will not describe 
feature structur,~s any fnrther (see \[Shieber 1985a\] for more details on fea- 
turc structures and an introduction to the unification based approach to 
grammars). 
2 Featm'e Structure Based Tree Adjoining 
Grammars (FTAG) 
The linguistic theory underlying TAG's is centered around the factor- 
ization of reeursion and localization of dependencies into the elementary 
trees. The "dependent" items usually belong to the same elementary 
tree 2. Thus, for example, the predicate and its arguments will be in the 
same tree, as will the filler and the gap. Our main goal in embedding 
TAG's in an unificational framework is to capture this localization of de- 
pendencies. Therefore, we would like to associate feature structures with 
the elementary trees (rather than break these trees into a CFG-like rule 
based systems, and then use some mechanism to ensure only the trees 
prodnced by the "lAG itself are generateda)~ In tbd':'feature structures 
2It is eometime~ possible for "dependent" iterem to belong to an elementary tree 
and the immediate auxiliary tree that is adjoined in it. 
aSuch a scheme wotdd be an alternate way of embedding TAG's in an unifieational 
framework. IIowever, it does not capture the linguistic intuitions tmderlying TAG's, 
and losc~ the attractive feature of localizing depende~tcles. 
associated with the elementary trees, we can state the constraints among 
the dependent nodes dircctly. IIence, in an initial tree corresponding to 
a simple sentence, wc can state that the main verb and the subject NP 
(which are part of the same initial tree) share the agreement feature. 
Thus, such checking, in many cases, can be precompiled (of course only 
after lexical insertion) and need not be done dynamically. 
2.1 General Schema 
Ill unification grammars, a feature structure is associated with a node 
in a derivation tree in order to describe that node and its realtion to 
featnres of other nodes in tile derivation tree. In a TAG, any node in an 
elementary tree is related to the other nodes in that trec in two ways. 
Feature structures written in FTAG using the standard matrix notation, 
describing a node, ~h can be made on the basis of: 
1. the relation of I 1 to its supertrce, i.e., tile view of the uode from the 
top. Let us call this feature structure as t,~. 
2. the rclation to its descendants, i.e., the view from below. This 
feature structure is called bo. 
Note that both the t,~ and b,~ feature structure hold of the node r l. In 
a derivation tree of a CFG based unification system, wc associate one 
featnre structure with a node (the unification of these two structures) 
since both the statements, t and b, together hold for the node, and uo 
further nodes are introduced between the node's supertrce and subtrec. 
This property is not trne in a TAG. On adjunction, at a node there is 
~o longer a single node; rather ~ul auxiliary trec replaces the node. Wc 
believe that this approach of ~sociating two statements with a node in 
the auxiliary tree is iu the spirit of TAG's because of the OA constraints 
in TAG's. A node with OA constraints cannot bc viewed as a single 
node and must be considered as something that has to be replaced by 
an auxiliary tree. t and b axe restrictions about tile auxiliary tree that 
must be adjoined at this node. Note that if the node does not have OA 
constraint then we should expect t and b to be compatible. For example, 
in the final sentential tree, this node will be viewed as a single entity. 
Thus, in general, with every internal node, ~, (i.e., where adjunction 
could take place), we associate two structures, tn and b n. With each 
terminal node, we would associate only one structure 4, 
xtroot t fo'~t X___-~ 
bfoot 
Figure 4: Feature structures and adjunction 
4It is posslblc to allow adjunctlons at nodes corresponding to pre-lexlcal items. 
For example, we may wish to obtain verb-clusters by adiunctions at nodes which are 
labelled ~s verbs. In such a c~se, we will have to associate two feature structures with 
pre.lexical nodes too. 
715 
Let Us now consider the case when adjoining takes place as showu in 
the figure 4. The notation we use is to write alongside each node, the 
t and b statements, with the t statement written above the b statement. 
Let us say that t~oot,b~oot aud tloo~,b/oo~ are the t and b statements of 
the root and foot nodes of the auxiliary tree used for adjunction at the 
node r/. Based on what t and b stand for, it is obvious that on adjnnction 
tim statements t,~ and troot hold of the node corresponding to the root of 
the anxifiary tree. Similarly, the statements b, and b/oot hold of the node 
corresponding to the foot of the auxiliary tree. Thus, ou adjunction, we 
unify t, with t~oot, and b,~ with b/oot. In fact, this adjunetion is permissible 
only if t,.oot and t o are cmnpatible as are b/oo~ and b,. If we do not adjoin 
at the node, 0, then we unify t s With b,. At the end of a derivation, the 
tree generated must not have any nodes with OA constraints. We cheek 
that by unifying the t and b feature structures of every node.• More details 
of the definition of FTAG may be found in \[Vijayashanker 1987\]. 
We now give an example of an initial tree and an auxiliary tree. We 
would like to note that, just as in a TAG, the elementary trees which 
are the domain of co-occurenee restrictions is available as a single unit 
during each step of the derivation. Thus, most of these co-occurence 
constraints can be eheckcd even before the tree is used in a derivation, 
and this checking need not be linked to the derivation process. 
2.2 Unification and Constraints 
Since we expect that there are linguistic reasons determining why some 
auxiliary tree can be adjoined at a tree and why some cannot, or why some 
nodes have OA constraint, we would like to express these constraints in 
the feature structm:es associated with nodes. Further, as described in 
Section 2.1, adjunctions will be allowed only if the appropriate feature 
structures can be unified. Thus, we expect to implement the adjoining 
constraints of TAG's simply by making declarative statements made in 
the feature structures associated with the nodes to ensure that only the 
appropriate trees get adjoined at a node. 
The adjoining constraints are implemented in FTAG as follows. No- 
tice, from Figure 4, t~ and troot, and b, and b.toa must be compatible for 
adjunction to occur. We hope to specify some feature-values in these t, b 
statements to specify the local constraints so that 
1. if some auxiliary tree should not adjoined at a node (because of its 
SA constraint) then some unification involved (tu with troop, or b/oo~ 
with b,~) in our attempt to adjoin this auxiliary tree will fail, and 
2. if a node has OA constraint, we should ensure that an appropriate 
auxiliary tree does get adjoined at that node. This is ensured if t, 
is incompatible with b,. 
The example, given in Figure 7, illustrates the implementation of both 
the OA and SA constraint. The view of the root node of a from below 
.suggests that b statement for this node makes the assertion that the value 
of the tense attribute is - (or untensed). However, the t statement should 
assert tense : + (since every complete sentence must be telised) 5. Thus, 
an auxiliary tree whose root node will correspond to a tensed sentence and 
whose foot node will dominate an untensed sentence can be adjoined at 
this node. Therefore, only those auxiliary trees whose main verb subcate- 
5t statement is more complicated than just "view from the top", t ~tatement is 
a statement about the node wlfile viewing the node from the top, and hence is a 
statement eoncenfing the entire subtree below this node (i.e., including the part due 
to an auxiliary tree adjoined at the node), and ho w it constrains the derivation of 
the nodes wlfich are its siblings alld ancestors, b remains the same as before, and 
is the statement about this node and the subtree below it, without considering the 
adjunctlon at this node. 
716 
S 
NP VP 
PRO to leave 
~ o:+\] 
NP VP ! vieS\[tense:'\] 
I I 
John tries 
Figure 6: Illustration of implementation of SA and QA constraints 
gorizes for an untensed sentence (or an infinitival clause) can be adjoined 
at the root node of this initial tree. This shows why only auxiliary tree 
such as fl can be adjoined, whereas an auxiliary tree corresponding to 
John thinks S can not be adjoined since the verb thinks subcategories for 
a tensed sentence. The example also serves to illustrate the implementa- 
tion of OA constraint at the root of a, since the t and b feature structures 
for this node are not unifiable. 
2.2.1 Comments on the Implementation of Constraints in FTAG 
In the TAG formalism, local constraints are specified by enumeration. 
However, specification by enumeration is not a linguistically attractive 
solution. In FTAG we associate with each node two feature structures 
which are declarations of linguistic facts about the node. The fact that 
only appropriate trees get adjoined is a corollary of the fact that only 
trees consistent with these declarations are acceptable trees in FTAG. As 
a result, in a FTAG, constraints are dynamically instantiated and are 
not pre-slpecified as in a TAG. This can be advmltageous and useful for 
economy of grammar specification. For example, consider the derivation 
of the sentence 
What do you think Mary thought John saw 
In the TAG formalism, we are forced to replicate some auxiliary trees. 
Consider the auxiliary tree fll in the TAG fragment in Figure 7. Since 
the intermediate phrase what Mary thought John saw is not a complete 
sentence, we will have to use OA constraints at the root of the auxiliary 
tree ill. However, tlfis root node should not have OA constraints when it 
is used in some other context; as in the case of the derivation of 
Mary thought John saw Peter 
We will need another auxiliary tree, fs, with exactly the same tree struc- 
ture as fll except that the root of/32 will not have an OA constraint. 
Further, the root nodes in c~1 and c~2 have SA constraints that allow 
for adjunetion only by fll and f~2 respectively: As seen in the Figure 8, 
corresponding to the FTAG fragment, we can make use of the fact that 
constraints are dynamically inatantiated and give only one specification 
of ill. When used in the derivation of 
What do you think Mary thought John saw 
troot inherits the feature inverted : + which it otherwise does not have, 
and broot inherits the feature inverted : -. Thus, the node which corre- 
sponds to root of ill, by the dynamic instantiation of the feature structure, 
gets an OA constraint. Note that there will not be any OA eoustraint in 
nodes of the final tree corresponding to 
What do you think Mary thought John saw. 
Also, the root of the auxiliary tree, corresponding to Mary thought S, 
does not get OA constraint, when this tree is used in the derivation of 
the sentence 
Mary thought John saw Peter. 
8 ./~--.. 
COMP~ 
I J"~ wh NP- ~P- 
dahri ~ NP. 
I I 
S 
NP_ VP- 
John ~t tlP- 
anw ~et o¢ 
AUX S 
do NP VP 
n v $ I I 
you think 
s ( B3} 
tiP_ V.P~ 
Mary ~z N I 
t hought 
olin} 
NP VP 
Mary v S 
I t bought 
S 
~" ~. \[ invert od: ~\] 
COMP_ S 
wh NP_ VP_ 
John ~ NP- 
I I 
~IOW a 
Figure 7: A TAG fragment 
/~.,a:q 
AUX 
I do 
ti 
NP VP 
I v//~sflnvorled:.\] you 
I think 
S 
NP_ VP- 
Mary v S 
I 
t bought 
Fignre 8: An FTAG fragment 
2.3 Some Possible Linguistic Stipulations in FTAG 
hi this section, we will discuss some possible stipulations for a FTAG 
granmmr, tIowever, at this stage, we do not want to consider these stip- 
ulations as a part of the formalism of FTAG. First, some of the linguistic 
issues pertaining to these stipulations have not yet been settled. Sec- 
ondly, ou~ ~irnary ¢o~cern ~'to sp~ify/tl,¢ FTA 9 formalism. ~,ther, 
if the form*lima haS t~) incorporate ~heie 4tip~ulatibns, it( can be done so, 
witbont ,lt~,ng tbe ~ochanlsm s,g~m0~n ly. 
The current linguistic theory u~derlying TAG's ...... that every 
foot node has *~ NA constraint. The justification of this stipulation is 
isinfilar to the projection principle in Chomsky's ~ransformation theory. 
!It is appealing to state that the adjunetion .operation does not alter the 
.grarmnatical relations defined by the intermediate tree structures. For 
~example, consider the following derivation of the ~ntence 
Ma~y thought John saw Bill hit Jill. 
If the derivation results in the intermediate tree corresponding to Mary 
thought Bill hit Jill, then we wofild expect to obtain 'the relation of Mary 
thinking that "Bill hit Jill". This relation is altered by the adjunction at 
the node corresponding to the foot node of the'auxiliary tree correspond- 
ing to Mary thought S. 
ff we wish to implement this stipulatio a, one solution is to insist that 
only one F-V statement is made with the foot node, i.e, the tloo~ and 
bloot are combined. The definition of adjunction can be suitably altered. 
The second stipulation involves the complexity of the feature structure 
associated with the nodes. So far, we have not placed any restrictions on 
the growth of these feature structures. One of the possible stipulations 
that are being considered from the point of view of linguistic relevance 
is to put a bound on the information content in these feature structures. 
This results in a bound on the size of feature structures and hence on 
the number of possible feature structures that can be associated with a 
node. An FTAG grammar, which incorporates this stipulation, will be 
equivalent to a TAG from the point of view of generative capacity but 
one with an enhanced descriptive capacity. 
Unbounded feature structures have been used to capture the subeat- 
~egorization phenomenon by having feature structures that act like stacks 
(and hence unbounded in size), llowever, in TAG's, the elementary trees 
give the subeategorization (Iomain. As noted earlier, the elements sub- 
categorized by the main vert~ in an elementary tree are part of the same 
elementary tree. Thus, with the feature structures associated with the 
elementary trees we can just point to the subcategorized elements and do 
not need any further devices. Note, that any stack based mechanism that 
might be needed for subeategorization is provided by the TAG formalism 
itself, in which the tree sets generated by TAG's have context free paths 
(unlike CFG's which have regular paths). This additional power provided 
by the TAG formalism has been used to an advantage in giving an account 
of West Germanic verb-raising \[Santorini 1986\]. 
3 A Calculus to Represent FTAG Gram- 
mars 
We will now consider a calculus to represent FTAG's by extending on the 
llogieal formulation oftbature structures given by Rounds and Kasper \[Rou 
Kasper et al. 1986\]. Feature structures in this logic (henceforth called lt- 
!K logic) are represented as formulae. The set of well-formed formulae in 
this logic is recursively defined as follows. 
e::= NIL 
TOP 
a 
I:el 
el A e2 
e~. V e2 
{pl ..... P.} 
where a is an atomic value, el,e2 are well-formed formulae. NIL and 
(TOP cl)nvey "no in(ormation" and "inconsistent information" respec- 
!~ively. ~aeh pl represents a path of the form li,1 : li,z .... : li,m re- 
ispectivel~y. This formula is interpreted as Pt .... = p,, and is used to 
iexpress reentrancy. 
Our representation of feature structures similar to the I/-K logie's 
:representation of feature structures and differs only in the clause for reen- 
\]traney. Given that we want to represent the grammar itself in our cMcu- 
lus, we call not represent reentrancy by a finite set of paths. For example, 
suppose we wish to mate that agreement features of a verb matches with 
,that of its subject (note in a TAG the verb and its subject are in the same 
elementary tree), tile two paths to be identified can not be stated until 
we obtain the final derived tree. To avoid this problem, we use a set of 
equations to specify the reentrancy. The set of equations have the form 
given by xi = ei for 1 < i < n, where ~1,... ,xn are variables, el,... ,en 
!are formulae which could involve these variables. 
717 
For exampl% the reentrant feature structure used in Section 1.2, is 
represented by the set of equations 
z = eat : S h l : y A 2 : (eat : VP h age : z A subject : y) 
y = cat : N P A agr : z 
We represent a set of equations, xi = ei for 1 <: i < n as 
rec ( Zh...,Xn >~( el,...,en ~. 
Let us now consider the representation of trees in FTAG and the 
feature structures that are a~so'ciated with the nodes. The elementary 
feature structure associated with each elementary tree encodes certain 
relationships between the nodes. Included among these relationships are 
the sibling and ancestor/descendent relationships; in short, the actual 
structure of the tree. Thus, associated with each node is a feature struc- 
ture which encodes the subtree below it. We use the attributes i E .hf to 
denote the i ~h child of a node. 
To understand the representation of the adjunction process, consider 
the trees given in Figure 4, and in particular, the node y. The feature 
structure associated with the node where adjunction takes place should 
reflect the feature structure after adjunction and as well as without ad- 
junction (if the constraint is not obligatory). Further, the feature struc- 
ture (corresponding to the tree structure below it) to be associated with 
the foot node is not knoWn bnt gets specified upon adjunetion. Thus, the 
bottom feature structure associated with the foot node, which is bloot be- 
fore adjunction, is instantiated on adjunction by unifying it with a feature 
structure for the tree that will finally appear below this node. Prior to 
adjunction, since this feature structure is not known, we will treat it asi 
a variable (that gets instantiated on adjunction). This treatment can be! 
obtained if we think of the auxiliary tree as corresponding to functional 
over feature structures (by A-abstracting the variable corresponding to i 
the feature structure for the tree that will appear below the foot node). 
Adjunction correponds to applying this function to th e •feature structure 
corresponding to the subtree below the node where takes place. 
We will formalize representation of FTAG as follows. If we do nott 
consider adjoining at the node y, the formula for "y will be of the form 
(...t, 1 Ab, A.../ 
Suppose the formula for the auxiliary tree # is of the form 
(t~oo~ A . . . bsoo,) 
tim tree obtained after adjunction at the node r I will the n be represente~ 
by the formula 
(...t, A (t,°°, A... bsoo,) A N A . . .) 
We would like to specify one formula with the tree % and use appropri- 
ate operation corresponding to adjunction by ~ or the case where we do 
not adjoin at ~. Imagining adjunction as function application where we~ 
consider auxiliary trees as functions, the representation of/3 is a function i 
say fz, of the form 
~f.(t,oo, A...(blo,, ^ f)) 
To allow tile adjunetion of ~ at the node ~, we have to represent T by 
(...t, A f#(bs) ^...) 
Then, corresponding to adjunction, we use function application to obtain 
the required formula. But note that if we do not adioin at ~l, we would 
like to represent ")" by the formula 
(...t, A b, A ~..) 
which can be obtained by representing T by 
718 
(...t,~ A Z(b,~) A...) 
where I is the identity function. Similarly, we inay have to attempt ad- 
junction at ~ by any auxiliary tree (SA constraints are handled by success 
or failure of unification). Thus, if/31,...,/3, form the set of auxiliary tree, 
we have a function, F, given by 
V = AL(Im(I) v... v/~.(/) V I(I)) = ~f.(lm(f) V... V l~(I) v f) 
and represent 7 by 
(...t, A F(b,) A...) 
Ill this way, we can represent tile elementary trees (and hence tile gram- 
mar) in an extended version of rt-K logic (to which we add A-abstraction 
and application). 
3,1 Representing Tree Adjoining Grammars 
We will now turn our attention to the actual representation of an FTAG 
grammar, having considered how the individual elementary trees are rep- 
resented. According to our discussion in the previous section, the auxiliary 
trees are represented as functions of the form Az.e where e is a term in 
FSTR which involves the variable ~. If/31,..., #n are the auxiliary trees 
of a FTAG, G, then we have equations of the form 
fl = ~x.el 
f. = Ax.e,~ 
el,...,e~ are encodings of auxiliary trees #h...,fl, as discussed above. 
These expressions obey the syntax which is defined ~ccursively as follows. 
e ::= NIL 
::= TOP 
::~ Cl A e 2 
::~ e I V g2 
::---- f(e) 
where x js a variable over feature structures and f is a function variable. 
In addition, as discussed above, we have another equation given by 
fo = Ax./I(x) V...V fn(~) 
The initial trees are represented by a set of equations of the form 
! xrn ~ ~ra 
where e~,.. ' ., e m are expressions which describe the initial trees at ,..., ~n 
Note that in the expressions el,..., e,, e~,.. ., e,,, wherever adjunction is 
possible, we use the function variable f0 as described above. The gram- 
mar is characterized by the structures derivable from any one of the initial 
trees. Therefore, we add 
~0 ---- Zt V... V ~tn 
Assuming that we specify reentrancy using the Variables Yl,...~ Yk and 
equations Yt : e~' for 1 _ i < k, an FTAG grammar is thus represented 
by the set of equations of the form 
.first (ree(xo, xl .... x,~, Yt .... , Yk, fo, 11 .... ,/,) 
(eo,e~,.. . ' 11 e" l ,era,el,"', k,g .... ,g,)) 
a.2 Semantics of FTACI 
So far, we have only considered only the syntax of the calcnlus used tbr 
representing fcatnre structures and FTAG grammars. Ia this see@m, we 
consider the mathematical modelling of the calculus. This can be used to 
show that the set of equations describing a grammar will always have a 
solution, which we can consider as the denotation of the grammar. 
Tire model that we present here is based on the work by llxnmds and 
Kssper \[Pmund, et al. 1986\] and in particular their notion ofsatisfiability 
of formulae. \[,st I" be the space of partial flmetions (with the parLial 
ordering E, the standard ordering on partial functions) defined by /" = 
(L .-~ F) + A where A is set of atoms and L is set of labels. This space 
has been characterized by Pereira and Sheiber \[Pereira ctal. 1984\]. Any 
expression e (which is not a hmction) can be thought w~ upward closed 
subset of F (the set of partial functions which satisfy the description 
el. Note that if n partial fimetion satisties a description then so will 
any function above it. We let U(F) stm\]d for the collection of upward 
closed subsets of F. Expressions are interpreted relative to an envirmnnent 
(since we have variables as cxpressions, wc need to consider environments 
which map era'tables to a member of U(F)). Functimm get interpreted as 
continuous functions in tim space U(/;') -~ U(F'), with the enviromncnt 
mapping fimetion variables to fimctions on U(P). Note that the ordering 
on U(F) is the inverse of set inclusion, since more functions satisfy the 
description of a more general featnre structure. 
Because of space limitations, we cannot go into the details of the 
interpretations function. \[{onghly, the interpretation is as follows. We 
interpret the expression a as the set containing just the atom "a"; the 
expressiou 1 : e is interl)reted as tire set of fnnctions which map / to an 
element iu the .':at denoted by e; eonjmmtion and disjunetion are treated 
as intersection snd union respectively except that we have to ensure that 
rely value assigned t<) a wtriable in one of the eonjunets is the same as the 
valne assigned to the same variable in the other conjnncg. 
Since the grammar is given by a set of equation;;, the denotation is 
given by tim least solution. This is obtaiued by considering the fimctiou 
corresponding to the set of equations in the standard way, and obtaining 
its least fixpoint. Details of these issues rnay be found in \[Vij ayashaaker i 9 
In \[Vijayashanker 1987\], we have shown that any set of equations has 
a solution. Thus, we can Live semantics for recursivc set of eqnatkms 
which may be used to describe cyclic feature structure. For example, we 
give the solution for equations such as 
x:: f :xAg:a 
As shown in \[V \]ayas ran mr 1987\], we can obtain the least lixedopoint by 
assuming the le~rst vahm for x (which is the cntirc set of partial fnnetions 
or the intcrl)retatkm of NIL) mrd obtaining better and better approxima-, 
lions. The least npper bound of these approximations (which will give the 
least fixed-point) corresponds to the reqnired cyclic structure, ;is desired. 
4 Conclusions and }~Nl~ure Work 
We have shown a method of embedding TAG's in a feature structmm 
based framewo¢k. This system takes advantage of the extended domain 
of locality of TAG's and allows lingusitic statements abont cooccurencc 
of features of dependent iterrLs to be stated within elententary trees. We 
have shown thst we can make a clearer statement of adjoining constraints 
in FTAG'a than in TAG's. The specification of local constraints in a 'tAG 
is by enmneration, which is not satisfactory from the liuguistic point of 
view. We show that in FTAG, we em~ avoid such specilications, instead 
the dedarative statements nrade about nodes are sufficient to mmure Ihat 
only the appropriate flees get adjoined at a node. Furthermore, we also 
illu.strate how duplication or iuformation can be aw~ided in FTAC's in 
comparisoJ~ with TA(Us. I~ cau bc shown that aualyses~ that require ex 
tensions of TA($'s using multi-component adjoining (simultaneous adjunc 
lion of a set of trees in distinct nodes of an dementary tree) ~ defined 
in \[\]oshi 1987, Kroeh 1987\], can be easily stated iu FTAG's. 
It is possible to parse an I,"\]?A(I grammar using the Earley-style parsel 
given by \[Sehabes et aL 1988\]. This l,;arley-style parser can extended 
in the same way that Sheiber extended the Earley parser lee PA'I3I: 
II \[Slfieber 1985b\]. The reason this extensi,~lt of the TAll parser to one fl)t 
I:'FAG is po,~;sible fi)llows from the lact that the treatment of haviJ,g the 
t and b feature structures fl)r every node in F'I)A(~ is compatible with the I 
characterization, adopted in the parsing algorithm in \[Schabes et al. :19881, 
of a node in le.rms of two subs\[rings. 
In \[Vii ayashanker 1987\], we haw~ prop osed a restr toted version (.f FTA G 
In a manaer similar to GPSG, we place a bound on the information con- 
tent or' feature structures associated with the nodes of trees used ill the 
grammar. The resulting system, 1U"TAG, g~nerates the same language as 
TAG's, and yet retains an increased descriptive and geaeraLive capacity 
due to the extended domain of locality o{ TAG's. 
Fiually, in this lml)er, we have brMly discussed a calculus to represent 
FTAG grammars. This cab:alas is an exteation of the llounds-Kasper 
logic for fi:ature structures, q'he extmltions deM with A abstraction ove~ ~ 
feature structures and flmetiou application, which is used to ehagacterizd 
auxiliary trees and the adjunctiml operation. \[Vijayashanker 19871 Lives 
a detailed description of this calculus and its semantics. 

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