1 Introduction 
VP Ellipsis in a DRT-implementation 
Johan Bos 
Department of Computational Linguistics, 
Faculty of Arts, University of Groningen, 
P.O.Box 716, 9700 AS Groningen. 
Email: sO615838@let.rug.al 
Klein \[Klein, 1986\] introduced Predicate-DRSs for 
the resolution of VP Ellipsis. In that approach a 
Predicate~DRS (henceforth PDRS) serves as the rep- 
resentation of a verb phrase, as will be shown in an 
example now. Consider: 
Nancy likes a cat. 
(1) Betty does too. 
This discourse is interpreted as meaning that Nancy 
and Betty both like a cat (though not necessarily 
the same cat). The source clause, Nancy likes a cat, 
parallels the target clause Betty does too, where the 
subjects are parallel elements. The phrase does too 
represents a trace of the VP in the target clause. 
Klein's treatment of (1) is shown in (2). 
(2) 
Xl Xs PI 
Xl "- Nancy 
\[Yl\] Y2 
PI(xl): cat(ys) 
like(y1 ,Ys) 
xs "- Betty Pdx2) 
In the second sentence of (1), a do-anaphor appears 
that must be linked to a marker which has already 
been introduced into the universe of the DtLS. The 
value of this marker, which is P1, as we can see in (2), 
is constrained by the conditions associated with the 
previous VP in the discourse \[Klein, 1986\]. Following 
Klein, we call P1 a predicate marker, and the Sub- 
DRS that is associated with Pt a Predicate-DRS. To 
the domain of P1, a distinguished reference marker 
Yl (indicated by square brackets) is added, which 
plays the role of the individual, in this case xl which 
is applied to the predicate. This application can also 
be shown as a lambda expression: 
(3) A Yl (cat(ys) A like (YhYs)) (xl) 
In (2) the condition Ps(xs) in the main DRS will 
apply the object xs to the predicate and solve the 
do-anaphor in (1). The scope of marker Ys is de- 
fined by the PDILS, instead of the main DRS, which 
allows that Nancy and Betty do not necessarily like 
the same cat. 
But this same feature introduces a problem for pro- 
noun resolution. This problem occurs when pro- 
nouns refer to indefinite NPs which are in the uni- 
verse of a PDRS and therefore inaccessible. Let us 
give an example by considering the DRS (5) as the 
translation of (4). 
Nancyt likes a cats. 
(4) She1 strokes its. 
(5) 
xl P1 P2 
xl = Nancy 
\[Yl\] Ys 
P I (xi)I cat (Ys) 
\[ like(yl,y2) 
\[Y3\] 
Ps(xl): stroke(y3,?) 
Since, in DRT, an anaphor can only refer to an- 
tecedents from its own domain or from universes 
that its DRS subordinates, the pronoun it cannot 
be anaphorically linked to the indefinite description 
a cat. This means, in the situation of (5), candi- 
date antecedents for it can only be found in the main 
D1LS, since Pz is subordinated to it. The desirable 
antecedent y2 in P1 is blockedJ 
A solution to the problem of the indefinite descrip- 
tions appearing in PDRSs, is to make them accessible 
in the main DRS. This paper shows, by slightly mod- 
ifying Klein's PDRSs, how that can be done, without 
losing their desirable characteristics. 
Firstly, we outline informally how indefinite descrip- 
tions in PDRSs are made accessible. Then we show 
how this idea relates to aspects like negation, disjunc- 
tion, quantification and the strict/sloppy identity of 
VP Ellipsis. Finally, we report about the implemen- 
tation under development. 
1Notice, that proper names and definite descriptions 
do not give rise to this problem. In DRT, these are 
usually added to the universe of the main DRS \[Kamp 
and Reyle, 1990\] or accommodated to it \[Van derSandt, 
1992\]. 
425 
2 A new approach to 
Predicate-DRSs 
By treating a PDRS just as an ordinary DRS, with 
the distinction that there is a correspondence be- 
tween the arguments which are applied to the PDRS 
and the members of its domain, it is possible to ex- 
tend the scope of reference markers in a PDRS to 
their superordinated DRS. The best way to show 
how this works is to look at a DRS for (4) in this 
new approach: 
(6) 
Xl X2 PI P2 
X 1 -- Nancy 
l Yl Y2 P1 (xl ,x2):l cat(y2) 
Ilike(yl,Y~) 
P2(xl) stroke(y3,x~) 
In (6), in the PDRS P1, Yl is linked to Xl, and y2 
to x2. So, the difference here to Klein's approach is 
that, besides the referent for the individual which is 
applied to the PDRS, all indefinite descriptions in 
the universe of the PDRS are associated with corre- 
sponding arguments as well. 2 A lambda expression 
for P1 in (6) is: 
(7) A Yl A Y2 (cat(y2) A like(yl,y2)) (xl) (x2) 
This treatment allows that we can refer to the indefi- 
nite cat, as is done in P2 of (6). An added advantage 
is that we maintain the original properties ofa PDtLS 
outlined previously. Note, that the number of argu- 
ments applied to a PDRS directly depends on the 
number of indefinite descriptions in the VP. Conse- 
quently, a VP with a ditransitive verb could yield 
two indefinite descriptions, as in (8). Optional rela- 
tive clauses can raise this number even higher (9). 
(8) Nancy gives a man an iron. 
Nancy likes a man who has an iron that 
(9) a woman gave him. 
3 Negation 
Concerning predicate negation, we will assume that 
the scope of negation does not embrace the subject 
(cf \[Kamp and Reyle, 1990\]). The approach we take 
2Therefore, it is not necessary to distinguish them 
with square brackets any more. Note that the agent cor- 
responds to the first referent in the PDRS. 
here is similar to standard DRT, because a new sub- 
ordinated DRS affixed with a negation symbol is in- 
troduced in case of negation. Let us consider (10): 
(I0) Nancy1 doesn't own a cats. 
* Shel beats it2. 
Here we simply negate the predicate by constructing 
the PDRS in a negated DRS. In (10) the pronoun it 
does not permit a link to the NP a cat, and this seems 
to be the case in general as well, because negation 
blocks anaphoric links. 8 
Thus, in the case of a negated VP, the indefinites are 
raised to the superordinated DRS which is the DRS 
for negation. This construction is figured in DRS 
(11) and causes exactly the result we wish: it cannot 
be linked to cat because the referent for cat, x2, is 
not accessible. 
(11) 
X1 P1 P2 
xl ---- Nancy Ix2 \[ 
-~ PI(Xl,X2): 
Yl Y~ 
cat(y2) 
own(yl ,Y2) 
l y3 
P2(xl): beat(y3,?) 
Now consider (12), where an anaphoric link between 
cat and it is permitted. At first sight, this sentence 
would appear as a counterexample to our character- 
ization of negation. But it is not, if we interpret the 
meaning of it as (13): 
(12) Either Nancy doesn't own a cat, 
or she beats it. 
(13) Either Nancy doesn't own a cat, 
or she does and she beats it. 
An interpretation of (12) as (13) permits the acces- 
sibility of cat in (12). In our DRT-framework with 
PDRSs we easily can obtain a DRS for (12), as the 
disjunction of two SubDRSs. Then, in one disjunct 
predicate negation takes place, while in the other the 
3However, \[Kamp and Reyle, 1990\] give as a possible 
counterexample to this generalization the discourse Jones 
does not llke a Porsche. He owns it, interpreting it by 
saying that there is some Porsche that Jones both dislikes 
and owns. According to me, such an interpretation seems 
only permitted if that Porsche is already uttered in the 
processed discourse. 
426 
do-anaphor is resolved, resulting in a accessibility for 
the indefinite NP a cat. 
(14) 
xl PI P2 
xl = Nancy 
X2 
Yl Y2 
"~ PI(Xl,X2): cat(y2) 
own(yl ,Y2) I 
V 
x3 
P1(xl ,x3) 
4 Quantification 
In this section we will see how the quantifiers every 
and no can be treated. We will demonstrate how 
quantification matches perfectly with our proposals 
about PDRSs and negation. Sentence (15) 
(15) Every woman likes a cat. 
involves applying the quantified NP every woman to 
the PDRS, visualized in DRS (16): 
(16) 
P1 
X1 
woman(xl) I 
X2 
--* PI(xI,x2): 
Yl Y2 
cat(y~) 
like(yl,y2) 
Of interest here is that the argument of P1 is the 
member of the antecedent DRS: xl. Also worth not- 
ing is that the referent of the indefinite a cat in P1 
is not raised to the main DRS but to the DRS that 
holds the consequent of the implication relation. In 
this case the NP a cat has narrow scope within the 
quantified phrase every woman, and therefore not 
accessible in the main DRS (as in standard DRT). 
In a similar way the quantifier no is interpreted, us- 
ing the logical equivalence of the formulae (17) and (18): 
(17)-,3z P(z) A Q(x) 
(18)YxP(z) ---*~Q(x) 
The traditional way to translate no in DRT is based 
on (17). 4 In this framework we use predicate nega- 
4Several proposals have been made to treat gener- 
alized quantifiers in DRT. Among them: \[Klein, 1986; 
Kamp and Reyle, 1990; geevat, 1991\]. 
tion combined with universal quantification, shown 
in (20), which is the translation for (19). 
(19) No woman likes a dog. 
(20) 
P1 Ix2 { 
xl 1 Yl Y2 
woman(xl) \]"" PI(Xl,X2) dog(y2) 
like(yl,y2) 
This way of dealing with quantification is exactly 
what we need for VP Ellipsis resolution. A discourse 
as in (19) could proceed with a sentence like: Bat 
Peter does, and he beats it, which is an example of a 
'missing antecedent' \[Hankamer and Sag, 1976\], since 
the pronoun it lacks an overt antecedent because the 
NP a dog is in the scope of negation and therefore 
not accessible. By generating a condition in D1LS 
(20) applying Peter to the PDRS PI, the 'missing' 
antecedent is found (21). 
iX3 X4 
X1 
(21) 
P1 P2 
woman(xi) 
x3 = Peter 
PI(X3,X4) 
P2(x3) t :;at(y3,x4) 
I x2( ) "~ PI Xl,X2 : Yl Y2 dog(y2) like(yl,y2) 
Summarizing so far, we have shown that PDRSs, 
with the ability to raise indefinite descriptions to its 
superordinated DRS, can be used quite effectively in 
our framework. Mainly, we distinguished two cases 
where referents of indefinite descriptions were not 
raised to the main DRS, but to a DIgS subordinated 
to the top level. The first case concerns predicate 
negation, where a negated DRS is superordinated to 
the PDRS involved. The second case concerns quan- 
tification, where the PDRS is subordinated to the 
consequent-DRS of the implication relation. 
5 Strict and Sloppy Readings 
This section shows how sloppy and strict readings 
arising in VP Ellipsis are obtained. Discourses like 
(22) are ambiguous as to whether Betty strokes 
427 
Nancy's cat (the strict reading) or Betty strokes 
Betty's cat (the sloppy reading). 
(22) Nancy strokes her cat. 
Betty does too. 
Following \[Van der Sandt, 1992\], presuppositions are 
accommodated to the preceding discourse. That is, if 
discourse does not provide an antecedent, one will be 
created. In processing the first sentence of (22), DRS 
(23) is obtained, where the presuppositional posses- 
sive construction her cat is paraphrased in a dashed 
DRS to indicate information for accommodation. 
(23) 
xl P1 
X 1 --" Nancy 
Yl 
stroke(y1 ,y~) 
Yy2 P i 
Zl Z2 ! 
P(Y,Y2): cat(z~) 1 
poss(zl ,z2) 
In the approach of \[Van der Sandt, 1992\] the 
anaphoric material in the dashed DRS is resolved 
after merging the DRS constructed for the sentence 
with the main DRS, resulting in a new DRS that 
contains no anaphoric material for accommodation 
still to be processed. This procedure is followed for 
(23) yielding DRS (24). 
xl x2 P1 P2 
xl = Nancy 
Z1 Z2 
P2(XI,X2) I cat(z2) 
(24) \[ poss(zl ,z2) 
PI(X1) t Y\[!!ir--!!!i!! ! !!!i!iii 
Discourse (22) provides one suitable antecedent for 
the possessor, namely Nancy, and Nancy possesses a 
cat is established in the DRS. But this gives us only 
the strict reading when in case of an elliptical VP in 
the proceeding discourse is referred to P1, which is 
the case in (22). 
To represent the sloppy reading, the anaphoric ma- 
terial in (23) that holds the presupposition must not 
be resolved at the stage of DRS-merging, but left 
there to provide accommodation another time (with 
other constraints, that depend on the antecedent of 
the possessor). In this way both the strict and sloppy 
reading are obtainable in case of VP Ellipsis. 
We show this proposal with our example (22), cor- 
responding with DRS (25). Similar to (24), the pre- 
supposition causes an antecedent to be created (i.e. 
Nancy possesses a cat), with this difference, that the 
anaphoric material is not resolved. The VP-anaphor 
finds as an antecedent PI: strokes her cat. The pre- 
suppositional material in the dashed DRS can now be 
accommodated to two different antecedents: Firstly, 
Nancy, where no antecedent has to be created for the 
possessive construction, resulting in the strict read- 
ing. Secondly, the newly introduced Betty, where in 
that case the presupposition Betty possesses a cat 
is accommodated and the sloppy reading can be de- 
rived. The latter is shown in (25): 
(25) 
xl x2 xa x4 P1 P2 P3 
xl : Nancy 
Zl Z2 
P2(Xl,X2) cat(z2) poss(zl,z2) 
Pl(xl): 
Yl 
stroke(y1 ,Y2) 
YY2 P 
Zl z2 
P(Y,Y2):I cat(z2) . 
I 
xa = Betty Zl z2 
Pa(x3,x4): cat(z2) 
poss(zl ,z~) 
Pl(Xa) 
If we compare this approach to the higher-order uni- 
fication approach to VP Ellipsis of \[Dalrymple et al., 
1991\], we can obtain all six readings of the compli- 
cated (26) generated by the equational analysis of 
\[Dalrymple et ai., 1991\]. 
ore John revised his paper before the 
~v/teacher did, and Bill did too. 
The reading of (26) where John, the teacher, and 
Bill all revised John's paper, is translated in a DRS 
with the presupposition that John possesses a paper 
428 
accommodated to the main DRS. The reading where 
John and Bill revised their own papers before the 
teacher revised John's paper, causes accommodation 
twice, once for John possesses a paper and once for 
Bill possesses a paper. The other readings can be 
obtained analogously. 
6 Implementation 
The PROLOO-implementation is a natural language 
processing system which parses simple discourses, 
The way DRSs are constructed in this system will 
be discussed concisely. 
The emphasis of the implementation lies on anaphora 
resolution (like do-anaphora and pronouns) in a do- 
main of a small fragment of English. A parse of a 
typical discourse is: 
> Mary likes a cat. 
> She does not beat it. 
> John does not either. 
drs : \[ xl x3 x6 p2 p5 \] 
\[ x l = mary 
p2(xl,x3):\[ y x4 \] 
\[ cat (x4) 
like(y,x4) \] 
not £ 3 
£ pS(xl):£ y \] 
\[ beat(y,x3) \] \] 
x6 = john 
not \[ \] 
£ pS(x~) \] \] 
This implementation differs from other PROLOG- 
implementations of DRT (e.g. the threading ac- 
count of \[Johnson and Klein, 1986\]) in the way it 
constructs DRSs. Following lasher, 1990\], DRSs 
are constructed in a bottom-up fashion, using A- 
conversion. 
Each lexical entry is associated with a SubDRS, rep- 
resenting the meaning of that entry. For instance, 
the lexical entries for a, man, and runs are: 
lex(ap 
det : \[agr=sing, 
def=ind, 
drsffi (X'P) "(X'Q) "drs( \[2, \[P ,Q\] )3 ). 
lex (mSll m 
noun: \[agr=sing, 
~s=X'~s ( \[X\], \[man(X) 3 ), 
gender--male, 
refffiX\] ). 
lex (runs, 
iv: \[agrfsing, 
drsfX'drs ( D, \[do(P ,X) :drs ( \[y\], \[run(y)\] )\] ), 
reffP\] ). 
As these entries make clear, a DtLS is constructed of 
a PROLOG term containing two lists, where the first 
one contains the discourse markers (i.e. the domain) 
and the second one the constraints (these are repre- 
sented as PROLOG terms). Furthermore, the lambda 
abstractor is constructed as the PROLOG operator '^' 
(this idea is taken from \[Pereira and Shieber, 1987\]). 
While parsing a sentence, the DtLS for that sen- 
tence is processed by A-conversion and merging, us- 
ing syntax rules of the following form 5 (as in \[Al- 
shawl, 1992\]): 
np:\[drs=Drs,agr=Agr .... \] ---> 
\[det:\[drs=A2"Drs .... \], 
noun:\[drs=Al,agr=Agr .... \], 
optrel:\[drs=Al'A2,...\]\]. 
The output of a sentence parse is a constructed DRS 
for that sentence, but with referring expressions (if 
any) still unresolved. This sentence-DRS then is 
merged with the ingoing DRS, representing the com- 
puted discourse so far. During this merge, the fol- 
lowing computing actions take place: 
• Computing of arguments for PDRSs; 
• Resolving of Pronouns and'VP Ellipsis; 
• Accommodation of Proper Names, Definite De- 
scriptions, and Possessive Constructions. 
An aid to these computations is a historylist com- 
puted during the sentence parse. This historylist 
contains all the items that are represented in the dis- 
course, extended with information that is not purely 
semantic, such as type and gender of certain sub- 
jects, but necessary for the computations mentioned 
above. 
This results in a new DRS, capturing the entire dis- 
course, which will be the ingoing DtLS for the merge 
after the next sentence is parsed. 
7 Conclusion 
By slightly changing Klein's treatment of Predicate- 
DRSs, that is making indefinite descriptions occur- 
ring in the scope of the VP accessible to the top level 
of the main DRS, we obtain a much better mecha- 
nism for handling VP Ellipsis in DRT without losing 
any old characteristics in the theory. Furthermore, 
we proposed to use Van der Sandt's theory on pre- 
suppositions in a different way in our framework to 
SFor reasons of clarity, some information in these rules 
is omitted. 
429 
derive both strict and sloppy readings where possi- 
ble. 
This presentation is informal. Formal definitions of 
this approach, and a comprehensive description of 
the PROLOG-implementation can be found in the au- 
thor's Master thesis under preparation, to appear in 
August 1993. 
Acknowledgments 
I would like to thank Peter Blok, Gosse Bouma, 
Robin Cooper, Ronald Klopstra, John Nerbonne, 
Gertjan van Noord, Elni Rigas, and the referees for 
their helpful and supportive comments on earlier ver- 
sions of this paper. 
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