Ellipsis and Quantification: A Substitutional Approach 
Richard Crouch 
SRI International, Cambridge Computer Science Research Centre 
23 Millers Yard, Mill Lane, 
Cambridge, CB2 1RQ, UK 
rcecam, sri. tom 
Abstract 
The paper describes a substitutional ap- 
proach to ellipsis resolution giving compa- 
rable results to (Dalrymple et al., 1991), 
but without the need for order-sensitive in- 
terleaving of quantifier scoping and ellip- 
sis resolution. It is argued that the order- 
independence results from viewing seman- 
tic interpretation as building a description 
of a semantic composition, instead of the 
more common view of interpretation as ac- 
tually performing the composition. 
1 Introduction 
Dalrymple, Shieber and Pereira (1991) (hence- 
forth, DSP) give an equational treatment of ellip- 
sis via higher-order unification which, amongst other 
things, provides an insightful analysis of the interac- 
tions between ellipsis and quantification. But it suf- 
fers a number of drawbacks, especially when viewed 
from a computational perspective. 
First, the precise order in which quantifiers are 
scoped and ellipses resolved determines the final in- 
terpretation of elliptical sentences. It is hard to see 
how DSP's analysis could be implemented within a 
system employing a pipelined architecture that, say, 
separates quantifier scoping out from other reference 
resolution operations--this would seem to preclude 
the generation of some legitimate readings. Yet 
many systems, for good practical reasons, employ 
this kind of architecture. 
Second, without additional constraints, DSP 
slightly overgenerate readings for sentences like 
(1) 3ohn revised his paper before the teacher did, 
and so did Bill. 
Kehler (1993a) has convincingly argued that this 
problem arises because DSP do not distinguish be- 
tween merely co-referential and co-indexed (in his 
terminology, role-linked) expressions. 
Third, though perhaps less importantly, higher- 
order unification going beyond second-order match- 
ing is required for resolving ellipses involving quart- 
tification. This increases the computational com- 
plexity of the ellipsis resolution task. 
This paper presents a treatment of ellipsis which 
avoids these difficulties, while having essentially the 
same coverage as DSP. The treatment is easily im- 
plementable, and forms the basis of the ellipsis res- 
olution component currently used within the Core 
Language Engine (Alshawi et al., 1992). 
Ellipsis interpretations are represented as simple 
sets of substitutions on semantic representations of 
the antecedent. The substitutions can be built up 
in an order-independent way (i.e. before, after or 
during scoping), and without recourse to higher- 
order unification. The treatment is similar to the 
discourse copying analysis of (Kehler, 1993a), and 
to the substitutional treatment suggested by Kamp 
within Discourse Representation Theory, described 
in (Gawron and Peters, 1990). However, we extend 
the notion of strict and sloppy identity to deal with 
more than just pronouns. In doing so, we readily 
deal with phenomena like scope parallelism. 
While the treatment of ellipsis is hopefully of 
some value in its own right, a more general con- 
clusion can be drawn concerning the requirements 
for a computational theory of semantics. Briefly, 
the standard view within formal semantics, which 
DSP inherit, identifies semantic interpretation with 
composition: interpretation is the process of taking 
the meanings of various constituents and composing 
them together to form the meaning of the whole. 
This makes semantic interpretation a highly order- 
dependent affair; e.g. the order in which a functor 
is composed with its arguments can substantially af- 
fect the resulting meaning. This is reflected in the 
order-sensitive interleaving of scope and ellipsis res- 
olution in DSP's account. In addition, composition 
is only sensitive to the meanings of its components. 
Typically there is a many-one mapping from com- 
positions onto meanings. So, for example, whether 
two terms with identical meanings are merely co- 
referential or are co-indexed is the kind of informa- 
tion that may get lost: the difference amounts to 
two ways of composing the same meaning. 
The alternative proposed here is to view seman- 
229 
tic interpretation as a process of building a (possibly 
partial) description of the intended semantic compo- 
sition; i.e. (partial) descriptions of what the mean- 
ings of various constituents are, and how they should 
be composed together: While the order in which 
composition operations are performed can radically 
affect the outcome, the order in which descriptions 
are built up is unimportant. In the case of ellip- 
sis, this extra layer of descriptive indirection per- 
mits an equational treatment of ellipsis that (i) is 
order-independent, (ii) can take account composi- 
tional distinctions that do not result in meaning dif- 
ferences, and also (iii) does not require the use of 
higher-order unification for dealing with quantitiers. 
The paper is organised as follows. Section 2 de- 
scribes the substitutional treatment of ellipsis by 
way of a few examples presented in a simplified ver- 
sion of Quasi Logical Form (QLF) (Alshawi and 
Crouch, 1992; Alshawi et el., 1992). Section 3 
gives the semantics for the notation, and argues that 
QLF is best understood as providing descriptions 
of semantic compositions. Section 4 raises some 
open questions concerning the determination of par- 
allelism between ellipsis and antecedent, and other 
issues. Section 5 concludes. 
2 Ellipsis Substitutions 
This section illustrates the substitutional treatment 
of ellipsis through a small number of examples. For 
presentation purposes we only sketch the intended 
semantics of the simplified QLF notation used, and 
a more detailed discussion is deferred until section 3. 
2.1 Simple VP E1Hpsis 
A simple, uninteresting example to fix some nota- 
tion: 
(2) John slept. So did Mary 
We represent the first sentence, ignoring tense, as a 
(resolved) QLF 
(3) \[+j\]: sleep( term(+j, ~, )ly.name(y, 'John'), 
Ay.y = j_smith)) 
The noun phrase John gives rise to an existentially 
quantified term, uniquely identified by the index 
-I-j. The texan expression has four arguments: an 
index, a determiner/quantifier, an explicit restric- 
tion, and an additional contextually derived restric- 
tion. In this case, the quantifier ranges over objects 
that are named 'John' and are further restricted to 
be identical to some (contextually salient) individ- 
ual, denoted by j.smith. Prior to reference resolu- 
tion, the contextual restriction on the term would 
be an uninstantiated meta~variable; resolution con- 
sists of instantiating mete-variables to contextually 
1This is similar to' Nerbonne's (1991) constraint- 
based semantics, except that he builds descriptions of 
logical forms, not semantic compositions. 
appropriate values. The scope of the term is in- 
dicated by the scope node I-l-j\] : prefixing the for- 
mula sleep(term(+j,...)). Again, prior to reso- 
lution this scope node would be an uniustantiated 
mete-variable. 
A generalized quantifier representation equivalent 
to the above is 
(4) 3(~y.(name(y, 'John') ^ y = j.smith), 
,~=.sleep(z)) 
The index in the scope node means that to semanti- 
cally evaluate the QLF, you get hold of the quanti- 
tier, restriction and contextual restriction of the cor- 
responding term. This forms a (generalized) quanti- 
fier expression, whose body is obtained by discharg- 
ing all occurrences of the term and it index to a 
variable, and abstracting over the variable. Terms 
and indices not dischargeable in this manner lead to 
uninterpretable QLFs (Alshawi and Crouch, 1992). 
We represent the elliptical sentence, again abbre- 
viated, as a (partially resolved) QLF: 
(5) ?P( tex~(+m, 3, ~y.name(y, 'Mary'), 
~y.y = m.,jones)) 
?P is an unresolved mete-variable. To resolve the 
ellipsis, it needs to be instantiated to some contex- 
tually salient predicate. 
Along similar lines to DSP, we can set up an equa- 
tion to determine possible values for ?p2: 
(6) ?P(tex~(+j,...)) = \[+j\]:sleep(term(+j .... )) 
That is, we are looking for a predicate that when 
applied to the subject term of the ellipsis antecedent 
returns the antecedent. The interpretation of the 
ellipsis is then given by applying this predicate to 
the subject of the ellipsis. 
The equation (6) is solved by setting ?P to some- 
thing that takes a term T as an argument and sub- 
stitutes T for tema(+j,...) and the index of T for +j 
throughout the ellipsis antecedent (the RHS of (6)): 
(7) ?P-" T^(\[+j\]:sleep(tena(+j,...)) 
J {te~(+j,...)/T, +j/idx_of(T)}) 
Here T^(...) is a form of abstraction; for now 
it will do no harm view it as a form of ~- 
abstraction, though this is not strictly accurate. 
The substitutions are represented using the notation I {oldlnew,...}. 
Applying this value for ?P in the ellipsis (5), we 
get 
(8) \[+j\]:sleep(tern(+j,...)) I 
{tez~t(-I-j .... )/term(q-m,...), +j/+m} 
Ellipsis resolution thus amounts to selecting an an- 
tecedent and determining a set of substitutions to 
apply to it. For reasons that will be explained 
shortly, it is important that resolution does not ac- 
tually carry out the application of the substitutions. 
2Terms shown abbreviated, i.e. tex~(Tj .... ) instead 
of tern(Wj, 3, ~y.name(y,'John'), ~y.y = j_maith). 
230 
However, were we to do this in this particular case, 
where the antecedent (3) is fully resolved, we would 
successfully capture the intended interpretation of 
the ellipsis, namely: 
(9) \[+in\] :sleep( l;erm(-I-m, 3, Ay.name(y, 'Mary'), 
Ay.y -- rn.Jones)) 
Note that the substitutions are not applied in the 
conventional order; viz. first replace +j by +m 
throughout (3) and then replace term(+j .... ) by 
term(+m,...). The first substitution would ensure 
that there was no term(+j,...) for the second sub- 
stitution to replace. The order in which substitu- 
tions apply instead depends on the order in which 
the expressions occur when making a top down pass 
through (3), such as one would do when applying 
semantic evaluation rules to the formula. 
Note also that the term index substitution applies 
to the scope node, so that \[+j\]: is replaced by \[+in\]:. 
This ensures that the term for Mary in the ellipsis 
gets a parallel scope to the term for John in the an- 
tecedent. Scope parallelism may not be significant 
where proper names are concerned, but is impor- 
tant when it comes to more obviously quantifica- 
tional terms (section 2.3). 
2.2 Evaluative Substitutions 
The meaning of an ellipsis is composed in essentially 
the same way, and from the same components, as the 
meaning of its antecedent. However, some changes 
need to be made in order to accommodate new ma- 
terial introduced by the ellipsis. The substitutions 
specify what these changes are. In the example dis- 
cussed above, the meaning of the ellipsis is built 
up in the same way as for the antecedent, except 
that whenever you encounter a term corresponding 
to 'John' or something dependent/co-indexed with 
it, you it is treated as though it were the term for 
'Mary' or dependent/co-indexed with it. 
This means that the substitutions act as directives 
controlling the way in which QLF expressions within 
their scope are evaluated. They are not syntactic 
operations on QLF expressions -- they are part of 
the QLF object language. 
The reason that substitutions are not 'applied' im- 
mediately upon ellipsis resolution is as follows. At 
the time of deciding on the ellipsis substitutions, 
the precise composition of the antecedent may not 
yet have been determined. (For instance the scopes 
of quantifiers or the contextual restrictions on pro- 
nouns in the antecedent may not have been resolved; 
this will correspond to the presence of uninstantiated 
meta-variables in the antecedent QLF.) The ellipsis 
should follow, modulo the substitutions, the same 
composition as the antecedent, whatever that com- 
position is eventually determined to be. It makes 
no sense to apply the substitutions before the an- 
tecedent is fully resolved, though it does make sense 
to decide what the appropriate substitutions should 
be. 
In practical terms what this amounts to is ex- 
ploiting re-entraney in QLFs. The elliptical QLF 
will contain a predicate formed from the antecedent 
QLF plus substitutions. Any uninstantiated recta- 
variables in the antecedent are thus re-entrant in the 
ellipsis. Consequently, any further resolutions to the 
antecedent are automatically imposed on the ellipsis. 
This would not be the ease if the substitutions were 
treated as syntactic operations on QLF to be applied 
immediately: some re-entrant meta-variables would 
be substituted out of the ellipsis, and those remain- 
ing would not be subject to the substitutions (which 
would have already been applied) when they were 
eventually instantiated. 
2.3 Scope Parallelism 
It was noted above that substitutions on term in- 
dices in scope nodes ensures scope parallelism. This 
is now illustrated with a more interesting example 
(adapted from Hirshbfihler as cited by DSP). 
(10) A Canadian flag hung in front of every house, 
and an American flag did too. 
The antecedent has two possible seopings: a single 
Canadian flag in front of all the houses, or each house 
with its own flag. Whichever seeping is given to the 
antecedent, a parallel seeping should be given to the 
ellipsis. 
A simplified QLF for (10) is 
(11) ?SI: and(?S2: hang(term(+c, B,...), ter~(+h,V,...)), 
3, ...))) 
where the indices +c, +a and +h are mnemonic for 
Canadian flag, American flag and house. Taking the 
first conjunct as the antecedent, we can set up an 
equation 
(12) ?S2:hang(term(+e...), term(+h...)) 
= )) 
the solution to which iss 
(13) ?P = T ^ (?S2:hang(term(+e...), term(+h...)) 
\[ {terffi(+c...)IT, +c/idx.of(T)} 
This make the elliptical conjunct equivalent to 
(14) ?S2:hang(term(÷c...), term(+h...)) 
I ...), +c/+a} 
The scope node, ?S2 can be resolved to \[+h, +el 
('every house' takes wide scope), or \[+e,+h\] ('a 
Canadian flag' takes wide scope). Whichever resolu- 
tion is made, the substitution of +a for +e ensures 
parallel scoping in the ellipsis for 'an American flag'. 
Cashing out the substitutions for the first case, we 
have 
3In the next section we place some extra constraints 
on possible solutions, but these aren't strictly relevant 
here. 
231 
(15) D:and(\[+h, +c\]:hang(tera(+c, 3,...), 
term(+h,V,...)), 
\[+h, +a\]:hang(t erm(+a, 3,...), 
term(-bh,V,...))) 
There is another scoping option which instantiates 
?$1 to \[q-h\], i.e. gives 'every house' wide scope over 
both antecedent and ellipsis. In this case the two 
terms, term(+h...) in ellipsis and antecedent are 
both discharged (i.e. bound) at the scope node ?$1, 
rather than being separately bound at the two copies 
of ?$2 
(16) \[+h\]:and(\[+c\]:hang(term(+c, 3,...), 
term(+h,V,.. 
\[-ka\]:hang(t erm(-ka, 3,.. :/!' 
(This has equivalent truth-conditions to (15)). 4 
Besides illustrating scope parallelism, this is an 
example where DSP have to resort to higher-order 
unification beyond second-order matching. But no 
such increase in complexity is required under the 
present treatment. 
2.4 Strict and Sloppy Identity 
The notion of strict and sloppy identity is usu- 
ally confined to pronominal items occurring in an- 
tecedents and (implicitly) in ellipses. 5 A standard 
example is 
(17) John loves his mother, and Simon does too. 
On the strict reading, Simon and John both love 
John's mother. The implicit pronoun has been 
strictly identified with the pronoun in the antecedent 
to pick out the same referent, John. On the sloppy 
reading Simon loves Simon's mother. The implicit 
pronoun has been sloppily identified with its an- 
tecedent to refer to something matching a similar 
description, i.e. the subject or agent of the loving 
relation, Simon. 
The sentence 
(18) John read abook he owned, and so did Simon. 
has three readings: John and Simon read the same 
book; John and Simon both read a book belonging 
to John, though not necessarily the same one; John 
reads one of John's books and Simon reads one of 
Simon's books. 
Intuitively, the first reading arises from strictly 
identifying the elliptical book with the antecedent- 
book. The second arises from strictly identifying 
the pronouns, while sloppily identifying the books. 
4If -Fc is given wide scope over antecedent and el- 
lipsis, the QLF is rendered uninterpretable, which is as 
required. As detailed in section 3, scoping +c discharges 
the term and its index by substituting a variable for it. 
But the ellipsis substitution overrides this, substituting 
a new term and index, -Fa. But there is no way of dis- 
charging them. 
s Also to pronouns of laziness. 
The third from sloppily identifying both the books 
and the pronouns. In the literature, the first reading 
would not be viewed as a case of strict identity. But 
this view emerges naturally from our treatment of 
substitutions, and is arguably a more natural char- 
acterisation of the phenomena. 
We need to distinguish between parallel and non- 
parallel terms in ellipsis antecedents. Parallel terms, 
like John in the example above, are those that cor- 
respond terms appearing explicitly in the ellipsis. 
Non-paraUel terms are those that do not have an 
explicit parallel in the ellipsis. (Determining which 
terms are parallel/non-parallel is touched on in sec- 
tion 4.) 
For parallel terms, we have no choice about the 
ellipsis substitution. We replace both the term and 
its index by the corresponding term and index from 
the ellipsis. But for all non-parallel terms we have a 
choice between a strict or a sloppy substitution, s 
A sloppy substitution involves substituting a new 
term index for the old one. This has the effect of 
reindexing the version of the term occurring in the 
ellipsis, so that it refers to the same kind of thing as 
the antecedent term but is not otherwise linked to 
it. 
A strict substitution substitutes the term by its 
index. In this way, the version of the term occurring 
in the ellipsis is directly linked to antecedent term. 
To illustrate, an abbreviated QLF for the an- 
tecedent John read a book he owned is 
(19) .~S: 
read( 
tezm(+j...) 
tera(+b, 3, 
Ay.book(y)A 
own(te~a(+h.., rft(+j)), y), ...)) 
Here, we have left the scope node as an uninstan- 
tinted meta-variable ?S. The pronominal term +h 
occurs in the restriction of the book term +b. The 
pronoun has been resolved to have a contextual re- 
striction, rft(+j), that co-indexes it with the subject 
term. Here, 'fit(.)' is a function that when applied 
to an entity-denoting expression (e.g. a variable or 
constant) returns the property of being identical to 
that entity; when it applies to a term index, it re- 
turns an E-type property contextually linked to the 
term. 
The ellipsis can be represented as 
(20) ?P(term(+s, 3, Ay.name(y,'Simon'),...)) 
which is conjoined with the antecedent. 
The three readings of (18) are illustrated below, 
listing substitutions to be applied to the antecedent 
SThis is true of the non-paraliel tera(-Fh .... ) in ex- 
ample (11); but this added complication does not affect 
the basic account of scope parallelism given earlier. 
232 
and cashing out the results of their application, 
though omitting scope. 
(21) Strict book 
{+j/+s, 
term(+b,...)/+b, ...} 
read( term(+s,...), +b) 
(a) Since all reference to the term +h is removed by 
the strict substitution on the term in which it occurs, 
it makes no difference whether the pronoun is given 
a strict or a sloppy substitution. (b) Strict substi- 
tution for the book leaves behind an occurrence of 
the index +b in the ellipsis. For the QLF to be 
interpretable, it is necessary to give the antecedent 
book term wide scope over the ellipsis in order to 
discharge the index. 
(22) Sloppy book, strict pronoun 
{+j/+s, 
-bb/+bl, 1;erm(-l-h,...)/-t-h} 
read( terffi(+s...) 
t ex'm(q-bl, :t, au.book(u)^ 
own(+h, y) ...)) 
As above, the antecedent pronoun is constrained to 
be given wide scope over the ellipsis, on pain of 
the index +h being undischargeable. (Pronouns, 
like proper names, are treated as contextually re- 
stricted quantifiers, where the contextual restriction 
may limit the domain of quantification to one indi- 
vidual.) 
(23) Sloppy book, sloppy pronoun 
{+j/+s, 
+b/+bl,-t-h/+hl} 
read( ts~m(+s...) 
term(+bl, B, 
)~y.book(y)A 
own(term(+hl...rft(+s)), y), ...)) 
The index substitution from the primary term re- 
indexes the contextual restriction of the pronoun. It 
becomes coindexed with +s instead of +j. 
DSP's account of the first reading of (18) is sig- 
nificantly different from their account of the last 
two readings. The first reading involves scoping the 
book quantifier before ellipsis resolution. The other 
two readings only scope the quantifier after resolu- 
tion, and differ in giving the pronoun a strict or 
a sloppy interpretation. In our account the choice 
of strict or sloppy substitutions for secondary terms 
can constrain permissible quantifier scopings. 7 But 
the making of these choices does not have to be in- 
terleaved in a precise order with the scoping of quan- 
tifiers. 
Moreover, the difference between strict and sloppy 
readings does not depend on somehow being able to 
distinguish between primary and secondary occur- 
rences of terms with the same meaning. In DSP's 
representation of the antecedent of (18), both NPs 
'John' and 'he' give rise to two occurrences of the 
same term (a constant, j). The QLF representation 
is able to distinguish between the primary and the 
secondary, pronominal, reference to John. 
2.5 Other Phenomena 
Space precludes illustrating the substitutional ap- 
proach through further examples, though more are 
discussed in (Alshawi et al., 1992; Cooper et al., 
1994b). The coverage is basically the same as DSP's: 
Antecedent Contained Deletion: A sloppy 
substitution for every person that Simon did in the 
sentence John greeted every person that Simon did 
results in re-introducing the ellipsis in its own reso- 
lution. This leads to an uninterpretable cyclic QLF 
in much the same way that DSP obtain a violation 
of the occurs check on sound unification. 
Cascaded Ellipsis: The number of readings ob- 
tained for John revised his paper before the teacher 
did, and then Simon did was used as n benchmark 
by DSP. The approach here gets the four readings 
identified by them as most plausible. With slight 
modification, it gets a fifth reading of marginal plau- 
sibility. The modification is to allow (strict) substi- 
tutions on terms not explicitly appearing in the el- 
lipsis antecedent -- i.e. the implicit his paper in the 
second ellipsis when resolving the third ellipsis. 
We do not get a sixth, implausible reading, pro- 
vided that in the first clause his is resolved as being 
coindexed with the term for John; i.e. that John 
and his do not both independently refer to the same 
individual. Kehler blocks this reading in a similar 
manner. DSP block the reading by n more artificial 
restriction on the depth of embedding of expressions 
in logical forms; they lack the means for distinguish- 
ing between coindexed and merely co-referential ex- 
pressions. 
Multiple VP Ellipsis Multiple VP ellipsis (Gar- 
dent, 1993) poses problems at the level of determin- 
ing which VP is the antecedent of which ellipsis. But 
at the level of incorporating elliptical material once 
TThe converse also holds. Giving aa antecedent term 
wide scope over the ellipsis renders the choice of a strict 
or & sloppy substitution for it in the ellipsis immate- 
rial. During semantic evaluation of the QLF, dischaxgo 
ing the antecedent through scoping will substitute out 
all occurrence of the term Lad its index before ellipsis 
substitutions are applied. Note though that this order 
dependence applies at the level of evaluating QLFs, not 
constructing and resolving them. 
233 
the antecedents have been determined, it appears to 
offer no special problems. 
Other Forms of Ellipsis: Other forms of eb 
lipsis, besides VP-ellipsis can be handled substitu- 
tionally. For example, NP-ellipsis (e.g. Who slept? 
John.) is straightforwardly accommodated. PP- 
ellipsis (e.g. Who left on Tnesdayf And on Wednes- 
day?) requires substitutions for form constructions 
in QLF (not described here) representing preposi- 
tional phrases. 
2.6 Comparisons 
The use of terms and indices has parallels to propos- 
als due to Kehler and Kamp (Kehler, 1993a; Gawron 
and Peters, 1990). Kehler adopts an analysis where 
(referential) arguments to verbs are represented as 
related to a Davidsonian event via thematic role 
functions, e.g. agent(e)--john). Pronouns typically 
refer to these functions, e.g. he-agent(e). In VP 
ellipsis, strict identity corresponds to copying the 
entire role assignment from the antecedent. Sloppy 
identity corresponds to copying the function, but ap- 
plying it to the event of the ellided clause. 
For Kamp, strict identity involves copying the dis- 
course referent of the antecedent and identifying it 
with that of the ellided pronoun. Sloppy identity 
copies the conditions on the antecedent discourse ref- 
erent, and applies them to the discourse referent of 
the ellided pronoun. 
Neither Kamp nor Kehler extend their copy- 
ing/substitution mechanism to anything besides 
pronouns, as we have done. In Kehler's case, it is 
hard to see how his role assignment functions can 
be extended to deal with non-referential terms in the 
desired manner. DRT's use of discourse referents to 
indicate scope suggests that Kamp's treatment may 
be more readily extended in this manner; lists of dis- 
course referents at the top of DRS boxes are highly 
reminiscent of the index lists in scope nodes. 
3 Semantic Evaluation 
Figure 1 defines a valuation relation for the QLF 
fragment used above, derived from (Alshawi and 
Crouch, 1992; Cooper et al., 1994a). If a QLF ex- 
pression contains uninstantiated recta-variables, the 
valuation relation can associate more than one value 
with the expression. In the case of formulas, they 
may be given both the values true and false, corre- 
sponding to the formula being true under one possi- 
ble resolution and false under another. A subsump- 
tion ordering over QLFS, ~, is employed in the eval- 
uation rules, in effect to propose possible instantia- 
tions for meta-variables (the rule fragment only al- 
lows for scope meta-variables, but (Cooper et al., 
1994a) describes the more general case where other 
kinds of meSa-variable are permitted). A partially 
instantiated QLF therefore effectively specifies a set 
of possible evaluations (or semantic compositions). 
Definition of \])(QLF, M, g, Subs, v) 
where QLF is a QLF expression 
M is a model, (O, F) 
g is an assignment of values to variables 
Subs is a set of substitutions 
v is a value assigned to the QLF expression 
1. Constant symbols, c: V( c, M, g, Subs, v) iff F(c) = v 
(where F is the interpretation function for non- 
logical constants provided by M) 
2. Variables, z: V(z, M, g, Subs, v) iff g(z) = v 
3. Reinterpretation: 
)2(QLF,, M, g, Subs, v) iff V(QLF~, M, g, Subs, v) 
where QLF1/ QLF 2 E Subs 
4. Merging reiaterpretations: 
V( QL~Subsl, M, g, Subs~, v) if 
V( QLF, M, g, Subs1 ~ Subs~, v) 
5. Abstraction: 
V()~z.~b, M, g, Subs, h) if 
~b _~ ~b' and h is such that Y(~*, M, g~k, Subs, v) iff 
h(k, .) 
6. Application: 
V(p(,,, ..... ,,.), M, g, Subs, P(A1 ..... A.)) if 
p(~, ..... ,,.) _~ p'(,d ..... ,.% 
i)(p', M, g, Subs, P), 
V(a'~, M, g, Subs, At), 
..., and 
V(a'n, M, g, Subs, A,) 
7. a-Application: 
r(X^~b (T), M, g, Subs, v) if 
~}(~b I {X/T}, M, g, Subs, v) 
8. Scoped formula: 
V( Scope:~, M, g, Subs, v) if 
"IJ( Q'( R', ~'), M, g, Subs, v) 
where 
a)~ is a formula containing the term, To, 
texa(I0, Qo, Ro, P0) 
b) newcxpr(To, Subs) = T = term(I, C, R, Q, P) 
c) Scope ~_ \[z .... \] 
d) R' is  =.(andCR(z), P'(=)) I {q=}) 
e) ~' is Xz.(\[...\]:~ \[ {TIz,I/x}) 
Operations on substitutions: 
• Subs1 bJ Subs~ combines two sets of substitutions. 
This is like set union, except that where Subs1 and 
Subs~ both substitute for a particular item, the sub- 
stitution from Subsl is retained and not that in 
Subs~. 
• newezpr(Old, Subs) returns New if Old/New E 
Subs and otherwise Old. 
Figure 1: QLF Evaluation Rules 
234 
As the QLF becomes more instantiated, the set of 
possible evaluations narrows towards a singleton. 
It is also possible for a QLF to be uninterpretable; 
to specify no possible evaluation. Thus, no rules are 
given for evaluating terms or their indices in isola- 
tion. They must first be discharged by the scop- 
ing rule, which substitutes the terms and indices 
by A-bound variables. Inappropriate scoping may 
leave undischarged and hence uninterpretable terms 
and indices (which accounts for the so-called free- 
variable and vacuous quantification constraints on 
scope (Alshawi and Crouch, 1992)). The substi- 
tutions employed by the evaluation rule for scop- 
ing achieve a similar effect to the introduction and 
discharging of quantifier assumptions in Pereira's 
(1990) categorial semantics. 
The non-deterministic nature of evaluation and 
the role of substitutions draws us to conclude that 
ellipsis substitutions operate on (descriptions of) the 
semantic compositions, not the results of such com- 
positions. 
4 Parallelism and Inference 
Selecting ellipsis antecedents and parallel elements 
within them is an open problem (Priist, 1992; Prfist 
et al., 1994; Kehler, 1993b; Grover et al., 1994). Our 
approach to parallelism is perhaps heavy-handed, 
but in the absence of a clear solutions, possibly more 
flexible. The QLFs shown above omitted category 
information present in terms and ~orms. s Categories 
are sets of feature value equations containing syntac- 
tic information relevant to determining how unin- 
stantiated meta-variables can be resolved. 
Tense in VP-ellipsis illustrates how categories can 
be put to work. In 
(24) I enjoyed it. And so will you 
the ellipsis is contained within a form expression 
whose category is 
vp_ellipsis It ense=inf ,modalffivill ,perf ectffi_, 
progressive=_,pol=pos .... \] 
This states the syntactic tense, aspect and polarity 
marked on the ellipsis (underscores indicate lack of 
specification). The category constrains resolution to 
look for verb phrase/sentence sources, which come 
wrapped in forms with categories like 
\[t ease=past, modalffino, pexf ectffino, 
progressive--no ,polffipos .... \] 
Heuristics simi\]ar to those described by Hardt (1992) 
may be used for this. The category also says that, 
for this kind of VP match 9, the term in the an- 
tecedent whose category identifies it as being the 
subject should be treated as parallel to the explicit 
term in the ellipsis. 
As this example illustrates, tense and aspect on 
ellipsis and antecedent do not have to agree. When 
Sforns axe described in (Alshawi and Crouch, 1992). 
9Not all VP ellipses have VP antecedents. 
this is so, the antecedent and ellipsis categories are 
both used to determine what fozm should be sub- 
stituted for the antecedent form. This comprises 
the restriction of the antecedent form and a new 
category constructed by taking the features of the 
antecedent category, unless overridden by those on 
the ellipsis--a kind of (monotonic) priority union 
(Grover et ai., 1994) except using skeptical as op- 
posed to credulous default unification (Carpenter, 
1993). When a new category is constructed for the 
antecedent, any tense resolutions also need to be un- 
done, since the original ones may no longer be appro- 
priate for the revised category. One thus merges the 
category information from source and antecedent to 
determine what verb phrase form should be substi- 
tuted for the original. In this case, it will have a 
category 
vp \[tensefinf, modalfeill, perfectffino, 
progressive=no ,polfneg .... \] 
A more general question is whether all ellipses in- 
volve recompositions, with variants, of linguistic an- 
tecedents. There are cases where a degree of infer- 
ence seems to be required: 
(25) We spent six weeks living in France, eating 
French food and speaking French, as we did in 
Austria the year before. 
(one must apply the knowledge that Austrians speak 
German to correctly interpret the ellipsis). Pulman's 
(1994) equational treatment of context-dependency 
suggests one method of dealing with such cases. But 
it remains to be seen how readily the equations used 
for ellipsis here can be integrated into Pulman's 
framework. 
5 Conclusions: Interpretation as 
Description 
The substitutional treatment of ellipsis presented 
here has broadly the same coverage as DSP's higher- 
order unification treatment, but has the computa- 
tional advantages of (i) not requiring order-sensitive 
interleaving of different resolution operations, and 
(ii) not requiring greater than second-order match- 
ing for dealing with quantifiers. In addition, it cures 
a slight overgeneration problem in DSP's account. 
It has been claimed that these advantages arise 
from viewing semantic interpretation as a process 
of building descriptions of semantic compositions. 
To conclude, a few further arguments for this view, 
that are independent of any particular proposals for 
dealing with ellipsis. 
Order-Independence: One of the reasons 
for the computational success of unification-based 
syntactic formalisms is the order-independence of 
parser/generator operations they permit. If one 
looks at the order-sensitive nature of the operations 
of semantic compositions, they provide a poor start- 
ing point for a treatment of semantics enjoying simi- 
lar computational success. But semantic interpreta~ 
235 
tion, viewed as building a description of the intended 
composition, is a better prospect. 
Context-Sensitivity: The truth values of many ~ 
?) sentences undeniably depend on context. 
ntext-dependence may enter either at the inter- 
pretive mapping from sentence to meaning and/or 
the evaluative mapping from meaning (and the 
world) to truth-values. 
Intrepretation Evaluation 
sentence ~ meaning ,\[--~value \[# 
? / 
CONTEXT 
The more that context-dependence enters into the 
interpretive mapping (so that meanings are corre- 
spondingly more context-independent), the harder 
it is to maintain a principle of strict composition- 
ality in interpretation. The syntactic structure un- 
derspecifies the intended composition, so that the 
meanings of some constituents (e.g. pronouns) and 
the mode of combination of other (e.g. quantifiers) 
are not fully specified. Further contextual informa- 
tion is required to fill the gaps. Again, interpretation 
seen as description building sits easily with this. 
Preserving Information: Focusing exclusively 
on the results of semantic composition, i.e. mean- 
ings, can ignore differences in how those meanings 
were derived that can be linguistically significant 
(e.g. co-referential vs co-indexed terms). If this in- 
formation is not to be lost, some way of referring to 
the structure of the compositions, as well as to their 
results, seems to be required. 
Acknowledgements 
The initial implementation of the work described 
here was carried out as part of the CLARE project, 
DTI IED4/1/1165. The writing of this paper was 
supported in part by the FraCaS project, LRE 62- 
051. I would especially like to thank Hiyan Alshawi 
and Steve Pulman for help and advice on topics re- 
lating to this paper. I have also benefited from con- 
versations with Claire Grover, Ian Lewin and Mas- 
simo Poesio. 
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