THE RESOLUTION OF QUANTIFICATIONAL AMBIGUITY IN THE TENDUM SYSTEM 
Harry Bunt 
Computational Linguistics Research Unit 
Dept. of Language and Literature, Tilburg University 
P.O.Box 90153, 5000 LE Tilburg 
The Netherlands 
ABSTRACT 
A method is described for handling the 
ambiguity and vagueness that is often found 
in quantifications - the semantically complex 
relations between nominal and verbal 
constituents. In natural language certain 
aspects of quantification are often left 
open; it is argued that the analysis of 
quantification in a model-theoretic framework 
should use semantic representations in which 
this may also be done. This paper shows a form 
for such a representation and how "ambiguous" 
representations are used in an elegant and 
efficient procedure for semantic analysis, 
incorporated in the TENDUM dialogue system. 
The quantification ambi~uit\[ explosion 
problem 
Quantification is a complex phenomenon 
that occurs whenever a nominal and a verbal 
constituent are combined in such a way that 
the denotation of the verbal constituent is 
predicated of arguments supplied by the 
(denotation of the) nominal constituent. 
This gives rise to a number of questions such 
as (i) What objects serve as predicate 
arguments? (2) Of how many objects is the 
predicate true? (3) How many objects are 
considered as potential arguments of the 
predicate? 
When we consider these questions for a 
sentence with a few noun phrases, we readily 
see that the sentence has a multitude of 
possible interpretations. Even a sentence 
with only one NP such as 
(I) Five boats were lifted 
has a variety of possible readings, depending 
on whether the boats were lifted individually, 
collectively, or in groups of five, and on 
whether the total number of boats involved 
is exactly five or at least five. For a 
sentence with two numerically quantified 
NPs, such as 'Three Russians visited five 
Frenchmen', Partee (1975) distinguished 8 
readings depending on whether the Russians 
and the Frenchmen visited each other indivi- 
dually of collectively and on the relative 
scopes of the quantifiers. Partee's analysis 
is in fact still rather crude; a somewhat 
more refined analysis, which distinguishes 
group readings and readings with equally wide 
scope of the quantifiers, leads to 30 inter- 
pretations (Bunt, in press). 
This presents a problem for any attempt 
at a precise and systematic description of 
semantic structures in natural language. On 
the one hand an articulate analysis of 
quantification Js needed for obtaining the 
desired interpretations of every sentence, 
while on the other hand we do not want to end 
up with dozens of interpretations for every 
sentence. 
To some extent this "ambiguity explosion 
problem" is an artefact of the usual method 
of formal semantic analysis. In this method 
sentences are translated into formulae of a 
logical language, the truth conditions of 
which are determined by model-theoretic in- 
terpretation rules. Now one might want to 
consider a sentence like (i) not as ambiguous, 
but only as saying that five boats were lifted, 
w~thout specifying how they were lifted. But 
translation of the sentence into a logical 
representation forces one to be specific. That 
is, the logical representation language 
requires distinction between such interpreta- 
tions as represented by (2) (individual 
reading) and (3) (group reading): 
(2) ~({x e BOATS: LIFTED(x)}) = 5 
(3) 3 x E{ y C BOATS:~ (y) = 5} : LIFTED(x) 
In other words, the analysis framework forces 
us to make distinctions which we might not 
always want to make. 
To tackle this problem, I have devised a 
method of representing quantified expressions 
in a logical language with the possibility of 
leaving certain quantification aspects open. 
This method has been implemented in the TENDUM 
dialogue system, developed jointly at the 
Institute for Perception Research in Eindhoven 
and the Computational Linguistics Research 
Unit at Tilburg University, Department of 
Linguistics (Bunt, 1982; ~983; Bunt & thoe 
Schwartzenberg, 1982;). This method is not 
only of theoretical interest, but also pro- 
vides a computationally efficient treatment 
of quantification. 
Ambiguity resolution 
In a semantic analysis system which 
translates natural language expressions into 
formal representations, all disambiguation 
takes place during this translation. 
130 
This applies both to purely lexical ambiguities and 
to structural ambiguities. For lexical disambigua- 
tion this means that a lexical item has several 
translations in the representation language (RL), 
which are all produced by a dictionary lookup at 
the beginning of the analysis. The generation of 
semantic representations for sentences that display 
both lexical and structural ambiguity thus takes 
place as depicted in Fig. i: 
" ~ Z;\];~;;Z 
NL ~ RL ........ model \ " ~ ......... / 
\ ~-"-~ ;;;Z;;; 
• ~ ......... / 
dictionary application of interpre- 
lookup grammar rules tation 
Fig. i Longer arrows indicate larger amount of 
processing. 
Since the lexical ambiguities considered here are 
purely semantic, the same grammar rules will be 
applicable to all the lexical interpretations 
(assuming that the grammar does not contain world 
knowledge to filter out those interpretations that 
are meaningless in the discourse domain under 
consideration). Since the amount of processing 
involved in the application of grammar rules is 
very large compared to that of translating a lexi- 
cal item to its RL instances, this set-up is not 
very efficient. In the PHLIQAI question-answering 
system (Bronnenberg et al., 1980) the syntactic/ 
semantic and lexical processing stages were there- 
fore reversed, so that disambiguation takes place 
as depicted in Fig. 2: 
NL • :::::::: oO0Ol 
/ ........ / 
;;-_222;2 / 
application of dictionary Interpre- 
grammar rules lookup ration 
Fig. 2 Longer arrows indicate larger amount of 
processing. 
In this setup an intermediate representation 
language is u~ed which is identical to RL except 
that is has an ambiguous constant for every content 
word of the natural language. 
It turns out that semantic analysis along 
these lines can be formulated entirely in terms of 
the traditional model-theoretic framework (Bunt, 
in press), therefore this method is appropriately 
called two-level model-theoretic semantics. This 
method has been implemented in the TENDUM system, 
with an intermediate representation language that 
contains ambiguous constants corresponding to 
quantification aspects, in addition to ambiguous 
constants corresponding to nouns, verbs, etc. 
Quantification aspects 
The different aspects of quantification are 
closely related to the semantic functions of 
determiners. These functions depend on their 
syntactic position in a determiner sequence. A 
full-fledged basic noun phrase has the layout: 
(4) pre- + central + post- + head 
determiner determiner determiner noun 
(see Quirk et al., 1972, p.146). For example, in 
the NP 
(5) All my four children 
the centraldeterminer 'my' restricts the range of 
reference of the head noun 'children' to the set 
of my children; the predeterminer 'all' indicates 
that a predicate, combined with the noun phrase to 
form a proposition, is associated with all the 
members of that set, and the postdeterminer 'four' 
expresses the presupposition that the set consists 
of four elements. This set is determined by the 
central determiner plus the denotation of the head 
noun; I will call it the source of the quantifica- 
tion. In the case of an NP without central 
determiner the source is the denotation of the head 
noun. For the indication of the quantity or 
fraction of that part of the source that is invol- 
ved in a predication I will use the term source 
involvement. 
Quantification owes its name to the fact that 
source involvement is often made explicit by means 
of quantitative (pre-)determiners like 'five', 
'many', 'all',or 'two liters of'. Obviously, source 
involvement is a central aspect of quantification. 
Another important aspect of quantification is 
illustrated by the following sentences: 
(6a) The chairs were lifted by all the boys 
(6b) The chairs were lifted by each of the boys 
These sentences differ in that (6b) says 
unambiguously that every one of the boys lifted the 
chairs, whereas (6a) is unspecific as to what each 
individual boy did: it only says that the chairs 
were lifted and that all the boys were involved in 
the lifting, but it does not specify, for instance, 
whether every one of the boys lifted the chairs or 
all the boys together lifted the chairs. The 
quantifiers 'all' and 'each (of)' thus both 
indicate complete involvement of the source, but 
differ in their determination of how a predicate 
('lifted the chairs') is applied to the source. 
'Each' indicates that the predicate is applied to 
the individual members of the source; 'all' leaves 
open whether the predicate is applied to individual 
members, to groups of meubers, or to the sources 
as a whole. To designate the way in which a pre- 
dicate is applied to, or "distributed over", the 
source of a quantification, I use the term 
distribution. A way of expressing the distribution 
of a quantification is by specifying the class of 
objects that the predicate is applied to, and how 
this class is related to the source. In the 
distributive case this class is precisely the : 
131 
source; in the collective case it is the set 
having the source as its only element. I will 
refer to the class of objects that the predicate is 
applied to as the domain of the quantification. The 
distribution of a quantification over an NP 
denotation can be viewed as specifying how the 
domain can be computed from the source. Where 
domain = source I will speak of individual distri- 
bution, where domain = {source} of collective 
distribution. 
Individual and collective are not the only 
possible distributions. Consider the sentence 
(7) All these machines assemble 12 parts. 
This sentence may describe a situation in which 
certain machines assemble sets of twelve parts, 
i.e. a relation between individual machines and 
groups of twelve parts. If PARTS is the set denoted 
by 'parts', the direct object quantification domain 
is ~I~(PARTS), the subset of ~(PARTS) containing 
only £~ose subsets of PARTS that have twelve 
members. I call this type of distribution group 
distribution. In this case the numerical quantifier 
indicates group size. 
A slightly different form of "group 
quantification" is found in the sentence 
(8) Twelve men conspired. 
In view of the collective nature of conspiring, it 
would seem that 'twelve' should again be inter- 
preted as indicating group size, so that the 
sentence may be represented by 
(9) B x E ~12(MEN): CONSPIRE(x) 
However, as the existential quantifier brings out 
clearly, this interpretation would leave open the 
possiblity that several groups of 12 men conspired, 
which is probably not what was intended. The more 
plausible interpretation, where exactly one group 
of 12 men conspired, I will call the strong group 
readinq of the sentence, and the other one the 
weak group reading. On the strong group reading 
the quantifier 'twelve' has a double function: it 
indicates both source involvement and group size. 
In a sentence like 
(i0) The crane lifted the tubes 
there is no indication as to whether the tubes were 
lifted one by one (individual distribution), two by 
two (weak group distribution with group size 2), 
one-or-two by one-or-two (weak group distribution 
with group size I-2), ..., or all in one go 
(collective distribution). The quantification is 
unspecific in this respect. In such a case I will 
say that the distribution is unspecific. If S is 
the source of the quantification, the domain is in 
this case the set consisting of the elements of S 
and the plural subsets of S. 
Distribution and source involvement are the 
two central aspects of quantification that I will 
focus on here. 
Quantification in two-level model-theoretic 
semantics 
Consider a non-intensional verb, denoting a 
one-place predicate P (a function from individuals 
to truth values), which is combined with a noun 
phrase with associated source S (a set of indivi- 
duals). The quantification then predicates the 
source involvement of the set of those elements of 
the quantification domain, defined by S and the 
distribution, for which P is true. This can be 
represented by a formula of the following form: 
(ii) S-INVOLVEMENT({xeQUANT.DOMAIN: P(x) } ) 
For example, consider the representation of the 
readings of sentence (I) 'Five boats were lifted', 
with individual, collective, and weak and strong 
group distribution: 
(12a) (Az:~z)=5) ({x ~ BOATS: LIFTED(x)}) 
(12b) (~z:~(z)>l) ({x 6 ~(BOATS) : LIFTED(x)}) 
(12c) (Az:~z)=l) ({x q~(BOATS): LIFTED(x)}) 
(12d) (Az:~z)=5) (UBoATSD({X e BOATS U ~+ (BOATS) : 
LIFTED(x) }) ) 
where~+(S) denotes the set of plural subsets of S. 
The notation U (D) is used to represent the set of S ,1 
those members of S occuring in D"; the precise 
definition is: 
(13) Us(D) = {xES: xED v (B yED: x6y)} 
Note that in all cases the quantification domain is 
closely related to the source in a way determined 
by the distribution. I have claimed above that the 
distribution can be construed as a function that 
computes the quantification domain, given the 
source. Indeed, this can be acomplished by meads 
of a function of two arguments, one being the 
source and the other the group size, in the case 
of a group distribution. A little bit of formula 
manipulation readily shows that all the formulas 
(12a-d) can be cast in the form 
(14) (lz: N(Us(Z))) ({xed(k,S): P(x) } ) 
where S represents the quantification source, 
~z: N(U_ (z))) the source involvement, k the group 
size, an~ d the "distribution function" computing 
the quantification domain. (For technical details 
of this representation see Bunt, in press). The 
most interesting point to note about this represen- 
tation is that the distribution of the quantifica- 
tion, which in other treatments is always reflec- 
ted in the syntactic structure of the representa- 
tion, corresponds to a term of the representation 
language here. For this term we substitute 
expressions like ~k,S:~k(S)) to obtain a particu- 
lar interpretation. 
I will now indicate how representations of 
the form (14) are constructed in the TENDUM system. 
The construction of quantification 
representation in the TENDUM system 
The TENDUM system uses a gra~nar consisting 
of phrase-structure rules augmented with semantic 
rules that construct a representation of a rewrit- 
ten phrase from those of its constituents (see 
Bunt, 1983). For the sentence 'Five boats were 
lifted' this works as follows. 
The number 'five' is represented in the 
lexicon as an item of syntactic category'number' 
with representation '5'. To this item, a rule 
applies that constructs a syntactic structure of 
category'numera~ with representation 
132 
(Ay:~ (y)=5), which I abbreviate as FIVE. TO this 
structure a rule applies that constructs a 
syntactic structure of category 'determiner with 
representation 
(15) (AX: (AP: FIVE(Ux({XEd(FIVE,X): P(x) } )))) 
A rule constructing a syntactic structure of cate- 
gory'noun phrase" from a determiner and a nominal 
(inthe simplest case: a noun) applies to 'five' and 
'boats', combining their representations by 
applying (15) as a function to the noun representa- 
tion BOATS. After l-conversion, this results in 
(16) (AP: FIVE(t)BOATS( {xEd(FIVE, BOATS): P(x)}))) 
A rule constructing a sentence from a noun phrase 
and a verb applies to 'five boats' and 'were 
lifted', combining their representations by 
applying (16) as a function to the verb representa- 
tion LIFTED. After l-conversion, this results in 
(17) : 
(17) FIVE~3BOATs({XEd(FIVE , BOATS): P(x)} )) 
NOW suppose the sentence is interpreted relative 
to a domain of discourse where we have such boats 
and lifting facilities that it is impossible for 
more than one boat to be lifted at the same time. 
This is reflected in the fact that the RL predicate 
LIFTED r is of such a type that it can only apply to 
individual boats. Assuming that the ambiguous 
constant BOATS has the single instance BOATS and r 
that LIFTED has the single instance 
(Az: LIFTED (z)), the instantiation rules, con- 
strained byrthe type restrictions of RL, will 
produce the representation: 
(18) FIVE(UBOAT S ({xEBOATSr: LIFTEDr(X) } )) 
r 
(For the instantiation process see Bunt, in press, 
chapter 7.) This is readily seen to be 
equivalent to the more familiar form: 
(19) #( {xEBOATS : LIFTED (x)}) = 5 r r 
If, in addition to, or instead of the distributive 
reading we want to generate another reading of the 
sentence, then we extend or modify the instantia- 
tion function for LIFTED accordingly. 
This shows how the analysis method generates 
the representations of only those interpretations 
which are relevant in a given domain of discourse, 
and does so without generating intermediate 
representations as artefacts of the use of a 
logical representation language. 
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