A Formal Semantics for Generating and Editing Plurals 
Paul Piwek 
ITRI - University of Brighton 
Watts Building, Moulsecoomb, 
Brighton BN2 4GJ 
UK 
Paul.Piwek@itri.brighton.ac.uk 
Abstract 
Wc present a formal semantics for an object- 
oriented formalism which allows for the represen- 
tation of plura, l objects (such as 'Three N', 'Most 
of the N', 'Some N',...). The semantics is given in 
terms of it mapping to a variant of Discourse Rep- 
resentation Theory. It is motivated by its suitability 
lkw natural  generation and interactive edit- 
ing of the representations. 
1 Introduction 
A natural  generator typically generates a 
noun plnase l'rom a representation consisting of an 
object with one or more attributes (cf. Reiter & 
l)ale, 2000). Usually this representation is sup- 
plemented with inl'ormation concerning the context 
in which the noun phrase has io be realized (e.g., 
the set of distractors, whether tile object is in fo- 
cus, etc.). \];or instance, the lil,ICUP, l{ system (Dale, 
1992) deals with reference to plural objects by hav- 
ing the l'ollowing three attributes on physical ob- 
jects: structure, whose wflue can be either a set or 
individual, cardinalio, which in case of a set records 
the numbers of elements which the set has, and con- 
stituents which in case of a set contains the elements 
of the set. 
Our proposal is intended to extend the representa- 
tions proposed in (Dale, 1992)) Most importantly, 
wc replace the attribute cardinalily with the more 
general attribute quant (for quantifier) whose value 
is a quantilier such as 'most', 'few', '5', '< 6' (at 
most 6), etc. Furthermole, we introduce the new at- 
tribute parl_of which takes its a value an ob.jecl of 
which the object in question is a part. ~ 
~Note that we are dealing with the generation of plurals 
from (logically) structured data as opposed lo raw data as in, 
e.g., Stone (1999). 
2We use the mcfcologicat 'part of' relation as an alternative 
It} "subset' For details, see the next section. 
The object-oriented (00) forlnalism in which we 
implement tile al'orelnentioned attributes is an ex- 
tension of standard oo formalisms. It is known 
as Scoped Semantic Networks (SSN; Kibble et al., 
1999; Power, 1999). 3 An SSN consists of a net- 
work of objects together with a mapping o1' these 
objects to a set o1' logical contexts. This makes it 
possible to represent universal quantification, im- 
plication, negation and other logical operators. In 
particulal; Power (1999) shows how an SSN can be 
mapped into a I)iscourse Representation Structure 
(DRS; Kamp & Reyle, 1993), lhus providing a for- 
lnal semantic interpretation of SSNs. 
In tiffs paper, we provide a mapping of SSNs with 
plural objects to an adapted version of Discourse 
Represemation Theory (I)RT). The mapping is pro- 
vided to obtain t%rmal truth conditions for the SSNs. 
Such a lnaPlfing provides us with a mathenmlically 
precise characterization el' the information which is 
represented by a SSN ill terms of its truth-condilions. 
This is useful if we want to automatically lnanipu- 
lale lhe information which is represented by means 
of an SSN. For example, we can formally define 
whether some piece of information is aheady im- 
plicit in some other piece of information; in other 
words, we can deline a notion of logical conse- 
quence. Related to this is the possibility to use the 
semantics in order to test the consistency of the in- 
formatiou conveyed by an SSN. For tlmt purpose, we 
can do so-called model checking: an SSN is consis- 
lent if we can construct a model -that is, a logically 
possible state of the world- in which tile SSN is true 
according to our truth-conditional semantics. 
We do not provide a direct formal semantics for 
SSN, but rather map it to a more convenient log- 
ical l'ormalistn, i.e., I)P,T. The main reason for 
tiffs approach is that phenomena which we will be 
modelling in this paper, i.e. (plural) reference and 
aScc also, e.g., Sowa (1984). 
607 
anaphora, have been studied extensively within I)RT 
(see, e.g., Kamp & Reyle, 1993; Krahmer & Van 
Deemter, 1998; Piwek, 1997). Fnrthermore, we be- 
lieve that the adaptation of DRT that we propose is 
of interest in its own right. 
The mapping which we provide from SSNs with 
plural objects to DRSs requires some modifications 
to standard DRT with plurals (Kamp & Reyle, 1993: 
Chapter 4). For networks with only singular objects, 
there is a straightforward mapping of the objects in 
a network to the discourse referents which populate 
a DRS. Things are different for networks with plural 
objects. Consider: 
(1) Susan has found most books which Bill needs. 
The DP, S for this sentence is: 
(2) 
y 
book(y) 
need(bill,y) found(susan,y) 
Intuitively, the meaning of this condition is that: fi')r 
most y which satisfy the conditions to the le/'t of the 
diamond, it holds that they also sati,@~ the condition 
on the right. Note, that the representation contains 
no plural discourse referent corresponding to the Nt' 
'most books which Bill needs'. The 'y' in this repre- 
sentation is a referent for singular individuals. This 
might make one wonder how it is possible in stan- 
dard DRT to refer back to plural individuals as in: 
(3) Susan has found most books which Bill needs. 
They were on her desk. 
For this purpose, there is a so-called abstraction op- 
eration (Kamp & Reyle, 1993:313) with which we 
can obtain a discourse referent for the set of books 
which Bill needs and Susan Jbund. In more tech- 
nical terms, the set is obtained by the summation 
of the values which 'y' can take. Thus there is no 
direct way of mapping a plural object in a seman- 
tic network (which represent the interpretation of an 
NP) to a plural discourse referent in the correspond- 
ing DRS. For this reason we have chosen to adapt 
the DP, T formalism, so that plural noun phrases do 
directly colTelate with plural discourse referents. 
We now proceed as follows. In Section 2, we 
specify the mapping from SSNs to our version of 
DRT. In the next section (Section 3), we describe an 
application which uses the SSNs with plurals. We 
finish this paper with a conclusions section (Section 
4). 
2 From SSNs to DRSs 
In this section, we provide a mapping from SSNs 
into discourse representation structures (DRSs) with 
plurals. We start out by specifying the target of the 
mapping, i.e., plural DRT. 
DRSs with Plurals Following Kamp & Reyle 
(1993), we treat singular objects and sets of objects 
as entities of the same kind. Both am considered 
to be individuals: atomic and non-atomic individ- 
uals, respectively. Thus, the model theory follows 
the models which Link (1983) provides for count 
nounsfl The idea is that the denotation of an NP 
which contains a count noun can be uniquely subdi- 
vided into atomic parts (as opposed to the denotata 
of mass nouns). The domain for NPs is structured by 
a prot-whole relation which satisfies the axioms of 
upper semilattices (for background information on 
these lattices see Kamp & Reyle, 1993:398-406). 
In formal terms, a model is defined as follows: 
A model _/14 is a quintuple (Lt, g, Pred, @mrzt, Name) 
which consist of: 
(1) A domain of individuals with the structure of a com- 
plete, free, atomic upper scmilattice H = (U, C) with 
zero; 
(II) A domain of eventualities with the structure of a 
complete, free, atomic upper semilattice g = @7, C); 
(III) A function Pred mapping predicates P to their ex- 
tensions in k//, such that 
(III.1) for tim relations representing thematic roles, such 
as agent and patiertt, I@ed assigns a set of tuples (c, a), 
wherecCEandaGU. 
(III.2) for eventuality predicates, Prod(P) C_ E. 
(I11.3) For object type predicates, Prod(P) C U. 
(IV) A function Qua~tt mapping determiners DEW to 
their corresponding interpretations, i.e., a set consisting 
of tuples {a, b) (where a, b C U). 
(V) A function Name mapping constants to members of 
U. in particular, the constants c/,, where P is a predi- 
cate are mapped to ®Pred(P), i.e., the supremum, also 
known as the sum, of the interpretation of P. 
Notice that in our models there are separate domains 
for objects and eventualities (i.e., states and events). 
4Fora critical discussion and alternative to Link (1983), see 
for instance Landman (1989). 
608 
The relations agent and patient have an eventual- 
ity as their first argument and an object as second 
argument (cf. Parsons, 1990). agent(e,o) is to be 
interpreted as: object o is the agent of eventuality e. 
Furtherlnore, there are predicates applying to even- 
tualities and others applying to objects. 
For our purposes, the most interesting part of the 
definition is the function Q~ta,~,t; which maps deter- 
miners to their respective interpretations. We take 
the interpretation of a determiner to be a set of tu- 
pies, where each tuple consist of a pair of (plural) in- 
dividuals. For instance, take the deterlniner 'most'. 
Q'~m, nt, maps it to the following interpretation: '5 
(4) Q~ga~,t(Most) = {(r, c) : r c c & r is a non- 
atomic entity of M & kl -> } 
Thus 'most' corresponds to the set of all tuples of 
individuals, such that the first individual is a non- 
atomic part of the second one and the cardinality 
of the first is greater than or equal to the cardinal- 
ity of the second divided by two. Henceforth, we 
will call the second individual the context individual 
(cf. Westerstfihl, 1985). Given a noun phrase, such 
as 'most birds', the first individual is intended as 
the interpretation of the entire noun phrase whereas 
the second individual plays the role of the con- 
text against which the noun phrase is interpreted. 
The context individual can be restricted by extra- 
linguistic circumstances (e.g., the siluation in wlaich 
a noun phrase is produced) and by linguistic means 
(as in 'most of the birds on the beach', where 'the 
birds on the beach' supplies the contextual individ- 
ual). 
Let us focus on the DRS condition which is inter- 
preted in the models in terms of @m,~,t. This con- 
dition functions as a substitute for the duplex condi- 
tions of standard DRT 6 The condition in question is: 
'51tere we follow Ihe 'more than half' interpretation of 
'most' common fi'om the literature on GEneralized Quantiliers 
(see, e.g, I?,arwise & Cooper, 1981; Keenan & Westerstahl, 
1997). This interpretation is not entirely unproblematic; see, 
for instance, (Kamp & P, eyle, 1993). Our use of the interpre- 
tation is, however, solely for illustrative purposes. We can also 
accommodate for alternative mappings fur Q~u~nt(Most). 
Similarly we cannot go into detailed discussions of other quan- 
tifiers such as, for instance, 'many' (of. Lappin, 1988). 
6Within the conlines of this paper it is impossible to give a 
full formal delinition of our version of plural I)RT, thcrelore we 
focus on the aforementioned condition. The other definitions 
closely lollow those in Kamp & P, eyle, 1993: 425-427, 677- 
6'79). 
If z is a discern;re referent and t is a discourse re\[er- 
ent or constant, then DETt(:c) is a condition. 
The verification condition for this condition is: 
(5) M ~f DETt(:C) (if" 
(11 II II t IIAJ'f> 
Let us illustrate these definitions with a simple ex- 
ample. Consider: 
(6) At most two men walk. 
The NP 'At most two men' introduces a plural dis- 
course referent X, together with a number of condi- 
tions on that referent. Additionally, the verb 'walk' 
supplies a condition to the effect that all the mem- 
bers of X walk. 7,s 
(7) 
X 
AT_MOST_2c ....... (X) 
man(z) 
walk(z) 
walk*(X) 
The first condition says that X consists of a subset of 
the set of all men (cm,,,~, alternatively, we could use 
a set of contextually given men) and that X should 
consist of at most 2 individuals belonging to that 
set. '° The implicative condition is there to make sure 
there is no other set apart from X with (other) men 
who are also walking. Such a closure condition is 
particularly useful for the direct representation of 
monotonically decreasing quantifiers. ~° A quantor 
Q is monotonically decreasing if and only if for all 
7For cxpository reasons, we have left out explicit represEn- 
tations of events in this example. But, see the next section for a 
DP, S with plurals and events. 
8Note that when a predicate in a condition is marked with 
a '*', this means that the prcdicate is interpreted distributively 
over the atomic parts of the objects in its denotation. 
"JWe assume that: @Umt(AT_MOST_2) = {(r, c) : r C c 
& I,'1 < 2} 
mln Van Eijck (1983), an allemative approach is proposed 
within a fl'amework which also allows for the direct representa- 
tion of plural referents in DRT. lie proposes to reanalyse mono- 
tonically decreasing quantiliers in terms of negation and mono- 
tonically increasing ones. This, however, means that WE no 
longer have a direct correlation between plural discourse ref- 
erents and monotonically decreasing quantifiers. Furthermore, 
it prevents such quantifiers from any anaphoric uptake as in 
'Fewer than ten students took the test. They all passed it'. 
609 
X,Y,Z it holds that: if QXY and Z ~ Y, then 
QXZ. Thus, for instance, (a) 'At most two meu 
walk and talk' does not imply that (b) 'At most two 
men walk'. If we would represent (a) without the 
closure condition (i.e., there is a set of at most two 
men and each of them walks and talks), then (b) (i.e., 
there is a set q\[" at most two men and each of them 
walks) would follow fi'om (a). However, if we add 
to the representation of (a) that there are no other 
sets of men who walk and talk and to the represen- 
tation of (b) that that there are no other sets of men 
who walk, then (a) no longer follows fiom (b); the 
additional information in (a) that there are no other 
sets elmen who both walk and talk, does not entail 
that there are no other sets o/'men who walk. 
Seeped Semantic Networks A seeped semantic 
network (SSN) is a triple (D, L, f), consisting of a 
typed DAG (Directed Acyclic Graph) D, a sef of log- 
ical contexts L and a function f which assigns a log- 
ical context (which are treated as primitive objects 
separate from those in the DAG) to each of the ob- 
jects in the DAG. In the DAG, there are objects which 
correspond with logical operators, such as implica- 
tion and negation, and non-logical objects, such as 
physical objects and events. The function f, which 
assigns logical contexts to objects in a typed DAG 
D, satisfies the following constraints: 
(I) The root object and all the objects which are direct 
descendants of a logical operator are assigned a unique 
logical context. These contexls inherit the partial order- 
ing (in the DAG) of the objects with which they are asso- 
ciated. Furthermore, this set of logical contexts consti- 
tutes the range of f. 
(II) Logical operators which have not been assigned a 
context by clause 1. are mapped to the logical context of 
their nearest ancestor to which clause 1. applies. 
(III) Objects which arc not assigned to a logical context 
by the clauses 1. and 2. are assigned to a logical context 
in accordance with DRT's accessibility rules. 
Consider, for instance, the following sentence: 
(8) If a man is happy, then he whistles. 
We can represent this sentence by means of the SSN 
in Figure 1. In this representation, the dots repre- 
sent objects, the circles represent logical contexts 
(an object inside a circle belongs to the correspond- 
ing logical context), the solid arrows represent at- 
tributes and the dotted arrows represent that the ob- 
ject fi'om which the arrow originates belongs to the 
context to which the arrow points. 
There is a straightforward procedure for mappiug 
a SSN into a I)RS: 
(I) Logical contexts are mapped into boxes, where the 
nesting of the boxes is isomorphic to the partial ordering 
of the corresponding logical contexts. 
(II) Objects are inserted into the box which corresponds 
with their logical context, except for logical operators. 
The latter are mapped onto the appropriate operators on 
the boxes of their directly subordinate objects. 
(III) Typing statements T(z) of a non-logical object are 
added to the same box as the object z itself. 
(IV) Attributions/{(.% !/), where z and !/are non-logical 
objects, are added to the same box as z. 
:~= ~ implication %% 
happy(e) ~ • )whistle 
4 #. 
e'~'O0~ ""0: 4~ 
• man 
Figure 1" Network for (8) 
By applying these rules, we obtain the following 
DP, S for the SSN in Figure 1 : 
(9) 
xe 
happy(e) 
man(x) 
agent(e,x) 
e ~ 
=> whistle(e') 
agem(e',x) 
Note how the three circles in the SSN correspond 
with the three boxes of the DRS. Furthermore, the 
discourse referent z colresponds to the object in the 
SSN of the type man and inhabits the same box as 
the conditions which correspond to the object of 
type happy and the attribute agent. 
SSNs with Plurals In this section, we describe an 
extension of SSNs for countable plural objects. This 
extension requires no changes to the format of SSNs. 
Rathel, we introduce a number of special-purpose 
610 
attributions and types. Subsequently, we specify 
their mapping to appropriate terms in a DRS. 
We introduce two attributes on cotmlable objects: 
(I) quant. The wdue of this feature is reslricted to an 
oltiect of the type det_type. Examples of tlle subtypes of 
dcl, d, ype arc 2, > 1, < 3, all,.f>w, etc. 
(11) parl,_of. The value of this feature is restricted to 
countable objects. 
The lnapping of SSNs which include these special- 
purpose attributions and types to a l)P,s is defined as 
follows: 
(1) For typing statements T(x), where T is a subtype of 
del,_type: ignore the statement 7'(x) and the object x; 
(H) For attributions quant(x,y) such that ~z : 
p(,,rt_of(:,:,z) & z is an a,,cho,'& Tt(x) & 7~(y), add 
to the box in which also x lives the lbllowing condition: 
.r = T2(c7~). Note that in this case T~ is subtype of 
&:t_type,. The role of contextual individual is played by 
(:7,~, i.e., a constant which denotes lhe supremum of the 
denotation of TI. Furthermore, we add a closure condi- 
tion; 
(I\]tl) For attributions q'uant(:r,y) such that ~z : 
part_of(x, z) & T1 (x) & 7)(y) add to the box in which 
also :r lives the following condition: x = 5/)(z) .Further- 
more, we add a closure condition; 
(IV) Otherwise apply the standard mapping rules for 
SSNs (see the previous section). 
Consider, lbr instance, the (phual) SSN for lhe sen- 
tence 'At most two men walk' in Figure (2). 
:, • ) walk -.f 
: 133 
V 
• mar 
, 
at most 2 
Figure 2: Network for 'At most two men wall<' 
This SSN contains only one logical context which is 
inhabited by the objects of type man and walk. The 
object of type man is possibly plural: its quant at- 
tribute points to an object of type at.anost_2. The 
value of the other attribute, i.e., part_oJ; is not in- 
stantiated in this case. This is represented by means 
of the empty box. When we apply the rules for map- 
ping SSNs to DRSS, we obtain the following repre- 
sentation: 
(lO) 
at 
AT_MOST_2 c' ...... (X) 
man(X) 
walk(c) 
agent(e,X) 
z e I 
man(z) z c X 
agent(e',z) => 
walk(e') ~e.'~ 
The first four conditions correspond to the types of 
the nodes and the attributes of the SSN. They are 
followed by the closure condition. 
3 Editing Plurals 
In tiffs section, we describe how plural SSNs can be 
used for WYSIWYM editing (Power et al., 1998). 1~ 
WYSIWYM stallds for What Yott See \]s What Yott 
Meant. it is a technology for directly manipulat- 
ing knowledge representations using natural lan- 
guage feedback. WYSIWYM \]las been used in var- 
ious systems for (multilingual) document authoring 
and query formulation. The proposal which is pre- 
sented in tiffs paper has been inaplemented as part 
of the M ILF, query-answering system (e.g., Piwek ct 
al., 2000). 
The basic idea underlying WYSIWYM editing can 
be presented by means of a simple diagram. 
W~- " update ,~'(!llt't'{ll(! 
Feedback text with anchors 
\[ 
/ / 
.~ select, paste, VIeW cut, copy 
Figure 3: The editing cycle 
11SCC also: 
http://www.itri.l~righton.ac.uk/resea,'ch.htnfl:ff:WYSlWYM 
611 
Figure 3. represents the editing cycle. Given a 
Semantic Network (SN) in a knowledge base (KB), 
the system generates a description of the SN in the 
form of a 'feedback text' containing 'auchors' rep- 
reseuting places where the knowledge base can be 
extended. Each anchor is associated with pop-up 
menus, which present the possible editing opera- 
tions on the SN. On the basis of the operation that 
the user selects, the knowledge base is updated and 
a new feedback text is generated from the slew con- 
tents of the SN. 
• conjunction 
fitted_with •/~- -~ • conjunction 
carrierS°lid bulkl~ bilge'~l • pump • 
v 
1 3 
• purpose 
states 
equipment fire. fighting 
Figure 4: Network underlying (11) 
Let us slow go through an example of editing plurals 
as it is supported by our prototype system. Let us 
join in at a point where the network in figure 4 has 
been constructed. 12 This network is presented to the 
user by means of the following feedback text: 
(ll) A solid bulk carries" is fitted with three bilge 
pumps. Some equipment is used fox" firefight- 
ing. Some states. 
copy 
copy some 
cut 
Figure 5: Pop-up menu on 'three bilge pumps' 
The spans in bold face indicate where tile network is 
still incomplete. Other spans of text represent spe- 
cific objects in the network. For instance, the span 
'three bilge pumps' is associated with a plural ob- 
ject of the type 'bilge pump'. When the user clicks 
12In order to keep the example transparent, not all informa- 
tion in the network has been represented. Attribute names on 
the edges, attributes without a value which arc not expressed in 
the feedback text and the mapping fi'om objects to their logical 
contexts have been ommited. 
on this span, tile menu of Figure 5. pops up. Let 
us assume that the user selects 'copy'. In that case, 
the object which is associated with the span is saved 
in a buffer. Subsequently, the user can click on the 
span 'Some equipment'. This causes tile following 
menu to pop up: 
I insert new 
paste 
Now, file user can paste the object from tile buffer 
into tlle location in tile network which is associated 
with 'Some equipment'. This gives rise to the net- 
work in figure 6 and the following feedback text: 
(12) A solid bulk carrier is fitted with three bilge 
pumps. They ax'e used for firefighting. Some 
states, 
• conjunction 
fittedwith • • conjunction 
carrierS°lid bulk/~ bilge Zl • ~purpose pump • Z.. 
states 
fire_fighting 
v 
1 3 
Figure 6: Network underlying (12) 
Note that now tile first attributes of both 'fitted_with' 
aud 'purpose' point to the same object. In the feed- 
back text, this is expressed by using a pronoun for 
the second reference to the object. 
Van Deemter and Power (1998) originally defined 
the 'copy' operation for singular objects. When we 
move to plurals, alternatives to a simple copy op- 
eration become available. Here, we want to dis- 
cuss one of those operations, i.e., copying part of 
an object, instead of the entire object. Let us return 
to (l 1). Suppose that the user had chosen 'copy 
some' on the menu of Figure 5. The effect would 
have been that a new object would have been cre- 
ated in the buffer with its attribute 'part_of' pointing 
to the object conesponding to 'three bilge pumps' 
(its 'quant' attribute would still have to be filled 
ill). Pasting this object into tile location marked by 
'Some equipment' would have yielded the follow- 
ing result: 
612 
(13) A solid bulk carrier is fitted with three bilge 
pumps. Some number of them is used for fire- 
lighting. Some states. 
Note that the text contains an anchor for the yet to 
be specified value of the 'quant' attribute. Clicking 
on the anchor activates the following menu: 
Selection of 'one' yields the following text, which 
is generated from the network in Figure 7: 
(14) A solid bulk carrier is fitted with three bilge 
pumps. One of them is used for firefighting. 
Some states. 
* conjunction 
fitted with • / 
f 
solid bulk / 
carrier 
V 
1 
-k • conjunction 
,, bilge J-- 
bilge, pumpz~ °purp°se \[& 
pump~ Z. ~,~..o~ 4 I states 
• fire_fighting 
t 
0 
3 
Figure 7: Network underlying (14) 
4 Conclusions 
In this papel, we have described some editing oper- 
ations on object-oriented networks with plural ob- 
jects and provided a Deeise formal interpretation 
for these networks in terms of a version of Dis- 
course Representation Theory. The networks which 
we have used are an extension of commonly used 
oo networks for natural  generation. In 
particulm, our networks cover quantificational plu- 
ral noun phrases such 'most N', 'few N', etc. 
Acknowledgements The research reported in this 
paper was carried out as part of the EC Esprit funded 
CLIME proiect (EP 25.414). Thanks are due to Lynne 
Cahill, Roger Evans and Neil Tipper for stimulating co- 
of, eration within the CLIME temn at the University of 
Brighton. Furthermore, i would like to thank Alexander 
Boer, Kces vail Deemtcr, Rodger Kibble, Richard Power 
and two anonynlous COLING reviewers for commenting 
on earlier versions of tiffs paper. 

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