On Identifying Sets 
Matthew Stone 
Department of Computer Science & 
,Center for Cognitive Science 
RutgersUniversity 
110 Frelinghuysen Road, Piscataway NJ 08854-8019 
mdstone @cs. rutgers, edu 
Abstract 
A range of research has explored the problem of 
generating referring expressions that uniquely iden- 
tify a single entity from the shared context. But 
what about expressions that identify sets of enti- 
ties? In this paper, I adapt recent semantic re- 
search on plural descriptions--using covers to ab- 
stract collective and distributive readings and us- 
ing sets of assignments to represent dependencies 
among references--to describe a search problem for 
set-identifying expressions that largely mirrors the 
search problem for singular referring expressions. 
By structuring the search space only in terms of 
the words that can be added to the description, the 
proposal defuses potential combinatorial explosions 
that might otherwise arise with reference to sets. 
1 Introduction 
Natural  interaction lends itself to tasks like 
generalization, abstraction, comparison, and sum- 
marization which call for SETS of objects to be 
picked out using definite referring expressions. 
For example, consider the concrete function of 
describing the elements of a figure. In (1 b), we find 
a pair of elements from (1 a); imagine that some no- 
table relationship leads to an intention to identify 
and comment on just THOSE elements. 
• . . , . " 
I 
(1)a ........... 
• . . . 
The intersecting dotted segments. 
116 
As (lc) records, these are the intersecting dotted 
segments of (1 a), and can be designated as such. 
Or again, we find distinguished in (2b) five ele- 
ments of (2a), which might hold some independent 
interest. So we can and should identify these ele- 
ments, and (2c), the squares clustered at the lower 
left, will do the trick. 
(2) a 
b 
c 
\[\] \[\] 
0    
0 \[\] O0 
\[30 \[\] 00\[ 3 
\[\] \[\] \[\] \[\] 
\[\] \[\] 
\[\] 
\[\] \[\] 
The squares clustered at the lower left. 
Concrete problems like those of (1) and (2) cast 
into relief the potential difficulty of identifying sets. 
The world provides sets with embarrassing abun- 
dance, yet we are able to call attention to any of these 
sets at will, and to describe it at will both by prop- 
erties its members .have on their .own---,the mem- 
bers individually may be square or dotted--and by 
properties or relationships that the set enjoys as a 
collection--the set as a whole may be intersecting, 
or clustered in a certain region• 
Reference to sets is more general than picturesque 
examples like (1) and (2) might suggest• Linguistic 
research suggests that covert reference to SETS OF 
SALIENT ALTERNATIVES plays a pervasive and fun- 
damental role in mediating between the meanings of 
sentences and their interpretations in context. Con- 
sider (3), for'example. 
(3) a 
b 
C 
Only \[Mary\] F passed. 
Well, \[I\]F passed. 
Another student passed. 
According to (Rooth, 1992), the inte!rpretation of fo- 
cusingadverbs such as ~anly~relates,an.'instance ~-to 
a set of alternatives C: the adverb describes a prop- 
erty that makes ~ unique in C. Thus in (3a) Mary is 
unique among some set C of individuals in passing. 
Likewise, scalar implicature, as illustrated in (3b), 
depends on distinguishing one claim--my passing, 
say--as the strongest claim that can be supported 
among some salient set of claims C--students in the 
class passing, for (3b). Rooth formalizes the focus 
marking in these examples as contributing a presup- 
position that helps to identify these alternative sets. 
More generally, a range of lexical items, including 
the morpheme other from (3c), carry discourse pre- 
suppositions that relate their referent to salient al- 
ternatives from the context--like the students we 
accommodate in understanding (3c) (Bierner and 
Webber, 2000). 
Overtly, all the examples in (3) involve singu- 
lar noun phrases that specify isolated individuals. 
Nevertheless, representing and reasoning about ref- 
erence to sets is required for faithful account of how 
such sentences are interpreted, and thus how such 
sentences can achieve the communicative goals of a 
system for natural  generation (NLG). 
So how are expressions that refer to sets to be con- 
structed? In this paper, I will argue that identifying 
sets of individuals is not as forbidding as it may at 
first appear. The extensive literature in NLG on sin- 
gular references starting with (Dale and Haddock, 
1991) tells us what to do. We must use the INTER- 
PRETATION of provisional descriptions in context 
to assemble a combination of descriptive elements 
which identifies the intended target. Take (2), where 
we used the descriptive elements square, clustered 
somewhere, and at thel lower :left:. Tracking the in- 
cremental interpretation of these descriptors should 
lead to a sequence like that in (4). 
\[\] \[\] 
O0 
(4) a 
b The squares. 
\[\] \[\] 
\[\] \[\] 
\[\] \[\] \[\] \[\] 
117 
C 
d 
\[\] 
O0 
\[\] \[\] 
\[\] \[\] 
O 0 \[\] 
• \[\] • k • 
The squares clustered somewhere. 
\[\] \[\] 
\[\] 
e \[\] \[\] 
f The squares clustered at the lower left. 
This high-level story leaves us on familiar ground. 
The project of this paper is to realize this high- 
level story in formal terms. I begin in Section 2 by 
framing the problem of singular noun phrase gener- 
ation more precisely. The sequel extends this frame- 
work with a formal account of plural interpreta- 
tion and generation. Section 3 introduces the two 
independently-motivated observations from formal 
semantics which form the basis of this account. 
The ASSIGNMENT-SET semantics for reference 
to plurals provides a way to evoke and describe 
collections with variables that range only over 
individuals (van den Berg, 1993; van den Berg, 
1996). By using the assignment-set semantics, 
we can dispense with explicit collections in for- 
malizing an interpretation such as that schema- 
tized in (4c); we represent only the individuals 
involved. 
The COVER semantics for predications about 
pluralities provides a simple scheme of im- 
plicit quantification to abstract collective and 
distributive predication (Gill•n, 1987; Verkuyl 
and van der Does, 1991; Schwarzschild, 1994; 
Schwarzschild;. 1996). :T.he cover semantics of- 
fers an elegant, and convenient, definition of 
what it means for the set distinguished in (4c) to 
be characterized as the squares clustered some- 
where. 
Section 4 presents the computational model of plu- 
ral descriptions based on these principles. In keep- 
ing with (4), this model simply and naturally extends 
," the models used to generate-singular references. In 
particular, as (4) suggests, this model continues to 
structure the search space for generation in terms of 
the words that can be added to the description and to 
arrive at corresponding interpretations by constraint 
satisfaction over individuals. In so doing, the pro- 
posal defuses the potential combinatorial explosions 
that might otherwise arise with reference to sets. 
• ,. .: 2 Background 
At a high level, we can characterize generation pro- 
cedures like that of (Dale and Haddock, 1991 ) or its 
successors as manipulating linguistic data structures 
that link together FORMS, MEANINGS and INTER- 
PRETATIONS. (5) illustrates such a data structure, 
as it might be entertained in identifying a uniquely 
identifiable element of (2a). 
(5) a F: /the square in the upper left/ 
b M: {square(x), in(x, r), upper-left(r) } 
g3x 
......... r 
c I: 
(5a) proposes the form the square in the upper left-- 
a syntactic structure represented to some degree of 
abstraction. (5b) records the semantics for the de- 
scription as a set of constraints--each constraint is 
an atomic formula with free variables that speci- 
fies the requirement that some lexical meaning con- 
tributes to the description; the variables are place- 
holders for the discourse entities that the descrip- 
tion identifies. And (5c) anticipates how the hearer 
could process the description, by outlining the pos- 
sible candidate referents for it; in (5c) we find the 
element of the figure which x must represent, along 
with the corresponding (vaguely delimited) region r 
in the upper left where x is located. 
Any data structure linking form, meaning and 
interpretation combines two kinds of information. 
Form and meaning are related by_the generator's 
model of linguistic resources. In the concrete case, 
this model is a grammar; LEXICAL semantics de- 
termines the separate constraints that can go into a 
description and COMPOSITIONAL semantics deter- 
mines how these constraints can share variables and 
so describe common objects. Meaning and interpre- 
tation, meanwhile, are related by a model of the con- 
text in which the form is to be uttered. In interpreting 
referring expressions,we appeal "to a CONTEXT SET 
enumerating the salient individuals at some point in 
the discourse and a COMMON GROUND listing the 
118 
instances of constraints that can be presumed to be 
mutually known at that point. To determine the in- 
terpretation from the meaning, we must instantiate 
the free variables to individuals in the context set 
and match the instantiated constraints against the el- 
ements:~of the1 common.ground., In, practice, con- 
straint satisfaction heuristics (Mackworth, 1987) are 
required to accomplish the process of instantiation 
and matching with any hope of efficiency. 
With an understanding of what data structures 
such as (5) represent and how to carry out reasoning 
over them, solving descriptive problems becomes a 
matter of search. In practice, this search is typi- 
cally managed quite simply: for example, (Dale and 
Haddock, 1991) select transitions among states ac- 
cording to a greedy heuristic, while (Dale and Re- 
iter, 1995) select alternatives by exploring differ- 
ent kinds of constraints in a fixed order. In any 
case, the search starts with a structure defining an 
empty description, which means nothing and could 
refer to anything. Structures are then extended and 
considered in turn until the interpretation satisfies 
the system's goals (for example because it allows 
only a specified value, the intended referent, for a 
particular variable). The process of extension sim- 
ply consists of deriving a more elaborate form with 
a richer meaning using the generator's linguistic 
resources--it is useful to think of obtaining this by 
carrying out a step of derivation in a lexicalized 
grammar (Stone and Doran, 1997)--and then con- 
sulting the model of the context to obtain an updated 
interpretation. 
To extend these data structures to sets, we cannot 
introduce set variables and maintain the alternative 
candidate set values those variables might ultimately 
refer to--for one thing, there are just too many sets 
to represent an interpretation this way. 
3 An intuition and some semantics 
Here. is a suggestion: REINTERPRET_data.structures 
like (5) as compatible with descriptions of collec- 
tions as well as singletons. This should have some 
intuitive appeal. After all, we always thought that 
a form like (5a) abstracted out details of syntax and 
morphology; there's no difficulty then in seeing it 
as short for a family of singular and plural expres- 
sions like/the square(s) in the upper left~. Similarly, 
the interpretation is already defined in terms of a set 
ofinstances that satisfy the description; why not use 
this as THE set that the description refers to? 
The problem is the meaning. We have to allow 
for both DISTRIBUTIVE predicates, which character- team, so both the collective and distributive readings 
ize collections based on properties of the individuals are false. 
involved, and COLLECTIVE predicates, which de- We will follow Schwarzschild's proposal most 
scribe collections that jointly participate in some re- closely. Schwarzschild argues that we establish that 
lation. If we have collective predicates, how can we a linguistic predicate applies to a plural argument 
get away without explicit set variables which could., ~ by reeoyeringa.salient~cover of:~that:~gumentf~r0m ~,- 
take on any set as a possible value? the context. A cover here means a set of plurali- 
Van den Berg's treatmen t of dependent plurals in 
dynamic semantics provides the first half of the an- 
swer(van den Berg, 1993; van den Berg, 1996). Van 
den Berg's starting observation is that discourse can 
set up and maintain dependencies between the indi- 
viduals in one set and the individuals in another. 
(6) a Every man loves a woman. 
b They prove this by giving them flowers. 
In (6) for example, the first sentence introduces a set 
of men and a set of women, where each man in the 
one set is related to a woman in the other set (by 
love); the second sentence builds on that relation- 
ship, indicating another connection (of giving) be- 
tween each man and the corresponding woman. 
For van den Berg, data like (6) show that dis- 
courses describe sets of CASES generally. Each case 
involves a sequence of entities that stand in vari- 
ous relationships to one another, sometimes directly 
as individuals and sometimes indirectly through 
their membership in larger, related groups. Some 
sentences in discourse aggregate cases together, to 
express relationships that hold collectively among 
groups. Other sentences, like (6), zoom in on in- 
dividual cases, and describe distributive properties 
which hold of isolated individuals. In zooming in on 
cases, rather than individuals, these sentences main- 
tain and extend the dependencies and other relation- 
ships that define a case. 
The second half of the answer derives from 
the observation, made in (Gillon, 1987; Verkuyl 
and van der Does, 1991; Schwarzschild, 1994; 
Schwarzschild, 1996),,-that :the collective and dis- 
tributive readings of plurals represent only the ex- 
tremes in a larger space of readings. Take (7): 
(7) Rogers, Hammerstein and Hart wrote 
musicals. 
This sentence is true, but only in virtue of the joint 
action of Rogers and Hammerstein in writing some 
musicals and thezioint--ac~fion..of.Rogers and Hart in , . 
writing other musicals. As a matter of fact, the three 
never wrote a musical individually or as a single 
ties whose union or sum is the overall plural argu- 
ment. Given the cover, the overall plural predication 
holds just in case the basic property denoted by the 
predicate is true (collectively)of each of the sets (or 
CELLS) in the cover. For example, the sets consist- 
ing of Rogers and Hammerstein and of Rogers and 
Hart form the salient cover of Roger, Hammerstein 
and Hart in (7); the example is true because each of 
the cells in this cover directly enjoys the property of 
having written a musical. 
Schwarzschild's covering proposal and van den 
Berg's assignment-set proposal are perfectly com- 
patible. Following van den Berg, we interpret dis- 
courses in terms of sets of cases, where these cases 
spell out dependencies among related individuals. 
But now, following Schwarzschild, we zoom in on 
those cases flexibly, by covering them. Sometimes 
we consider all the cases together and describe rela- 
tionships among aggregated groups; sometimes we 
consider cases separately and describe individuals 
distributively; and sometimes, as in (7), we take an 
intermediate step and cluster the cases into some 
salient subgroups. 
Now let us return to (4c), repeated as (8a), and 
consider informally what this proposal amounts to: 
\[\] 
~\[\] 
\[\] \[\] 
\[\] \[\] 
(8) a \[\] \[\] \[\] \[\] 
b The squares clustered.somewhere. 
The assignment-set cover semantics fits the descrip- 
tion to the figure this way. As in (5c), the figure 
schematizes a set of cases; here each case involves 
two entities, a square and the location of the cluster 
to which the square belongs. The description applies 
because we can look at the individual cases to see 
that we have squares, and we can group the cases to- 
gether by. regioninto a cover so,that ineach cell the 
squares are indeed clustered at the location. 
At this point, some formalism is required to pro- 
119 
ceed with the development. We'll use assignment 
variables like g to range over cases; gx is the value 
of g for variable x.t Interpretations are defined in 
terms of sets of cases, naturally; we'll use F to range 
over a set of cases and write F(x) for {gx:g E F}. 
Most constraints will involve several variables; we 
can abstract this in terms of a sequence Of {,~iable~'x .... 
and the tuple of collections that those variables take 
on across a set of cases, F(x). (We can define this 
explicitly as F(x) = G where Gxi = F(xi).) 
Now, consider an atomic constraint F(x). In gen- 
eral, F(x) will have multiple known instances, and 
each instance will relate collections of individuals 
to one another. Thus the common ground will asso- 
ciate F(x) with a set of tuples of sets, which we write 
as ~F(x)~. An interpretation F will fit one of those 
instances directly iff F(x) E ~F(x)~. In this case we 
say F(x) DESCRmES F. 
For example, consider the constraint 
clustered(x,r). Let us say a set X is clustered 
around R if R is a singleton spatial location {r} and 
X is a group of sufficient cardinality and density 
located together at r. Then we might find three 
tuples of \[\[clustered(x, r)\]\] in the explicit depiction 
of (8). If we define Fl as in (9) then clustered(x,r) 
describes Fi. 
(9) { (x, r) : x a square in the lower left region r} 
Of course, we are principally interested in the 
ability to zoom in to particular cases, using covers. 
We represent a cover using a reflexive binary rela- 
tion that links each assignment to any assignment 
in its cell. Given such a relation C, the constraint 
@cP--read "covered by C, p"--says that p is true 
on each of the cells of the cover specified by C. We 
will only consider the case where p is an atomic con- 
straint F(x). 
(10) @cF(x) DESCRIBES F ifffor all g E F, 
Then with C defined as in (12), @cclustered(x, r) 
describes Fi U F2 U F3. 
(12) C=(F, xr,)u(r2xr2)u(r3xr3). 
Observe the close connection between this formal 
.jiidgment'and 'the-info,mat-disenssion' of"(8) pre- 
sented earlier. We have a set of cases involving a 
square and the location of the cluster to which the 
square belongs; we cover the cases together by re- 
gion and find that the resulting groups define a spec- 
ified cluster at a specified location. 
Schwarzschild's proposal is that the salient cover 
C is supplied from context. In the case of definite 
reference to tuples F, we can regard the tuples in any 
predicate as defining the appropriate salient cover 
for plural predication; any tuples that help to iden- 
tify F must be prominent parts of the shared context. 
Meanings of referring expressions should therefore 
appeal to a condition @ p which describes F iff there 
is a C for which @cP describes F. 
Clearly, if @p describes F and @p describes F' 
then @p describes FU I '~. This in turn entails that 
any condition ©p describes a maximal set of cases 
from the current context; the same goes for any con- 
junction of conditions of this form. We can treat 
this set of cases as the interpretation of a description. 
In particular, consider a description L that consists 
of a list of constraints @Li(x) formulated in terms 
of a tuple of variables x and atomic conditions on 
those variables Li(x). Assume a context set D defin- 
ing a domain of salient individuals, so that candidate 
cases to interpret L are given by I" := {a : ai E D}. 
The development thus far leads us to define the IN- 
TERPRETATION of I_,---l(L)--as: 
(13) I(L) := maxrcrVi: @Li(x) describes F 
Drawing on our running discussion, we 
can apply this definition to the description 
F(x) describes {h E F: C(g,h)}. 
Continuing from (9), define F2 and F3 in (1 ta) and 
(11 b) respectively. 
(ll)a {(x.r) : x a square in the center top r} 
b {(x,r) :x a square in the lower right r} 
I I adopt the notation throughout that v is a tuple and v i is 
component i of v, where components may be indexed equiva- 
................ L . ~_ . .{@square(x),@clustered(x,r) } and the 
context schematized by (2a). Of course, we find 
I(L) = Fi U F2 U F3. The fully distributive cover 
shows that the square condition is satisfied; the 
cover of (12) shows that the clustered condition 
is satisfied. Meanwhile, no further cases can be 
considered without adding either a circle or the 
unclustered square. 
The reader will already have recognized I(L) = lently by variables or numbers..Lower case Roman letters are 
for ordinary individuals and tuples thereof; upl~er case Ronl.an FI U:F2I'3 F 3 as~the set of cases:that-goes'with (8a). 
letters are for sets of individuals and tuples thereof; upper case Thus, we have reconciled the informal picture of (4) 
Greek letters are for sets of tuples, with the concrete data structures of form, meaning 
120 
and interpretation that NLG demands. For (8) we 
can now read (14). 
(14)a F: /the squares(s) clustered __ / 
b M: {(~) square(x), (~) cluster(x, r)} 
\] ..... 
c I: 
\[\] \[\] 
\[\] 
\[\] \[\] 
\[\] \[\]" 
\[\] 
\[\] \[\] 
4 Computing referring expressions 
At this point, we have an understanding of what 
kinds of representations we can use to describe the 
derivation of plural referring expressions. But we 
still must devise appropriate reasoning methods for 
these representations. The problem is the subject of 
this section. 
4.1 Collective Constraints 
The first step is to formulate a constraint-satisfaction 
heuristic that accounts for cover-constraints on col- 
lections. In general, constraint-satisfaction heuris- 
tics provide a technique for approximating the inter- 
pretation of a description. The key notion is that of a 
CONSTRAINT NETWORK for a description L, which 
determines a tuple C of CONSTRAINED VALUES. 
This tuple specifies a generous set of possible val- 
ues Ci for each variable xi in x; it is obtained by con- 
servatively eliminating values that are determined to 
be inconsistent with L according to heuristic tests. 
For example, the usual arc-consistency heuristic for 
a constraint over individuals K(x) is to eliminate a 
value v for variable z unless some g E ~K(x)~ has 
g: -- v and gk E Ck for all k. 
We will adapt this to the case of cover constraints 
with the following test of consistency. An individ- 
ual value v for a variable xi maintains its member- 
ship in C i in the.presence~oLa,collective,constraint 
©Lj(x) whenever v belongs to a SUBSET Gi of Ci 
which participates directly in the relation denoted by 
Lj(x) with sets of possible values for the other vari- 
ables. This criterion is spelled out formally in the 
definition in (15). 
(15) Value v for variable xi is 
COVER-CONSISTENT (C-CONSISTENT) 
with constraint (~)Lj(x)~under, constrained 
values C if there is an G E ~Lj(x)\]\] with 
v E Gi and Gt. C Ck for all k. 
121 
All values of xi 
with constraint QLj(x) may be deleted from Ci, 
as they will not satisfy the constraint. Doing so 
makes Ci ARC c-consistent with respect to (~)Lj(x), 
and provides the basic step in a network-based arc- 
.consistency .constraint-satisfier. As with. ordinary 
constraint satisfaction, we arrive at a final tuple of 
values for x by starting with an initial tuple Co of 
values---often an assignment Dx giving each vari- 
able D--and a queue of arcs linking each Ci with any 
QLj(x) that constrains it. Until the queue is empty, 
we select an arc and enforce the arc c-consistency 
by pruning Ci; if Ci changes we requeue all arcs 
that might no longer be arc c-consistent after the 
deletion. I will refer to the final tuple of values as 
P(L; Co), for the PLURAL constraint network on de- 
scription L and domains Co. 
The properties of this algorithm are in line with 
ordinary constraint satisfaction. The output will 
not provide all and only solutions to the constraints 
without further assumptions about the constraints. 
However, we can show, as usual, that the network 
converges on consistent values for variables in the 
ordinary linguistic case where the constraint graph 
is a tree--a semantic property, that there are no dis- 
joint sets of constraints that overlap on the same 
two variables, that follows under plausible assump- 
tions about the derivation of semantics from a tree- 
structured syntax. We can show further that these 
values, together with the tuples in \[\[Li(x)~ that cover 
these values, determine precisely the collection of 
assignments I(L). 
4.2 Search for Referring Expressions 
The second step is to formalize the task of construct- 
ing a description as a state-space search task. Sup- 
pressing details of form for exposition, each state is 
a tuple E as set out in (16). 
(16) E---- (L,r,R,x,P(L;R, Dx),P(L;Dr,Dx)) 
The state represents: 
(17)a a description L; 
b a tuple r of distinguished free variables in 
the description for which we must identify 
specific intended values; 
c a tuple R of sets describing the value Ri 
which we intend for the corresponding 
-variable ri; .... . • - 
d the remaining free variables of the 
description x; 
in Cithat are-not c-consistent_ 
e a constraint network P(L; R, Dx) describing 
the values for all the free variables in the 
description, on the assumption that the 
distinguished variables take on the values we 
intend; and 
f a constraint network p(L; Dr, Dx) 
describing the values for all the free 
variables in the description, on the 
assumption that the distinguished variables 
may, like other variables, take on any values 
from the context set. 
The distinction between the variables whose in- 
tended reference is fixed and those for which it is de- 
rived as a byproduct of the search process is due to 
Horacek (Horacek, 1995; Horacek, 1996); the dis- 
tinction derives increased importance when relating 
one collection to another as the choice of collections 
need not give rise to explicit branching in search. 
The initial state involves an empty description and 
so has the form given in (18). 
(18) Y~= (®,r,R, 0,P(Q;R),P(Q;Dr)) 
A state such as (16) represents a final state that suc- 
cessfully resolves the generation task when each 
variable x from r and x is associated with the same 
set Cx in both P(L;R,Dx) and P(L;Dr,Dx). This 
simply means that the hearer's interpretation of the 
referring expression matches the speaker's intended 
interpretation. 
At any state Z, the grammar defines a set of con- 
straints of the form (~) L(rx; y) that could potentially 
be added to the description to obtain L~--L is some 
domain relation, r and x name the old variables 
from L while y names fresh variables. Of course, 
we want to restrict our attention to constraints that 
are compatible with our intended interpretation. To 
achieve this restriction, we begin by computing the 
new constraint network C ~ = P(L~;R;Dxy). We 
check, whenever R assigns a value to x, that Rx C C' x. 
If this test admits .the_new constraint, the newstate 
obtained from state E is computed as in (19). 
(19) (L', r, R, xy, P(L'; R, Dxy), P(L'; Dr, Dxy)} 
4.3 An Example 
I return to (1) to provide an illustration of the final 
scheme; the goal is to identify the segments in (20f), 
R, from among those in (20a). I use figures and ref- 
erences to figures, .in .place-of=.eonstraint, networks; 
the description uses the variable r. The states pro- 
ceed, perhaps, thus: 
(20)a 
b 
C 
° • ° ° • • ° 
(o, r,g, l), (20a), (20f)) 
({ Qsegment(r) . ;r,R, 0, (20/1), (20ff)) 
" ° , . 
* ~%'. 
°•°• • 
• • ° • 
d 
e ({Qsegment(r), Qintersecting(r)}, 
r,R, (), (20d), (20f)) 
• , . . . 
• ° ° 
g ({Qsegment(r),Qintersecting(r), 
Qdotted(r) }, r, R, {), (20f), (20f) } 
5 Closing thoughts 
Descriptions of sets obviously have much in com- 
mon with expressions that describe a single entity 
from the shared context• In particular, adopting the 
standard view of NLG as goal-directed activity (Ap- 
pelt, 1985; Dale, 1992; Moore, 1994; Moore and 
Paris, 1993), singular and plural descriptions agree 
both in the kinds of intentions that they can achieve 
and the stages of generation at which they can be for- 
mulated. We cannot expect a single process to be re- 
sponsible for set descriptions across all intentions or 
stages of NLG. 
For example, as with a singular description, a 
description of a set may appeal to properties that 
play a role in the argument the speaker is trying to 
make, and may therefore address goals above and 
beyond simple identification of discourse entities. 
(Se e .(Donellan, ..! 966;: Kx~0nfeld, 1986) on the dis- .- 
tinction.) (Green et al., 1998a; Green et al., 1998b) 
show how such descriptions may be represented and 
formulated in NLG at a high-level process of con- 
tent or rhetorical planning. At the same time, plu- 
rals and singulars are alike in offering resources for 
reference--such as pronouns, one-anaphora or ag- 
gregated expressions--that bypass explicit descrip- 
tion altogether• The use of these resources may be 
.... ~quite-closety dependent onthe surface 'form being 
generated and so could reflect a relatively late deci- 
sion in the generation process (Dale and Haddock, 
122 
1991; Reiter, 1994; Dalianis, 1996). 
These complexities notwithstanding, we can ex- 
pect many descriptions of sets, like descriptions 
of individuals, to be formulated from scratch to 
achieve purely referential goals during the SEN- 
TENCE PLANNING. plaase: of .NLG, io:.he:tween ~gon=. 
tent planning and surface realization (Rainbow and 
Korelsky, 1992; Reiter, 1994). I have shown that 
using covers to abstract collective and distributive 
readings--and using sets of assignments to repre- 
sent plural references--yields a search space for 
this problem which largely mirrors that for singu- 
lars, and which avoids computation and search over 
sets of collections. Although sets proliferate explo- 
sively, it is no surprise that the search space for plu- 
rals set up by (19) is, like that for singulars, ulti- 
mately defined by the sequences of elements that 
make up descriptions. NLG involves search to use 
words effectively--choices of words should be the 
only decisions a referring expression generation sys- 
tem has to make. 
Acknowledgments 
This paper benefits from the comments of anonymous 
referees and from discussions with Kees van Deemter, 
Roger Schwarzschild, Bonnie Webber, the Edinburgh 
generation group, and the participants of the GNOME 
workshop where a preliminary version was presented; it 
was supported by a postdoctoral fellowship from RuCCS. 

References 
Douglas Appelt. 1985. Planning English Sentences. 
Cambridge University Press, Cambridge England. 
Gann Bierner and Bonnie Webber. 2000. Inference 
through alternative-set semantics. Journal of Lan- 
guage and Computation. 
Robert Dale and Nicholas Haddock. 1991. Content de- 
termination in the generation of referring expressions. 
Computational Intelligence, 7(4): 252-265. 
Robert Dale and Ehud Reiter. 1995. Computational in- 
terpretations of the Gricean maxims in the generation 
of referring expressions. Cognitive Science, 18:233- 
263. 
Robert Dale. 1992. Generating Referring E~pressions: 
Constructing Descriptions in a Domain of Objects and 
Processes. MIT Press, Cambridge MA. 
Hercules Dalianis. 1996. Concise Natural Language 
Generation .from Formal Specifications. Ph.D. thesis, 
Royal Institute of Technology. Stockholm. 
K. Donellan. 1966. Reference and definite description. 
Philosophical Review, 75:281-304. 
Brendan-Gillon. 1987. The readings of plural noun 
phrases in english. Linguistics and Philosophy, 
10(2): 199-299. 
Nancy Green, Giuseppe Carenini, Stephan Kerpedjiev, 
Steven Roth, and Johanna Moore. 1998a. A media- 
independent content  for integrated text and 
graphics generation. In CVIR '98- Workshop on Con- 
tent Visualization and lntermedia Representations. 
Nancy Green, Giuseppe Carenini, and Johanna Moore. 
...... ~ v~ t998b~ k,p~ineipted~representafiort . of attributive de- 
scriptions for generating integrated text and informa- 
tion graphics presentations. In Proceedings of In- 
ternational Natural Language Generation Workshop, 
pages 18-27. 
Helmut Horacek. 1995. More on generating referring ex- 
pressions. In Proceedings of the Fifth European Work- 
shop on Natural Language Generation, pages 43-58, 
Leiden. 
Helmut Horacek. 1996. A new algorithm for generating 
referring expressions. In ECAI 8, pages 577-581. 
Amichai Kronfeld. 1986. Donellan's distinction and a 
computational model of reference. In Proceedings of 
ACL, pages 186.--191. 
Alan Mackworth. 1987. Constraint Satisfaction. In 
S.C. Shapiro, editor, Encyclopedia of Artificial Intel- 
ligence, pages 205-211. John Wiley and Sons. 
Johanna D. Moore and C6cile L. Paris. 1993. Plan- 
ning text for advisory dialogues: capturing intentional 
and rhetorical information. Computational Linguis- 
tics, 19(4):651-695. 
Johanna Moore. 1994. Participating in Explanatory Di- 
alogues. MIT Press, Cambridge MA. 
Owen Rainbow and Tanya Korelsky. 1992. Applied text 
generation. In ANLP, pages 40-47. 
Ehud Reiter. 1994. Has a consensus NL generation 
architecture appeared, and is it psycholinguistically 
plausible? In Seventh International Workshop on Nat- 
ural Language Generation, pages 163-170, June. 
Mats Rooth. 1992. A theory of focus interpretation. Nat- 
ural Language Semantics, 1 ( 1 ): 75- I 16. 
Roger Schwarzschild. 1994. Plurals, presuppositions, 
and the sources of distributivity. Natural Language 
Semantics. 2:201-248. 
Roger Schwarzschild. 1996. Pluralities. Kluwer, Dor- 
drecht. 
Matthew Stone and Christine Doran. 1997. Sentence 
planning as description using tree-adjoining grammar. 
:: In ProceedingsofACL, pages_198-205. 
M. H. van den Berg. 1993. Full dynamic plural logic. 
In K. Bimb6 and A. Mfit6, editors, Proceedings of the 
Fourth Symposium on Logic and Language, Budapest. 
M. H. van den Berg. 1996. Generalized dynamic quanti- 
tiers. In J. van der Does and J. van Eijk, editors, Quan- 
tifiers, Logic and Language. CSLI. 
Henk Verkuyl and Jaap van der Does. 1991. The seman- 
tics of plural noun phrases. Preprint, ITLI. Amster- 
dam. 
