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<Paper uid="W00-1416">
  <Title>On Identifying Sets</Title>
  <Section position="2" start_page="0" end_page="121" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> Natural language interaction lends itself to tasks like generalization, abstraction, comparison, and summarization which call for SETS of objects to be picked out using definite referring expressions.</Paragraph>
    <Paragraph position="1"> 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 notable relationship leads to an intention to identify and comment on just THOSE elements.</Paragraph>
    <Paragraph position="3"> The intersecting dotted segments.</Paragraph>
    <Paragraph position="4">  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 elements of (2a), which might hold some independent interest. So we can and should identify these elements, and (2c), the squares clustered at the lower left, will do the trick.</Paragraph>
    <Paragraph position="5">  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 abundance, yet we are able to call attention to any of these sets at will, and to describe it at will both by properties its members .have on their .own---,the members 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 fundamental role in mediating between the meanings of sentences and their interpretations in context. Consider (3), for'example.</Paragraph>
    <Paragraph position="6">  Well, \[I\]F passed.</Paragraph>
    <Paragraph position="7"> Another student passed.</Paragraph>
    <Paragraph position="8"> According to (Rooth, 1992), the inte!rpretation of focusingadverbs 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 presupposition that helps to identify these alternative sets. More generally, a range of lexical items, including the morpheme other from (3c), carry discourse presuppositions that relate their referent to salient alternatives from the context--like the students we accommodate in understanding (3c) (Bierner and Webber, 2000).</Paragraph>
    <Paragraph position="9"> Overtly, all the examples in (3) involve singular noun phrases that specify isolated individuals. Nevertheless, representing and reasoning about reference 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 language generation (NLG).</Paragraph>
    <Paragraph position="10"> So how are expressions that refer to sets to be constructed? 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 singular 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 incremental interpretation of these descriptors should lead to a sequence like that in (4).</Paragraph>
    <Paragraph position="11">  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 generation more precisely. The sequel extends this framework with a formal account of plural interpretation and generation. Section 3 introduces the two independently-motivated observations from formal semantics which form the basis of this account.</Paragraph>
    <Paragraph position="12"> 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 formalizing an interpretation such as that schematized in (4c); we represent only the individuals involved.</Paragraph>
    <Paragraph position="13"> The COVER semantics for predications about pluralities provides a simple scheme of implicit quantification to abstract collective and distributive predication (Gill*n, 1987; Verkuyl and van der Does, 1991; Schwarzschild, 1994; Schwarzschild;. 1996). :T.he cover semantics offers an elegant, and convenient, definition of what it means for the set distinguished in (4c) to be characterized as the squares clustered somewhere. null Section 4 presents the computational model of plural descriptions based on these principles. In keeping with (4), this model simply and naturally extends ,&amp;quot; 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 proposal defuses the potential combinatorial explosions that might otherwise arise with reference to sets.</Paragraph>
    <Paragraph position="14"> * ,. .: 2 Background At a high level, we can characterize generation procedures 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).</Paragraph>
    <Paragraph position="15"> (5) a F: /the square in the upper left/</Paragraph>
    <Paragraph position="17"> (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 description as a set of constraints--each constraint is an atomic formula with free variables that specifies the requirement that some lexical meaning contributes to the description; the variables are placeholders for the discourse entities that the description identifies. And (5c) anticipates how the hearer could process the description, by outlining the possible 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.</Paragraph>
    <Paragraph position="18"> Any data structure linking form, meaning and interpretation combines two kinds of information.</Paragraph>
    <Paragraph position="19"> Form and meaning are related by_the generator's model of linguistic resources. In the concrete case, this model is a grammar; LEXICAL semantics determines the separate constraints that can go into a description and COMPOSITIONAL semantics determines how these constraints can share variables and so describe common objects. Meaning and interpretation, meanwhile, are related by a model of the context in which the form is to be uttered. In interpreting referring expressions,we appeal &amp;quot;to a CONTEXT SET enumerating the salient individuals at some point in the discourse and a COMMON GROUND listing the  instances of constraints that can be presumed to be mutually known at that point. To determine the interpretation from the meaning, we must instantiate the free variables to individuals in the context set and match the instantiated constraints against the elements:~of the1 common.ground., In, practice, constraint satisfaction heuristics (Mackworth, 1987) are required to accomplish the process of instantiation and matching with any hope of efficiency.</Paragraph>
    <Paragraph position="20"> 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 typically managed quite simply: for example, (Dale and Haddock, 1991) select transitions among states according to a greedy heuristic, while (Dale and Reiter, 1995) select alternatives by exploring different 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 simply 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 consulting the model of the context to obtain an updated interpretation.</Paragraph>
    <Paragraph position="21"> 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.</Paragraph>
    <Paragraph position="22"> 3 An intuition and some semantics Here. is a suggestion: REINTERPRET_data.structures like (5) as compatible with descriptions of collections 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 expressions 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 answer(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 individuals in one set and the individuals in another.</Paragraph>
    <Paragraph position="23"> (6) a Every man loves a woman.</Paragraph>
    <Paragraph position="24"> b They prove this by giving them flowers.</Paragraph>
    <Paragraph position="25"> 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 relationship, indicating another connection (of giving) between each man and the corresponding woman.</Paragraph>
    <Paragraph position="26"> For van den Berg, data like (6) show that discourses describe sets of CASES generally. Each case involves a sequence of entities that stand in various 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 individual cases, and describe distributive properties which hold of isolated individuals. In zooming in on cases, rather than individuals, these sentences maintain and extend the dependencies and other relationships that define a case.</Paragraph>
    <Paragraph position="27"> 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 distributive readings of plurals represent only the extremes in a larger space of readings. Take (7): (7) Rogers, Hammerstein and Hart wrote musicals.</Paragraph>
    <Paragraph position="28"> 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 argument. 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 consisting 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.</Paragraph>
    <Paragraph position="29"> Schwarzschild's covering proposal and van den Berg's assignment-set proposal are perfectly compatible. Following van den Berg, we interpret discourses in terms of sets of cases, where these cases spell out dependencies among related individuals.</Paragraph>
    <Paragraph position="30"> But now, following Schwarzschild, we zoom in on those cases flexibly, by covering them. Sometimes we consider all the cases together and describe relationships 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.</Paragraph>
    <Paragraph position="31"> Now let us return to (4c), repeated as (8a), and consider informally what this proposal amounts to:  The assignment-set cover semantics fits the description 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 together by. regioninto a cover so,that ineach cell the squares are indeed clustered at the location.</Paragraph>
    <Paragraph position="32"> At this point, some formalism is required to pro- null 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}.</Paragraph>
    <Paragraph position="33"> 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 general, F(x) will have multiple known instances, and each instance will relate collections of individuals to one another. Thus the common ground will associate 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.</Paragraph>
    <Paragraph position="34"> 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.</Paragraph>
    <Paragraph position="35"> (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.</Paragraph>
    <Paragraph position="36"> We represent a cover using a reflexive binary relation that links each assignment to any assignment in its cell. Given such a relation C, the constraint @cP--read &amp;quot;covered by C, p&amp;quot;--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 constraint F(x).</Paragraph>
    <Paragraph position="37">  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 region and find that the resulting groups define a specified cluster at a specified location.</Paragraph>
    <Paragraph position="38"> 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 identify 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.</Paragraph>
    <Paragraph position="39"> Clearly, if @p describes F and @p describes F' then @p describes FU I '~. This in turn entails that any condition (c)p describes a maximal set of cases from the current context; the same goes for any conjunction 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 defining a domain of salient individuals, so that candidate cases to interpret L are given by I&amp;quot; := {a : ai E D}. The development thus far leads us to define the IN-</Paragraph>
    <Paragraph position="41"> Drawing on our running discussion, we can apply this definition to the description F(x) describes {h E F: C(g,h)}.</Paragraph>
    <Paragraph position="42"> Continuing from (9), define F2 and F3 in (1 ta) and</Paragraph>
    <Paragraph position="44"> 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</Paragraph>
    <Paragraph position="46"> 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.</Paragraph>
    <Paragraph position="47"> 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  and interpretation that NLG demands. For (8) we can now read (14).</Paragraph>
    <Paragraph position="48">  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.</Paragraph>
    <Section position="1" start_page="120" end_page="121" type="sub_section">
      <SectionTitle>
4.1 Collective Constraints
</SectionTitle>
      <Paragraph position="0"> The first step is to formulate a constraint-satisfaction heuristic that accounts for cover-constraints on collections. In general, constraint-satisfaction heuristics provide a technique for approximating the interpretation of a description. The key notion is that of a CONSTRAINT NETWORK for a description L, which determines a tuple C of CONSTRAINED VALUES.</Paragraph>
      <Paragraph position="1"> This tuple specifies a generous set of possible values Ci for each variable xi in x; it is obtained by conservatively eliminating values that are determined to be inconsistent with L according to heuristic tests.</Paragraph>
      <Paragraph position="2"> 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.</Paragraph>
      <Paragraph position="3"> We will adapt this to the case of cover constraints with the following test of consistency. An individual value v for a variable xi maintains its membership in C i in the.presence~oLa,collective,constraint (c)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 variables. This criterion is spelled out formally in the definition in (15).</Paragraph>
      <Paragraph position="4"> (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.</Paragraph>
      <Paragraph position="5">  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 variable 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 description L and domains Co.</Paragraph>
      <Paragraph position="6"> 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.</Paragraph>
      <Paragraph position="7"> 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 disjoint sets of constraints that overlap on the same two variables, that follows under plausible assumptions 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).</Paragraph>
    </Section>
    <Section position="2" start_page="121" end_page="121" type="sub_section">
      <SectionTitle>
4.2 Search for Referring Expressions
</SectionTitle>
      <Paragraph position="0"> The second step is to formalize the task of constructing a description as a state-space search task. Suppressing details of form for exposition, each state is a tuple E as set out in (16).</Paragraph>
    </Section>
  </Section>
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