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Piwek, Paul (2000). A Formal Semantics for Editing and Generating Plurals. In: Procs of COLING 2000 pp.607–613.

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A Formal Semantics for Generating and Editing Plurals

Paul Piwek

ITRI – University of BrightonWatts Building, Moulsecoomb,

Brighton BN2 4GJUK

Paul.Piwek@itri.brighton.ac.uk

Abstract

We present a formal semantics for an object-oriented formalism which allows for the represen-tation of plural objects (such as ‘Three N’, ‘Mostof the N’, ‘Some N’,...). The semantics is given interms of a mapping to a variant of Discourse Rep-resentation Theory. It is motivated by its suitabilityfor natural language generation and interactive edit-ing of the representations.

1 Introduction

A natural language generator typically generates anoun phrase from a representation consisting of anobject with one or more attributes (cf. Reiter &Dale, 2000). Usually this representation is sup-plemented with information concerning the contextin which the noun phrase has to be realized (e.g.,the set of distractors, whether the object is in fo-cus, etc.). For instance, the EPICURE system (Dale,1992) deals with reference to plural objects by hav-ing the following three attributes on physical ob-jects: structure, whose value can be either a set orindividual, cardinality which in case of a set recordsthe numbers of elements which the set has, and con-stituents which in case of a set contains the elementsof the set.

Our proposal is intended to extend the representa-tions proposed in (Dale, 1992).� Most importantly,we replace the attribute cardinality with the moregeneral attribute quant (for quantifier) whose valueis a quantifier such as ‘most’, ‘few’, ‘5’, ‘� �’ (atmost 6), etc. Furthermore, we introduce the new at-tribute part of which takes as a value an object ofwhich the object in question is a part.�

�Note that we are dealing with the generation of pluralsfrom (logically) structured data as opposed to raw data as in,e.g., Stone (1999).

�We use the mereological ‘part of’ relation as an alternativeto ‘subset’ For details, see the next section.

The object-oriented (OO) formalism in which weimplement the aforementioned attributes is an ex-tension of standard OO formalisms. It is knownas Scoped Semantic Networks (SSN; Kibble et al.,1999; Power, 1999).� An SSN consists of a net-work of objects together with a mapping of theseobjects to a set of logical contexts. This makes itpossible to represent universal quantification, im-plication, negation and other logical operators. Inparticular, Power (1999) shows how an SSN can bemapped into a Discourse Representation Structure(DRS; Kamp & Reyle, 1993), thus providing a for-mal semantic interpretation of SSNs.

In this paper, we provide a mapping of SSNs withplural objects to an adapted version of DiscourseRepresentation Theory (DRT). The mapping is pro-vided to obtain formal truth conditions for the SSNs.Such a mapping provides us with a mathematicallyprecise characterization of the information which isrepresented by a SSN in terms of its truth-conditions.This is useful if we want to automatically manipu-late the information which is represented by meansof an SSN. For example, we can formally definewhether some piece of information is already im-plicit in some other piece of information; in otherwords, we can define a notion of logical conse-quence. Related to this is the possibility to use thesemantics in order to test the consistency of the in-formation conveyed by an SSN. For that purpose, wecan do so-called model checking: an SSN is consis-tent if we can construct a model –that is, a logicallypossible state of the world– in which the SSN is trueaccording to our truth-conditional semantics.

We do not provide a direct formal semantics forSSN, but rather map it to a more convenient log-ical formalism, i.e., DRT. The main reason forthis approach is that phenomena which we will bemodelling in this paper, i.e. (plural) reference and

�See also, e.g., Sowa (1984).

In: Proceedings of COLING 2000, Saarbruecken, Germany.

anaphora, have been studied extensively within DRT(see, e.g., Kamp & Reyle, 1993; Krahmer & VanDeemter, 1998; Piwek, 1997). Furthermore, we be-lieve that the adaptation of DRT that we propose isof interest in its own right.

The mapping which we provide from SSNs withplural objects to DRSs requires some modificationsto standard DRT with plurals (Kamp & Reyle, 1993:Chapter 4). For networks with only singular objects,there is a straightforward mapping of the objects ina network to the discourse referents which populatea DRS. Things are different for networks with pluralobjects. Consider:

(1) Susan has found most books which Bill needs.

The DRS for this sentence is:

(2)y

book(y)need(bill,y)

��

������

ymost found(susan,y)

Intuitively, the meaning of this condition is that: formost y which satisfy the conditions to the left of thediamond, it holds that they also satisfy the conditionon the right. Note, that the representation containsno plural discourse referent corresponding to the NP‘most books which Bill needs’. The ‘y’ in this repre-sentation is a referent for singular individuals. Thismight 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 wecan obtain a discourse referent for the set of bookswhich Bill needs and Susan found. In more tech-nical terms, the set is obtained by the summationof the values which ‘y’ can take. Thus there is nodirect way of mapping a plural object in a seman-tic network (which represent the interpretation of anNP) to a plural discourse referent in the correspond-ing DRS. For this reason we have chosen to adaptthe DRT formalism, so that plural noun phrases dodirectly correlate with plural discourse referents.

We now proceed as follows. In Section 2, wespecify the mapping from SSNs to our version ofDRT. In the next section (Section 3), we describe an

application which uses the SSNs with plurals. Wefinish this paper with a conclusions section (Section4).

2 From SSNs to DRSsIn this section, we provide a mapping from SSNsinto discourse representation structures (DRSs) withplurals. We start out by specifying the target of themapping, i.e., plural DRT.

DRSs with Plurals Following Kamp & Reyle(1993), we treat singular objects and sets of objectsas entities of the same kind. Both are consideredto be individuals: atomic and non-atomic individ-uals, respectively. Thus, the model theory followsthe models which Link (1983) provides for countnouns.� The idea is that the denotation of an NPwhich contains a count noun can be uniquely subdi-vided into atomic parts (as opposed to the denotataof mass nouns). The domain for NPs is structured bya part-whole relation which satisfies the axioms ofupper semilattices (for background information onthese lattices see Kamp & Reyle, 1993:398–406).In formal terms, a model is defined as follows:

A model M is a quintuple hU � E � P red�Quant�Nameiwhich consist of:

(I) A domain of individuals with the structure of a com-plete, free, atomic upper semilattice U � hU��i withzero;(II) A domain of eventualities with the structure of acomplete, free, atomic upper semilattice E � hE��i;(III) A function Pred mapping predicates P to their ex-tensions in M , such that(III.1) for the relations representing thematic roles, suchas agent and patient, Pred assigns a set of tuples he� ai,where e � E and a � U .(III.2) for eventuality predicates, Pred�P � � E.(III.3) For object type predicates, Pred�P � � U .(IV) A function Quant mapping determiners DET totheir corresponding interpretations, i.e., a set consistingof tuples ha� bi (where a� b � U ).(V) A functionName mapping constants to members ofU . In particular, the constants cP , where P is a predi-cate are mapped to �Pred�P �, i.e., the supremum, alsoknown as the sum, of the interpretation of P .

Notice that in our models there are separate domainsfor objects and eventualities (i.e., states and events).

�For a critical discussion and alternative to Link (1983), seefor instance Landman (1989).

In: Proceedings of COLING 2000, Saarbruecken, Germany.

The relations agent and patient have an eventual-ity as their first argument and an object as secondargument (cf. Parsons, 1990). agent(e,o) is to beinterpreted as: object o is the agent of eventuality e.Furthermore, there are predicates applying to even-tualities and others applying to objects.

For our purposes, the most interesting part of thedefinition is the function Quant which maps deter-miners to their respective interpretations. We takethe interpretation of a determiner to be a set of tu-ples, where each tuple consist of a pair of (plural) in-dividuals. For instance, take the determiner ‘most’.Quant maps it to the following interpretation:�

(4) Quant�Most� � fhr� ci � r � c & r is a non-atomic entity of M & jrj � jcj

�g

Thus ‘most’ corresponds to the set of all tuples ofindividuals, such that the first individual is a non-atomic part of the second one and the cardinalityof the first is greater than or equal to the cardinal-ity of the second divided by two. Henceforth, wewill call the second individual the context individual(cf. Westerståhl, 1985). Given a noun phrase, suchas ‘most birds’, the first individual is intended asthe interpretation of the entire noun phrase whereasthe 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 situation in whicha noun phrase is produced) and by linguistic means(as in ‘most of the birds on the beach’, where ‘thebirds 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 Quant. This con-dition functions as a substitute for the duplex condi-tions of standard DRT� The condition in question is:

�Here we follow the ‘more than half’ interpretation of‘most’ common from the literature on Generalized Quantifiers(see, e.g, Barwise & Cooper, 1981; Keenan & Westerståhl,1997). This interpretation is not entirely unproblematic; see,for instance, (Kamp & Reyle, 1993). Our use of the interpre-tation is, however, solely for illustrative purposes. We can alsoaccommodate for alternative mappings for Quant�Most�.Similarly we cannot go into detailed discussions of other quan-tifiers such as, for instance, ‘many’ (cf. Lappin, 1988).

�Within the confines of this paper it is impossible to give afull formal definition of our version of plural DRT, therefore wefocus on the aforementioned condition. The other definitionsclosely follow those in Kamp & Reyle, 1993: 425–427, 677–679).

If x is a discourse referent and t is a discourse refer-ent or constant, then Dett�x� is a condition.

The verification condition for this condition is:

(5) M j�f Dett�x� iffhk x kM�f � k t kM�f i � Quant�DET�.

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)

X

at most �cman(X)z

man(z)walk(z)

� z � X

walk*(X)

The first condition says that X consists of a subset ofthe set of all men (cman, alternatively, we could usea set of contextually given men) and that X shouldconsist of at most 2 individuals belonging to thatset. The implicative condition is there to make surethere is no other set apart from X with (other) menwho are also walking. Such a closure condition isparticularly useful for the direct representation ofmonotonically decreasing quantifiers.� A quantorQ is monotonically decreasing if and only if for all

�For expository reasons, we have left out explicit represen-tations of events in this example. But, see the next section for aDRS with plurals and events.

�Note that when a predicate in a condition is marked witha ‘*’, this means that the predicate is interpreted distributivelyover the atomic parts of the objects in its denotation.

We assume that: Quant�at most �� � fhr� ci � r � c& jrj � �g

�In Van Eijck (1983), an alternative approach is proposedwithin a framework which also allows for the direct representa-tion of plural referents in DRT. He proposes to reanalyse mono-tonically decreasing quantifiers in terms of negation and mono-tonically increasing ones. This, however, means that we nolonger 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’.

In: Proceedings of COLING 2000, Saarbruecken, Germany.

X�Y�Z it holds that: if QXY and Z � Y , thenQXZ . Thus, for instance, (a) ‘At most two menwalk and talk’ does not imply that (b) ‘At most twomen walk’. If we would represent (a) without theclosure condition (i.e., there is a set of at most twomen and each of them walks and talks), then (b) (i.e.,there is a set of at most two men and each of themwalks) would follow from (a). However, if we addto the representation of (a) that there are no othersets of men who walk and talk and to the represen-tation of (b) that that there are no other sets of menwho walk, then (a) no longer follows from (b); theadditional information in (a) that there are no othersets of men who both walk and talk, does not entailthat there are no other sets of men who walk.

Scoped Semantic Networks A scoped semanticnetwork (SSN) is a triple hD�L� fi, consisting of atyped 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 objectsseparate from those in the DAG) to each of the ob-jects in the DAG. In the DAG, there are objects whichcorrespond with logical operators, such as implica-tion and negation, and non-logical objects, such asphysical objects and events. The function f , whichassigns logical contexts to objects in a typed DAGD, satisfies the following constraints:

(I) The root object and all the objects which are directdescendants of a logical operator are assigned a uniquelogical context. These contexts 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 acontext by clause 1. are mapped to the logical context oftheir nearest ancestor to which clause 1. applies.(III) Objects which are not assigned to a logical contextby the clauses 1. and 2. are assigned to a logical contextin 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 SSNin 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 from which the arrow originates belongs to thecontext to which the arrow points.

There is a straightforward procedure for mappinga SSN into a DRS:

(I) Logical contexts are mapped into boxes, where thenesting of the boxes is isomorphic to the partial orderingof the corresponding logical contexts.(II) Objects are inserted into the box which correspondswith their logical context, except for logical operators.The latter are mapped onto the appropriate operators onthe boxes of their directly subordinate objects.(III) Typing statements T �x� of a non-logical object areadded to the same box as the object x itself.(IV) Attributions R�x� y�, where x and y are non-logicalobjects, are added to the same box as x.

Figure 1: Network for (8)

By applying these rules, we obtain the followingDRS for the SSN in Figure 1:

(9)

x e

happy(e)man(x)

agent(e,x)

�

e’

whistle(e’)agent(e’,x)

Note how the three circles in the SSN correspondwith the three boxes of the DRS. Furthermore, thediscourse referent x corresponds to the object in theSSN of the type man and inhabits the same box asthe conditions which correspond to the object oftype happy and the attribute agent.

SSNs with Plurals In this section, we describe anextension of SSNs for countable plural objects. Thisextension requires no changes to the format of SSNs.Rather, we introduce a number of special-purpose

In: Proceedings of COLING 2000, Saarbruecken, Germany.

attributions and types. Subsequently, we specifytheir mapping to appropriate terms in a DRS.

We introduce two attributes on countable objects:

(I) quant. The value of this feature is restricted to anobject of the type det type. Examples of the subtypes ofdet type are �, � �, � �, all, few, etc.(II) part of . The value of this feature is restricted tocountable objects.

The mapping of SSNs which include these special-purpose attributions and types to a DRS is defined asfollows:

(I) For typing statements T �x�, where T is a subtype ofdet type: ignore the statement T �x� and the object x;(II) For attributions quant�x� y� such that �z �part of�x� z� & z is an anchor& T��x� & T��y�, addto the box in which also x lives the following condition:x � T��cT��. Note that in this case T� is subtype ofdet type. The role of contextual individual is played bycT� , i.e., a constant which denotes the supremum of thedenotation of T�. Furthermore, we add a closure condi-tion;(III) For attributions quant�x� y� such that �z �part of�x� z� & T��x� & T��y� add to the box in whichalso x lives the following condition: x � T��z� .Further-more, we add a closure condition;(IV) Otherwise apply the standard mapping rules forSSNs (see the previous section).

Consider, for instance, the (plural) SSN for the sen-tence ‘At most two men walk’ in Figure (2).

Figure 2: Network for ‘At most two men walk’

This SSN contains only one logical context which isinhabited by the objects of type man and walk. Theobject of type man is possibly plural: its quant at-tribute points to an object of type at most 2. Thevalue of the other attribute, i.e., part of, is not in-

stantiated in this case. This is represented by meansof the empty box. When we apply the rules for map-ping SSNs to DRSs, we obtain the following repre-sentation:

(10)

X e

at most �cman(X)man(X)walk(e)

agent(e,X)z e�

man(z)agent(e�,z)walk(e�)

� z � Xe� � e

The first four conditions correspond to the types ofthe nodes and the attributes of the SSN. They arefollowed by the closure condition.

3 Editing Plurals

In this section, we describe how plural SSNs can beused for WYSIWYM editing (Power et al., 1998).��

WYSIWYM stands for What You See Is What YouMeant. It is a technology for directly manipulat-ing knowledge representations using natural lan-guage feedback. WYSIWYM has been used in var-ious systems for (multilingual) document authoringand query formulation. The proposal which is pre-sented in this paper has been implemented as partof the MILE query-answering system (e.g., Piwek etal., 2000).

The basic idea underlying WYSIWYM editing canbe presented by means of a simple diagram.

update

select, paste,cut, copy

generate

view

Feedback text with anchors

Figure 3: The editing cycle

��See also:http://www.itri.brighton.ac.uk/research.html�WYSIWYM

In: Proceedings of COLING 2000, Saarbruecken, Germany.

Figure 3. represents the editing cycle. Given aSemantic Network (SN) in a knowledge base (KB),the system generates a description of the SN in theform of a ‘feedback text’ containing ‘anchors’ rep-resenting places where the knowledge base can beextended. Each anchor is associated with pop-upmenus, which present the possible editing opera-tions on the SN. On the basis of the operation thatthe user selects, the knowledge base is updated anda new feedback text is generated from the new con-tents of the SN.

Figure 4: Network underlying (11)

Let us now go through an example of editing pluralsas it is supported by our prototype system. Let usjoin in at a point where the network in figure 4 hasbeen constructed.�� This network is presented to theuser by means of the following feedback text:

(11) A solid bulk carrier is fitted with three bilgepumps. Some equipment is used for firefight-ing. Some states.

copycopy somecut

Figure 5: Pop-up menu on ‘three bilge pumps’

The spans in bold face indicate where the network isstill 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

��In order to keep the example transparent, not all informa-tion in the network has been represented. Attribute names onthe edges, attributes without a value which are not expressed inthe feedback text and the mapping from objects to their logicalcontexts have been ommited.

on this span, the menu of Figure 5. pops up. Letus assume that the user selects ‘copy’. In that case,the object which is associated with the span is savedin a buffer. Subsequently, the user can click on thespan ‘Some equipment’. This causes the followingmenu to pop up:

insert newpaste

Now, the user can paste the object from the bufferinto the location in the network which is associatedwith ‘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 bilgepumps. They are used for firefighting. Somestates.

Figure 6: Network underlying (12)

Note that now the first attributes of both ‘fitted with’and ‘purpose’ point to the same object. In the feed-back text, this is expressed by using a pronoun forthe second reference to the object.

Van Deemter and Power (1998) originally definedthe ‘copy’ operation for singular objects. When wemove 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 ofan object, instead of the entire object. Let us returnto (11). Suppose that the user had chosen ‘copysome’ on the menu of Figure 5. The effect wouldhave been that a new object would have been cre-ated in the buffer with its attribute ‘part of’ pointingto the object corresponding to ‘three bilge pumps’(its ‘quant’ attribute would still have to be filledin). Pasting this object into the location marked by‘Some equipment’ would have yielded the follow-ing result:

In: Proceedings of COLING 2000, Saarbruecken, Germany.

(13) A solid bulk carrier is fitted with three bilgepumps. Some number of them is used for fire-fighting. Some states.

Note that the text contains an anchor for the yet tobe specified value of the ‘quant’ attribute. Clickingon the anchor activates the following menu:

somemostfewonetwo

Selection of ‘one’ yields the following text, whichis generated from the network in Figure 7:

(14) A solid bulk carrier is fitted with three bilgepumps. One of them is used for firefighting.Some states.

Figure 7: Network underlying (14)

4 ConclusionsIn this paper, we have described some editing oper-ations on object-oriented networks with plural ob-jects and provided a precise formal interpretationfor these networks in terms of a version of Dis-course Representation Theory. The networks whichwe have used are an extension of commonly usedOO networks for natural language generation. Inparticular, our networks cover quantificational plu-ral noun phrases such ‘most N’, ‘few N’, etc.

Acknowledgements The research reported in thispaper was carried out as part of the EC Esprit fundedCLIME project (EP 25.414). Thanks are due to LynneCahill, Roger Evans and Neil Tipper for stimulating co-operation within the CLIME team at the University ofBrighton. Furthermore, I would like to thank Alexander

Boer, Kees van Deemter, Rodger Kibble, Richard Powerand two anonymous COLING reviewers for commentingon earlier versions of this paper.

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