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<Paper uid="C00-2088">
  <Title>A Formal Semantics for Generating and Editing Plurals</Title>
  <Section position="3" start_page="607" end_page="610" type="metho">
    <SectionTitle>
2 From SSNs to DRSs
</SectionTitle>
    <Paragraph position="0"> 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.</Paragraph>
    <Paragraph position="1"> DRSs with Plurals Following Kamp &amp; 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 individuals, 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 subdivided 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 &amp; Reyle, 1993:398-406).</Paragraph>
    <Paragraph position="2"> 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 complete, 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 extensions 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.</Paragraph>
    <Paragraph position="3"> (III.2) for eventuality predicates, Prod(P) C_ E. (I11.3) For object type predicates, Prod(P) C U.</Paragraph>
    <Paragraph position="4"> (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).</Paragraph>
    <Paragraph position="5"> (V) A function Name mapping constants to members of  U. in particular, the constants c/,, where P is a predicate are mapped to (r)Pred(P), i.e., the supremum, also known as the sum, of the interpretation of P.</Paragraph>
    <Paragraph position="6"> Notice that in our models there are separate domains for objects and eventualities (i.e., states and events).  The relations agent and patient have an eventuality 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 eventualities and others applying to objects.</Paragraph>
    <Paragraph position="7"> For our purposes, the most interesting part of the definition is the function Q~ta,~,t; which maps determiners to their respective interpretations. We take the interpretation of a determiner to be a set of tupies, where each tuple consist of a pair of (plural) individuals. 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 &amp; r is a non-atomic entity of M &amp; kl -&gt; } 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 cardinality 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 context 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 individual). null Let us focus on the DRS condition which is interpreted in the models in terms of @m,~,t. This condition functions as a substitute for the duplex conditions 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 &amp; Cooper, 1981; Keenan &amp; Westerstahl, 1997). This interpretation is not entirely unproblematic; see, for instance, (Kamp &amp; P, eyle, 1993). Our use of the interpretation 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 quantifiers 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 &amp; P, eyle, 1993: 425-427, 6776'79). null If z is a discern;re referent and t is a discourse re\[erent or constant, then DETt(:c) is a condition.</Paragraph>
    <Paragraph position="8"> The verification condition for this condition is:  (5) M ~f DETt(:C) (if&amp;quot; (11 II II t IIAJ'f&gt; Let us illustrate these definitions with a simple example. Consider: (6) At most two men walk.</Paragraph>
    <Paragraph position="9">  The NP 'At most two men' introduces a plural discourse referent X, together with a number of conditions on that referent. Additionally, the verb 'walk' supplies a condition to the effect that all the mem-</Paragraph>
    <Paragraph position="11"> 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. 'deg 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. ~deg A quantor  a '*', this means that the prcdicate is interpreted distributively over the atomic parts of the objects in its denotation. &amp;quot;JWe assume that: @Umt(AT_MOST_2) = {(r, c) : r C c &amp; I,'1 &lt; 2} mln Van Eijck (1983), an allemative approach is proposed within a fl'amework which also allows for the direct representation of plural referents in DRT. lie proposes to reanalyse monotonically decreasing quantiliers in terms of negation and monotonically increasing ones. This, however, means that WE no longer have a direct correlation between plural discourse referents 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'.  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\[&amp;quot; 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 representation 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.</Paragraph>
    <Paragraph position="12"> 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 logical contexts L and a function f which assigns a logical context (which are treated as primitive objects separate from those in the DAG) to each of the objects in the DAG. In the DAG, there are objects which correspond with logical operators, such as implication 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 ordering (in the DAG) of the objects with which they are associated. Furthermore, this set of logical contexts constitutes the range of f.</Paragraph>
    <Paragraph position="13"> (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.</Paragraph>
    <Paragraph position="14"> (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.</Paragraph>
    <Paragraph position="15"> Consider, for instance, the following sentence: (8) If a man is happy, then he whistles.</Paragraph>
    <Paragraph position="16"> We can represent this sentence by means of the SSN in Figure 1. In this representation, the dots represent objects, the circles represent logical contexts (an object inside a circle belongs to the corresponding logical context), the solid arrows represent attributes and the dotted arrows represent that the object fi'om which the arrow originates belongs to the context to which the arrow points.</Paragraph>
    <Paragraph position="17"> 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.</Paragraph>
    <Paragraph position="18"> (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.</Paragraph>
    <Paragraph position="19"> (III) Typing statements T(z) of a non-logical object are added to the same box as the object z itself.</Paragraph>
    <Paragraph position="20"> (IV) Attributions/{(.% !/), where z and !/are non-logical objects, are added to the same box as z.</Paragraph>
    <Paragraph position="21">  By applying these rules, we obtain the following DP, S for the SSN in Figure 1 :</Paragraph>
    <Paragraph position="23"> 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.</Paragraph>
    <Paragraph position="24"> 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.</Paragraph>
    <Paragraph position="25"> Rathel, we introduce a number of special-purpose  attributions and types. Subsequently, we specify their mapping to appropriate terms in a DRS.</Paragraph>
    <Paragraph position="26"> 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, &gt; 1, &lt; 3, all,.f&gt;w, etc.</Paragraph>
    <Paragraph position="27"> (11) parl,_of. The value of this feature is restricted to countable objects.</Paragraph>
    <Paragraph position="28"> 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 :</Paragraph>
    <Paragraph position="30"> to the box in which also x lives the lbllowing condition: .r = T2(c7~). Note that in this case T~ is subtype of &amp;: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 condition; null (I\]tl) For attributions q'uant(:r,y) such that ~z : part_of(x, z) &amp; T1 (x) &amp; 7)(y) add to the box in which also :r lives the following condition: x = 5/)(z) .Furthermore, we add a closure condition; (IV) Otherwise apply the standard mapping rules for SSNs (see the previous section).</Paragraph>
    <Paragraph position="31"> Consider, lbr instance, the (phual) SSN for lhe sentence 'At most two men walk' in Figure (2).</Paragraph>
    <Paragraph position="32">  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 attribute points to an object of type at.anost_2. The value of the other attribute, i.e., part_oJ; is not instantiated in this case. This is represented by means of the empty box. When we apply the rules for mapping SSNs to DRSS, we obtain the following representation: null</Paragraph>
    <Paragraph position="34"> The first four conditions correspond to the types of the nodes and the attributes of the SSN. They are followed by the closure condition.</Paragraph>
  </Section>
  <Section position="4" start_page="610" end_page="612" type="metho">
    <SectionTitle>
3 Editing Plurals
</SectionTitle>
    <Paragraph position="0"> 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 manipulating knowledge representations using natural language feedback. WYSIWYM \]las been used in various systems for (multilingual) document authoring and query formulation. The proposal which is presented in tiffs paper has been inaplemented as part of the M ILF, query-answering system (e.g., Piwek ct al., 2000).</Paragraph>
    <Paragraph position="1"> The basic idea underlying WYSIWYM editing can be presented by means of a simple diagram.</Paragraph>
    <Paragraph position="2"> W~- &amp;quot; update ,~'(!llt't'{ll(!  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' represeuting places where the knowledge base can be extended. Each anchor is associated with pop-up menus, which present the possible editing operations 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 contents of the SN.</Paragraph>
    <Paragraph position="3">  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&amp;quot; is fitted with three bilge pumps. Some equipment is used fox&amp;quot; firefighting. Some states.</Paragraph>
    <Paragraph position="4">  The spans in bold face indicate where tile network is still incomplete. Other spans of text represent specific objects in the network. For instance, the span 'three bilge pumps' is associated with a plural object of the type 'bilge pump'. When the user clicks 12In order to keep the example transparent, not all information 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.</Paragraph>
    <Paragraph position="5"> 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 network 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,  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.</Paragraph>
    <Paragraph position="6"> Van Deemter and Power (1998) originally defined the 'copy' operation for singular objects. When we move to plurals, alternatives to a simple copy operation become available. Here, we want to discuss 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 created 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 following result:  (13) A solid bulk carrier is fitted with three bilge  pumps. Some number of them is used for firelighting. Some states.</Paragraph>
    <Paragraph position="7"> 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.</Paragraph>
    <Paragraph position="8"> Some states.</Paragraph>
  </Section>
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