<?xml version="1.0" standalone="yes"?>
<Paper uid="P99-1017">
  <Title>Using aggregation for selecting content when generating referring expressions</Title>
  <Section position="4" start_page="0" end_page="128" type="metho">
    <SectionTitle>
2 The aggregation-based metaphor
</SectionTitle>
    <Paragraph position="0"> Aggregation in generation has hitherto generally consisted of lists of more or less ad hoc, or case-specific rules that group together paxticulax pre-specified configurations (cf. Dalianis and Hovy (1996) and Shaw (1998)); however Bateman et al. (1998) provide a more rigorous and generic foundation for aggregation by applying results from data-summarization originally developed for multimedia information presentation (Kamps, 1997). Bateman et al. set out a general purpose method for constructing aggregation lattices which succinctly represent all possible structural aggregations for any given data set. 1 The application of the aggregation-based metaphor to RE-content determination is motivated by the observation that if something is a 'potential distractor' for some intended referent, then it is equally, under appropriate conditions, a candidate for aggregation together with the intended referent. That 1'Structural' aggregation refers to opportunities for grouping inherent in the structure of the data and ignoring additional opportunities for grouping that might be found by modifying the data inferentially.</Paragraph>
    <Paragraph position="1">  is, what makes something a distractor is precisely the same as that which makes it a potential co-member of some single grouping created by structural aggregation. To see this, consider the following simple example discussed by Dale and Reiter (1995) consisting of three objects with various properties (re-represented here in a simple association list format): 2 (ol (type dog) (size small) (color (02 (type dog) (size large) (color (03 (type cat) (size small) (color To successfully refer to the first object ol, sufficient information must be given so as to 'rule out' the possible distractors: therefore, type alone is not sufficient, since this fails to rule out o2, nor is any combination of size or color sufficient, since these fail to rule out 03. Successful RE's are 'the small dog' or 'the black dog' and not 'the small one', 'the dog', or 'the black one'.</Paragraph>
    <Paragraph position="2"> Considering the data set from the aggregation perspective, we ask instead how to refer most succinctly to all of the objects ol, o2, o3.</Paragraph>
    <Paragraph position="3"> There are two basic alternatives, indicated by bracketing in the following: 3  1. (A (small black and a large white) dog) and (a small black cat).</Paragraph>
    <Paragraph position="4"> 2. (A small black (dog and cat)) and (a large white dog).</Paragraph>
    <Paragraph position="5">  The former groups together ol and o2 on the basis of their shared type, while the latter groups together ol and o3 on the basis of their shared size and color properties. Significantly, these are just the possible sources of distraction that Dale and Reiter discuss.</Paragraph>
    <Paragraph position="6"> The set of possible aggregations can be determined from an aggregation lattice corresponding to the data set. We construct the lattice using methods developed in Formal Concept Analysis (FCA) (Wille, 1982). For the example at hand, the aggregation lattice is built up as follows. The set of objects is considered as a relation table where the columns represent the object attributes and their values, and the rows 2This style of presentation is not particularly perspicuous but space precludes providing intelligible graphics, especially for the more complex situations used as examples below. In case of difficulties, we recommend quickly sketching the portrayed situation as a memory aid.</Paragraph>
    <Paragraph position="8"> represent the individual objects. Since the attributes (e.g., 'color', 'size', etc.) can take multiple values (e.g., 'large', 'small'), this representation of the data is called a multivalued context. This is then converted into a one-valued context by comparing all rows of the table pair-wise and, for each attribute (i.e., each column in the table) entering one distinguished value (e.g., T or 1) if the corresponding values of the attributes compared are identical, and another distinguished value (nil or 0) if they are not.</Paragraph>
    <Paragraph position="9"> The one-valued context for the objects ol-o3 is thus: object pairs type size color</Paragraph>
    <Paragraph position="11"> This indicates that objects ol and o2 have equal values for their type attribute but otherwise not, while ol and 03 have equal values for both their size and color attributes but not for their type attributes. The one-valued context readily supports the derivation of formal concepts. A formal concept is defined in FCA as an extension-intension pair (A,B), where the extension is a subset A of the set of objects and the intension is a subset B of the set of attributes. For any given concept, each element of the extension must accept all attributes of the intension. Visually, this corresponds to permuting any rows and columns of the one-valued context and noting all the maximally 'filled' (i.e., containing l's or T's) rectangles. A 'subconcept' relation, '&lt;FCA', is defined over the set of formal concepts thus: (A, B) &lt;FCA (A*, B*) iff A C A* ~=~ B* C B The main theorem of FCA then shows that &lt;FCA induces a complete lattice structure over the set of formal concepts. The resulting lattice for the present example is shown in Figure 1.</Paragraph>
    <Paragraph position="12"> Each node is shown labeled with two pieces of information: the intension and the extension.</Paragraph>
    <Paragraph position="13"> The intensions consist simply of the sets of properties involved. The representations of the extensions emphasize the function of the nodes in the lattice--i.e., that the indicated objects (e.g., ol and o2 for the leftmost node) are equal with respect to all the attributes contained in the intension (e.g., type for the leftmost node).</Paragraph>
    <Paragraph position="14">  This lattice may be construed as an aggregation lattice because the functional redundancies that are captured are precisely those redundances that indicate opportunities for structurally-induced aggregation. The leftmost node shows that the attribute type may be aggregated if we describe ol together with o2, and the right-most node shows that {color, size} may be aggregated when describing ol and o3.</Paragraph>
    <Paragraph position="15"> Now, given the equivalence between aggregation possibilities and 'distractors', we can also use the lattice to drive RE-content determination. Assume again that we wish to refer to object ol. In essence, a combination of attributes must be selected that is not subject to aggregation; any combination susceptible to aggregation will necessarily 'confuse' the objects for which the aggregation holds when only one of the objects, or co-aggregates, is mentioned.</Paragraph>
    <Paragraph position="16"> For example, the rightmost node shows that an RE with the content size&amp;color(ol), e.g., 'the small black thing', confuses ol and o3. To select attributes that are appropriate, we first examine the minimal nodes of the lattice to see if any of these do not 'impinge' (i.e., have no aggregation consequences: we make this more precise below) on the intended referent. In this case, however, all these nodes do mention ol and so no strong preference for the RE-content is delivered by the data set itself. This appears to us to be the correct characterization of the reference situation: precisely which attributes are selected should now be determined by factors not attributable to 'distraction' but rather * by more general communicative goals involving discourse and the requirements of the particular language. The resulting attribute combinations are then checked against the aggregation lattice for their referential effectiveness in a manner reminiscent of the incremental approach of previous algorithms. Selection of type is not sufficient but the addition of either color or size is (type~zcolor = +- and type~size=l).</Paragraph>
    <Paragraph position="17"> The reference situation is quite different when we wish to refer to either o2 or o3. For both of these cases there exists a non-impinging node (the right and leftmost nodes respectively). This establishes immediate attribute preferences based on the organizational properties of the data. Content-determination for o2 should include at least size or color ('the white thing', 'the large thing') and for o3 at least type ('the cat'). These RE's are minimal.</Paragraph>
  </Section>
  <Section position="5" start_page="128" end_page="132" type="metho">
    <SectionTitle>
3 Examples of aggregation-driven
</SectionTitle>
    <Paragraph position="0"> RE-content determination In this section, we briefly summarize some more significant examples of RE-content determination using aggregation. Length limitations will require some shortcuts to be taken in the discussion and we will not follow up all of the alternative RE's that can be motivated.</Paragraph>
    <Section position="1" start_page="128" end_page="129" type="sub_section">
      <SectionTitle>
3.1 Minimal descriptions
</SectionTitle>
      <Paragraph position="0"> Dale and Reiter (1995) consider a number of variant algorithms that deviate from full brevity in order to achieve more attractive computational behavior. The first variant they consider relies on a 'Greedy Heuristic' (Dale, 1989; Johnson, 1974); they illustrate that this algorithm sacrifices minimality by constructing an RE for object ol in the context of the following properties concerning a set of seven cups of varying size (large, small), color (red, green, blue) and material (paper, plastic):  The greedy algorithm produces 'the large red plastic cup' although the true minimum description is 'the large red cup'.</Paragraph>
      <Paragraph position="1"> The aggregation-based approach to the same data set provides an interesting contrast in result. The aggregation lattice for the data is given in Figure 2. The lattice is constructed as before: first by converting the multivalued context of the original data set to a one-valued context and then by imposing the subconcept</Paragraph>
      <Paragraph position="3"> cups' example relation over the complete set of formal concepts. The nodes of the lattice are also labeled as before, although we rely here on the formal properties of the lattice to avoid redundant labeling. For example, the two sets of attribute equalities given for node 1 (one relating o2 and o3, the other relating o6 and o7) apply to both color (inherited from node 2) and size (inherited from node 4); we do not, therefore, repeat the labeling of properties for node 1. Similarly, and due to the bidirectionality inherent in the subconcept definition, the attribute equalities of node 1 are also 'inherited' upwards both to node 2 and to node 4. The attribute equalities of node 4 therefore include contributions from both node 1 and node 6. We will generally indicate in the labeling only the additional information arising from the structure of the lattice, and even then only when it is relevant to the discussion. So for node 4 we indicate that ol, o5, o6 and o7 now form a single attribute equality set made up of three contributions: one from node 1 (o6 and o7) and two from node 6. Their combination in a single set is only possible at node 4 because node 4 is a superconcept of both node 1 and node 6. The other attribute equality set for node 1 (o2 and o3) does not add further information at node 4 and so is left implicit in node 4's labeling. The labeling or non-labeling of redundant information has of course no formal consequences for the information contained in the lattice.</Paragraph>
      <Paragraph position="4"> To determine RE-content appropriate for referring to object ol, we again look for minimal (i.e., nearest the bottom) concepts, or aggregation sets, that do not 'impinge' on ol. The only node satisfying this requirement is node 1. This tells us that the set of possible co-aggregates for ol with respect to the properties {size &amp; color} is empty, which is equivalent to stating that there are no objects in the data set which might be confused with ol if size&amp;color(ol) forms the RE-content. Thus, 'the large red cuP' may be directly selected, and this is precisely the true minimal RE for this data set.</Paragraph>
    </Section>
    <Section position="2" start_page="129" end_page="131" type="sub_section">
      <SectionTitle>
3.2 Relational descriptions: restricting
</SectionTitle>
      <Paragraph position="0"> recursion One early extension of the original REalgorithms was the treatment of data sets involving relations (Dale and Haddock, 1991). Subsequently, Horacek (1995) has argued that the extension proposed possesses several deficits involving both the extent of coverage and its behavior. In particular, Horacek notes that &amp;quot;it is not always necessary that each entity directly or indirectly related to the intended referent and included in the description be identified uniquely&amp;quot; (p49). Partially to handle such situations, Horacek provides a further related algorithm that is intended to improve on the original and which he illustrates in action with reference to a rather more complex situation involving two tables with a variety of cups and bottles on them. One table (tl) has two bottles and a cup on it, another (t2) has only a cup. Information is also given concerning the relative positions of the cups and bottles.</Paragraph>
      <Paragraph position="1"> The situation that Horacek identifies as problematic occurs when the reference task is to refer to the table tl and the the RE-algorithm has decided to include the bottles that are on this table as part of its description. This is an appropriate decision since the presence of these bottles is the one distinguishing feature of the selected table. But it is sufficient for the identification of tl for bottles to be mentioned at all: there is no need for either or both of the bottles to be distinguished more specifically. An RE-algorithm should therefore avoid attempting this additional, unnecessary reference task. To form an aggregation lattice for this fact set, we extend our data representation to deal with relations as well as attributes. This is limited to 'reifying' the relations and labeling them with 'instance variables' as commonly done in input expressions for generation systems (Kasper, 1989). For convenience, we also at this point fold in the type information di- null rectly as would be normal for a typed semantic representation. This gives the set of facts g7g12 shown at the top of Figure 3. 4 Once the data set is in this form, aggregation lattice construction may proceed as described above; the result is also shown in Figure 3. This lattice reflects the more complex reference situation represented by the data set and its possible aggregations: for example, node 7 shows that the facts {g7, g8, gg, gl0} may be aggregated with respect to both arg2type ('table': node 5) and pred ('on': node 6). Node 3, in contrast, shows that the two distinct sets {g9, gl0} and {g7, g8} (again inherited upwards from node 2) may both individually (but not collectively) also be aggregated with pred, arg2type, and additionally with argltype ('cup': node 4).</Paragraph>
      <Paragraph position="2"> We first consider the reference task described by Horacek, i.e., identifying the object tl. Now that we are dealing with relations, the ob* jects to be referred to generally occur as values of 'attributes'--that is, as entries in the data table--rather than as entire rows. In order to construct an appropriate RE we need to find relations that describe the intended referent and which do not allow aggregation with other rela4Note that this is then isomorphic to a set of SPL specifications of the form (g7 / on :argl (bl / bottle) :arg2 (tl / table)), etc.</Paragraph>
      <Paragraph position="3"> tions describing other conflicting referents. We also need to indicate explicitly that the RE-content should not avail itself of the literal instance variables: these are to remain internal to the lattice and to RE-construction so that individuals remain distinct. We therefore distinguish been 'public' and 'private' attributes: public attributes are available for driving linguistic expression, private attributes are not. If we were not to impose this distinction, then referring expressions such as 'the table tl' would be seen as appropriate and probably minimal descriptions! 5 An aggregation set that does hot involve a private attribute will be called a public concept.</Paragraph>
      <Paragraph position="4"> The first step in constructing an RE is now to identify the relations/events in which the intended referent is involved--here {g7, g8, gg}-and to specify the positions (both private and public) that the referent holds in these. We call the set of potentially relevant relations, the reference information source set (ares).</Paragraph>
      <Paragraph position="5"> In the present case, the same argument position is held by the intended referent t l for all RISS-members, i.e., privately arg2 and publicly arg2type: Next, we proceed as before to  find a non-impinging, minimal aggregate set.</Paragraph>
      <Paragraph position="6"> However, we can now define 'non-impinging' more accurately. A non-impinging node is one for which there is at least one public superconcept fulfilling the following condition: the required superconcept may not bring any RISSnon-member together as co-aggregate with any RISS-member drawn from the originating aggregation set with respect to the specified public attribute of the intended referent.</Paragraph>
      <Paragraph position="7"> By these definitions both the minimal nodes of the lattice are non-impinging. However, node 2 is more supportive of minimal RE's and we will only follow this path here; formal indications of minimality are given by the depth and number of paths leading from the node used for aggregation to the top of the aggregation lattice (since any resulting description then combines discriminatory power from each of its chains of superconcepts) and the number of additional facts that are taken over and above the original RISS-members. Node 2 is therefore the 'default' choice simply given a requirement of brevity, although the generation process is free to ignore this if other communicative goals so decide.</Paragraph>
      <Paragraph position="8"> There are two public superconcepts for node 2: both of nodes 7 and 3 inherit arg2type from node 5 but do not themselves contain a private attribute. Of these only node 7 brings one of the originating RIss-members (i.e., g7 and g8 from node 2) into an aggregation set with a RISS non-member (gl0). Node 2 is therefore non-impinging via node 3. The attributes that may be aggregated at node 2 are arg2 (node 2 &lt;EVA 8), arg2type (2 &lt;FCA 5), pred (2 &lt;FCA 6) and argltype (2 &lt;:FCA 4). Since this includes arg2, the private position of the intended referent, we know that the data set does not support aggregation for g7 and g8 with respect to any other distracting value for arg2, and so g7 and g8, both collectively and individually, are appropriate and sufficient RE's for tl. * Rendering these in English would give us: g7 or g8 'the table with a bottle on it' g? plus g8 'the table with some bottles on it' The precise rendering of the bottles depends on other generator decisions; important here is only the fact that it is known that we do not need to uniquely identify which bottles are in question. More identifying information for argl  if an aggregation with other arg2's (e.g., other tables) were possible, but it is not, and so the type information is already sufficient to produce an RE with no unwanted aggregation possibilities. The aggregation-based approach will not, therefore, go on to consider further facts unless there is an explicit communicative intention to do so.</Paragraph>
    </Section>
    <Section position="3" start_page="131" end_page="132" type="sub_section">
      <SectionTitle>
3.3 Relational descriptions: when
</SectionTitle>
      <Paragraph position="0"> further information is necessary In this final example we show that the behavior above does not preclude information being added when it is in fact necessary. We show this by adapting Horacek's set of facts slightly to create a different aggregation lattice; we move one of the bottles (b2) over to the other table t2, placing it to the right of the cup. We show the modified facts and the new aggregation lattice in Figure 4. Here a few concepts have moved in response to the revised reference situation: for example, arg2type (node 3) is now a direct subconcept of pred indicating that in the revised data set there is a functional relationship between the two attributes: all co-aggregates with respect to arg2type are necessarily also co-aggregates with respect to pred. In the previous example this did not hold because there were also facts with shared pred and non-shared arg2type (facts gll and g12: node 6).</Paragraph>
      <Paragraph position="1">  We will again attempt to refer to the table t 1 to compare the results with those of the previous subsection. To begin, we have a RISS of {gT, gg} with the intended referent in arg2 (private) and arg2type (public) as before. We then look for non-impinging, most-specific nodes. Here, nodes 4 and 5 are both impinging. Node 4 is impinging in its own right since it sanctions aggregation of both the RIss-members it mentions with non-members with respect to arg2type (node 3) and argltype (node 6); this deficit is then inherited upwards. Node 5 is impinging by virtue of its first and only available public superconcept, node 3, which sanctions as co-aggregates {gT, g8 ~, gg, gl0} with respect to arg2type. Neither node 4 nor node 5 can therefore support appropriate RE's. Only node 2 is non-impinging, since it does not sanction aggregation involving arg2type or arg2, and is the only available basis for an effective RE with the revised data set.</Paragraph>
      <Paragraph position="2"> To construct the RE we take the RISS-member of node 2 (i.e., gT) and consider it and the aggregations it sanctions as candidate material. Node 2 indicates that g7 may be aggregated with gll with respect to argltype; such an aggregation is guaranteed not to invoke a false referent for argl because it is non-impinging. Moreover, we can infer that g? alone is insufficient since nodes 3 and 4 indicate that g7 is a co-aggregate with facts with non-equal argl values (e.g., gSr), and so aggregation is in fact necessary. The RE then combines:  to produce 'the table on which a bottle is to the left of a cup'. This is the only RE that will identify the required table in this highly symmetri* cal context. No further information is sought because there are no further aggregations possible with respect to arg2 and so the reference is unique; it is also minimal.</Paragraph>
    </Section>
  </Section>
class="xml-element"></Paper>