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<Paper uid="P02-1013">
  <Title>Generating Minimal Definite Descriptions</Title>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
2 The incremental approach
</SectionTitle>
    <Paragraph position="0"> Dale and Reiter's incremental algorithm (cf. Figure 1) iterates through the properties of the target entity (the entity to be described) selecting a property, adding it to the description being built and computing the distractor set i.e., the set of elements for which the conjunction of properties selected so far holds. The algorithm succeeds (and returns the selected properties) when the distractor set is the singleton set containing the target entity. It fails if all properties of the target entity have been selected and the distractor set contains more than the target entity (i.e. there is no distinguishing description for the target).</Paragraph>
    <Paragraph position="1"> This basic algorithm can be refined by ordering properties according to some fixed preferences and thereby selecting first e.g., some base level category in a taxonomy, second a size attribute third, a colour attribute etc.</Paragraph>
    <Paragraph position="3"> To generate the UID a5 a2 , do:  1. Initialise: a6 := a0 , a5 a2 := a7 . 2. Check success:  tractor set a43 which initially is equal to the set of individuals present in the context. It then incrementally selects a property a44 that is true of the target set (a45a47a46a49a48a50a48a44a52a51a50a51 ) but not of all elements in the distractor set (a43a49a53a46a54a48a50a48a44a55a51a50a51 ). Each selected property is thus used to simultaneously increment the description being built and to eliminate some distractors. Success occurs when the distractor set equals the target set. The result is a distinguishing description (DD, a description that is true only of the target set) which is the conjunction of properties selected to reach that state.</Paragraph>
    <Paragraph position="4"> a0 : the domain; a56a58a57 a0 , the set to be described; a1a60a59 , the properties true of the set a56 (</Paragraph>
    <Paragraph position="6"> the set of properties that are true of a24 ); To generate the distinguishing description a5 a59 , do:  1. Initialise: a6 := a0 , a5 a59 := a7 . 2. Check success: If a6a9a8 a56 return a5 a59 elseif a1a60a59 a8a20a7 then fail else goto step 3.</Paragraph>
    <Paragraph position="7"> 3. Choose propertya15 a16 a18 a1a60a59 s.t. a56a58a57a9a68a69a68a15 a16a71a70a69a70 and a6a73a72a57a9a68a69a68a15 a16a50a70a74a70 4. Update: a5 a59 := a5 a59 a34 a10a30a15a35a16a42a12a75a37a28a6 := a6a14a21 a68a69a68a15a35a16 a70a69a70 , a1 a59 := a1 a59 a41</Paragraph>
    <Paragraph position="9"> ; if this is successful then stop, otherwise go to phase 2.</Paragraph>
    <Paragraph position="10">  erties of the form a1a14a80a55a1a82a81 with a1 a37 a1a82a81 a18 a1</Paragraph>
    <Paragraph position="12"> To generalise this algorithm to disjunctive and negative properties, van Deemter adds one more level of incrementality, an incrementality over the length of the properties being used (cf. Figure 3).</Paragraph>
    <Paragraph position="13"> First, literals are used i.e., atomic properties and their negation. If this fails, disjunctive properties of length two (i.e. with two literals) are used; then of length three etc.</Paragraph>
  </Section>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3 Problems
</SectionTitle>
    <Paragraph position="0"> We now show that this generalised algorithm might generate (i) epistemically redundant descriptions and (ii) unnecessarily long and ambiguous descriptions. null Epistemically redundant descriptions. Suppose the context is as illustrated in Figure 4 and the target set is a83a85a84a87a86a85a88a89a84a91a90a93a92 .</Paragraph>
    <Paragraph position="1"> pdt secr treasurer board-member member</Paragraph>
    <Paragraph position="3"> members and not treasurers&amp;quot; To build a distinguishing description for the target set a83a85a84a87a86a85a88a89a84a91a90a101a92 , the incremental algorithm will first look for a property a44 in the set of literals such that (i) a83a85a84a87a86a85a88a89a84a91a90a101a92 is in the extension of P and (ii) a44 is not true of all elements in the distractor set a43 (which at this stage is the whole universe i.e., a83a85a84 a86 a88a89a84 a90 a88a89a84a91a102a93a88a89a84a91a103a35a88a89a84a91a104a17a88a89a84a91a105a93a92 ). Two literals satisfy these criteria: the property of being a board member and that of not being the treasurer2 Suppose the incremental algorithm first selects the board-member property thereby reducing the distractor set to a83a85a84 a86 a88a89a84 a90 a88a89a84a106a102a17a88a89a84a107a103a35a88a89a84a106a104a93a92 . Then a108 treasurer is selected which restricts the distractor set to a83a85a84a109a86a75a88a89a84a106a90a17a88a89a84 a103 a88a89a84 a104 a92 . There is no other literal which could be used to further reduce the distractor set hence properties of the form a44a47a110a111a44a55a112 are used. At this stage, the algorithm might select the property a113a91a114a116a115a117a110a73a118a93a119a85a120a67a121 whose intersection with the distractor set yields the target set a83a85a84 a86 a88a89a84 a90 a92 . Thus, the description produced is in this case: board-member a122a123a108 treasurer a122a125a124a126a113a91a114a32a115a17a110a127a118a93a119a85a120a67a121a116a128 which can be phrased as the president and the secretary who are board members and not treasurers whereas the minimal DD the president and the secretary would be a much better output.</Paragraph>
    <Paragraph position="4"> 2Note that selecting properties in order of specificity will not help in this case as neither president nor treasurer meet the selection criterion (their extension does not include the target set).</Paragraph>
    <Paragraph position="5"> One problem thus is that, although perfectly well formed minimal DDs might be available, the incremental algorithm may produce &amp;quot;epistemically redundant descriptions&amp;quot; i.e. descriptions which include information already entailed (through what we know) by some information present elsewhere in the description.</Paragraph>
    <Paragraph position="6"> Unnecessarily long and ambiguous descriptions.</Paragraph>
    <Paragraph position="7"> Another aspect of the same problem is that the algorithm may yield unnecessarily long and ambiguous descriptions. Here is an example. Suppose the context is as given in Figure 5 and the target set is</Paragraph>
    <Paragraph position="9"> The most natural and probably shortest description in this case is a description involving a disjunction with four disjuncts namely a44a55a136a39a110a20a44a27a137a138a110a20a110a82a139a140a110a142a141 which can be verbalised as the Pitbul, the Pooddle, the Holstein and the Jersey.</Paragraph>
    <Paragraph position="10"> This is not however, the description that will be returned by the incremental algorithm. Recall that at each step in the loop going over the properties of various (disjunctive) lengths, the incremental algorithm adds to the description being built any property that is true of the target set and such that the current distractor set is not included in the set of objects having that property. Thus in the first loop over properties of length one, the algorithm will select the property a143 , add it to the description and update the distractor set to a43a145a144a146a48a50a48a69a143a147a51a50a51a149a148 a83a85a84a109a86a85a88a89a84a106a90a17a88a89a84 a102 a88a89a84 a103 a88a89a84 a104 a88a89a84 a105 a88a89a84a39a150a93a88a89a84a91a151a93a88a89a84a91a129a17a88a89a84a87a86a28a130a93a92 . Since the new distractor set is not equal to the target set and since no other property of length one satisfies the selection criteria, the algorithm proceeds with properties of length two. Figure 6 lists the properties a44 of length two meeting the selection criteria at that stage (a83a85a84 a104 a88a89a84 a105 a88a89a84a91a129a93a88a89a84a109a86a28a130a93a92a153a152a154a48a50a48a44a52a51a50a51 and</Paragraph>
    <Paragraph position="12"> The incremental algorithm selects any of these properties to increment the current DD. Suppose it selects a159 a110a162a43 . The DD is then updated to a143 a122a163a124 a159 a110a147a43a164a128 and the distractor set to a83a85a84 a102 a88a89a84 a103 a88a89a84 a104 a88a89a84 a105 a88a89a84a39a150a93a88a89a84a91a151a17a88a89a84a106a129a17a88a89a84a109a86a28a130a93a92 . Except for a160a162a110a165a43 and a108a82a160a54a110 a159 which would not eliminate any distractor, each of the other property in the table can be used to further reduce the distractor set. Thus the algorithm will eventually build the description</Paragraph>
    <Paragraph position="14"> ducing the distractor set to a83a85a84a106a102a17a88a89a84a91a104a17a88a89a84a106a105a17a88a89a84 a151 a88a89a84 a129 a88a89a84 a86a28a130 a92 .</Paragraph>
    <Paragraph position="15"> At this point success still has not been reached (the distractor set is not equal to the target set).</Paragraph>
    <Paragraph position="16"> It will eventually be reached (at the latest when incrementing the description with the disjunction a44a55a136a106a110a14a44a27a137a117a110a9a110a82a139a172a110a9a141 ). However, already at this stage of processing, it is clear that the resulting description will be awkward to phrase. A direct translation from the description built so far (a143 a122a173a124 a159 a110a174a43a164a128a123a122 a124a36a139a162a110a14a108a123a45a123a128a175a122a176a124a171a141a22a110a9a108a82a158a163a128 ) would yield e.g., (1) The white things that are big or a cow, a Holstein or not small, and a Jersey or not medium size Another problem then, is that when generalised to disjunctive and negative properties, the incremental strategy might yield descriptions that are unnecessarily ambiguous (because of the high number of logical connectives they contain) and in the extreme cases, incomprehensible.</Paragraph>
    <Paragraph position="17"> 4 An alternative based on set constraints One possible solution to the problems raised by the incremental algorithm is to generate only minimal descriptions i.e. descriptions which use the smallest number of literals to uniquely identify the target set.</Paragraph>
    <Paragraph position="18"> By definition, these will never be redundant nor will they be unnecessarily long and ambiguous.</Paragraph>
    <Paragraph position="19"> As (Dale and Reiter, 1995) shows, the problem of finding minimal distinguishing descriptions can be formulated as a set cover problem and is therefore known to be NP hard. However, given an efficient implementation this might not be a hindrance in practice. The alternative algorithm I propose is therefore based on the use of constraint programming (CP), a paradigm aimed at efficiently solving NP hard combinatoric problems such as scheduling and optimization. Instead of following a generate-and-test strategy which might result in an intractable search space, CP minimises the search space by following a propagate-and-distribute strategy where propagation draws inferences on the basis of efficient, deterministic inference rules and distribution performs a case distinction for a variable value.</Paragraph>
    <Paragraph position="20"> The basic version. Consider the definition of a distinguishing description given in (Dale and Reiter, 1995).</Paragraph>
    <Paragraph position="21"> Let a121 be the intended referent, and a43 be the distractor set; then, a set a177 of attribute-value pairs will represent a distinguishing description if the following two conditions hold: C1: Every attribute-value pair in a177 applies to a121 : that is, every element of a177 specifies an attribute value that a121 possesses.</Paragraph>
    <Paragraph position="22"> C2: For every member a120 of a43 , there is at least one element a178 of a177 that does not apply to a120 : that is, there is an a177 in a177 that specifies an attribute-value that a120 does not possess. a178 is said to rule out a120 .</Paragraph>
    <Paragraph position="23"> The constraints (cf. Figure 7) used in the proposed algorithm directly mirror this definition.</Paragraph>
    <Paragraph position="24"> A description for the target set a45 is represented by a pair of set variables constrained to be a subset of the set of positive(i.e., properties that are true of all elements in a45 ) and of negative (i.e., properties that are true of none of the elements in a45 ) properties  of a45 respectively. The third constraint ensures that the conjunction of properties thus built eliminates all distractors i.e. each element of the universe which is not in a45 . More specifically, it states that for each distractor a120 there is at least one property a44 such that either a44 is true of (all elements in) a45 but not of a120 or a44 is false of (all elements in) a45 and true of a120 . The constraints thus specify what it is to be a DD for a given target set. Additionally, a distribution strategy needs to be made precise which specifies how to search for solutions i.e., for assignments of values to variables such that all constraints are simultaneously verified. To ensure that solutions are searched for in increasing order of size, we distribute (i.e. make case distinctions) over the cardinality of the output description a192a44 a181a185 a187 a44 a183a185 a192 starting with the lowest possible value. That is, first the algorithm will try to find a description a188a36a44 a181a185 a88a25a44 a183a185 a190 with cardinality one, then with cardinality two etc. The algorithm stops as soon as it finds a solution. In this way, the description output by the algorithm is guaranteed to always be the shortest possible description.</Paragraph>
    <Paragraph position="25"> Extending the algorithm with disjunctive properties. To take into account disjunctive properties, the constraints used can be modified as indicated in  That is, the algorithm looks for a tuple of sets such that their union a45a198a86 a187a142a199a75a199a75a199a93a187 a45a106a200 is the target set a45 and such that for each set a45a106a201 in that tuple there is a basic  conjunctive description.</Paragraph>
    <Paragraph position="26"> As before solutions are searched for in increasing order of size (i.e., number of literals occurring in the description) by distributing over the cardinality of the resulting description.</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
5 Discussion and comparison with related
</SectionTitle>
    <Paragraph position="0"> work Integration with surface realisation As (Stone and Webber, 1998) clearly shows, the two-step strategy which consists in first computing a DD and second, generating a definite NP realising that DD, does not do language justice. This is because, as the following example from (Stone and Webber, 1998) illustrates, the information used to uniquely identify some object need not be localised to a definite description. null (2) Remove the rabbit from the hat.</Paragraph>
    <Paragraph position="1"> In a context where there are several rabbits and several hats but only one rabbit in a hat (and only one hat containing a rabbit), the sentence in (2) is sufficient to identify the rabbit that is in the hat. In this case thus, it is the presupposition of the verb &amp;quot;remove&amp;quot; which ensures this: since x remove y from z presupposes that a212 was in a213 before the action, we can infer from (2) that the rabbit talked about is indeed the rabbit that is in the hat.</Paragraph>
    <Paragraph position="2"> The solution proposed in (Stone and Webber, 1998) and implemented in the SPUD (Sentence Planning Using Descriptions) generator is to integrate surface realisation and DD computation. As a prop-erty true of the target set is selected, the corresponding lexical entry is integrated in the phrase structure tree being built to satisfy the given communicative goals. Generation ends when the resulting tree (i) satisfies all communicative goals and (ii) is syntactically complete. In particular, the goal of describing some discourse old entity using a definite description is satisfied as soon as the given information (i.e. information shared by speaker and hearer) associated by the grammar with the tree suffices to uniquely identify this object.</Paragraph>
    <Paragraph position="3"> Similarly, the constraint-based algorithm for generating DD presented here has been integrated with surface realisation within the generator</Paragraph>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
INDIGEN (http://www.coli.uni-sb.de/
</SectionTitle>
    <Paragraph position="0"> cl/projects/indigen.html) as follows.</Paragraph>
    <Paragraph position="1"> As in SPUD, the generation process is driven by the communicative goals and in particular, by informing and describing goals. In practice, these goals contribute to updating a &amp;quot;goal semantics&amp;quot; which the generator seeks to realise by building a phrase structure tree that (i) realises that goal semantics, (ii) is syntactically complete and (iii) is pragmatically appropriate.</Paragraph>
    <Paragraph position="2"> Specifically, if an entity must be described which is discourse old, a DD will be computed for that entity and added to the current goal semantics thereby driving further generation.</Paragraph>
    <Paragraph position="3"> Like SPUD, this modified version of the SPUD algorithm can account for the fact that a DD need not be wholy realised within the corresponding NP - as a DD is added to the goal semantics, it guides the lexical lookup process (only items in the lexicon whose semantics subsumes part of the goal semantics are selected) but there is no restriction on how the given semantic information is realised.</Paragraph>
    <Paragraph position="4"> Unlike SPUD however, the INDIGEN generator does not follow an incremental greedy search strategy mirroring the incremental D&amp;R algorithm (at each step in the generation process, SPUD compares all possible continuations and only pursues the best one; There is no backtracking). It follows a chart based strategy instead (Striegnitz, 2001) producing all possible paraphrases. The drawback is of course a loss in efficiency. The advantages on the other hand are twofold.</Paragraph>
    <Paragraph position="5"> First, INDIGEN only generates definite descriptions that realize minimal DD. Thus unlike SPUD, it will not run into the problems mentioned in section 2 once generalised to negative and disjunctive properties. null Second, if there is no DD for a given entity, this will be immediately noticed in the present approach thus allowing for a non definite NP or a quantifier to be constructed instead. In contrast, SPUD will, if unconstrained, keep adding material to the tree until all properties of the object to be described have been realised. Once all properties have been realised and since there is no backtracking, generation will fail.</Paragraph>
    <Paragraph position="6"> N-ary relations. The set variables used in our constraints solver are variables ranging over sets of integers. This, in effect, means that prior to applying constraints, the algorithm will perform an encoding of the objects being constrained - individuals and properties - into (pairwise distinct) integers. It follows that the algorithm easily generalises to n-ary relations. Just like the proposition red(a119a17a86 ) using the unary-relation &amp;quot;red&amp;quot; can be encoded by an integer, so can the proposition on(a119 a86 a88a11a119 a90 ) using the binaryrelation &amp;quot;on&amp;quot; be encoded by two integers (one for on( a88a11a119a85a90 ) and one for on(a119a35a86a161a88 ). Thus the present algorithm improves on (van Deemter, 2001) which is restricted to unary relations. It also differs from (Krahmer et al., 2001), who use graphs and graph algorithms for computing DDs - while graphs provides a transparent encoding of unary and binary relations, they lose much of their intuitive appeal when applied to relations of higher arity.</Paragraph>
    <Paragraph position="7"> It is also worth noting that the infinite regress problem observed (Dale and Haddock, 1991) to hold for the D&amp;R algorithm (and similarly for its van Deemter's generalisation) when extended to deal with binary relations, does not hold in the present approach.</Paragraph>
    <Paragraph position="8"> In the D&amp;R algorithm, the problem stems from the fact that DD are generated recursively: if when generating a DD for some entity a119a17a86 , a relation a121 is selected which relates a119a17a86 to e.g., a119a85a90 , the D&amp;R algorithm will recursively go on to produce a DD for a119a85a90 . Without additional restriction, the algorithm can thus loop forever, first describing a119a35a86 in terms of a119a161a90 , then a119a85a90 in terms of a119a17a86 , then a119a35a86 in terms of a119a85a90 etc. The solution adopted by (Dale and Haddock, 1991) is to stipulate that facts from the knowledge base can only be used once within a given call to the algorithm.</Paragraph>
    <Paragraph position="9"> In contrast, the solution follows, in the present algorithm (as in SPUD), from its integration with surface realisation. Suppose for instance, that the initial goal is to describe the discourse old entity a119a17a86 . The initially empty goal semantics will be updated with its DD say, a83a101a214a67a137a93a215a127a178a89a124a30a214a75a128a13a88a25a137a93a216a63a124a30a214a161a88a89a115a25a128a11a92 .  This information is then used to select appropriate lexical entries i.e., the noun entry for &amp;quot;bowl&amp;quot; and the preposition entry for &amp;quot;on&amp;quot;. The resulting tree (with leaves &amp;quot;the bowl on&amp;quot;) is syntactically incomplete hence generation continues attempting to provide a description for a115 . If a115 is discourse old, the lexical entry for the will be selected and a DD computed say, a83a85a115a221a227a3a214a67a178a36a119a77a124a228a115a89a128a13a88a25a137a93a216a63a124a30a214a101a88a89a115a25a128a11a92 . This then is added to the current goal semantics yielding the goal semantics a83a85a115a221a227a3a214a67a178a36a119a77a124a228a115a89a128a13a88a11a214a67a137a93a215a127a178a89a124a30a214a75a128a13a88a25a137a93a216a63a124a30a214a101a88a89a115a89a128a11a92 which is compared with the semantics of the tree built so far i..e.,  Since goal and tree semantics are different, generation continue selecting the lexical entry for &amp;quot;table&amp;quot; and integrating it in the tree being built.</Paragraph>
    <Paragraph position="10">  At this stage, the semantics of that tree is a83a85a115a79a227a40a214a67a178a30a119a116a124a228a115a25a128a13a88a11a214a67a137a101a215a23a178a89a124a30a214a38a128a13a88a25a137a93a216a63a124a30a214a101a88a89a115a25a128a11a92 which is equivalent to the goal semantics. Since furthermore the tree is syntactically and pragmatically complete, generation stops yielding the NP the bowl on the table.</Paragraph>
    <Paragraph position="11"> In sum, infinite regress is avoided by using the computed DDs to control the addition of new material to the tree being built.</Paragraph>
    <Paragraph position="12"> Minimality and overspecified descriptions. It has often been observed that human beings produce overspecified i.e., non-minimal descriptions. One might therefore wonder whether generating minimal descriptions is in fact appropriate. Two points speak for it.</Paragraph>
    <Paragraph position="13"> First, it is unclear whether redundant information is present because of a cognitive artifact (e.g., incremental processing) or because it helps fulfill some other communicative goal besides identification. So for instance, (Jordan, 1999) shows that in a specific task context, redundant attributes are used to indicate the violation of a task constraint (for instance, when violating a colour constraint, a task participant will use the description &amp;quot;the red table&amp;quot; rather than &amp;quot;the table&amp;quot; to indicate that s/he violates a constraint to the effect that red object may not be used at that stage of the task).</Paragraph>
    <Paragraph position="14"> More generally, it seems unlikely that no rule at all governs the presence of redundant information in definite descriptions. If redundant descriptions are to be produced, they should therefore be produced in relation to some general principle (i.e., because the algorithm goes through a fixed order of attribute classes or because the redundant information fulfills a particular communicative goal) not randomly, as is done in the generalised incremental algorithm.</Paragraph>
    <Paragraph position="15"> Second, the psycholinguistic literature bearing on the presence of redundant information in definite descriptions has mainly been concerned with unary atomic relations. Again once binary, ternary and disjunctive relations are considered, it is unclear that the phenomenon generalises. As (Krahmer et al., 2001) observed, &amp;quot;it is unlikely that someone would describe an object as &amp;quot;the dog next to the tree in front of the garage&amp;quot; in a situation where &amp;quot;the dog next to the tree&amp;quot; would suffice.</Paragraph>
    <Paragraph position="16"> Implementation. The ideas presented in this paper have been implemented within the generator INDIGEN using the concurrent constraint programming language Oz (Programming Systems Lab Saarbr&amp;quot;ucken, 1998) which supports set variables ranging over finite sets of integers and provides an efficient implementation of the associated constraint theory. The proof-of-concept implementation includes the constraint solver described in section 4 and its integration in a chart-based generator integrating surface realisation and inference. For the examples discussed in this paper, the constraint solver returns the minimal solution (i.e., The cat and the dog and The poodle, the Jersey, the pitbul and the Holstein) in 80 ms and 1.4 seconds respectively. The integration of the constraint solver within the generator permits realising definite NPs including negative information (the cat that is not white) and simple conjunctions (The cat and the dog).</Paragraph>
  </Section>
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