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<Paper uid="P90-1013">
  <Title>THE COMPUTATIONAL COMPLEXITY OF AVOIDING CONVERSATIONAL IMPLICATURES</Title>
  <Section position="3" start_page="0" end_page="98" type="metho">
    <SectionTitle>
2. Referring Expression Model
</SectionTitle>
    <Paragraph position="0"> The referring-expression model used in this paper is a variant of Dale's (1989) model for full definite noun phrase referring expressions. Dale's model is applicable in situations in which the speaker intends to refer to an object that the speaker and hearer are mutually aware of, and the speaker has no other communicative goal besides identifying the referred-to object. 2 The model assumes that objects belong to a taxonomy class (e.g., Chair) and possess values for various attributes (e.g., Color:Brown). 3 Referring expressions are represented as a classification and a set of attribute-value pairs: the classification is syntactically realized as the head noun, while the attribute-value pairs are syntactically realized as NP modifiers. Successful referring expressions are required to be distinguishing descript/ons, i.e., descriptions that contain a classification and a set of attributes that are true of the object being referred to, but not of any other object in the current discourse context. 4 More formally, and using a somewhat different terminology from Dale, let a component be either a classification or an attribute-value pair. A classification component will be written class:Class; an attribute-value pair component will be written Attribute:Value. Then, given a target object, denoted Target, and a set of contrasting objects in the current discourse context, denoted Excluded, a set of components will represent a successful referring expression (a distinguishing description, in Dale's terminol2 Appelt (1985) presented a more complex rderringexpression model that covered situations where the hearer was not already aware of the referred-to object, and that allowed the speaker to have more complex communicative goals. A similar laalysis to the one presented in this paper could in principle be done for Appelt's model, but it would be substantially more difficult, in part because the model is more complex, and in pa~t because Appeh did not separate his 'content detcrminatiou' subsystem frona his planner and his sudaee-form generator.</Paragraph>
    <Paragraph position="1">  distinguishing descril0dons of minimal cardinality; this is discussed in footnote 7.</Paragraph>
    <Paragraph position="2"> ogy) if the set, denoted RE, satisfies the following constraints: 1) Every component in RE applies to Target: that is, every component in RE is either a classification that subsumes Target, or an attribute-value pair that Target possesses.</Paragraph>
    <Paragraph position="3"> 2) For every member E of Excluded, there is at least one component in RE that does not apply toE.</Paragraph>
    <Paragraph position="4"> Example: the current discourse context contains objects A, B, and C (and no other objects), and these objects have the following classifications and attributes (of which both the speaker and the hearer are aware):  In this context, the referring expressions {class:Table} (&amp;quot;the table&amp;quot;) and {class:Table, Material:Wood, Color:Brown} (&amp;quot;the brown wooden table&amp;quot;) both successfully refer to object A, because they match object A but no other object. Similarly, the referring expressions {class:Chair, Color:Brown} (&amp;quot;the brown chair&amp;quot;) and {class:Chair, Material:Wood, Color:Brown} (&amp;quot;the brown wooden chair&amp;quot;) both successfully refer to object B, because they match object B, but no other object. The referring expression {class:Chair} (~the chair&amp;quot;), however, does not successfully refer to object B, because it also matches object C.</Paragraph>
  </Section>
  <Section position="4" start_page="98" end_page="99" type="metho">
    <SectionTitle>
3. Conversational Implicature
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="98" end_page="99" type="sub_section">
      <SectionTitle>
3.1. Grice's Maxims and Their Interpretation
</SectionTitle>
      <Paragraph position="0"> Grice (1975) proposed four maxims of conversation that speakers needed to obey: Quality, Quantity, Relevance, and Manner. For the task of generating referring expressions as formalized in Section 2, these maxims can be interpreted as follows: Quality: The Quality maxim requires utteranees to be truthful. In this context, it requires referring expressions to be factual descriptions of the referred-to object. This condition is already part of the definition of a successful referring expression, and does not need to be restated as a conversational implicature constraint.</Paragraph>
      <Paragraph position="1"> Quantity: The Quality maxim requires utterantes to contain enough information to fulfill the speaker's communicative goal, but not more information. In this context, it requires referring expressions to contain enough information to enable the hearer to identify the referred-to object, but not more information. Therefore, referring expressions should be successful (as defined in Section 2), but should not conrain additional elements that are unnecessary for fulfilling the referring goal.</Paragraph>
      <Paragraph position="2"> Relevance: The Relevance maxim requires utterances to be relevant to the discourse. In this context, where the speaker is assumed just to have the communicative goal of identifying an object to the hearer, the maxim prohibits referring expressions from containing elements that do not help distinguish the target object from other objects in the discourse context. Irrelevant elements are also unnecessary elements, so the Relevance maxim may be considered to be a special case of the Quantity maxim, at least for the referring-expression generation task as formalized in Section 2.</Paragraph>
      <Paragraph position="3"> Manner: The Brevity submaxim of the Manner maxim requires a speaker to use short utterances if possible. In this context it requires the speaker to use a short referring expression if such a referring expression exists. The analysis of the other Manner submaxims is left for future work.</Paragraph>
      <Paragraph position="4"> An additional source of conversational implicatm'e was proposed by Cruse (1977) and Hirschberg (1985), who hypothesized that. implicatures might arise from the failure to use basic-level classes (Rosch 1978) in an utterance. In this paper, such implicatures are generalized by assuming that there is a lexical-preference hierarchy among the lexical classes (classes that can be realized with single lexical units) known to the hearer, and that the use of a lexical class in an utterance implicates that no preferred lexical class could have been used in its place. In summary, conversational implicature considerations require referring expressions to be brief, to not contain unnecessary elements, and to use lexically-preferred classes whenever possible. The following requests illustrate how violations of these principles in referring expressions may lead to unwanted conversational implicatares:  3a) &amp;quot;Wait for me by the pine.&amp;quot; ({class:Pine})  3b) &amp;quot;Wait for me by the tree that has pinecones.&amp;quot; ({class:Tree, Seed-type :Pinecone } ) 3c) &amp;quot;Wait for me by the 50-foot-high pine.&amp;quot; ({class:Pine, Height:50-feet } ) 3d) ~Wait for me by the sugar pine.&amp;quot;</Paragraph>
      <Paragraph position="6"> If there were only two trees in the hearer's immediate surroundings, a pine and an oak, then all of the above utterances would be successful referring expressions that enabled the hearer to pick out the object being referred to (assuming the hearer could recognize pines and oaks). In such a situation, however, utterance (3b) would violate the brevity principle, and thus would implicate that the tree could not be described as a &amp;quot;pine&amp;quot; (which might lead the hearer to infer that the tree was not a real pine, but some other tree that happened to have pinecones). Utterance (3c) would violate the no-unnecessary-elements principle, and thus would implicate that it was important that the tree was 50 feet tall (which might lead the hearer to infer that there was another pine tree in the area that had a different height). Utterance (3d) would violate the lexical-preference principle, and thus would implicate that the speaker wished to emphasize that the tree was a sugar pine and not some other kind of pine (which might lead the hearer to infer that the speaker was trying to impress her with his botanical knowledge). A speaker who only wished to tell the hearer where to wait, and did not want the hearer to make any of these implicatures, would need to use utterance (3a), and to avoid utterances (3b), (3c), and (30).</Paragraph>
    </Section>
    <Section position="2" start_page="99" end_page="99" type="sub_section">
      <SectionTitle>
3.2. Formalizing Conversational Implicature
Through Preference Rules
</SectionTitle>
      <Paragraph position="0"> The brevity, no-unnecessary-elements, and lexical-preference principles may be formalized by requiring a description to be a maximal element under a preference function of the set of successful referring expressions. More formally, let D be the set of successful referring expressions, and let &gt;&gt; be a preference function that prefers descriptions that are short, that do not contain unnecessary elements, and that use lexically preferred classes. Then, a referring expression is considered free of false implicatures if it is a maximal element of D with respect to &gt;&gt;. In other words, a description B in D is free of false implicatures if there is no description A in D, such that A &gt;&gt; B. This formalization is similar to the partially ordered sets that Hirschberg (1985) used to formalize scalar implicatures: D and &gt;&gt; together form a partially ordered set, and the assumption is that the use of an element in D carries the conversational implicature that no higher-ranked element in D could have been used.</Paragraph>
      <Paragraph position="1"> The overall preference function &gt;&gt; will be decomposed into separate preference rules that cover each type of implicature: &gt;&gt;B for brevity, &gt;&gt;u for unnecessary elements, and &gt;&gt;t. for lexical prefereuce. &gt;&gt; is then defined as the disjunction of these preference rules, i.e., A &gt;&gt; B if A &gt;&gt;s B, A &gt;&gt;v B, or A &gt;&gt;L B. The assumption will be made in this paper that there are no conflicts between preference rules, i.e., that it is never the case that A is preferred over B by one preference rule, but B is preferred over A by another preference rule. 5 Therefore, &gt;&gt; will be a partial order if &gt;&gt;B, &gt;&gt;v, and &gt;&gt;n are partial orders. null</Paragraph>
    </Section>
    <Section position="3" start_page="99" end_page="99" type="sub_section">
      <SectionTitle>
3.3. Computational Tractability
</SectionTitle>
      <Paragraph position="0"> Computational complexity considerations are used in this paper to determine exactly how the nounnecessary-elements, brevity, and lexical-preference principles should be formalized as preferenee rules. Sections 4, 5, and 6 examine various preference rules that might plausibly be used to formalize these implicatures, and reject preference rules that make the generation task NP-Hard. This is justified on the grounds that computer NLG systems should not be asked to solve NP-Hard problems. 6 Human speakers and hearers are also probably not very proficient at solving NP-Hard problems, which suggests that it is unlikely that NP-Hard preference rules have been incorporated into language.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="99" end_page="99" type="metho">
    <SectionTitle>
4. Brevity
</SectionTitle>
    <Paragraph position="0"> Grice's submaxim of brevity states that utterauces should be kept brief. Many NLG researchers (e.g., Dale 1989; Appelt 1985: pages 117-118) have suggested that this means generation systems need to produce the shortest possible utterance. This will be called the Full Brevity preference rule. Unfortunately, it is NP-Hard to find the shortest successful referring expression (Section 4.1). Local Brevity (Section 4.2) is a weaker version of the brevity submaxim that can be incorporated into a polynomial-time algorithm for generating successful referring expressions.</Paragraph>
    <Section position="1" start_page="99" end_page="99" type="sub_section">
      <SectionTitle>
4.1. Full Brevity
</SectionTitle>
      <Paragraph position="0"> The Full Brevity preference rule requires the generation system to generate the shortest successful referring expression. Formally, A &gt;&gt;FB B if length(A) &lt; length(B). The task of finding a maximal element of &gt;&gt;FB, i.e., of finding the shortest successful referring expression, is NP-Hard. This result holds for all definitions of length the author has examined (number of open-class words, number of words, number of characters, number of components). null To prove this, let Target-Components denote those components (classifications and attribute-value pairs) of Target that are mutually known by the speaker and the hearer. For each Tj in Target-Components, let Rules-Out(Tj) be the members of Excluded that do not possess Tj (so, the presence of Tj in a referring expression 'rules out' these members). Then, consider a potential referring expression, RE = {Ct ..... C,}. RE will be a successful referring expression if and only if a) Every Ci is in Target-Components b) The union of Rules-Out(Ci), for all Ci in RE, is equal to Excluded.</Paragraph>
      <Paragraph position="1"> For example, if the task was referring to object B in the example context of Section 2, then Target-Components would be {class:Chair, Material:Wood, Color:Brown}, Excluded would be {A, C}, and</Paragraph>
      <Paragraph position="3"> brown chair&amp;quot;) would be a successful referring expression for object B in this context.</Paragraph>
      <Paragraph position="4"> If description length is measured by number of components, 7 finding the minimal length referring expression is equivalent to solving a minimum set cover problem, where Excluded is the set being covered, and the Rules-Out(Tj) are the covering sets. Unfortunately, finding a minimal set cover is an NP-</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="99" end_page="99" type="metho">
    <SectionTitle>
7 Dale's (1989) minimal distinguishing descriptions are,
</SectionTitle>
    <Paragraph position="0"> in the terminology of this paper, successful referring expressions that are maximal under Full Brevity when number of components is used as the measure of description length.</Paragraph>
    <Paragraph position="1"> Therefore, finding a minimal distinguishing description is an NP-Hard problem. The algorithm Dale used was essentially equivalent to the greedy heuristic for minimal set cover (Johnson 1974); as such it ran quickly, but did not always find a tree minimal distinguishing description.</Paragraph>
    <Paragraph position="2"> Hard problem (Garey and Johnson 1979), and thus solving it is in general computationally intractable (assuming that P ~ NP).</Paragraph>
    <Paragraph position="3"> Similar proofs will work for the other definitions of length mentioned above. On an intuitive level, the basic problem is that finding the shortest description requires searching for the global minimum of the length function, and this global minimum (like many global minima) may be very expensive to locate.</Paragraph>
    <Section position="1" start_page="99" end_page="99" type="sub_section">
      <SectionTitle>
4.2. Local Brevity
</SectionTitle>
      <Paragraph position="0"> The Local Brevity preference rule is a weaker interpretation of Grice's brevity submaxim. It states that it should not be possible to generate a shorter successful referring expression by replacing a set of components by a single new componenL Formally, &gt;&gt;us is the transitive closure of &gt;&gt;us', where A &gt;&gt;us,</Paragraph>
      <Paragraph position="2"> length(A) &lt; length(B). The best definition of length(A) is probably the number of open-class words in the surface realization of A.</Paragraph>
      <Paragraph position="3"> Local brevity can be checked by selecting a potential new component, finding all minimal sets of old components whose combined length is greater than the length of the new component, performing the substitution, and checking if the result is a suecessful referring expression. This can be done in polynomial time if the number of minimal sets is polynomial in the length of the description, which will happen if (non-zero) upper and lower bounds are placed on the length of any individual component (e.g., the surface realization of every component must use at least one open-class word, but no more than some fixed number of open-class words).</Paragraph>
      <Paragraph position="4"> element is defined: detecting unnecessary words in referring expressions is NP-Hard (Section 5.1), but unnecessary components can always be found in polynomial time (Section 5.2).</Paragraph>
      <Paragraph position="5"> 5.1. No Unnecessary Words The No Unnecessary Words preference rule forbids referring expressions from containing unnecessary words. Formally, A &gt;&gt;ow B if A's surface form uses a subset of the words used by B's surface form. There are several variants, such as only considering open-class words, or requiring the words in B to be in the same order as the corresponding words in A. All of these variants make the generation problem NP-Hard.</Paragraph>
      <Paragraph position="6"> The formal proofs are in Reiter (1990b). Intuitively, the basic problem is that any preference that is stated solely in terms of surface forms must deal with the possibility that new parses and semantic interpretations may arise when the surface form is modified. This means that the only way a generation system can guarantee that an utterance satisfies the No Unnecessary Words rule is to generate all possible subsets of the surface form, and then run each subset through a parser and semantic interpreter to check if it happens to be a successful referring expression.</Paragraph>
      <Paragraph position="7"> The number of subsets of the surface form is exponential in the size of the surface form, so this process will take exponential time.</Paragraph>
      <Paragraph position="8"> To illustrate the 'new parse' problem, consider two possible referring expressions:  4a) &amp;quot;the child holding a pumpkin&amp;quot; 4b) &amp;quot;the child holding a slice of pumpkin pie&amp;quot; 5. No Unnecessary Elements  The Gricean maxims of Quantity and Relevance prohibit utterances from containing elements that are unnecessary for fulfilling the speaker's communicative goals. The undesirability of unnecessary elements is further supported by the observation that humans find pleonasms (Cruse 1986) such as &amp;quot;a female mother&amp;quot; and &amp;quot;an unmarried bachelor&amp;quot; to be anomalous. The computational tractability of the no-unnecessary-elements principle depends on how 8 This is a set formula, where &amp;quot;-* means set-difference and &amp;quot;size&amp;quot; means nmnher of members. The formula requires A to have exactly one COmlx~ent that is not present in B; B can have an ~oitra W number of components that are not present in A.</Paragraph>
      <Paragraph position="9">  If utterances (4a) and (4b) were both successful referring expressions (i.e., the child had a pumpkin in one hand, and a slice of pumpkin pie in the other), then (4a) &gt;&gt;ow (4b) under any of the variants mentioned above. However, because utterance (4a) has a different syntactic structure than utterance (4b), the only way the generation system could discover that (4a) &gt;&gt;vw (4b) would be by constructing utterance (4b)'s surface form, removing the words &amp;quot;slice,&amp;quot; &amp;quot;of,&amp;quot; and &amp;quot;pie&amp;quot; from it, and analyzing the reduced surface form.</Paragraph>
      <Paragraph position="10"> This problem, of new parses and semantic interpretations being uncovered by modifications to the surface form, causes difficulties whenever a preference rule is stated solely in terms of the surface form. Accordingly, such preference rules should be avoided.</Paragraph>
    </Section>
    <Section position="2" start_page="99" end_page="99" type="sub_section">
      <SectionTitle>
5.2. No Unnecessary Components
</SectionTitle>
      <Paragraph position="0"> The No Unnecessary Components preference rule forbids referring expressions from containing unnecessary components. Formally, A &gt;&gt;uc B if A uses a a subset of the components used by B.</Paragraph>
      <Paragraph position="1"> Unnecessary components can be found in polynomial time by using a simple incremental algorithm that just removes each component in turn, and checks if what is left constitutes a successful referring expression.</Paragraph>
      <Paragraph position="2"> The key algorithmic difference between No Unnecessary Components and No Unnecessary Words is that this simple incremental algorithm will not work for the No Unnecessary Words preference rule. This is because there are cases where removing any single word from an utterance's surface form wifl leave an unsuccessful (or incoherent) referring expression (e.g., imagine removing just &amp;quot;slice&amp;quot; from utterance (4b)), but removing several words will uncover a new parse that corresponds to a successful referring expression. In contrast, if B is a successful referring expression, and there exists another suecessful referring expression A that satisfies components(A) c components(B) (and hence A is preferred over B under the No Unnecessary Components preference rule), then it will be the case that any referring expression C that satisfies components(A) c components(C) c components(B) will also be successful. This means that the simple algorithm can always produce A from B by incremental steps that remove a single component at a time, because the intermediate descriptions formed in this process will always be successful referring expressions. Therefore, the simple incremental algorithm will always find unnecessary components, but may not always find unnecessary words.</Paragraph>
    </Section>
  </Section>
  <Section position="7" start_page="99" end_page="102" type="metho">
    <SectionTitle>
6. Lexlcal Preference
</SectionTitle>
    <Paragraph position="0"> If the attribute values and classifications used in the description are members of a taxonomy, then they can be realized at different levels of specificity.</Paragraph>
    <Paragraph position="1"> For example, the object in the parking lot outside the author's window might be called &amp;quot;a vehicle,&amp;quot; &amp;quot;a motor vehicle,&amp;quot; &amp;quot;a car,&amp;quot; &amp;quot;a sports car,&amp;quot; or &amp;quot;a Porsche.&amp;quot; The Lexical Preference rule assumes there is a lexical-preference hierarchy among the taxonomy's lexical classes (classes that can be realized with single lexical units). The rule states that utterances should use preferred lexical classes whenever possible. Formally, A &gt;&gt;t. B if for every component in A, that is a component in B that has the same structure,  and the lexieal class used by the A component is equal to or lexically preferred over the lexical class used by the B component.</Paragraph>
    <Paragraph position="2"> The lexical-preference hierarchy should, at minimum, incorporate the following preferences: i) Lexical class A is preferred over lexical class B if A's realization uses a subset of the open-class words used in B's realization. For example, the class with realization ``vehicle&amp;quot; is preferred over the class with realization &amp;quot;motor vehicle.&amp;quot; ii) Lexical class A is preferred over lexical class B if A is a basic-level class, and B is not. For example, if car was a basic-level class, then &amp;quot;a car&amp;quot; would be preferred over ``a vehicle&amp;quot; or ``a porsche. &amp;quot;9 In some cases these two preferences may conflict; this is discussed in Section 7.2.</Paragraph>
    <Paragraph position="3"> Utterances that violate either preference (i) or preference (ii) may implicate unwanted implicatures. Preference rule (ii) has been discussed by Cruse (1977) and Hirschberg (1985). Preference rule (i) may be considered to be another application of the Gricean maxim of quantity, and is illustrated by the following utterances:  If utterances (5a) and (5b) were both successful referring expressions (e.g., if the speaker possessed only one ear), then the use of utterance (5b) would implicate that the speaker wished to emphasize that his vehicle was a sports car, and not some other kind of car.</Paragraph>
    <Paragraph position="4"> From an algorithmic point of view, referring expressions that are maximal under the lexical-preference criteria can be found in polynomial time if the following restriction is imposed on the lexical-preference hierarchy: Restriction: If lexical class A is preferred over lexical class B, then A must either subsume B or be subsumed by B in the class taxonomy.</Paragraph>
    <Paragraph position="5"> For example, it is acceptable for car to be preferred over vehicle or Porsche, but it is not acceptable for car to be preferred over gift (because car neither subsumes nor is subsumed by g~ft).</Paragraph>
    <Paragraph position="6"> If the above reslriction holds, a variant of the simple incremental algorithm of Section 5.2 may be used to implement lexical preference: the algorithm simply attempts each replacement that lexical preference suggests, and checks if this results in a successful referring expression. If the restriction does not hold, then the simple incremental algorithm may fall, and obeying the Lexical Preference rule is in fact N-P-Hard (the formal proof is in Reiter (1990b)).</Paragraph>
  </Section>
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