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<Paper uid="P88-1021">
  <Title>A Practical Nonmonotonic Theory for Reasoning about Speech Acts</Title>
  <Section position="5" start_page="171" end_page="171" type="metho">
    <SectionTitle>
2 Perrault's Default Theory
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="171" end_page="171" type="sub_section">
      <SectionTitle>
of Speech Acts
</SectionTitle>
      <Paragraph position="0"> As an alternative to monotonic theories, Perrault has proposed a theory of speech acts, based on an extension of Reiter's default logic \[11\] extended to include-defanlt-rule schemata. We shall summarize Perrault's theory briefly as it relates to informing and belief. The notation p =~ q is intended as an abbreviation of the default rule of inference, p:Mq q Default theories of this form are called normal. Every normal default theory has at least one extension, i.e., a mutually consistent set of sentences sanctioned by the theory.</Paragraph>
      <Paragraph position="1"> The operator Bz,t represents Agent z's beliefs at time t and is assumed to posess all the properties of the modal system weak $5 (that is, $5 without the schema Bz,t~ D ~b), plus the following axioms:</Paragraph>
      <Paragraph position="3"> In addition, there is a default-rule schema stating that, if p =~ q is a default rule, then so is B~,~p =~ Bx,tq for any agent z and time t.</Paragraph>
      <Paragraph position="4"> Perrault could demonstrate that, given his theory, there is an extension containing all of the desired conclusions regarding the beliefs of the speaker and hearer, starting from the fact that a speaker utters a declarative sentence and the hearer observes him uttering it. Furthermore, the theory can make correct predictions in cases in which the usual preconditions of the speech act do not obtain. For example, if the speaker is lying, but the hearer does not recognize the lie, then the heater's beliefs are exactly the same as when the speaker tells the truth; moreover the speaker's beliefs about mutual belief are the same, but he still does not believe the proposition he uttered m that is, he fails to be convinced by his own lie.</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="171" end_page="172" type="metho">
    <SectionTitle>
3 Problems with Perrault's
</SectionTitle>
    <Paragraph position="0"> Theory A serious problem arises with Perrault's theory concerning reasoning about an agent's ignorance. His theory predicts that a speaker can convince himself of any unsupported proposition simply by asserting it, which is clearly at odds with our intuitions. Suppose that it is true of speaker s that ~Bs,tP. Suppose furthermore that, for whatever reason, s utters P. In the absence of any further information about the speaker's and hearer's beliefs, it is a consequence of axioms (1)-(5) that Bs,~+IBh,~+IP. From this consequence and the belief transfer rule (4) it is possible to conclude B,,~+IP. The strongest conclusion that can be derived about s's beliefs at t + 1 without using  this default rule is B,,t+I&amp;quot;~B,,~P, which is not sufficient to override the default.</Paragraph>
    <Paragraph position="1"> This problem does not admit of any simple fixes. One clearly does not want an axiom or default rule of the form that asserts what amounts to &amp;quot;ignorance persists&amp;quot; to defeat conclusions drawn from speech acts. In that case, one could never conclude that anyone ever learns anything as a result of a speech act. The alternative is to weaken the conditions under which the default rules can be defeated. However, by adopting this strategy we are giving up the advantage of using normal defaults. In general, nonnormal default theories do not necessarily have extensions, nor is there any proof procedure for such logics.</Paragraph>
    <Paragraph position="2"> Perrault has intentionally left open the question of how a speech act theory should be integrated with a general theory of action and belief revision. He finesses this problem by introducing the persistence axiom, which states that beliefs always persist across changes in state. Clearly this is not true in general, because actions typically change our beliefs about what is true of the world. Even if one considers only speech acts, in some cases * one can get an agent to change his beliefs by saying something, and in other cases not. Whether one can or not, however, depends on what belief revision strategy is adopted by the respective agents in a given situation. The problem cannot be solved by simply adding a few more axioms and default rules to the theory. Any theory that allows for the possibility of describing belief revision must of necessity confront the problem of inconsistent extensions. This means that, if a hearer initially believes -~p, the default theory will have (at least) one extension for the case in which his belief that -~p persists, and one extension in which he changes his mind and believes p. Perhaps it will even have an extension in which he suspends belief as to whether p.</Paragraph>
    <Paragraph position="3"> The source of the difficulties surrounding Perranlt's theory is that the default logic h e adopts is unable to describe the attitude revision that occurs in consequence of a speech act. It is not our purpose here to state what an agent's belief revision strategy should be. Rather we introduce a framework within which a variety of belief revision strategies can be accomodated efficiently, and we demonstrate that this framework can be applied in a way that eliminates the problems with Perranlt's theory.</Paragraph>
    <Paragraph position="4"> Finally, there is a serious practical problem faced by anyone who wishes to implement Perfault's theory in a system that reasons about speech acts. There is no way the belief transfer rule can be used efficiently by a reasoning system; even if it is assumed that its application is restricted to the speaker and hearer, with no other agents in the domain involved. If it is used in a backward direction, it applies to its own result. Invoking the rule in a forward direction is also problematic, because in general one agent will have a very large number of beliefs (even an infinite number, if introspection is taken into account) about another agent's beliefs, most of which will be irrelevant to the problem at hand.</Paragraph>
  </Section>
  <Section position="7" start_page="172" end_page="173" type="metho">
    <SectionTitle>
4 Hierarchic Autoepistemic
Logic
</SectionTitle>
    <Paragraph position="0"> Autoepistemic (AE) logic was developed by Moore \[I0\] as a reconstruction of McDermott's nonmonotonic logic \[9\]. An autoepistemic logic is based on a first-order language augmented by a modal operator L, which is interpreted intuitively as self belief. A stable ezpansio, (analogous to an extension of a default theory) of an autoepistemic base set A is a set of formulas T satisfying the following conditions:  1. T contains all the sentences of the base theory A 2. T is closed under first-order consequence 3. If ~b E T, then L~b E T 4. If C/ ~ T, then --L~b 6 T  Hierarchic autoepistemic logic (HAEL) was developed in response to two deficiences of autoepistemic logic, when the latter is viewed as a logic for automated nonmonotonic reasoning. The first is a representational problem: how to incorporate preferences among default inferences in a natural way within the logic. Such preferences arise in many disparate settings in nonmonotonic reasoning -- for example, in taxonomic hierarchies \[6\] or in reasoning about events over time \[12\]. To some extent, preferences among defaults can be  encoded in AE logic by introducing auxiliary information into the statements of the defaults, but this method does not always accord satisfactorily with our intuitions. The most natural statement of preferences is with respect to the multiple expansions of a particular base set, that is, we prefer certain expansions because the defaults used in them have a higher priority than the ones used in alternative expansions.</Paragraph>
    <Paragraph position="1"> The second problem is computational: how to tell whether a proposition is contained within the desired expansion of a base set. As can be seen from the above definition, a stable expansion of an autoepistemic theory is defined as a fixedpoint; the question of whether a formula belongs to this fixedpoint is not even semidecidable. This problem is shared by all of the most popular nonmonotonic logics. The usual recourse is to restrict the expressive power of the language, e.g., normal default theories \[11\] and separable circumscriptive theories \[8\]. However, as exemplified by the difficulties of Perrault's approach, it may not be easy or even possible to express the relevant facts with a restricted language.</Paragraph>
    <Paragraph position="2"> Hierarchical autoepistemic logic is a modification of autoepistemic logic that addresses these two considerations. In HAEL, the primary structure is not a single uniform theory, but a collection of subtheories linked in a hierarchy. Snbtheories represent different sources of information available to an agent, while the hierarchy expresses the way in which this information is combined. For example, in representing taxonomic defaults, more specific information would take precedence over general attributes. HAEL thus permits a natural expression of preferences among defaults. Furthermore, given the hierarchical nature of the subtheory relation, it is possible to give a constructive semantics for the autoepistemic operator, in contrast to the usual self-referential fixedpoints. We can then arrive easily at computational realizations of the logic.</Paragraph>
    <Paragraph position="3"> The language of HAEL consists of a standard first-order language, augmented by a indexed set of unary modal operators Li. If ~b is any sentence (containing no free variables) of the first-order language, then L~ is also a sentence. Note that neither nesting of modal operators nor quantifying into a modal context is allowed. Sentences without modal operators are called ordinary.</Paragraph>
    <Paragraph position="4"> An HAEL structure r consists of an indexed set of subtheories rl, together with a partial order on the set. We write r~ -&lt; rj if r~ precedes rj in the order. Associated with every subtheory rl is a base set Ai, the initial sentences of the structure. Within A~, the occurrence of Lj is restricted by the following condition: If Lj occurs positively (negatively) in (6) Ai, then rj _ r~ (rj -&lt; ri).</Paragraph>
    <Paragraph position="5"> This restriction prevents the modal operator from referring to subtheories that succeed it in the hierarchy, since Lj~b is intended to mean that ~b is an element of the subtheory rj. The distinction between positive and negative occurrences is simply that a subtheory may represent (using L) which sentences it contains, but is forbidden from representing what it does not contain.</Paragraph>
    <Paragraph position="6"> A complez stable e~pansion of an HAEL structure r is a set of sets of sentences 2~ corresponding to the subtheories of r. It obeys the following conditions (~b is an ordinary sentence):  1. Each T~ contains Ai 2. Each Ti is closed under first-order consequence null 3. If eEl, and ~'j ~ rl, then Lj~b E~ 4. If C/ ~ ~, and rj -&lt; rl, then -,Lj ~b E 5. If ~ E Tj, and rj -&lt; vi, then ~bE~.</Paragraph>
    <Paragraph position="7">  These conditions are similar to those for AE stable expansions. Note that, in (3) and (4), 2~ contains modal atoms describing the contents of subtheories beneath it in the hierarchy. In addition, according to (5) it also inherits all the ordinary sentences of preceeding subtheories.</Paragraph>
    <Paragraph position="8"> Unlike AE base sets, which may have more than one stable expansion, HAEL structures have a unique minimal complex stable expansion (see Konolige \[7\]). So we are justified in speaking of &amp;quot;the&amp;quot; theory of an HAEL structure and, from this point on, we shall identify the subtheory r~ of a structure with the set of sentences in the complex stable expansion for that subtheory.</Paragraph>
    <Paragraph position="9"> Here is a simple example, which can be interpreted as the standard &amp;quot;typically birds fly&amp;quot; default  scenario by letting F(z) be &amp;quot;z flies,&amp;quot; B(z) be &amp;quot;z is a bird,&amp;quot; and P(z) be &amp;quot;z is a penguin.&amp;quot;</Paragraph>
    <Paragraph position="11"> Theory r0 contains all of the first-order consequences of P(a), B(a), LoP(a), and LoB(a).</Paragraph>
    <Paragraph position="12"> -~LoF(a) is not in r0, hut it is in rl, as is LoP(a), -LooP(a), etc. Note that P(a) is inherited by rl; hence L1P(a) is in rl. Given this, by first-order closure &amp;quot;,F(a) is in rl and, by inheritance, LI&amp;quot;,F(a) is in r2, so that F(a) cannot be derived there. On the other hand, r2 inherits &amp;quot;,F(a) from rl.</Paragraph>
    <Paragraph position="13"> Note from this example that information present in the lowest subtheories of the hierarchy percolates to its top. More specific evidence, or preferred defaults, should be placed lower in the hierarchy, so that their effects will block the action of higher-placed evidence or defaults.</Paragraph>
    <Paragraph position="14"> HAEL can be given a constructive semantics that is in accord with the closure conditions.</Paragraph>
    <Paragraph position="15"> W'hen the inference procedure of each subtheory is decidable, an obvious decidable proof method for the logic exists. The details of this development are too complicated to be included here, but are described by Konolige \[7\]. For the rest of this paper, we shall use a propositional base language; the derivations can be readily checked.</Paragraph>
  </Section>
  <Section position="8" start_page="173" end_page="176" type="metho">
    <SectionTitle>
5 A HAEL Theory of Speech
Acts
</SectionTitle>
    <Paragraph position="0"> We demonstrate here how to construct a hierarchic autoepistemic theory of speech acts. We assume that there is a hierarchy of autoepisternic subtheories as illustrated in Figure i. The lowest subtheory, ~'0, contains the strongest evidence about the speaker's and hearer's mental states. For example, if it is known to the hearer that the speaker is lying, this information goes into r0.</Paragraph>
    <Paragraph position="1"> In subtheory vl, defaults are collected about the effects of the speech act on the beliefs of both hearer and speaker. These defaults can be overridden by the particular evidence of r0. Together r0 and rl constitute the first level of reasoning about the speech act. At Level 2, the beliefs of the speaker and hearer that can be deduced in rl are used as evidence to guide defaults about nested beliefs, that is, the speaker's beliefs about the heater's beliefs, and vice versa. These results are collected in r2. In a similar manner, successive levels contain the result of one agent's reflection upon his and his interlocutor's beliefs and intentions at the next lower level. We shall discuss here how Levels r0 and rl of the HAEL theory are axiomatized, and shall extend the axiomatization to the higher theories by means of axiom schemata.</Paragraph>
    <Paragraph position="2"> An agent's belief revision strategy is represented by two features of the model. The position of the speech act theory in the general hierarchy of theories determines the way in which conclusions drawn in those theories can defeat conclusions that follow from speech acts. In our model, the speech act defaults will go into the subtheory rl, while evidence that will be used to defeat these defaults will go in r0. In addition, the axioms that relate rl to r0 determine precisely what each agent is willing to accept from 1&amp;quot;0 as evidence against the default conclusions of the speech act theory.</Paragraph>
    <Paragraph position="3"> It is easy to duplicate the details of Perrault's analysis within this framework. Theory r0 would contain all the agents' beliefs prior to the speech act, while the defaults of rl would state that an agent believed the utterance P if he did not believe its negation in r0. As we have noted, this analysis does not allow for the situation in which the speaker utters P without believing either it or its opposite, and then becomes convinced of its truth by the very fact of having uttered it m nor does it allow the hearer to change his belief in -~P as a result of the utterance.</Paragraph>
    <Paragraph position="4"> We choose a more complicated and realistic expression of belief revision. Specifically, we allow an agent to believe P (in rl) by virtue of the utterance of P only if he does not have any evidence (in r0) against believing it. Using this scheme, we can accommodate the hearer's change of belief, and show that the speaker is not convinced by his own efforts.</Paragraph>
    <Paragraph position="5"> We now present the axioms of the HAEL theory for the declarative utterance of the proposition P.</Paragraph>
    <Paragraph position="6"> The language we use is a propositional modal one  for the beliefs of the speaker and hearer. Agents s and h represent the speaker and hearer; the subscripts i and f represent the initial situation and the situation resulting from the utterance, respectively. There are two operators: \[a\] for a's belief and {a} for a's goals. The formula \[hI\]c~, for example, means that the hearer believes ~b in the final situation, while {si}C/ means that the speaker intended ~b in the initial situation. In addition, we use a phantom agent u to represent the content of the utterance and certain assumptions about the speaker's intentions. We do not argue here as to what constitutes the correct logic of these operators; a convenient one is weak $5.</Paragraph>
    <Paragraph position="7"> The following axioms are assumed to hold in all subtheories.</Paragraph>
    <Paragraph position="8"> \[u\]P, P the propositional content of ut- (8) terance \[~\]C/ D \[~\]{s~}\[hAC/ (9) \[a\]{a}C/ ~. {a}C/, where a is any (10) agent in any situation. null The contents of the u theory are essentially the same for all types of speech acts. The precise effects upon the speaker's and heater's mental states is determined by the propositional content of the utterance and its mood. We assume here that the speaker utters, a simple declarative sentence, (Axiom 8), although a similar analysis could be done for other types of sentences, given a suitable representation of their propositional content. Propositions that are true in u generally become believed by the speaker and hearer in rl, provided that these propositions bear the proper relationship to their beliefs in r0. Finally, the speaker in$ends to bring about each of the beliefs the hearer acquires in rl, also subject to the caveat that it is consistent with his beliefs in to.</Paragraph>
    <Paragraph position="9"> Relation between subtheories: ro -~ n (11) Speaker's beliefs as a consequence of the speech act: in AI: \[u\]C/ A -~L0-~\[sl\]C/ D \[s/\]C/ (12) Level 1  Hearer's beliefs as a consequence of the speech act: in AI:</Paragraph>
    <Paragraph position="11"> The asymmetry between Axioms 12 and 13 is a consequence of the fact that a speech act has different effects on the speaker's and hearer's mental states. The intuition behind these axioms is that a speech act should never change the speaker's mental attitudes with regard to the proposition he utters. If he utters a sentence, regardless of whether he is lying, or in any other way insincere, he should believe P after the utterance if and only if he believed it before. However, in the bearer's case, whether he believes P depends not only on his prior mental state with respect to P, but also on whether he believes that the speaker is being sincere. ~iom 13 states that a hearer is willing to believe what a speaker says if it does not conflict with his own beliefs in ~, and if the utterance does not conflict with what the hearer believes about the speaker's mental state, (i.e., that the speaker is not lying), and if he believes that believing P is consistent with his beliefs about the speaker's prior intentions (i.e., that the speaker is using the utterance with communicative intent, as distinct from, say, testing a microphone).</Paragraph>
    <Paragraph position="12"> As a first example of the use of the theory, consider the normal case in which A0 contains no evidence about the speaker's and bearer's beliefs after the speech act. In this event, A0 is empty and A1 contains Axioms 8-1b. By the inheritance conditions, 1&amp;quot;1 contains -~L0-,\[sl\]P , and so must contain \[s/\]P by axiom 12. Similarly, from Axiom 13 it follows that \[h/\]P is in rl. Further derivations lead to {sl}\[hl\]P , {si}\[hl\]{si}\[hy\]P , and so on.</Paragraph>
    <Paragraph position="13"> As a second example, consider the case in which the speaker utters P, perhaps to convince the hearer of it, but does not himself believe either P or its negation. In this case, 1&amp;quot;0 contains -~\[sf\]P and -~\[sl\]-~P , and ~'1 must contain Louis tiP by the inheritance condition. Hence, the application of Axiom 12 will be blocked, and so we cannot conclude in ~'1 that the speaker believes P. On the other hand, since none of the antecedents of Axiom 13 are affected, the hearer does come to believe it.</Paragraph>
    <Paragraph position="14"> Finally, consider belief revision on the part of the hearer. The precise path belief revision takes depends on the contents of r0. If we consider the hearer's belief to be stronger evidence than that of the utterance, we would transfer the heater's initial belief \[hl\]~P to \[h/\]'-,P in ~'0, and block the default Axiom 13. But suppose the hearer does not believe --P strongly in the initial situation. Then  we would transfer (by default) the belief \[h\]\]~P to a subtheory higher than rl, since the evidence furnished by the utterance is meant to override the initial beliefi Thus, by making the proper choices regarding the transfer of initial beliefs in various subtheories, it becomes possible to represent, the revision of the hearer's beliefs.</Paragraph>
    <Paragraph position="15"> This theory of speech acts has been presented with respect to declarative sentences and representative speech acts. To analyze imperative sentences and directive speech acts, it is clear in what direction one should proceed, although the required augmentation to the theory is quite complex. The change in the utterance theory that is brought about by an imperative sentence is the addition of the belief that the speaker intends the hearer to bring about the propositional content of the utterance. That would entail substituting the following effect for that stated by Axiom 8: \[u\]{s/}P, P the propositional con- (14) tent of utterance One then needs to axiomatize a theory of intention revision as well as belief revision, which entails describing how agents adopt and abandon intentions, and how these intentions are related to their beliefs about one another. Cohen and Levesque have advanced an excellent proposal for such a theory \[4\], but any discussion of it is far beyond the scope of this article.</Paragraph>
  </Section>
  <Section position="9" start_page="176" end_page="177" type="metho">
    <SectionTitle>
6 Reflecting on the Theory
</SectionTitle>
    <Paragraph position="0"> When agents perform speech acts, not only are their beliefs about the uttered proposition affected, but also their beliefs about one another, to arbitrayr levels of reflection.</Paragraph>
    <Paragraph position="1"> If a speaker reflects on what a hearer believes about the speaker's own beliefs, he takes into account not only the beliefs themselves, but also what he believes to be the hearer's belief revision strategy, which, according to our theory, is reflected in the hierarchical relationship among the theories. Therefore, reflection on the speech-actunderstanding process takes place at higher levels of the hierarchy illustrated in Figure 1. For example, if Level 1 represents the speaker's reasoning about what the hearer believes, then Level 2 rep- null resents the speaker's reasoning about the heater's beliefs about what the speaker' believes.</Paragraph>
    <Paragraph position="2"> In general, agents may have quite complicated theories about how other agents apply defaults.</Paragraph>
    <Paragraph position="3"> The simplest assumption we can make is that they reason in a uniform manner, exactly the same as the way we axiomatized Level 1. Therefore, we extend the analysis just presented to arbitrary reflection of agents on one another's belief by proposing axiom schemata for the speaker's and heater's beliefs at each level, of which Axioms 12 and 13 are the Level 1 instances. We introduce a schematic operator \[(s, h)n\] which can be thought of as n levels of alternation of s's and h's beliefs about each other. This is stated more precisely as \[(8, h),,\]C/ (is) n times Then, for example, Axiom 12 can be restated as</Paragraph>
  </Section>
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