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<Paper uid="C94-2190">
  <Title>Towards a Dynamic Theory of Belief-Sharing in Cooperative Dialogues</Title>
  <Section position="2" start_page="0" end_page="7166" type="metho">
    <SectionTitle>
2 Dynamic Maintenance
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="0" end_page="1164" type="sub_section">
      <SectionTitle>
Shared Beliefs
of
2.1 DRS
</SectionTitle>
      <Paragraph position="0"> However cooperative, real-world dialogues are fraught with hedges, understatements, or even white lies, which would necessitate introducing a distinction between what is literally conveyed by an utterance, and its real intent on the part of both speaker and hearer.</Paragraph>
      <Paragraph position="1"> In this study, however, we restrict ourselves to those cases without such complications, and assume that an utterance reflects the speaker's intent in a straight manner, and is taken as such by the hearer. The content of an utterance is represented in tile following style:</Paragraph>
      <Paragraph position="3"> We call K discourse representation structure (DRS), {a, b, x,y,z .... } K's domain (UK), the elements of UI&lt; discourse referents, tile boxed area below tile unbroken line K's condition part (CA-), and CK's elements conditions. K is represented as (UK,CK}. The broken line divides CK into the self-referentiN part SRP (above the line), and the dialogue database DB(K) (below the line). A condillon is the result of an n(_&gt; 0) times application of Bel(ct,.) to a first-order formula p. Bel(c~,.) is called a belief operator, where c~ designates the utterer. Given C/ as a condition, Bel(ce, C/) reads &amp;quot;the partMpant a believes C/.&amp;quot; n is called the rank of C/ with regard to its embedding within belief operators. Conditions of rank O are called bare formulas, while those with a rank greater than 0 belief formulas. K represents the shared beliefs formed through a dialogue between the two participants a and b. The conditions in SRP indicate a recursive embedding of self-referentiN belief sturueture with regard to eo,nmon knowledge, and are assumed throughout the dialogue. By contrast, DB(K) is empty when a dialogue starts off. Thus, at the outset of a dialogue, the DRS Ko = ({a, b}, {Bel(a, Ko), Bel(b, K0)}). As an utterante is made, new discourse entities may be introduced, making it necessary to add new conditions to DB(K) and sometimes to retract or negate part of the conditions in DB(K). With the progress of the dialogue, the DRS changes front K0 ~ K1 ~ ... =~ Kn =:~ ...</Paragraph>
      <Paragraph position="4"> Since only cooperative dialogues are considered, the goal is to arrive at a DRS in which no contradictory beliefs arc held by the participants. But this goN is not Mways achieved. We Nso assume that at certain points of a dialogue, the participants can hold contradictory beliefs, and that tile same pariticipant  Call hold contradictory beliefs at dill?rent points of a dialogue, whereas the s;Hne particip~mt cannot hold contradictory beliefs C/tt any partic:ular point.</Paragraph>
      <Paragraph position="5"> In what follows, we just indicate DB(K) unless otherwise noted.</Paragraph>
    </Section>
    <Section position="2" start_page="1164" end_page="1164" type="sub_section">
      <SectionTitle>
2.2 How shared belieig are registered
</SectionTitle>
      <Paragraph position="0"> An utterance made by ~ 1)~rticipmlt in a. diMogue is transformed into a condition(s) and registered in DB(K), folh)wing the constrMnts stetted beh)w.</Paragraph>
      <Paragraph position="1"> First, discourse referents m'e taken to be epistemological entities without counterparts in snrfacc senten('es, but introduced into the DRS by the participants of ~ diMogue, and of which prot)erties corresponding to surface linguistic expressions are l)redi null cared. Thus, an utterance (2) a: Sato is a student is not anMyzed ~s (3) student(Snto) but as (4) Sato(x), student(x)  with the discourse referent a&amp;quot; introdu(:ed into U~C/ 1)y a, and the predicates corresponding to expressions in the utterauce.</Paragraph>
      <Paragraph position="2"> Second, an utterance is registered not in the form of a bare formula., but in the fl)rm of a 1)elief forlnula indicating the t)elief agent. (4), tbr example, is registered as  (5) .Bcl(a, Sato(x) ), Bel(a, student(x)) because at (2), b has not agreed with or opl)osed a's utterance. Note theft (5) is nevertheless ;t shared belief t~t this point. Suppose (6) is uttered folh)wing upon (2): (6) b: Yes, he is.</Paragraph>
      <Paragraph position="3"> This utterance is interpreted as (7) Bet(b, Sato(x)), BelCh, stu.dent(x)) and so registered in DB(K). At this t,oint, both (5) and (7) are shared beliefs, which me,ms (4) is a belief shared by a and b. This transition is tbrumlated as the axiom of shared belief: (8) The axiom of shared belief Whet, DUCK) cont~ns P,q(,*,v), ,~nd 13~l(b, v), DB(K') obtained from DB(K) 1)y the substitu null tion of p for them is equivahmt to DB(K).</Paragraph>
      <Paragraph position="4"> DB(K) can bc derived fl'om DB(K') without using this axiom, since K tt~us the self-referential part SRP. But the converse does not hold. The ttxiom of shared belief Mlows the rank of shared beliefs to be zero, while the conditions in general are initially registered with a rtmk higher than zero.</Paragraph>
      <Paragraph position="5"> Third, there is involved a step of identification in the transition front b's utterance of (6) to the condition (7). Just as the discourse referent x was introduced by a's utterance of (2), b introduces a (listinct discourse referent y, in terms of which  (9) BelCh, S.to(y)), Bed(b, student(y)) is registered in DB(K). We ~uSSulne that a and b  ~gree to tiu', identity of x and y ~Lt this point. To sum up, in dialogue (2), (6), DB(K) is composed of (5) ahme when (2) is uttered, hut is extended by the utterance of (6) as follows:  0()) wt(,,,:,: = y),B,~t(~,.~ = v), Bed(a, Sato( x ) ), Bel( a, Sato(y) ), ~l(b, s,,to(,) ), ~l(b, S~to(v) ), Bel(a, student(:r) ), Bet(a, student(y)), Bel( b, student(a,)), BelCh, student(y)).</Paragraph>
      <Paragraph position="6">  By applying the axioin of shared belief, mid x =: y, we obta.in (11) Sat@c), student(x) By contr~st, (12) l.a: Satoisastudent.</Paragraph>
      <Paragraph position="7"> 2.b: No, he is an otfice clerk now.</Paragraph>
      <Paragraph position="8"> can only h~vc its DB(K) reduced to (13) Sato(:,:), lJel(a, ,st~dcm.t(x)), Bel(h,oJlice_cle, rk(x)). 3 Diachronic analysis of dialogue null  In this section, we consider the changes DRS's undergo in the course, of ~t dialogue. In (2), (6) in the previous section, we saw a case where a DRS with nothing but shm'ed beliefs is successfiflly obtMned in one inning, so to speak, without incurring any conflict. We will look at the other three kinds of cases in which conflicts are treated in particular ways which tMlnit of formMization in terms of CMS.</Paragraph>
    </Section>
    <Section position="3" start_page="1164" end_page="1165" type="sub_section">
      <SectionTitle>
3.1 Direct solution of conflicts
</SectionTitle>
      <Paragraph position="0"> Consider the following dialogue.</Paragraph>
      <Paragraph position="1">  (1.4) 1. I~: Sato is ~t good guy. 2. b: By no means, he is a liar. 3. a: No kidding.</Paragraph>
      <Paragraph position="2"> Just after (14.2) is uttered, DB(K) looks as follows: (1~) w~t(.,:,; : :,j),Vel(b,. = y),  B a(c,, s~,to(.)), u~t(c~, st, to(v)), Uet(*,, S.to(.~-)), ~et(b, Sato(v ) ), .~t(,,,,oo(t0,,)), .c~l(t,,/i(,.(v)).</Paragraph>
      <Paragraph position="3">  The utterance of (14.3) is considered as the consequence of an inference such as this:  (16) 1. x = y 2. Sato(x) 3. Bet(a, good(x)) 4. Bel(b, liar(x))  is derived from (15), (16.3-4) do not bring about an inconsistency since they are belief formulas with different propositions inside. But obviously, a has drawn an inconsistency by taking off the belief operators, and carrying out the following inference.  (17) 1. x=y 2. liar(y) 3. liar(x) 1, 2 4. Vx(liar(x)-*-,good(x)) 5. -good(x) 3, 4 6. good(x) 7. \[\] 5,6 Suppose (14) is continued as follows: (18) a: I meain the Sato in tile linguistics department. b: ()it, I thougtit you were talking about the Sato in the AI department. The one you mean is indeed a good guy.</Paragraph>
      <Paragraph position="4"> (19) a: He does sometimes. But you can't dislike him.  b: I guess not.</Paragraph>
      <Paragraph position="5"> In this case, in order to avoid the conflict, one traces its causes, and retracts the weakest one (16.1) for (18), and (17.4) fro' (19), or replaces it by its negation. As a result, (18), for examle, is associated with</Paragraph>
    </Section>
    <Section position="4" start_page="1165" end_page="1165" type="sub_section">
      <SectionTitle>
3.2 Indirect solution of conflicts
</SectionTitle>
      <Paragraph position="0"> small coifference room today.</Paragraph>
      <Paragraph position="1"> 7.a: I see.</Paragraph>
      <Paragraph position="2"> The inference of (22) is formalized as follows:  (23) 1. Sato 2. Sato ~ 203 3. 203 4. office 5. office-* s.e.r 6. s.c.r 4, 5 7. s.c.r -* -~203 8. -1203 6, 7 9. \[\] 3~8  In this case, the conflict between (22.1) and (22.2) 1, 2 cannot be solved between themselves. (22.3) to (22.6) reflects the process of deciding which is to be preferred by tracing the source of each condition. That is~ when one cannot choose between two conflicting conditions Pl and p2 on their own account, one replaces Pl and p2 by ql, ql --* Pl and q2, q~ '~ P2~ respectively, and decide which of ql, q2 is to be preferred so that one can avoid the conflict by retracting the weaker condition in favor of the stronger.</Paragraph>
    </Section>
    <Section position="5" start_page="1165" end_page="1165" type="sub_section">
      <SectionTitle>
3.3 Conflicts ending in a draw
</SectionTitle>
      <Paragraph position="0"> Consider the following case.</Paragraph>
      <Paragraph position="1">  (24) 1.a: That's Muranishi over there. 2.b: No, it's Hokuto.</Paragraph>
      <Paragraph position="2"> 3.a: Really? This case is formalized ~s follows: (25) 1. x = y 2. Hokuto(y) 3. Hokuto(x) 1, 2 4. Muranishi(x) 5. Vx(Muranish.i(x)-*-,Hokuto(x)) 6. -~Hokuto(x) 4, 5 7. \[\] 3,6  As (24.3) indicates, there is no retractable belief in DB(K), which caused the diMog to end in a breakdown. null</Paragraph>
    </Section>
    <Section position="6" start_page="1165" end_page="7166" type="sub_section">
      <SectionTitle>
3.4 Formalization of diachronic analy-
</SectionTitle>
      <Paragraph position="0"> sis The processes of belief revision illustrated in 3.1 through 3.3 can be h)rmalized as in (27). First, we define some terms:  (26) i) I,et c~ be one of the partieilmnts a and b in a dialogue, and/3 the other.</Paragraph>
      <Paragraph position="1"> ii) Given p in DB(K), substitute Bel((e,p) and Bel(fl,p) for it. When Bel((e,p) is replaced by Bel((e, &amp;quot;~p), it is called p's self-denial by (r. When Bel(c~,p) is simply retracted, it is called p's selfwithdraw,'d by (e.</Paragraph>
      <Paragraph position="2"> iii) When Bel((~,p), Bel(/J,p), and p are substituted for by ~p, it is called p's strong-denial. When they are simply retracted, it is (:ailed p's strong-withdrawM.</Paragraph>
      <Paragraph position="3"> iv) Let E be a set of Horn-clauses, PI(E) the set of its prime implieants. When ~p {-~p~, ..., ~p,,} for any qV-~p~ V...V~p,~ e PI(E),  q is subordinate, ~o p.</Paragraph>
      <Paragraph position="4"> (27) Whenever a new condition is added to DB(K) in response to a dialogue ntove, the participant (e starts her CMS, calculates a way of resolving any contlict, and revises DB(K) dymunically: 1) a) When a condition is explicitly registered in DB(K), strip off its belief operator (if any), add it to CMS as an atomic formula.</Paragraph>
      <Paragraph position="5"> b) Add implicitly assumed conditionals such as Vx(Muranishi(x) --~ ~Hokuto(x)) to CMS as an atomic formula.</Paragraph>
      <Paragraph position="6"> c) Add the implicit inference ruh!s in the (lial()gue to CMS ~s a conditional formula. (E.g., the inference rule a, b/c eorresl)onds to the conditional formula c ~- a, b.) 2) Let E be tit(! set of CMS-cbmses obtained in 1). Change E into PI(E) (the set of its prime imI)licants).</Paragraph>
      <Paragraph position="7"> a) If PI(E) V El, then the dialogue suc(:eeds. Either terminate it, or go on to another.</Paragraph>
      <Paragraph position="8"> h) If PI(E) F D, mdess there is a retractabh~ or deniable assumption 1 &gt; in E, go to c). If dmre is, try to make either p's strong-denial or strongwithdrawal. If it fails, go to c). If successful, for all q such that q is subordinate to p, m~d(e q's self-withdrawal, and call the result E ~.</Paragraph>
      <Paragraph position="9"> A) If PI(E') V D, then the dialogue suceeds. Either terminate it, or go on to another.</Paragraph>
      <Paragraph position="10"> B)If PI(E') ~- ~, then S := E' all(L gO to b). e) If every assulnption p in E is well justi null fied, the dialogue fails. If any p has negotiable justifications q,,..., q,, replace p by p ~ql,...,%;ql,...,qn altd call the result E'. Set E :-- El, and go to b).</Paragraph>
    </Section>
  </Section>
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