Towards a Dynamic Theory of Belief-Sharing in Cooperative 
Dialogues 
Hisashi Komatsu Norihiro Ogata Akira Ishikawa 
The Toin Corporation University of Tsukuba Sophia University 
1 Introduction 
In this paper, we propose a dynamic theory of belief- 
sharing which dens with certain processes of forming 
and revising shared beliefs during cooperative dia- 
logues. 
Since Clark & MarshNl(1981) the problem of de- 
termination of the referents of referring expressions 
has been discussed in relation to mutual knowl- 
edge. In natural language processing, there haw~ 
also been several studies treating this problem of 
referent-determination in terms of mutuN knowledge 
(Perrault &: Cohen, 1981; Joshi, 1982; Nadathur & 
Joshi, 1983; Appelt, 1985). In this paper, we con- 
ceive referent-determination as a process of belicf- 
sllaring in dialogues, and propose a formal theory 
of dialogue in which referent-determination can be 
characterized as part of belief-sharing processes. We 
use Discourse Representation Theory (DRT) to model 
the characteristics of referents in discourse (Kamp, 
1981, 1990; Asher, 1993), and propose a model of 
dynamic maintenance of the mutual beliefs of tile 
participants in diNogues based on Clause Mainte- 
nance System (CMS) (Doyle, 1979; Levesquc, 1989; 
Poole, 1988; Reggia, 1983; de Kleer, 1986; Rciter & 
de Kleer, 1987). By this nmdel, we characterize the 
relationships between a diMogue process and its suc- 
cessfulness, which is mainly illustrated by examples 
of referent-determination but can be applied to any 
type of belief-sharing. 
2 Dynamic Maintenance 
Shared Beliefs 
of 
2.1 DRS 
However cooperative, real-world dialogues are fraught 
with hedges, understatements, or even white lies, 
which would necessitate introducing a distinction be- 
tween what is literally conveyed by an utterance, and 
its real intent on the part of both speaker and hearer. 
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 con- 
tent of an utterance is represented in tile following 
style: 
(1) K: a,b,x,y,z .... 
Bel(a, K) 
~cl( b, IO 
....................................... A(~:), 
B(v), C(~), ... 
We call K discourse representation structure 
(DRS), {a, b, x,y,z .... } K's domain (UK), the el- 
ements of UI< 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 con- 
dillon is the result of an n(_> 0) times application 
of Bel(ct,.) to a first-order formula p. Bel(c~,.) is 
called a belief operator, where c~ designates the ut- 
terer. Given ¢ as a condition, Bel(ce, ¢) reads "the 
partMpant a believes ¢." n is called the rank of 
¢ with regard to its embedding within belief opera- 
tors. Conditions of rank O are called bare formulas, 
while those with a rank greater than 0 belief formu- 
las. 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,n- 
mon knowledge, and are assumed throughout the dia- 
logue. 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 utter- 
ante is made, new discourse entities may be intro- 
duced, 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 =:~ ... 
Since only cooperative dialogues are considered, 
the goal is to arrive at a DRS in which no contra- 
dictory 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 
1164 
Call hold contradictory beliefs at dill?rent points of a 
dialogue, whereas the s;Hne particip~mt cannot hold 
contradictory beliefs ¢tt any partic:ular point. 
In what follows, we just indicate DB(K) unless 
otherwise noted. 
2.2 How shared belieig are registered 
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. 
First, discourse referents m'e taken to be epistemo- 
logical entities without counterparts in snrfacc sen- 
ten('es, but introduced into the DRS by the partic- 
ipants of ~ diMogue, and of which prot)erties corre- 
sponding to surface linguistic expressions are l)redi- 
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" introdu(:ed into U~¢ 1)y 
a, and the predicates corresponding to expressions in 
the utterauce. 
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 regis- 
tered 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 be- 
lief t~t this point. Suppose (6) is uttered folh)wing 
upon (2): 
(6) b: Yes, he is. 
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- 
tion of p for them is equivahmt to DB(K). 
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. 
Third, there is involved a step of identification in 
the transition front b's utterance of (6) to the condi- 
tion (7). Just as the discourse referent x was intro- 
duced 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 com- 
posed 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)). 
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. 
2.b: No, he is an otfice clerk now. 
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 dia- 
logue 
In this section, we consider the changes DRS's un- 
dergo 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 con- 
flict. 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. 
3.1 Direct solution of conflicts 
Consider the following dialogue. 
(1.4) 1. I~: Sato is ~t good guy. 
2. b: By no means, he is a liar. 
3. a: No kidding. 
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)). 
1165 
The utterance of (14.3) is considered as the conse- 
quence 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 dif- 
ferent 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 in- 
deed a good guy. 
(19) a: He does sometimes. But you can't dislike him. 
b: I guess not. 
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 
(20) Bel(a,-~x = y), 
B~l(a, Sato(x)), 
Bel(a, Sato(y)), 
Bel(b,-,x = y), 
nd(~,Sato(x)), 
~el(~,Sato(y)), 
Bel(a, LiD(x)), Bel(a, AiD(y)), 
Bd(a, Vx(li~,'(x) -, ~good(x))), 
Bel(b, LiD(x)) , Bel(b, AiD(y)), 
Bcl(~, Vx(liar(x) -* ~qood(x))), 
B~l( ~, good(x)), ~el(~, liar(y)), 
Bel(b, good(x)), Bcl(b, liar(y)). 
All Bel's can be taken off in (20), resulting in 
(21) ~x = y, Sato(x), Sato(y), 
LiD(x), AiD(y), Vx(liar(x) -* ~good(x) ), 
9ood(x),liar(y), 
which is shared by a and b. 
3.2 Indirect solution of conflicts 
Consider the following dialogue. 
(22) 1.a: Today's meeting is held at 203, isn't it? 
2.b: No, I heard it is at the small conference 
rooln, 
3.b: Who toht you that'? 
4.a: Sat() told me yesterday. 
5.b: That's strange. I'll call the office. 
6.b: They say it was changed fl'om 203 to the 
small coifference room today. 
7.a: I see. 
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 pre- 
ferred 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 re- 
places Pl and p2 by ql, ql --* Pl and q2, q~ '~ P2~ 
respectively, and decide which of ql, q2 is to be pre- 
ferred so that one can avoid the conflict by retracting 
the weaker condition in favor of the stronger. 
3.3 Conflicts ending in a draw 
Consider the following case. 
(24) 1.a: That's Muranishi over there. 
2.b: No, it's Hokuto. 
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 break- 
down. 
3.4 Formalization of diachronic analy- 
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: 
7166 
(26) i) I,et c~ be one of the partieilmnts a and b in a 
dialogue, and/3 the other. 
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, "~p), it is called p's self-denial by (r. When 
Bel(c~,p) is simply retracted, it is called p's self- 
withdraw,'d by (e. 
iii) When Bel((~,p), Bel(/J,p), and p are sub- 
stituted for by ~p, it is called p's strong-denial. 
When they are simply retracted, it is (:ailed p's 
strong-withdrawM. 
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. 
(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. 
b) Add implicitly assumed conditionals such as 
Vx(Muranishi(x) --~ ~Hokuto(x)) to CMS as 
an atomic formula. 
c) Add the implicit inference ruh!s in the (lial()gue 
to CMS ~s a conditional formula. (E.g., the in- 
ference 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). 
a) If PI(E) V El, then the dialogue suc(:eeds. Ei- 
ther terminate it, or go on to another. 
h) If PI(E) F D, mdess there is a retractabh~ or 
deniable assumption 1 > in E, go to c). If dmre 
is, try to make either p's strong-denial or strong- 
withdrawal. 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 ~. 
A) If PI(E') V D, then the dialogue suceeds. Ei- 
ther terminate it, or go on to another. 
B)If PI(E') ~- ~, then S := E' all(L gO to b). 
e) If every assulnption p in E is well justi- 
fied, the dialogue fails. If any p has nego- 
tiable justifications q,,..., q,, replace p by p ~- 
ql,...,%;ql,...,qn altd call the result E'. Set 
E :-- El, and go to b). 
4 Synchronic analysis of dia- 
logue 
Next, according to Ogata(1993), we consider a clas- 
sification which characterizes the degree of belief 
sharing for the t)articipmtts at a l)articular point of 
the conversation, and the eorre(:tness of the shared 
beliefs. 
(28) 1) The beliei~ are all shared by the participants: 
see (2), (~) above. DB(K) ,:o,,tai,s ~o eo,di- 
lions l)re, tixed with Bcl. Since the set of belie, fs 
of either partMpant is .considered to be consis- 
tent, I'I(E) V \[\] fi)r tile CMS corresponding to 
the DRS. 
2) There remain some conditions prefixed with 
Be.l in DB(K), but PI(E) y \[\] for the CMS cor- 
resl)on(ling to the DRS. A typical c~use, is when 
/~'s ~ussertions prol)erly inchtde (~'s t)eliefs and 
about the rest of ~'s assertions (~ has not been 
able to decide in one way or another. 
3) There remain some (:onditions prefixed with 
BeI in DB(K), and PI(E) I- E~. This is a case 
of breakdown ,'~s seen in (25). 
We call titese three ('~uses, respectively, 1) obser- 
w~tionally susscessful, 2) observ~Ltionally consis- 
tent, and 3) obserw~tionally unsuccessful. 
Take the ease of (2), (6) again. Tit(, dMogue was suc- 
cessfiflly terminated because the Sato a had in mind 
and the Sato b had in mind were both students. But 
suppose a's Sato was a student in the linguistics de- 
1)artm(mt, and b's Sato in the AI department, (.hat is, 
they were different persons. Or suppose a and b had 
the same Sato in rain(|, but that he was no longer a 
student ;~t the time. These two eases are obscrwttion- 
ally suc(:essfifl, hut the partMpants end a 1) with the 
wrong beliefs. In order to meet this gat) ~ we introduce 
a standard of correctness that might be eml)odied by 
God's viewpoint, the reality, or the conventions of 
the language community to which the 1)articipants 
belong. We call this standard the facts. The cate- 
gories in (28) are further broken down relative to the 
facts as in (29): 
Detine K' as the result of adding the Nets to tim 
DB(K) of a DRS K, and extending UI¢ accordiugly. 
Let PI(E I) be the set of prime implicants fi)r the CMS 
corresponding to DB(K/). The facts are a set of bare 
formub~s. Then (28) is subclassifled as follows: 
(29) 1) ol)servationally su(:c, essfui 
a) m(~') V u, b) m(s') ~- 
u. 
2) observationally consistent 
a) m(E') V El, 
1,) m(s') ~ u. 
3) obserwttionally unsuccessful 
PI(E I) F ~. 
We cM1 la) strongly successful, 2a) strongly consis- 
tent, and the rest (the cases where PI(N ~) F n) 
strongly unsuccessful. A comparison of (28) and (29) 
suggests the folh)wing implications whose converses 
do not hold: 
(3(I) a) strongly successfltl ~ observationally success- 
tiff 
1)) obserwt.tionally m~suc('.essful -~ strongly un- 
su(:cessful 
c) strongly consistent -~ observationally consis- 
tent 
1167 
4.1 Characterization of expressions 
referring to individuals 
We consider the problem of how the concepts of suc- 
cess introduced in the previous section might be ap- 
plied to the dialogues identifying the denotation of 
individual terms, especially proper nouns. 
(31) a: That's Sato over there. 
b: Yes, it is. 
DB(K) for (31) is 
(32) a) x = y, 
b) Sato(x), Sato(y). 
If a and b believe there is only one Sato in this situ- 
ation, (32b) becomes 
(33) Bel(a, tx.Bel(a, Sato(x)) = x), 
~el(b,,x.Bcl(~, Sato( x ) ) = ~ ), 
Bel(a, tx.Bcl(b, Sato(x)) = y), 
~el(b, ,~.Bel(b, Sato(x) ) = y), 
which gives rise to 
(34) tx.Bel(a, Sato(x)) = x, 
,x.Bel(b, Sato(x)) = y. 
From this, we obtain by (32a) 
(35) ,x.Bel(a, Sato(x)) = ,x.~el(~, Sato(x)). 
If (31) is a case of strong success in which "Sato" 
correctly refers to the unique Sato, DB(K t) contains 
(36) x = z, tx.Sato(x) = z. 
From (323), (34), and (36), we ca,, derive 
(37) tx.Sato(x)= 
tx.Bel(a, Sato(x)) = ,x.Bel(b, Sato(x)). 
In general, of an atomic formula T(x), we call 
tx.Bel(a, T(x)) a's intended referent, tx.Bel(b,T(x)) 
b's intended referent, and tx.T(x) the semanite refer- 
ent. Thus, a strongly successful dialogue with regard 
to an identification of an individual referent is a case 
where a's intended referent, b's intended referent, and 
the semantic referent all coincide. 
However, if in (31) the individual referred to is Ki- 
noshita rather than Sato, DB(K t) will contain 
(38) ~(z = x),,x.Sato(x) = z. 
If a and b have different Sato's in mind, DB(K ~) will 
contain 
(39) tx.Sato(x) = t, ~(~ = x). 
In either case, the result is strongly unsuccessflfl: 
(40) -~(tx.Sato(x) = tx.BeI(a, Sato(x) ) ), 
-~(tx.Sato(x) = tx.BeI(b, Sato(x))). 
A case of being observationally unsuccessful such as 
(24) will be 
(41) -~(tx.Bel(a, Sato(x)) = tx.Bel(b, Sato(x))). 
By indicating a's intended referent, b's intended refer- 
ent, and the semantic referent by Ta, Tb, and Tcom, 
respectively, we can summarize what has been dis- 
cussed above as follows: 
(42) strongly successful: Tcom = Ta = Tb, 
observationally successful: Ta = Tb, 
strongly unsuccessfld: Tcom ¢ Ta, 
Tcom ~ Tb, 
observationally unsuccessful: Ta ~ Tb. 
5 Conclusion 
In this paper, we proposed a system which combines 
DRS with CMS, and an Mgorithm for the dynamic re- 
vision of shared beliefs in cooperative dialogues. Fur- 
ther, the degree of success in dialogues was formal- 
ized. 
Still, the following problems remain to be solved: 
1) The treatment of background knowledge must be 
made precise. E.g., 'Sato ~ 203' in (23), or 
'Vx(Muranishi(x) -~ "~Hokuto(x))' in (25) is im- 
plicitely introduced into the inference without ex- 
planation of its origin. 
2) The translation procedure of an utterance into the 
condition of DRS must be formalized. 
3) It's necessary to give a semantic foundation to our 
systenl. 
4) hnplelnentation of a system which simulates our 
dialogue mechanism. 
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