REPRESENTING KNOWLEDGE ABOUT KNOWLEDGE AND MUTUAL KNOWLEDGE 
Sald Soulhi 
Equipe de Comprehension du Raisonnement Naturel 
LSI - UPS 
llg route de Narbonne 31062 Toulouse - FRANCE 
ABSTRACT 
In order to represent speech acts, in a 
multi-agent context, we choose a knowledge 
representation based on the modal logic of 
knowledge KT4 which is defined by Sato. Such 
a formalism allows us to reason about know- 
ledge and represent knowledge about knowled- 
ge, the notions of truth value and of defi- 
nite reference. 
I INTRODUCTION 
Speech act representation and the lan- 
guage planning require that the system can 
reason about intensional concepts like know- 
ledge and belief. A problem resolver must 
understand the concept of knowledge and know 
for example what knowledge it needs to achie- 
ve specific goals. Our assumption is that a 
theory of language is part of a theory of ac- 
tion (Austin \[4\] ). 
Reasoning about knowledge encounters the 
problem of intensionality. One aspect of this 
problem is the indirect reference introduced 
by Frege ~\] during the last century. Mc Car- 
thy \[15\] presents this problem by giving the 
following example : 
Let the two phrases : Pat knows Mike's tele- 
phone number (I) 
and Pat dialled Mike's te- 
lephone number (2) 
The meaning of the proposition "Mike's tele- 
phone number" in (I) is the concept of the 
telephone number, whereas its meaning in (2) 
is the number itself. 
Then if we have : "Mary's telephone number 
= Mike's telephone number", 
we can deduce that : 
"Pat dialled Mary's tele- 
phone number" 
but we cannot deduce that : 
"Pat knows Mary's telephone 
number", 
because Pat may not have known the equality 
mentioned above. 
Thus there are verbs like "to know", "to 
believe" and "to want" that create an "opaque" 
context. For Frege a sentence is a name, refe- 
rence of a sentence is its truth value, 
the sense of a sentence is the proposi- 
tion. In an oblique context, the refe- 
rence becomes the proposition. For exam- 
ple the referent of the sentence p in the 
indirect context "A knows that p" is a 
proposition and no longer a truth value. 
Me Carthy \[15\] and Konolige \[I I\] 
have adopted Frege's approach. They consi- 
der the concepts like objects of a first- 
order language. Thus one term will denote 
Mike's telephone number and another will 
denote the concept of Mike's telephone 
number. The problem of replacing equalities 
by equalities is then avoided because the 
concept of Mike's telephone number and the 
number itself are different entities. 
Mc Carthy's distinction concept/object 
corresponds to Frege's sense/reference or 
to modern logicians' intension/extension. 
Maida and Shapiro \[13\] adopt the same 
approach but use propositional semantic 
networks that are labelled graphs, and that 
only represent intenslons and not exten- 
sions, that is to say individual concepts 
and propositions and not referents and 
truth values. We bear in mind that a seman- 
tic network is a graph whose nodes repre- 
sent individuals and whose oriented arcs 
represent binary relations. 
Cohen E6\], being interested in speech 
act planning, proposes the formalism of 
partitioned semantic networks as data base 
to represent an agent's beliefs. A parti- 
tioned semantic network is a labelled graph 
whose nodes and arcs are distributed into 
spaces. Every node or space is identified 
by its own label. Hendrix ~9\] introduced 
it to represent the situations requiring 
the delimitation of information sub-sets. 
In this way Cohen succeeds in avoiding the 
problems raised by the data base approach. 
These problems are clearly identified by 
Moore FI7,18\]. For example to represent 
'A does-not believe P', Cohen asserts 
Believe (A,P) in a global data base, en- 
tirely separated from any agent's know- 
ledge base. But as Appelt ~\] notes, this 
solution raised problems when one needs to 
combine facts from a particular data base 
194 
with global facts to prove a single assertion. 
For example, from the assertion : 
know (John,Q) & know (John,P ~Q) 
where P~ Q is in John's data base and ~ know 
(John,Q) is in the global data base, it should 
be possible to conclude % know (John,P) but 
a good strategy must be found ! 
In a nutshell, in this first approach 
which we will call a syntactical one, an a- 
gent's beliefs are identified with formulas 
in a first-order language, and propositional 
attitudes are modelled as relations between 
an agent and a formula in the object langua- 
ge, but Montague showed that modalities can- 
not consistently be treated as predicates ap- 
plying to nouns of propositions. 
The other approach no longer considers 
the intenslon as an object but as a function 
from possible worlds to entities. For ins- 
tance the intension of a predicate P is the 
function which to each possible world W (or 
more generally a point of reference, see 
Scott \[23\] ) associates the extension of P 
in W. 
This approach is the one that Moore 
D7,18\] adopted. He gave a first-order axio- 
matization of Kripke's possible worlds seman- 
tics \[12\] for Hintikka's modal logic of know- 
ledge \[,0\]. 
The fundamental assumption that makes 
this translation possible, is that an attri- 
bution of any propositional attitude like 
"to know", "to believe", "to remember", "to 
strive" entails a division of the set of pos- 
sible worlds into two classes : the possible 
worlds that go with the propositional attitu- 
de that is considered, and those that are in- 
compatible with it. Thus "A knows that P" is 
equivalent to "P is true in every world com- 
patible with what A knows". 
We think that possible worlds language 
is complicated and unintuitive, since, rather 
than reasoning directly about facts that some- 
one knows, we reason about the possible worlds 
compatible with what he knows. This transla- 
tion also presents some problems for the plan- 
ning. For instance to establish that A knows 
that P, we must make P true in every world 
which is compatible with A's knowledge. This 
set of worlds is a potentially infinite set. 
The most important advantage of 
Moore's approach \[17,183 is that it gives 
a smart axiomatization of the interaction 
between knowledge and action. 
II PRESENTATION OF OUR APPROACH 
Our approach is comprised in the general 
framework of the second approach, but in- 
stead of encoding Hintikka's modal logic 
of knowledge in a first-order language, 
we consider the logic of knowledge propo- 
sed by Mc Carthy, the decidability of 
which was proved by Sato \[21\] and we pro- 
pose a prover of this logic, based on na- 
tural deduction. 
We bear in mind that the idea of u- 
sing the modal logic of knowledge in A.I. 
was proposed for the first time by Mc Car- 
thy and Hayes \[14\]. 
A. Languages 
A language L is a triple (Pr,Sp,T) 
where : 
.Pr is the set of propositional va- 
riables, 
.Sp is the set of persons, 
.T is the set of positive integers. 
The language of classical proposi- 
tional calculus is L = (Pr,6,~). SoCSp 
will also be denoted by 0 and will be 
called "FOOL". 
B. Well Formed Formulas 
The set of well formed formulas is 
defined to be the least set Wff such as : 
(W|) PrC Wff 
(W 2) a,b-~ Wff implies aD b eWff 
(W 3) S6_Sp,t 6.T,aeWff implles(St)a~_Wff 
The symbol D denotes "implication". 
(St)a means "S knows a at time t" 
<St>a (= % (St) ~ a) means "a is pos- 
sible for S at 
time t". 
{St}a (= (St)a V (St) % a) means 
"S knows whether 
a at time t". 
195 
C. Hilbert-type System KT4 
The axiom schemata for KT4 are : 
At. Axioms of ordinary propositional lo- 
gic 
A2. (St)a • a 
A3. (Ot)a ~ (Or) (St)a 
A4. (St) (a D b) ~ ((Su)a D(Su)b), where 
t 6 u 
A5. (St)a ~ (St) (St)a 
A6. If a is an axiom, then (St)a is an 
axiom. 
Now, we give the meaning of axioms : 
(A2) says that what is known is true, that 
is to say that it is impossible to have 
false knowledge. If P is false, we cannot 
say : "John knows that P" but we can say 
"John believes that P". This axiom is the 
main difference between knowledge and be- 
lief. 
This distinction is important for plan- 
ning because when an agent achieves his goals, 
the beliefs on which he bases his actions must 
generally be true. 
(A3) says that what FOOL knows at time t, 
FOOL knows at time t that anyone knows 
it at time t. FOOL's knowledge represents 
universal knowledge, that is to say all 
agents knowledge. 
(A4) says that what is known will remain 
true and that every agent can apply modus 
ponens, that is, he knows all the logical 
consequences of his knowledge. 
(A5) says that if someone knows something 
then he knows that he knows it. This a- 
xiom is often required to reason about 
plans composed of several steps. It will 
be referred to as the positive introspec- 
tive axiom. 
(A6) is the rule of inference. 
D. Representation of the notion of truth va- 
lue. 
We give a great importance to the repre- 
sentation of the notion of truth value of a 
proposition, for example the utterance : 
John knows whether he is taller than 
Bill (I) 
can be considered as an assertion that mentions 
the truth value of the proposition P = John is 
taller than Bill, without taking a position as 
to whether the latter is true or false. 
In our formalism (I) is represented 
by : 
{John} P 
This disjunctive solution is also adopted 
by Allen and Perrault D\]" Maida and Sha- 
piro \[13\] represent this notion by a node 
because the truth value is a concept (an 
object of thought). 
The representation of the notion of 
truth value is useful to plan questions : 
A speaker can ask a hearer whether a cer- 
tain proposition is true, if the latter 
knows whether this proposition is true. 
E. Representing definite descriptions in 
conversational systems : 
Let us consider a dialogue between 
two participants : A speaker S and a hea- 
rer H. The language is then reduced to : 
Sp = (O,H,S} and T = {l} 
Let P stand for the proposition : "The 
description D in the context C is unique- 
ly satisfied by E". 
Clark and Marshall \[5\] give examples that 
show that for S to refer to H to some en- 
tity E using some description D in a con- 
text C, it is sufficient that P is a mu- 
tual knowledge; this condition is tanta- 
mount to (O)P is provable. Perrault and 
Cohen \[20\] show that this condition is 
too strong. They claim that an infinite 
number of conjuncts are necessary for suc- 
cessful reference : 
(S) P& (S)(H) e& (S)(H)(S) e & ... 
with only a finite number of false conjuncts. 
Finally, Nadathur and Joshi ~9\] give the 
following expression as sufficient condition 
for using D to refer to E : 
(S) BD (S)(H) P & ~ ((S) BO(S)~(O)P) 
where B is the conjunction of the set of 
sentences that form the core knowledge of 
S and ~ is the inference symbole. 
III SCHOTTE - TYPE SYSTEM KT4' 
Gentzen's goal was to build a forma- 
lism reflecting most of the logical rea- 
sonings that are really used in mathemati- 
196 
cal proofs• He is the inventor of natural de- 
duction (for classical and intultionistic lo- 
gics). Sato ~|\] defines Gentzen - type sys- 
men GT4 which is equivalent to KT4. We consi- 
der here, schStte-type system KT4' \[22\] which 
is a generalization of S4 and equivalent to 
GT4 (and thus to KT4), in order to avoid the 
thinning rule of the system GT4 (which intro- 
duces a cumbersome combinatory). Firstly, we 
are going to give some difinitions to intro- 
duce KT4'. 
A. Inductive definition of positive and ne- 
gative parts of a formula F 
Logical symbols are ~ and V. 
a. F is a positive part of F. 
b. If % A is a positive part of F, then 
A is a negative part of F. 
c. If ~ A is a negative part of F, then 
A is a positive part of F. 
d. If A V B is a positive part of F, 
then A and B are positive parts of F. 
Positive parts or negative parts which do not 
contain any other positive parts or negative 
parts are called minimal parts. 
B. Semantic property 
The truth of a positive part implies the 
truth of the formula which contains this posi- 
tive part. 
The falsehood of a negative part implies 
the truth of the formula which contains this 
negative part. 
C. Notation 
F\[A+\] is a formula which contains A as a 
positive part 
F\[A-\] is a formula which contains A as a 
negative part. 
F\[A+,B-\] is a formula which contains A as 
a positive part and B as a negative 
part where A and B are disjoined (i. 
e, o~e is not a subformula of the o- ther). 
D. Inductive definition of F \[.j 
From a formula F \[A\], we build another 
formula or the empty formula F \[.\] by dele- 
ting A : 
a. If F \[A 3 ° A, then F\[.\] is the empty 
formula. 
c. If F G\[A V BJ or = G V AJ 
then . = G \[BJ. 
E. Axiom 
An axiom is any formula of the form 
F\[P+,P-\] where P is a propositional varia- 
ble. 
F. Inference rules 
(R!) F\[(A V B)j V ~ A, FI(A V B) \] 
v ~ B ~ FL(A V B) J -- 
(R2) F\[(St)A 3 V~A ~ FT(st)A~ 
(PO) ~(Su)A 1V ... V ~(Su)Am V 
~(Ou)B. V ... V ~(Ou)Bn V C 
where (Su)A I ..... (Su)Am, (Ou)B I , 
..., (Ou) B6 must appear as neg6- 
tire parts in the conclusion, and 
uK t 51c 9, F2\[C-\] F, v F2\[J 
(cut) 
G. Cut-elimlnation theorem (Hauptsatz) 
Any KT4' proof-figure can be trans- 
formed into a KT4' proof-figure with the 
same conclusion and without any cut as a 
rule of inference (hence, the rule (R4) 
is superfluous. The proof of this theo- 
rem is an extension of Sch~tte's one for 
$4'. This theorem allows derivations 
"without detour"• 
IV DECISION PROCEDURE 
A logical axiom is a formula of the 
form F\[P+,P-\]. A proof is an single-roo- 
ted tree of formulas all of whose leaves 
are logical axioms. It is grown upwards 
from the root, the rules (RI), (R2) and 
(R3) must be applied in a reverse sense. 
These reversal rules will be used as 
"production rules"• The meaning of each 
production expressed in terms of the pro- 
granting language PROLOG is an implication• 
It can be shown \[24J that the following 
strategy is a complete proof procedure : 
• The formula to prove is at the star- 
197 
ring node; 
• Queue the minimal parts in the given for- 
mula; 
• Grow the tree by using the rule (R|) in 
priority , followed by the rule (R2), then 
by the rule (R3). 
The choice of the rule to apply can be 
done intelligently. In general, the choice of 
(RI) then (R2) increases the likelihood to 
find a proof because these (reversal) rules 
give more complex formulas. In the case where 
(R3) does not lead to a loss of formulas, it 
is more efficient to choose it at first• The 
following example is given to illustrate this 
strategy : 
Example 
Take (A4) as an example and let Fo deno- 
tes its equivalent version in our language 
(Fo is at the start node) : 
Fo = ~(St)(~a V b) V ~(Su)a V (Su)b where 
t < u 
P~ denotes positive parts and P? denotes 
I negative parts l 
P+ = {~(St)(~ a V b), %(Su)a,(Su)b}; 2 
P = {(St)(~ a V b),(Su)a}; O 
By (R3) we have (no losses of formulas) : 
F l = ~(St)(% a V b) V %(Su)a V b 
÷ PI = {%(St)(~ a V b), ~(Su)a,b} 
F- = {(St)(% a V b),(Su)a} 
By (~2) we have : 
F~ = F~ V ~,(~a V b) 
P2 PI U {%(~a V b)} 
P2 = P7 U {~a V b} 
By (RI) we have : 
F~ = F~ V ~ a 
P3 P2 U {~ a,a} 
andP~ = P2 O {~ a} 
F 4 = F 2 V % b + + 
P4 = P2 ~ {~ b} 
P~= P2 U {b} 
+' P~ {b} F 4 is a logical axiom because P4 ~ = 
Finally, we have to apply (R2) to the last but 
one node : 
F 5 
F~ F~V~a 
P5 \[ P3 U {~ a} 
P5 = P3 iJ {a} 
is a logical axiom because P51~ F 5 =\[a} 
The generated derivation tree is then : 
I ÷ -- Fo,Po,Po 
I F,,P ,FT 
1 I 
, F2'P~'P2 j 
1 / 
+ -- 
F3,P3,P 3 
R 2 
+ - +;\] P5 = {a} F5'Pb'P5 P5 
I ÷ -- 
I F4'P4'P4 1 
rPV~4-- {b} 
Derivation tree 
198 
V ACKNOWLEDGMENTS 
We would like to express our sincerest 
thanks to Professor AndrOs Raggio who has gui- 
ded and adviced us to achieve this work. We 
would like to express our hearty thanks to 
Professors Mario Borillo, Jacques Virbel and 
Luis Fari~as Del Cerro for their encouragments. 
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