Download Group knowledge is not always distributed (neither is it always implicit)

Mathematical Social Sciences 38 (1999) 215–240
Group knowledge is not always distributed (neither is it
always implicit)q
Wiebe van der Hoek*, Bernd van Linder 1 , John-Jules Meyer
Utrecht University, Department of Computer Science, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands
Abstract
In this paper we study the notion of group knowledge in a modal epistemic context. Starting
with the standard definition of this kind of knowledge on Kripke models, we show that this
definition gives rise to some quite counter-intuitive behaviour. Firstly, using a strong notion of
derivability, we show that group knowledge in a state can always, but trivially, be derived from
each of the agents’ individual knowledge. In that sense, group knowledge is not really implicit, but
rather explicit knowledge of the group. Thus, a weaker notion of derivability seems to be more
adequate. However, adopting this more ‘local view’, we argue that group knowledge need not be
distributed over (the members of) the group: we give an example in which (the traditional concept
of) group knowledge is stronger than what can be derived from the individual agents’ knowledge.
We then propose two additional properties on Kripke models: we show that together they are
sufficient to guarantee ‘distributivity’, while, when leaving one out, one may construct models that
do not fulfil this principle.  1999 Elsevier Science B.V. All rights reserved.
Keywords: Group knowledge; Kripke models
1. Introduction
In the field of AI and computer science, the modal system S5 is a well-accepted and
by now familiar logic to model the logic of knowledge (cf. Halpern and Moses, 1985;
Meyer and van der Hoek, 1995b). Since the discovery of S5’s suitability for epistemic
logic, many extensions and adaptations of this logic have been proposed. In this paper
we look into one of these extensions, namely a system for knowledge of m agents
*Corresponding author. Tel.: 131-30-2531454; fax: 131-30-2513791.
E-mail address: [email protected] (W. van der Hoek)
q
This is an improved and extended version of van der Hoek et al. (1995).
1
Currently at ABN-AMRO Bank, Amsterdam.
0165-4896 / 99 / $ – see front matter  1999 Elsevier Science B.V. All rights reserved.
PII: S0165-4896( 99 )00013-X
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W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
containing a modality for ‘ group knowledge’. Intuitively, this ‘group knowledge’ (let us
write G for it) is not the knowledge of each of the agents in the group, but the
knowledge that would result if the agents could somehow ‘combine’ their knowledge.
The intuition behind this notion is best illustrated by an example. Let the formula w
denote the proposition that P ± NP. Assume that three computer scientists are working
on a proof of this proposition. Suppose that w follows from three lemmas: c1 , c2 and c3 .
Assume that scientist 1 has proved c1 and therefore knows c1 . Analogously, for agents 2
and 3 regarding c2 and c3 5 (c1 ∧ c2 ) → w, respectively. If these computer scientists
would be able to contact each other at a conference, thereby combining their knowledge,
they would be able to conclude w. This example also illustrates the relevance of
communication with respect to this kind of group knowledge: the scientists should
somehow transfer their knowledge through communication in order to make the
underlying implicit knowledge explicit.
In this paper, we try to make the underlying notions that together constitute G
explicit: What does it mean for a group of, say, m agents to combine their knowledge?
We start by giving and explaining a clear semantical definition of group knowledge, as
given by Halpern and Moses (1985). In order to make some of our points, we
distinguish between a global and a local notion of (deductive and semantic) consequence. Then we argue that the defined notion of group knowledge may show some
counter-intuitive behaviour. For instance, we show in Section 3 that, at the global level,
group knowledge does not add deductive power to the system: group knowledge is not
always a true refinement of individual knowledge. Then, in Section 4, we try to
formalize what it means to combine the knowledge of the members of a group. We give
one principle (called the principle of distributivity: it says that G that is derived from a
set of premises can always be derived from a conjunction of m formulae, each known by
one of the agents) that is trivially fulfilled in our setup. We also study a special case of
this principle (called the principle of full communication), and show that this property is
not fulfilled when using standard definitions in standard Kripke models. In Section 5 we
impose two additional properties on Kripke models: we show that they are sufficient to
guarantee full communication; they are also necessary in the sense that one can construct
models that do not obey one of the additional properties and that at the same time do not
verify the principle under consideration. In Section 6 we round off. Proofs of theorems
are provided in Appendix A.
2. Knowledge and group knowledge
Halpern and Moses (1985) introduced an operator to model group knowledge.
Initially, this knowledge in a group was referred to as ‘implicit knowledge’ and indicated
with a modal operator I. Since in systems for knowledge and belief, the phrase ‘implicit’
had already obtained its own connotation (cf. Levesque, 1984), later on the term
‘distributed knowledge’ (with the operator D) became the preferred name for the group
knowledge we want to consider here (cf. Halpern and Moses, 1992). Since we do not
want to commit ourselves to any fixed terminology we use the operator G to model this
kind of ‘group knowledge’. From the point of view of communicating agents, G-
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217
knowledge may be seen as the knowledge being obtained if the agents were fully able to
communicate with each other. Actually, instead of being able to communicate with each
other, one may also adopt the idea that the G-knowledge is just the knowledge of one
distinct agent, to whom all the agents communicate their knowledge [this agent was
called the ‘wise man’ by Halpern and Moses (1985); a system to model such
communication was proposed by van Linder et al. (1994)]. We will refer to this reading
of G (i.e., in a ‘send and receiving context’) as ‘a receiving agent’s knowledge’. We start
by defining the language that we use.
Definition 2.1. Let P be a non-empty set of propositional variables, and m [ N be
given. The language + is the smallest superset of P such that:
if w,c [ + then ¬w,(w ∧ c ),Ki w,Gw [ + (i # m).
The familiar connectives
∨ , → , ↔ are introduced
by definitional abbreviation in the
def
def
usual way; ¡ 5p ∨ ¬p for some p [ P and ' 5¬¡.
The intended meaning of Ki w is ‘agent i knows w ’ and Gw means ‘w is group
knowledge of the m agents.’ We assume the following ‘standard’ [the system below was
originally introduced by Halpern and Moses (1985)] inference system S5 m (G) for the
multi-agent S5 logic extended with the operator G.
Definition 2.2. The logic S5 m (G) has the following axioms:
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
any axiomatization for propositional logic
(Ki w ∧ Ki (w → c )) → Ki c
Ki w → w
Ki w → Ki Ki w
¬Ki w → Ki ¬Ki w
Ki w → Gw
(Gw ∧ G(w → c )) → Gc
Gw → w
Gw → GGw
¬Gw → G¬Gw
On top of that, we have the following derivation rules:
R1
R2
£w,£w → c ⇒ £c
£w ⇒ £Ki w, for all i # m.
In words, we assume a logical system (A1,R1) for rational agents. Individual
knowledge, i.e. the knowledge of one agent, is moreover supposed to be veridical (A3).
The agents are assumed to be fully introspective: they are supposed to have positive (A4)
as well as negative (A5) introspection. In the receiving agent’s reading, the axiom A6
may be understood as the ‘communication axiom’: what is known to some member of
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
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the group is also known to the receiving agent. Note how the axiom A6 and the rule R2
also induce the principle of necessitation on the operator G. The other axioms express
that the receiver has the same reasoning and introspection properties as the other agents
of the group. In the group knowledge reading, A6 declares what the members of the
group are; the other axioms enforce this group knowledge to obey the same properties
that are ascribed to the individual agents.
The derivability relation £ S5 m (G ) , or £ for short, is defined in the usual way. That is, a
formula w is said to be provable, denoted £w, if w is an instance of one of the axioms or
if w follows from provable formulae by one of the inference rules R1 and R2.
To relate this notion of group knowledge to the motivating example of Section 1, in
which the group is able to derive a conclusion w from the lemmas c1 , c2 and c3 , each
known by one of the agents, let us consider the following derivation rule which is to be
understood to hold for all w,ci , i # m:
R3£(c1 ∧ ? ? ? ∧ cm ) → w ⇒ £(K1 c1 ∧ ? ? ? ∧ Km cm ) → Gw.
Theorem 2.3. Let the logic S5 m (G)[R3 /A6] be as S5 m (G), with axiom A6 replaced by
the rule R3. Then, the two systems are equally strong: for all w,
S5 m (G)£w ⇔S5 m (G)[R3 /A6]£w.
From the proof of this theorem, as given by Meyer and van der Hoek (1995b) (p.
263), one can even derive that this equivalence is obtained for any multi-modal logic
that is normal with respect to G, i.e. for any logic X containing the axioms A7 and the
rules R1 and R29:£w ⇒ £Gw we have an equivalence between X 1 A6 and X 1 R3.
We now define two variants of provability from premises: one in which necessitation
on premises is allowed and one in which it is not.
Definition 2.4. Let c be some formula, and let F be a set of formulae. Using the
relation £ of provability within the system S5 m (G) we define the following two relations
£ 1 and £ 2 7 2 + 3 + : F £ 1 c (F £ 2 c ) ⇔ 'w1 , . . . ,wn with wn 5 c, and such that for all
1 # i # n:
•
•
•
•
either wi [ F
or there are j,k , i with wj 5 wk → wi
or wi is an S5 m (G) axiom
or wi 5 Kk s where s is such that
H
s 5 wj with j , i in the case of £ 1 ,
2
s
in the case of £ 2 .
From a modal logic point of view, one establishes:
5£ 1 w ⇔ 5£ 2 w,
such that £w is unambiguously defined as 5£ 1 w or 5£ 2 w, and
£(w1 ∧ . . . ∧ wn ) → c ⇔ w1 , . . . ,wn £ 2 c ⇒ w1 , . . . ,wn £ 1 c.
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
219
In other words, the deduction theorem holds for £ 2 , not for £ 1 . Note that the relation
£ is more liberal than £ 2 in the sense that £ 1 allows for necessitation on premises,
where £ 2 only applies necessitation to S5 m (G)-theorems. Our definition of £ 1 deviates
from standard terminology of premises: in classical logic, the formulas w1 , . . . ,wn are
called premises in a derivation of c if (w1 ∧ . . . ∧ wn ) → c is derivable. Thus, premises
in our sense in the context of £ 1 play the role of logical axioms [cf. Kaneko and
Nagashima (1997), where a related discussion on ‘logical’ versus ‘nonlogical axioms’
leads to a number of (in-)definability results in epistemic logic]. Also, one can prove the
following connection between the two notions of derivability from premises.
1
Theorem 2.5. Let hK1 , . . . ,Km j * be the set of sequences of epistemic operators. Then
hXw u X [ hK1 , . . . ,Km j * ,w [ F j£ 2 c ⇔F £ 1 c.
From Meyer and van der Hoek (1995a) (Lemma A1), for the one agent case without
group knowledge (i.e., S5 1 ), one even obtains
hKw u w [ F j£ 2 c ⇔ F £ 1 c.
From an epistemic point of view, when using £ 2 , we have to view a set of premises F
as a set of additional given (i.e., true) formulae [non-logical axioms, in the terminology
of Kaneko and Nagashima (1997)], whereas in the case of £ 1 , F is a set of known
formulae (‘logical axioms’). For instance, when describing a game and the knowledge
that is involved (see also Bacharach et al., 1997), it would be convenient to express the
mutual knowledge of the players as premises under the £ 1 notion of derivability.
Definition 2.6. A Kripke model } is a tuple } 5 kS,p,R 1 , . . . ,R m l where
1. S is a non-empty set of states,
2. p : S 3 P → h0,1j is a valuation to propositional variables per state,
3. for all 1 # i # m, R i 7 S 3 S is an (accessibility) relation. For any s [ S and i # m,
with R i (s) we mean ht [ S u R i stj.
We refer to the class of Kripke models as _m or, when m is understood, as _. The class
of Kripke models in which the accessibility relations R i are equivalence relations, is
denoted by 6 5 m (or by 6 5).
Definition 2.7. The binary relation * between a formula w and a pair },s consisting of
a model } and a state s in } is inductively defined by:
},s* p⇔p (s, p) 5 1,
},s*w ∧ c ⇔ },s*w and },s*c,
},s*¬w ⇔ },s±
u w,
},s*Ki w ⇔ },t*w for all t with (s,t) [ Ri ,
},s*Gw ⇔ },t*w for all t with (s,t) [ R 1 > . . . > Rm .
For a given w [ + and } [ 6 5, nw m 5 ht u },t*w j. For sets of formulae F, },s*F is
220
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
defined by: },s*F ⇔ },s*w for all w [ F. In this paper, a model } is always an
65-model.
The intuition behind the truth definition of G is as follows: if t is a world which is not
an epistemic alternative for agent i, then, if the agents would be able to communicate, all
the agents would eliminate the state t. This is justified by the idea that the actual, or real
state, is always an epistemic alternative for each agent (in 65-models, R i is reflexive; or,
speaking in terms of the corresponding axiom, knowledge is veridical). Using the
wise-man metaphor: this man does not consider any state which has already been
abandoned by one of the agents.
A formula w is defined to be valid in a model } iff },s*w for all s [ S; w is valid
with respect to 65 iff } *w for all } [ 6 5. The formula w is satisfiable in } iff
},s*w for some s [ S; w is satisfiable with respect to 65 iff it is satisfiable in some
} [ 6 5. A set of formulae F is valid with respect to 65 iff each formula w [ F is
valid with respect to 65. For a class of models # we define two relations between a set
2
of formulae and a formula: F * 1
# w iff ; } [ # ( } *F ⇒ } *w ) and F * # w iff
; } [ # ;s( },s*F ⇒ },s*w ). If the class symbol # is left out, we always mean
6 5m .
Theorem 2.8. (Soundness and Completeness.) Let w be some formula, and let F be a set
of formulae. The following soundness and completeness results hold.
1. £w ⇔ *w,
2. F £ 2 w ⇔ F * 2 w,
3. F £ 1 w ⇔ F * 1 w.
3. Group knowledge is not always implicit
Within the system S5 m (G), all agents are considered ‘equal’: if we make no additional
assumptions, they all know the same.
Lemma 3.1. For all i, j # m, we have: £Ki w ⇔ £Kj w.
Remark 3.2. One should be sensitive for the comment that Lemma 3.1 is a meta
statement about the system S5 m (G), and as such it should be distinguished from the
claim that one should be able to claim within the system S5 m (G) that all agents know the
same: i.e., we do not have (nor wish to have) that £Ki w ↔Kj w, if i ± j. Considering £w
as ‘w is derivable from an empty set of premises’ (cf. our remarks following Definition
2.4), Lemma 3.1 expresses ‘ When no contingent fact is known beforehand, all agents
know the same’. In fact, one easily can prove that ‘ When no contingent facts are given,
each agent knows exactly the S5 m (G)- theorems’, since we have, for each i # m: £w ⇔
£Ki w
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Interestingly, we are able to prove a property like that in Lemma 3.1 even when the G
operator is involved.
Theorem 3.3. Let X and Y range over hK1 ,K2 , . . . ,Km ,Gj. Then: £Xw ⇔ £Yw.
Theorem 3.3 has, for both the reading as group knowledge as well as that of a
receiving agent for G, some remarkable consequences. It implies that the knowledge in
the group is nothing else than the knowledge of any particular agent. We can also phrase
this in terms of implicit knowledge. First, let us agree upon what it means to say that
group knowledge is implicit:
Definition 3.4. Given a notion of derivability £ a (where we will typically take
a [ h 1 , 2 j), we say that group knowledge is strongly implicit (under £ a) if there is a
formula w for which we have £ a Gw, but for all i # m Z a Ki w. It is said to be weakly
implicit under £ a if there is a set of premises F for which F £ a Gw, but F Z a Ki w, for all
i # m.
Thus, Theorem 3.3 tells us that, using £ of S5 m (G), group knowledge is not strongly
implicit! Although counter-intuitive at first, Theorem 3.3 also invites one to reconsider
the meaning of £w as opposed to a statement F £ 1 w. When interpreting the case where
F 5 5 as ‘initially’ or, more loosely, ‘nothing has happened yet’ (in particular, when no
communication of contingent facts has yet taken place), it is perhaps not too strange that
all agents know the same and thus that all group knowledge is explicitly present in the
knowledge of all agents.
With regard to derivability from premises, we have to distinguish the two kinds of
derivation introduced in Definition 2.4.
Lemma 3.5. Let w be a formula, F a set of formulae and i some agent.
• F £ 1 Gw ⇔F £ 1 Ki w,
• F £ 2 Ki w ⇒ F £ 2 Gw,
• F £ 2 Gw £F £ 2 Ki w.
The first clause of Lemma 3.5 states that for derivability from premises in which
necessitation on premises is allowed, group knowledge is also not implicitly but
explicitly present in the individual knowledge of each agent in the group (thus F may be
considered an initial set of facts that are known to everybody of the group). The second
and third clause indicate that the notion of group knowledge is relevant only for
derivation from premises in which necessitation on premises is not allowed. In that case,
group knowledge has an additional value over individual knowledge. A further
investigation into the nature of group knowledge when necessitation on premises is not
allowed is the subject of the next section.
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
222
4. Group knowledge is not always distributed
In the previous section we established that group knowledge is in some cases not
implicit, but explicit. In particular, for provability per se and for derivation from
premises with necessitation on premises, group knowledge is rather uninteresting. The
only case where group knowledge could be an interesting notion on its own is that of
derivability from premises without necessitation on premises. For this case, we want to
formalize the notion of distributed knowledge. Although intuitively clear, a formalization of distributed knowledge brings a number of hidden parameters to the surface.
Informally, we say that the notion of group knowledge is distributed if the group
knowledge (apprehended as a set of formulae) equals the set of formulae that can be
derived from the union of the knowledge of the agents that together constitute the group.
The following quote is taken from a recent dissertation (Borghuis, 1994):
Implicit knowledge is of interest in connection with information dialogues: if we
think of the dialogue participants as agents with information states represented by
epistemic formulae, then implicit knowledge precisely defines the propositions the
participants could conclude to during an information dialogue . . .
Borghuis (1994) means with ‘implicit knowledge’ what we call ‘group knowledge’.
We will see in this section that using standard epistemic logic one cannot guarantee that
group knowledge is precisely that what can be concluded during an information
dialogue.
To do so, we will first formalize the notion of distributivity using the notions of
derivability used so far.
4.1. A deductive approach
To formalise what we mean with distributed group knowledge, let us have a second
look at derivation rule R3 first. It says that, if w follows from the conjunction of
w1 , . . . ,wm , then, if we also have K1 w1 ∧ ? ? ? ∧ Km wm , we are allowed to conclude that w
is known by the group, i.e. then also Gw. Now, we are inclined to call group knowledge
distributed if the converse also holds, that is, if everything that is known by the group
does indeed follow from the knowledge of the group’s individuals. Let us, for
convenience, say that group knowledge is perfect (over the individuals) if both directions
hold. What we now have to do is be precise about the phrase ‘follows from’ and about
the role that can be played by premises here. Let us summarize our first approximation
to the crucial notions in a semi-formal definition:
Definition 4.1 (first approximation).
1. We say that G represents complete group knowledge if we have that for every w and
w1 , . . . ,wm such that (K1 w ∧ ? ? ? ∧ Km wm ) can be derived, and also the implication
(w1 ∧ ? ? ? ∧ wm ) → w, then we also have that Gw can be derived.
2. The group knowledge represented by G is called distributed if for every w such that
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
223
Gw is derivable we can find w1 , . . . ,wm such that: (1) together they imply w and (2)
the conjunction (K1 w ∧ ? ? ? ∧ Km w ) is derivable.
3. If both items above hold we call the group knowledge represented by G perfect group
knowledge.
Observation 4.2. In S5 m (G), using £, G-knowledge is both complete group knowledge
and distributed.
The proof of Observation 4.2 shows that it holds for a trivial reason. For distributivity,
if £Gw, by Theorem 3.3, we also have £K1 w, . . . ,£Km w, and then distributivity follows
from £(w ∧ ? ? ? ∧ w ) → w. To show that G models complete group knowledge, one
reasons as follows: from £K1 w1 ∧ ? ? ? ∧ Km wm and £(w1 ∧ ? ? ? ∧ wm ) → w one obtains
£Gw1 ∧ ? ? ? ∧ Gwm and £G((w1 ∧ ? ? ? ∧ wm ) → w ) from which £Gw immediately follows.
So, to obtain distributivity in a non-trivial way, we should allow for premises.
However, we know from the previous section that we then have at least two notions of
derivability to consider. Unfortunately, as the following observation summarizes, even
when allowing premises, we trivially obtain distributivity.
Observation 4.3. If the notion of derivability used in Definition 4.1 is to be F £ 1 c or
F £ 2 c, we again obtain distributivity of G-knowledge in a trivial way.
Let us try to understand the source of Observation 4.3. For both w 5 1 , 2 , if
F £ w Gw, then we also have F £ w w, and hence F £ w (¡ ∧ ? ? ? ∧ ¡) → w. Obviously, we
also have F £ w (K1 ¡ ∧ ? ? ? ∧ Km ¡), justifying the claim that group knowledge is
distributed.
From the two Observations 4.2 and 4.3 above, we conclude that, in order to have an
interesting notion of distributed group knowledge, we have to allow for a subtle use of
premises and of notions of derivability. Let us therefore introduce the following
notation. Let k [ h0,1j be a characteristic function in the sense that, for every set F,
k ? F 5 5 if k 5 0, and F else. Now, we want to distinguish three types of derivability:
one to obtain the (K1 w1 ∧ ? ? ? ∧ Km wm ) formulas (the ‘knowledge conclusions’), one to
obtain (w1 ∧ ? ? ? ∧ wm ) → w (the ‘logical conclusions’) and one to obtain formulas of
type Gw (the ‘group knowledge conclusions’). For each type of derivability, we may
choose to allow for premises or not, and we also choose between £ 1 and £ 2 .
Definition 4.1. Let kk ,kl ,kg [ h0,1j and let K, L and G be variables over h 1 , 2 j. For
each tuple L 5 kkk ,K,kl ,L,kg ,Gl we now define the following notions: we say that
L-derivable group knowledge is complete if, for all sets of premises F and every
formula w :
kg ? F £ G Gw ⇐'w1 , . . . ,wm : [kk ? F £ K (K1 w1 ∧ ? ? ? ∧ Km wm ) & kl ? F £ L (w1 ∧ ? ? ?
∧ wm ) → w ];
(1)
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224
we say that L-derivable group knowledge is distributed, if for every set of premises F,
and every w :
kg ? F £ G Gw ⇒ 'w1 , . . . ,wm : [kk ? F £ K (K1 w1 ∧ ? ? ? ∧ Km wm ) & kl ? F £ L (w1 ∧ ? ? ?
∧ wm ) → w ].
(2)
If both (1) and (2) hold for a tuple L and every F and w, we say that the knowledge
represented by G is perfect with respect to L.
Thus, we have specialized our first attempt to define perfect group knowledge, in
notions depending on the choices in L. There are, in principle, 64 such notions, but the
following observations give a systematic account of them: the reader may also consult
the summarising Table A.1 in Appendix A.
Observation 4.4. For 49 possibilities for L 5 kkk ,K,kl ,L,kg ,Gl one can show that group
knowledge is indeed distributed—but the proofs show that this is only established in a
‘trivial’ manner.
In the observation above, ‘trivial’ refers to the fact distributivity is proven in a way
similar to that in Observations 4.2 and 4.3 – for an exhaustive proof, the reader is
referred to Appendix A. Those cases are summarized in Table A.1, with an a, b, c or d in
the ‘yes’ column.
Observation 4.5. For the 13 of the remaining 15 possibilities for L 5 kkk ,K,kl ,L,kg ,Gl
one can show that group knowledge is not distributed; they are summarized in Table A.1
with the items e, f and g in the ‘no’ column.
The remaining interesting cases are L 5 k1, 2 ,0,L,k1 ,1l which we take as a starting
point for the next section. Note that, in fact, this is only one case, since if kl 5 0, the
choice of L 5 2 , 1 is arbitrary. Also note that, apart from the technical results of the
observations above, this remaining L seems to make sense: we do not distinguish
between kk and kg ( 5 1), we take premises seriously but we only consider the case where
K 5 G 5 2 , so that group knowledge can be called implicit, according to Section 3.
Then, according to Observation 4.3 the only sensible choice for kl is 0. Let us formulate
the notion of distributivity for the remaining L’s:
F £ 2 Gw ⇒ 'w1 , . . . ,wm : F £ 2 (K1 w1 ∧ ? ? ? ∧ Km wm ) & £(w1 ∧ ? ? ? ∧ wm ) → w.
(3)
Thus, the conclusion w should be an S5 m (G)-consequence of the lemma’s w1 , . . . ,wm , for
Gw and (K1 w1 ∧ ? ? ? ∧ Km wm ) that are derivable using the same rules: using premises
but without applying necessitation to them.
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
225
4.2. A semantic approach
If we try to reformulate (3) semantically, we obtain
If Gw is true in every situation that verifies F, then there should be w1 , . . . ,wm such
that (w1 ∧ ? ? ? ∧ wm ) → w is valid and (K1 w1 ∧ ? ? ? ∧ Km wm ) is true in every situation
verifying F.
However, we think that the order of quantification in the phrase above deserves some
more attention. Consider F 5 hK1 p ∨ K2 pj. Indeed, we have F * 2 Gp, but does p
follow from formulas w1 ,w2 such that (K1 w1 ∧ K2 w2 ) is true in every situation },s for F ?
No, obviously not (see Observation 4.6 below or its proof in Appendix A); this is too
strong a requirement: instead of finding such w1 and w2 such that every situation },s for
F satisfies (K1 w1 ∧ K2 w2 ), however we are able to find such formula for every situation
that verifies F, since such a situation },s either satisfies K1 p or K2 p, so that we can
take (w1 ,w2 ) to be either ( p,¡) or (¡, p).
Observation 4.6. Although we have hK1 p ∨ K2 pj£ 2 Gp, there exist no w1 ,w2 with
£(w1 ∧ w2 ) → p and hK1 p ∨ K2 pj£ 2 (K1 w1 ∧ K2 w2 ).
Corollary 4.7. Group knowledge is not distributed in the sense of (3).
So, according to the latter Corollary, also the remaining choices for L do not yield a
notion of distributed knowledge. Let us thus push our argument about the order of
quantification one step further, by introducing a principle of ‘full communication’:
We say that group knowledge satisfies the principle of full communication if in
every situation in which Gw is true one can find w1 , . . . ,wm such that (K1 w1 ∧ ? ? ? ∧
Km wm ) also holds in that situation, and, moreover, (w1 ∧ ? ? ? ∧ wm ) → w is a valid
implication.
Phrased negatively, group knowledge does not satisfy the principle of full communication if there is a situation in which the group knows something, i.e. Gw, where w does
not follow from the knowledge of the individual group members. We will now show that
in the logic S5 m (G), group knowledge does not satisfy the principle of full communication, either.
First, note that the semantic counterpart of full communication reads as follows:
; },s,w 'w1 , . . . wm : [ },s*Gw ⇒ },s*(K1 w1 ∧ ? ? ? ∧ Km wm ) & (w1 ∧ ? ? ? ∧
wm )* 2 w ].
(4)
Definition 4.8. Let } be a Kripke model with state s, and i # m. The knowledge set of i
in },s is defined by: K(i, },s) 5 hw u },s*Ki w j.
Then, one easily verifies that (4) is equivalent to
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W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
},s*Gw ⇒ <K(i, },s)* 2 w.
i
(5)
Property (5) expresses that G-knowledge can only be true at some world s if it is
derivable from the set that results when putting all the knowledge of all the agents in s
together.
For multi-agent architectures in which agents have the possibility to communicate
(like, for instance, van Linder et al., 1994) principle (5) is rather relevant. This so-called
principle of full communication captures the intuition of fact discovery (cf. Halpern and
Moses, 1990) through communication. The principle of full communication formalizes
the intuitive idea that it is possible for one agent to become a wise man by
communicating with other agents: these other agents may pass on formulae from their
knowledge that the receiving agent combines to end up with the knowledge that
previously was implicit. As such, the principle of full communication seems highly
desirable a property for group knowledge. It is questionable whether group knowledge is
of any use if it cannot somehow be upgraded to explicit knowledge by a suitable
combination of the agents’ individual knowledge sets, probably brought together through
communication. Coming back to the example of the three computer scientists and the
question whether P ± NP, if there is no way for them to combine their knowledge such
that the proof results, it is not clear whether this statement should be said to be
distributed over the group at all.
Unfortunately, in the context of S5 m (G), the principle of full communication does not
hold. The following counterexample describes a situation in which group knowledge
cannot be upgraded to explicit knowledge (thereby answering a question raised by van
Linder et al., 1994).
Counterexample 4.9. Let the set P of propositional variables contain the atom q.
Consider a Kripke model } such that
• S 5 hx,x9,z,z9j,
• p (q,x) 5 p (q,x9) 5 1, p (q,z) 5 p (q,z9) 5 0 and for all p [ P, p ( p,x) 5 p ( p,x9) and
p ( p,z) 5 p ( p,z9) (the precise value, 0 or 1, of p does not matter),
• R 1 is given by the solid lines in Fig. 1, and R 2 is given by the dashed ones (reflexive
arrows are omitted).
Observation 4.10. It holds that (see Appendix A for a justification):
Fig. 1. A finite model for two agents.
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
227
• K(1, },x) 5 K(2, },x),
• },x*Gq,
• q[
⁄ K(1, },x) < K(2, },x) 5 K(1, },x).
From this last observation it follows that the principle of full communication does not
hold in this model: although the formula q is group knowledge, it is not possible to
derive this formula from the combined knowledge of the agents 1 and 2. Thus it is not
possible for one of these agents to become a wise man through receiving knowledge
from the other agent. In terms of Borghuis: the model of Counterexample 4.9 shows that
one can have group knowledge of an atomic fact q, although this group knowledge will
never be derived during a dialogue between the agents that are involved. The ‘reason’
for this in our example is that the agents have no means in their language to distinguish
between x and x9, or between z and z9. In Section 5 we will see that such a negative
result can already be obtained in a model in which there are no such ‘copies of the same
world’ (see also Fig. 3).
Corollary 4.11. In S5 m (G), group knowledge does not satisfy the principle of full
communication.
5. Models in which group knowledge is distributed
In this section we will characterize a class of Kripke models, in which G-knowledge
is always distributed, in the sense that they satisfy the principle of full communication.
To be more precise, the models that we come up with will satisfy
},s*Gw ⇔ <K(i, },s)* 2 w.
i
(6)
Note that the principle (6) indeed is that of (5), combined with the validity of axiom A6.
Definition 5.1. A Kripke model } 5 kS,p,R 1 , . . . ,R m l is called finite if S is finite;
moreover, it is called a distinguishing model if for all s,t [ S with s ± t, there is a ws 1 ,t 2
such that },s*ws 1 ,t 2 and },t±
u ws 1 ,t 2 .
When considering an S5 1 -model as an epistemic state of an agent, it is quite natural to
use only distinguishing models: such a model comprises all the different possibilities the
agent has. One can show that for S5, questions about satisfiability of finite sets of
formulae, and hence that of logical consequence of a finite set of premises, can be
decided by considering only finite distinguishing models. In fact, one only needs to
require such models to be distinguishing at the propositional level: any two states in
such a simple S5-model differ in assigning a truth value to at least one atom (see, e.g.,
Meyer and van der Hoek, 1995b, Section 1.7). In the multiple-agent case, this
distinguishing requirement needs to be lifted from the propositional level to the whole
language +.
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
228
The nice feature of finite distinguishing models is that sets of states can be named:
Lemma 5.2. Let } 5 kS, . . . l be a finite distinguishing model. Then:
;X , S'aX [ + ;x [ S ( },x*aX ⇔x [ X).
For a given set X, we call aX the characteristic formula for X.
Lemma 5.3. Let } 5 kS,p,R 1 , . . . ,R m l be a given finite distinguishing model, with s [ S.
Suppose that Z 7 S is such that for all z [ Z we have },z* z. Moreover, suppose
X1 , . . . ,Xn 7 S are such that R i (s) 7 (X1 < ? ? ? < Xn < Z). Then
},s*Ki (¬aX 1 → (¬aX 2 → ( . . . (¬aXn → z ) . . . ))).
If one tries to link up the two lemmas above with the principle of full communication,
one may take the following point of view. First of all, note that, given a state s, all
agents at least share one epistemic alternative, which is s itself. Now, consider the set of
epistemic alternatives for agent i in s, R i (s). Relative to agent j we can partition R i (s)
into two subsets: a set Yj 7 R i (s) of worlds that are considered as epistemic alternatives
for agent j, and a set Xj 5 Ri (s)\Yj of worlds that are possible alternatives for i, but not
for j. Let aXj and aYj be the characteristic formulas for Xj and Yj , respectively. Then,
agent i knows that the real world must be in either one of Xj and Yj , i.e. agent i knows
that ¬aXj → aYj , and agent j knows ¬aXj . Thus, agent j may inform agent i so to speak
which of i’s alternatives can be given up. Now, if all the agents h would communicate
this information Xh to agent i, then finally agent i would end up with those epistemic
alternatives that are considered possible by all of the agents. This argument is made
more precise in the (proof of) the following theorem.
Theorem 5.4. Let } be a finite distinguishing model. Then, in }, G-knowledge is
distributed, in the sense that it satisfies (6).
We finally observe that the two properties of Definition 5.1 are in some sense also
necessary for distributivity. Let us say that in } group knowledge satisfies the principle
of full communication if, for all states s and formulae w, Eq. (6) is satisfied. Then, we
know already from Fig. 1 and Observation 4.10 that there is a finite 6 5 2 model in which
this principle of full communication is not valid. One might wonder if there is a
distinguishing model that falsifies the same principle (this model should then, by virtue
of Theorem 5.4, be infinite). It is tempting to apply the following reasoning to the model
} of Fig. 1.
Take the canonical model } c for S5 m (G). There is a set G 5 hc u },x*c j which
is a world in this canonical model } c , where },x is as in Fig. 1. G is a maximal
consistent set, and hence, a member of the canonical model } c . Also, one has
} c ,G *(K1 a ↔K2 a ), for all a, } c ,G *¬K1 q and also } c ,G *Gq. Thus, } c ,G
demonstrates that the principle of full communication is violated in the canonical
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
229
model. This model } c moreover is infinite, and distinguishing, by definition ( its set of
states is the set of all maximal consistent sets).
However, we have to mention here that } c is in general not a model of the kind as
defined in Definition 2.6; the true definition of },s*Gw is ;t(s,t) [ R G ⇒ },t*w,
where R G can only be guaranteed to be some subset of R 1 > ? ? ? > R m . For more on
this, the reader is referred to van der Hoek and Meyer (1992). But still, we claim the
existence of an 6 5 2 model 1 that is distinguishing, in which G-knowledge is defined as
in our Definition 2.7, and in which the principle of full communication does not hold.
This model 1 is provided in Definition A.4 in Appendix A. Let us conclude this section
by summarizing the discussion of the last paragraphs:
Observation 5.5.
• There is a finite, not distinguishing model, in which group knowledge does not fulfil
the principle of full communication.
• There is a distinguishing model, not finite, in which group knowledge does not fulfil
the principle of full communication.
6. Conclusion
We have given formal definitions of what it means for group knowledge to be implicit
or to be distributed relative to a group. Basically, group knowledge is implicit, if there is
some formula known by the group, but not by any of its members. Here, one has to be
precise about the role to be played by premises, and about the notion of derivability that
is used to conclude that something is known. It appears that, under the standard notion
of derivability in S5 m (G), and when no premises are allowed, group knowledge is not
implicit. The same holds if one allows for applications of the Necessitation rule to
premises. Thus, the only case in which group knowledge may be implicit in a group is
the case in which premises describe a situation rather than the knowledge present
beforehand.
Next, we studied whether group knowledge is always distributed. We made the notion
of distributivity precise by saying that a formula is distributed (given some set of
premises) if it can somehow, using some notion of derivability and some related set of
premises, be derived from the union of the agents’ knowledge. We argued that there was
one way of choosing the parameters such that distributivity is not trivialised or directly
absent. We gave a semantic account of a variant of distributivity (called the principle of
full communication): we then were able to show that this principle was still not
guaranteed, in general. Then, we gave two properties of Kripke models that guarantee
that group knowledge does allow full communication; we also showed that both
properties are, in some sense, needed: giving up one of the properties enables one to
falsify the principle mentioned.
We think it quite remarkable that such an appealing and natural notion like group
knowledge as discussed in this paper can give rise to rather complicated behaviour or
230
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
properties. For instance, it is also well known that the canonical models for a logic like
S5 m (G) only have the property that the accessibility relation for G, say R G , is a subset of
R 1 > ? ? ? > R m , and cannot be guaranteed to be equal to this intersection. In other
words, this equality is not modally definable, although one can find a complete
axiomatization for those models that satisfy it. For more on this, the reader is referred to
van der Hoek and Meyer (1992). In that paper, the authors also show that definability of
R 1 > ? ? ? > R m 5 R G is achieved when using graded modal logic. But in this logic, the
assumption of distinguishability of a model seems to be unnatural; graded modal logic is
typically designed to be able to reason about numbers (and hence copies) of worlds.
Also, in van der Hoek and Meyer (1997) the same authors demonstrate that proving
an epistemic logic involving both group knowledge and common knowledge complete
with respect to S5 m (G)-like models can be quite complicated. Again, distinguishability
of the models has to be given up. To be more precise, given a state },s for c, the
technique of filtration in modal semantics often yields a finite distinguishing model } 9
with state s9, such that } 9,s9 still verifies c, which might suggest af first sight that finite
distinguishing models are always obtained. It is not guaranteed, however, that this model
} is in the appropriate class, i.e. if } is an 6 5m (G)-model, } 9 need not be.
Finally, in a recent manuscript (Lomuscio and Ryan, 1997) a rather surprising
property of group knowledge was mentioned in the context of so-called Interpreted
Systems, systems that are widely accepted as a general semantics for distributed systems
(this is also a major approach in Fagin et al., 1995). In such a system, one obtains
Gw ↔w, for all w. Since such a system is by definition distinguishing (the states usually
differ already on the objective level), in the finite versions the principle of full
communication would be automatically obtained, but, at the same time, it might be a too
strong, and perhaps optimistic principle: all facts about the world would be eventually
obtained by the agents, would they be able to fully communicate.
Acknowledgements
This work was partially supported by ESPRIT WG No. 23531 (FIREworks) and the
Human Capital and Mobility network EXPRESS. The contents of this paper evolved by
fruitful discussions with and comments from several people, and can thus be conceived
as a good example of a body of distributed knowledge, obtained by repeated and fruitful
communication. More particularly, Mark Ryan and Elias Thijsse gave useful comments
on previous versions of this paper. Frank de Boer gave helpful suggestions to simplify
the presentation of the model of Fig. 3. We also thank the participants of LOFT2 (the
Second Conference on Logic and the Foundations of the Theory of Games and
Decisions) for the inspiring discussion on this paper. Finally, the comments of two
anonymous referees of Mathematical Social Sciences helped to further improve our
presentation.
Appendix A. Proofs
Proof of Theorem 2.5. Our proof is inspired by that of Lemma A1 of Meyer and van
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
231
der Hoek (1995a), where a similar property is proven for an ‘S5 1 -like’ logic. Let us
abbreviate hX u X [ hK1 , . . . ,Km j 1 j to W (the set of non-empty words over the alphabet
hK1 , . . . Km j) and, for a set of premises F, let us write WF for hXw u X [ W,w [ F j. We
first prove the following lemma:
Lemma A.1. For all F,w,i # m: WF £ 2 c ⇔WF £ 2 Ki c.
Proof. The ‘⇐’ direction follows immediately from an application of R1 to Ki c and
axiom A3. The ‘ ⇒ ’ direction is proven by induction on the length l(p ) of a shortest
£ 2 -proof p 5 kc1 , . . . cl(p ) 5 c l of c from WF, i.e. we show that, for all n and all c,
If WF £ 2 c has a shortest proof of length n, then WF £ 2 Ki c.
(A.1)
• If l(p ) 5 1 then
1.1.either c [ WF, i.e. c 5 Xw for a sequence X 5 Kj 1 Kj 2 . . . Kj t [ W and w [ F.
Obviously, Ki Xw [ WF, and hence we also have WF £ 2 Ki c.
1.2.Or c is an axiom of S5 m (G). By necessitation, we also have £Ki c and thus
WF £ 2 Ki c.
• Suppose that (A.1) holds for all WF £ 2 a with l(p ) # k(k $ 1) and suppose now that
WF £ 2 c has a shortest proof p 5 kc1 , . . . ck 11 ( 5 c )l.
2.1.We do not have to consider the case that c [ WF, since then p would not be a
shortest proof. The same holds for the case c being an S5 m (G)-axiom.
2.2.Suppose c 5 Kj b with £b. By necessitation, we then also have £Ki b and we
prove Ki c 5 Ki Ki b by applying A4 and R1, yielding £Ki c and thus WF £ 2 Ki c.
2.3.The only other possibility is that ck is obtained by an application of modus ponens
to cj and cj → ck 11 , j # k. Using the induction hypothesis on the proofs for cj and
cj → ck11 , we obtain WF £ 2 Ki cj , WF £ 2 Ki (c → ck 11 ). Applying R1 and A2 to
the latter two conclusions yields WF £ 2 Ki ck 11 , i.e. WF £ 2 Ki c. j
We now reformulate and prove Theorem 2.5:
Theorem 2.5. Let hK1 , . . . Km j 1 be the set of non-empty sequences of epistemic
operators. Then hXw u X [ hK1 , . . . Km j 1 ,w [ F j£ 2 c ⇔F £ 1 c.
Proof. (‘ ⇒ ’): Suppose that WF £ 2 c. Since £ 1 extends £ 2 we then also have WF £ 1 c.
Moreover, we also have that F £ 1WF (meaning that F £ 1 Xw for any X [ W,w [ F ).
Combining F £ 1Wf and WF £ 1 c then yields F £ 1 c. (‘⇐’): Suppose that F £ 1 c, with
p 1 5 kc1 . . . ck ( 5 c )l a £ 1 -proof of c from F. Let `(p ) be the number of applications
of the Necessitation rule in a proof p to non-S5 m (G)-theorems. We prove, for all n and
all c :
If F £ 1 c has a proof p 1 with `(p 1 ) 5 n, then WF £ 2 c.
(A.2)
• If `(p 1 ) 5 0, we replace every step in p 1 of the form wj , which is justified by
232
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
wj [ F, by the three steps Ki wj ,Ki wj → wj ,wj to obtain a proof k (p 1 )2 for WF £ 2 w.
(Note that the resulting proof k (p 1 )2 indeed only uses premises from WF.)
• Suppose that (A.2) holds for all p 1 with `(p 1 ) # k, and we have a F £ 2 c with a
proof p 1 with `(p 1 ) 5 k 1 1, say p 1 5 kc1 . . . cr , cr 11 , . . . ck 11 ( 5 c )l, where
cr 11 is obtained from ch (h # r) as a result of the last Necessitation step to previous
formulas. So, cr 11 5 Kj ch , for some j # m. Thus, if p 9 1 is the proof for F £ 1 ch , then
`(p 9 1 ) # k. The induction hypothesis yields WF £ 2 ch . By Lemma A.1, we obtain
WF £ 2 Kj c. Note that the remaining part s 1 of the £ 1 -proof p 1 with s 1 5
k(ck 11 ( 5 c ) . . . c r 1 1( 5 Kj ch )l from F does not involve Necessitation to formulas
in this sequence, so `(s 1 ) 5 0. Thus, as above, we can transform it in a £ 2 -proof
k (s 1 )2) for Kj c from WF (replacing each assumption in s 1 of the form cv ( [ F )
by the triple K1 cv ( [ WF ),K1 cv → cv ,cv ). j
Proof of Theorem 2.8.
• This is a standard result in modal logic; see, for instance, Hughes and Cresswell
(1984) or Fagin et al. (1995) and Meyer and van der Hoek (1995b) for this specific
multi-agent epistemic logic S5 m (G). This property is also referred to as ‘weak
soundness and completeness’.
• This is the standard notion of adding premises in modal logic. Proofs of this ‘strong
soundness and completeness’ result can again be found in Hughes and Cresswell
(1984) or Fagin et al. (1995) and Meyer and van der Hoek (1995b).
• For the ⇒ -direction, one only has to check that applying Necessitation to premises
in F is * 1 -valid. The argument for ⇐ runs as follows. Suppose that we have F and
w for which F Z 1 w. By Theorem 2.5, we then have WF Z 2 w, where W is defined
as in the proof for Theorem 2.5 above. Thus, we find some Kripke model } and
world s for which },s*WF and },s±
u w. Using a standard argument in modal logic
(see, for instance, Theorem 2.12 of Meyer and van der Hoek (1995a)), one may
assume
that the model } is generated from s: for all w [ S there is a path
→
v 5 ks 5 w 1 ,w 2 , . . . w n 5 wl such that for each w i and w i 11 (1 # i , n) there is some
vi # m with R vi w i w i 11 . We now claim that } *F and } ±
u w, which is equivalent to
F±
u 1 w. To show that } *F, let c [ F and let w be
an
arbitrary element in the set
→
of states, S. We know that },s*WF. Let the path v be as above, and let, as above
again, for vi # m, R vi w i w i 11 . Now, Kv 1 Kv 2 . . . Kvn 21 c is an element of WF, so
},s*Kv 1 Kv 2 . . . Kvn 21 c. But then, also },w*c, which was to be proven. The fact
that } ±
u w follows immediately from the fact that },s±
u w. j
On Observations 4.4 and 4.5. Table A.1 should be read as follows: the first column
gives the name of the corresponding horizontal line; the second column give possible
tuple L and then, the ‘yes’ column give a reason why this tuple gives rise to the claim
that the corresponding notion of group knowledge indeed may be called distributed (see
the proof of Observation 4.4 below). The ‘no’ column, to the contrary, argues why the
notion cannot be called distributed (see the proof of Observation 4.5). Then, the same
exercise is done for the tuple L9 that is obtained from L by changing kk from 0 to 1.
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
233
Table A.1
Table summarizing Observations 4.4 and 4.5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
kk
K
kl
L
kg
G
Yes
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
a,b,c
a,c
No
e
e,f
a,b,c
a,b,c
e
e
c
c
b
f
c
c
b
b
a,b,c,d
a,c,d
e
e,f
a,b,c,d
a,b,c,d
e
e
c,d
c,d
b
f
c,d
c,d
b
b
kk
K
kl
L
kg
G
Yes
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
b,c
c
No
h
g
b,c
b,c
h
g
c
c
b
g
c
c
b
b
b,c
c
d
d
b,c
b,c
d
d
c
c
b,d
d
c
c
b,d
b,d
Proof of Observation 4.4. We claim that the following tuples L 5 kkk ,K,kl ,L,kg ,Gl make
the notion of group knowledge distributed – but in a trivial way.
(a) Tuples of type k0,K,0,L,0,Gl, i.e. no premises allowed. This was already observed
in Observation 4.2, and the argument is given also in Section 4.1.
(b) Suppose that kg 5 kl , and that G 5 L or G 5 2 ,L 5 1 . Then, if kg ? F £ G Gw,
then also kg ? F £ G w and, under the specified conditions, also kl ? F £ L (¡ ∧ ? ? ? ∧
¡) → w. Of course, one also has kk ? F £ K (K1 ¡ ∧ ? ? ? ∧ Km ¡), independent of kk and
K. Note how this item generalizes (the proof of) Observation 4.3.
(c) Tuples L for which kg 5 0 (note how this item generalizes item (a)). Under this L,
suppose that £Gw. Then, by Theorem 3.3, we have £(K1 w1 ∧ ? ? ? ∧ Km wm ) and hence
kk ? F £ K (K1 w1 ∧ ? ? ? ∧ Km wm ). Also, one has kl £ L (w ∧ ? ? ? ∧ w ) → w. Thus, by this
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W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
item, we have judged half of all the possibilities to obtain distributed group
knowledge as trivial.
(d) Here, K 5 1 and kg 5 kk . The crucial observation now is that if kg ? F £ G Gw,
then kk ? F £ K Ki w. Obviously, we also have kl ? £ L (w ∧ ? ? ? ∧ w ) → w.
Proof of Observation 4.5. The following tuples L 5 kkk ,K,kl ,L,kg ,Gl make the notion of
group knowledge to be not distributed.
(e) kk 5 kl 5 0, kg 5 1. Let F 5 hGpj, where p is an atom. Suppose that we would
have w1 , . . . ,wm for which £(K1 w1 ∧ ? ? ? ∧ Km wm ) and £(w1 ∧ ? ? ? ∧ wm ) → p. Then
(using A3 and R1) we would also have £p, which we do not.
(f) kg 5 1, G 5 1 and, as a first subcase, suppose kk 5 0 and L 5 2 . Let F 5
h p,Gp → qj. We have F £ 1 Gq. Suppose for this subcase that there are w1 , . . . wm with
£ K (K1 w1 ∧ ? ? ? ∧ Km wm ) and h p,Gp → qj£ 2 (w1 ∧ ? ? ? ∧ wm ) → q. Then we would
have h p,Gp → qj£ 2 q. But the fact that h p,Gp → qj±
u 2 q is easily verified, and then
2
Theorem 2.8 yields h p,Gp → qj£ q.
(g) As a second subcase to the previous item, suppose, on top of kg 5 1, G 5 1 , that
kk 5 1, K 5 2 and that either kl 5 0 or L 5 2 . As in the previous item, we then
have that if there would be w1 , . . . wm for which h p,Gp → qj£ K (K1 w1 ∧ ? ? ? ∧ Km wm ),
and kl ? h p,Gp → qj£ K (w1 ∧ ? ? ? ∧ wm ) → q, then we would also have h p,Gp → qj£ 2 q,
which we do not, as argued above.
(h) The remaining tuples are L 5 kkk ,K,kl ,L,kg ,Gl 5 k1, 2 ,0,L,1, 2 l, with L 5 2 2
, 1 . That they do not guarantee distributivity of group knowledge follows from
Observation 4.6.
Proof of Observation 4.6. In Observation 4.6 we claimed the following: Although we
have hK1 p ∨ K2 pj£ 2 Gp, there exist no w1 ,w2 with £(w1 ∧ w2 ) → p and hK1 p ∨
K2 pj£ 2 (K1 w1 ∧ K2 w2 ). Consider the model of Fig. 2 (reflexive arrows are omitted; R 1 is
denoted with the solid line, R 2 with a dashed one). Suppose that there would be formulas
w1 and w2 of the type described above. Since we both have },s*K1 p ∨ K2 p and
},t*K1 p ∨ K2 p, we have },s*(K1 w1 ∧ K2 w2 ) and },t*(K1 w1 ∧ K2 w2 ). Thus, we have
Fig. 2. The model proving Observation 4.6.
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
235
},u*w1 ∧ w2 . However, this is not compatible with £(w1 ∧ w2 ) → p, thus such w1 and w2
do not exist. j
Proof of Corollary 4.7. For F 5 hK1 p ∨ K2 pj we have F £ 2 Gp, but we also saw that
there are no w1 and w2 such that £(w1 ∧ w2 ) → p and for which F £ 2 (K1 w1 ∧ K2 w2 ), which
is sufficient to disprove distributivity for the tuples L 5 k1, 2 ,0,L,1, 2 l, L 5 1 , 2 . j
Proof of Observation 4.10. Observation 4.10 claimed about 1 that:
1. K(1, },x) 5 K(2, },x),
2. },x*Gq,
3. q [
⁄ K(1, },x) < K(2, },x) 5 K(1, },x).
Proof. We first state and prove the following lemma.
Lemma A.2. For all u [ hx,zj, and all w : },u*w ⇔ },u9*w.
Proof of Lemma A.2. Using induction on w. For atoms in P this follows by the
definition of the valuation p. The proof for the connectives ∧ , ∨ and ¬ is
straightforward. So let us assume the statement has been proven for formula c and
consider w 5 K2 c. If },x*K2 c, we have that },u*c for u 5 x,z. The induction
hypothesis then yields that },u9*c for u 5 x,z, and thus },x9*K2 c. Similar arguments
hold for the other direction, as well as for the world z in the model, and also for
w 5 K1 c,Gc. End of Proof of Lemma A.2. j
Now, the first item of Observation 4.10 is proven as follows. Suppose that c [
K(2, },x), that is, },x*K2 c. Thus, by definition of K2 , we have },u*c for u 5 x,z.
Then, by the previous lemma, we also have that z9 verifies c. But then, obviously, we
have },x*K1 c. To see the second item, observe that (x,v) [ R 1 > R 2 ⇔v 5 x. And,
since q is true in x, we have },x*Gq. Finally, since q is false in z9, we have that K1 q is
false at x, so q [
⁄ K(1, },x). The equality K(1, },x) < K(2, },x) 5 K(1, },x) follows
immediately from the first item. j
Proof of Lemma 5.2. This is easily first obtained for singletons X 5 hxj. Since
} 5 kS, . . . l is finite, we can enumerate the worlds that differ from x as s 1 ,s 2 , . . . s n . Let
ax be (wx 1,s 12 ∧ wx 1,s 22 ∧ . . . ∧ wx 1,s 2n ). Then, obviously },s*ax ⇔s 5 x. Subsequently,
one puts aX 5 k x[X ax . j
Proof of Lemma 5.3. Let t [ R i (s). We have to show that },t*¬aX 1 → (¬aX 2
→ ( . . . (¬aXn → z ) . . . )) (*). Note that, for arbitrary x [ S and formulas b, we have
( },x*¬aXj → b )⇔(x [
⁄ Xj ⇒ },x* b ).
So,
(*)
is
equal
to
t[
⁄ X1 ⇒ (t [
⁄ X2 ⇒ ( . . . (t [
⁄ Xn ⇒ },t* z ) . . . )) (**). Now, (**) is true because t [
(X1 < ? ? ? < Xn < Z) and, for all x [ S, x [ Z ⇒ },x* z. j
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W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
Proof of Theorem 5.4. For the ‘⇐’ direction, suppose < i K(i, },s)* 2 w, and let t be
such that (s,t) [ R 1 > . . . R m . Then, for every b [ < i K(i, },s)* 2 w we have },t* b
and, thus, },t*w. By definition of t, we may conclude that },s*Gw. For the
‘ ⇒ ’-direction, suppose },s*Gw. Let us consider R 1 (s): we can cover it with possibly
overlapping regions: let Z 7 R 1 (s) be R 1 (s) > ? ? ? > R m (s), and for every agent i, let
Xi 5 hx [ R 1 (s) u x [
⁄ R i (s)j. Thus, Xi is the set of worlds considered possible by agent 1,
but not by agent i. If the agents were able to communicate, agent i could tell agent 1 that
each x [ Xi is not a good epistemic alternative, so that the agent 1 can eliminate all
worlds x that appear in some set Xi , thus ending up in exactly those alternatives in
R 1 (s) > ? ? ? > R m (s). Formally, we put Lemma 5.3 to work. Note that R 1 (s) 7 (X1 < ? ?
? < Xm < Z). By Lemma 5.2, we find a characteristic formula aXi for every Xi , and, by
definition of Z, we have },z*w, for all z [ Z. Then, Lemma 5.3 guarantees that
},s*K1 (¬aX 1 → (¬aX 2 → ( . . . (¬aXn → w ) . . . ))) and, by definition of Xi and aXi , we
also have },s*Ki ¬aXi (for, if },s±
u Ki ¬aXi there would be a t [ R i (s) with },t*aXi ,
i.e.
t [ Xi ,
which
contradicts
the
definition
of
Xi ).
Thus,
h¬aX 1 → (¬aX 2 → ( . . . (¬aXn → w ) . . . ),¬aX 1 ,¬aX 2 , . . . ¬aXm j 7 < i K(i, },s), and thus
< i K(i, },s)* 2 w. j
Proof of Observation 5.5.
• The proof of this claim is provided by the model of Fig. 1.
• This follows from Theorem A.5, presented below.
Before formally introducing our next Kripke model, we need some preliminary notions
and notation.
Definition A.3. We now assume that the set of atomic propositions P consists of an
atom q together with infinitely atoms pi (i $ 1). In the following, we will use infinite
binary sequences for the representation of worlds in our model. Here, a binary sequence
is either an infinite sequence s 5 ks 1 ,s 2 , . . . l, or a finite sequence s 5 ks 1 ,s 2 , . . . ,s n l with
each s i [ h0,1j. In the finite case, we say that the length l(s) of s is n. Let us denote the
set of all the infinite sequences by IBS, and the finite sequences by FBS. It is convenient
to define, for a finite sequence u 5 ku 1 , . . . ,u n l and an arbitrary binary sequence v the
concatenation u;v of u and v, as u;v 5 ku 1 , . . . u n ,v1 ,v2 , . . . l. Also, for a finite sequence
u, we define u 1 5 u, and u n 11 5 u;u n (n . 1). For any finite sequence u of length n,
u 5 ku 1 , . . . u n l, let the infinite sequence u v [ IBS be ku 1v ,u 2v , . . . l, where u iv 5 u i mod n .
Another way of defining u v would be as the solution of the equivalence u v 5 u;u v .
Thus, k0,1,1l v 5 k0,1,1,0,1,1,0,1,1, . . . l, k1l v 5 k1,1,1, . . . l and k0l v 5 k0,0,0, . . . l. We
call a sequence t a suffix of a given sequence u iff for some finite sequence v we have
u 5 v;t (v may be the empty sequence e ). Finally, the equivalence relation ; s on IBS
holds between sequences with an equal suffix, that is, u ; s t iff there is some v [ IBS
and some u 1 ,t 1 [ FBS such that u 5 u 1 ;v and t 5 t 1 ;v. Given this relation ; s , for any
u [ IBS, [u] 5 ht [ IBS u u ; s tj.
Definition A.4. Let 1 5 kS,p,R 1 ,R 2 l be the Kripke model of Fig. 3, that is, S 5 S1 <
S2 < S3 < S4 7 IBS, where S1 5 [k1l v ], S2 5 [k0l v ], S3 5 [k0,0,1l v ], and S4 5 [k1,1,0l v ].
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
237
Fig. 3. An infinite distinguishing model for two agents.
We have chosen the worlds x, z, x9 and z9 as named members of the sets S1 , S2 , S3 and
S4 , respectively. We can choose for these worlds the representatives of the equivalence
class that defines the Si that contains that world: then, for instance, we get x 5 k1l v ,
x9 5 k0,0,1l v , and so on. Then, we put p (u)(q) 5 1 iff u [ S1 < S3 , and p (u)( pi ) 5 1 iff
u i 5 1. Thus, for instance, in each state u from S1 , q is true, and, although some of the
p’s may be false in such a world in S1 , there is some index k u [ N, for which all pi ’s are
true, if i $ k u , since u ; s k1l v . The relations R 1 and R 2 are indicated by the straight and
dotted lines, respectively, and formally defined as follows. R 1 is the equivalence relation
with the two equivalence classes S1 < S4 and S2 < S3 , whereas the equivalences classes
of R 2 are S1 < S2 and S3 < S4 .
Note that the model 1 is a distinguishing infinite model. The relation between the
model 1 just defined and the model } of Fig. 1 is already partly suggested by the
names of the distinguished worlds. For instance, in both cases, x and x9 are worlds in
which q is false, and which are both R 1 and R 2 related to a world of z-type, but not
R 1 > R 2 -related. However, in 1, it is not the case that x and x9 verify the same formulas,
which is imposed by the mere fact that 1 is distinguishing (in this case, note that
1,x* p1 whereas 1,x9*¬p1 ). However, we will be able to prove that, although x and
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W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
x9 are not equal, the cluster S1 is ‘similar’ to S3 , whereas S2 is in the same sense related
to S4 . We are now ready to state the main result about the model 1.
Theorem A.5. Let 1 be the model defined as in Definition A.4, let x [ S1 , z [ S2 ,
x9 [ S3 and z9 [ S4 . Then:
1. K(1, 1,x) 5 K(2, 1,x),
2. },x*Gq,
3. q [
⁄ K(1, },x) < K(2, },x) 5 K(1, },x).
The proof of Theorem A.5 proceeds by combining a number of lemmas. First we need
an additional definition.
Definition A.6. Let n [ N, n $ 1. The language +n consists of the formulas of which the
only propositional atoms are q, p1 , . . . , pn . Also, two sequences u,v [ IBS are nequivalent, written u ; n v if for all i # n, u i 5 vi .
Now we can state the lemma that guarantees that the clusters S1 and S3 behave
similarly (as do S2 and S4 ).
Lemma A.7. Let n $ 1 be given. Then, for every world in S1 there is an n-equivalent
world in S3 , and vice versa. The same relation holds between S2 and S4 . Put more
formally, we have:
;n(;x [ S1 'x9 [ S3 (x ; n x9) & ;x9 [ S3 'x [ S1 (x9 ; n x)) & ;n(;z [ S2 'z9 [
S4 (z ; n z9) & ;z9 [ S4 'z [ S2 (z9 ; n z)).
Proof. Note that a cluster Si contains all sequences with a common suffix. So, as an
example, take x [ S1 . We know that x 5 kx 1 ,x 2 , . . . ,x n l;v;k1l v for some finite sequence
v. But, then, obviously, by putting x9 5 kx 1 ,x 2 , . . . ,x n l;v;k0,0,1l v we have that x9 [ S3
and x9 ; n x. j
Lemma A.8. Let n $ 1. Then:
;x [ S1 ;x9 [ S3 (x ; n x9⇔;w [ +n ( 1,x*w ⇔ 1,x9*w )),
& ;x [ S2 ;z9 [ S4 (z ; n z9⇔;w [ +n ( 1,z*w ⇔ 1,z9*w )).
(9)
(10)
Proof. For the main ‘⇐’ of (9), suppose that 1,x*w ⇔ 1,x9*w, for all w. We can now
in particular take w 5 F (x) 5 n 1#i #n ki (x i , pi ), where ki (x i , pi ) 5 pi if x i 5 1, and
ki (xi , pi ) 5 ¬pi else. This choice F (x) for w exactly encodes the first n bits of x, and,
since by assumption 1,x9*F (x), we must have x9 ; n x. A similar argument proves ‘⇐’
in (10). The other direction is proven simultaneously for (9) and (10), by induction on
the complexity of w. Let us concentrate on the first conjunct and assume that x ; n x9
and, for the base case, that w is a propositional atom. Then either it is q (which is
W. van der Hoek et al. / Mathematical Social Sciences 38 (1999) 215 – 240
239
falsified by x as well as by x9) or it is some pi with i # n and, since x ; n x9 these two
worlds also agree on pi . The proof for conjunctions and negations is straightforward, so
let us now assume that w 5 K1 c, for some c [ +n with 1,x*K1 c. To reach a
contradiction, assume that 1,x9*¬K1 c. It means that there is some z [ S2 with
1,z*¬c. Using the induction hypothesis on (10), we find some z9 [ S4 for which
1,z9*¬c. However, we have R 1 xz9, which indeed contradicts 1,x*K1 c. We conclude
that 1,x9*K1 c. j
Proof of Theorem A.5.
1. Suppose c [
⁄ K(1, 1,x). Then, either there is an s [ S1 with 1,s*¬c and, in that
case, we have c [
⁄ K(1, 1,x) or there is a z9 [ S4 with 1,z9*¬c. Let n be such that
c [ +n . By definition of the model 1, there is a z [ S2 , with z ; n z9. By Lemma A.8,
we find that 1,z*¬c, and thus, by definition of R 2 , c [
⁄ K(2, 1,x).
2. We have 1,x*Gq since q is true in the cluster S1 , which are exactly the R 1 > R 2 accessible worlds from x.
3. The atom q is not an element of K(1, 1,x), since q is not true in z9 [ S4 , with R 1 xz9.
Proof of Observation 5.5.
• This is proven by the model of Fig. 1 and Observation 4.10.
• This follows from the model of Fig. 3 and Theorem A.5.
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