Group knowledge is not always distributed

222 W . van der Hoek et al. Mathematical Social Sciences 38 1999 215 –240

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 formaliza- tion 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 w , . . . ,w , then, if we also have K w ∧ ? ? ? ∧ K w , we are allowed to conclude that w 1 m 1 1 m m 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 w , . . . ,w such that K w ∧ ? ? ? ∧ K w can be derived, and also the implication 1 m 1 m m w ∧ ? ? ? ∧ w → w, then we also have that Gw can be derived. 1 m 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 w , . . . ,w such that: 1 together they imply w and 2 1 m the conjunction K w ∧ ? ? ? ∧ K w is derivable. 1 m 3. If both items above hold we call the group knowledge represented by G perfect group knowledge. Observation 4.2. In S5 G, using £, G-knowledge is both complete group knowledge m 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 £K w, . . . ,£K w, and then distributivity follows 1 m from £w ∧ ? ? ? ∧ w → w. To show that G models complete group knowledge, one reasons as follows: from £K w ∧ ? ? ? ∧ K w and £w ∧ ? ? ? ∧ w → w one obtains 1 1 m m 1 m £Gw ∧ ? ? ? ∧ Gw and £Gw ∧ ? ? ? ∧ w → w from which £Gw immediately fol- 1 m 1 m lows. 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. 1 Observation 4.3. If the notion of derivability used in Definition 4.1 is to be F £ c or 2 F £ 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 w w w F £ Gw, then we also have F £ w, and hence F £ ¡ ∧ ? ? ? ∧ ¡ → w. Obviously, we w also have F £ K ¡ ∧ ? ? ? ∧ K ¡, justifying the claim that group knowledge is 1 m 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 K w ∧ ? ? ? ∧ K w formulas the ‘knowledge conclusions’, one to 1 1 m m obtain w ∧ ? ? ? ∧ w → w the ‘logical conclusions’ and one to obtain formulas of 1 m type Gw the ‘group knowledge conclusions’. For each type of derivability, we may 1 2 choose to allow for premises or not, and we also choose between £ and £ . Definition 4.1. Let k ,k ,k [ h0,1j and let K, L and G be variables over h 1 , 2 j. For k l g each tuple L 5 kk ,K,k ,L,k ,Gl we now define the following notions: we say that k l g L-derivable group knowledge is complete if, for all sets of premises F and every formula w : G K L k ? F £ Gw ⇐ w , . . . ,w : [k ? F £ K w ∧ ? ? ? ∧ K w k ? F £ w ∧ ? ? ? g 1 m k 1 1 m m l 1 ∧ w → w]; 1 m 224 W . van der Hoek et al. Mathematical Social Sciences 38 1999 215 –240 we say that L-derivable group knowledge is distributed, if for every set of premises F, and every w : G K L k ? F £ Gw ⇒ w , . . . ,w : [k ? F £ K w ∧ ? ? ? ∧ K w k ? F £ w ∧ ? ? ? g 1 m k 1 1 m m l 1 ∧ w → w]. 2 m 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 kk ,K,k ,L,k ,Gl one can show that group k l g 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 kk ,K,k ,L,k ,Gl k l g 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,k ,1l which we take as a starting 1 point for the next section. Note that, in fact, this is only one case, since if k 5 0, the l 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 k and k 5 1, we take premises seriously but we only consider the case where k g 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 k is 0. Let us formulate l the notion of distributivity for the remaining L’s: 2 2 F £ Gw ⇒ w , . . . ,w : F £ K w ∧ ? ? ? ∧ K w £w ∧ ? ? ? ∧ w → w. 1 m 1 1 m m 1 m 3 Thus, the conclusion w should be an S5 G-consequence of the lemma’s w , . . . ,w , for m 1 m Gw and K w ∧ ? ? ? ∧ K w that are derivable using the same rules: using premises 1 1 m m 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 w , . . . ,w such 1 m that w ∧ ? ? ? ∧ w → w is valid and K w ∧ ? ? ? ∧ K w is true in every situation 1 m 1 1 m m verifying F. However, we think that the order of quantification in the phrase above deserves some 2 more attention. Consider F 5 hK p ∨ K p j. Indeed, we have F Gp, but does p 1 2 follow from formulas w ,w such that K w ∧ K w is true in every situation },s for F ? 1 2 1 1 2 2 No, obviously not see Observation 4.6 below or its proof in Appendix A; this is too strong a requirement: instead of finding such w and w such that every situation },s for 1 2 F satisfies K w ∧ K w , however we are able to find such formula for every situation 1 1 2 2 that verifies F, since such a situation },s either satisfies K p or K p, so that we can 1 2 take w ,w to be either p,¡ or ¡, p. 1 2 2 Observation 4.6. Although we have hK p ∨ K p j£ Gp, there exist no w ,w with 1 2 1 2 2 £w ∧ w → p and hK p ∨ K p j£ K w ∧ K w . 1 2 1 2 1 1 2 2 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 w , . . . ,w such that K w ∧ ? ? ? ∧ 1 m 1 1 K w also holds in that situation , and, moreover, w ∧ ? ? ? ∧ w → w is a valid m m 1 m implication . Phrased negatively, group knowledge does not satisfy the principle of full communica- tion 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 G, group knowledge does not satisfy the principle of full communica- m tion, either. First, note that the semantic counterpart of full communication reads as follows: ;},s,w w , . . . w : [},sGw ⇒ },sK w ∧ ? ? ? ∧ K w w ∧ ? ? ? ∧ 1 m 1 1 m m 1 2 w w]. 4 m Definition 4.8. Let } be a Kripke model with state s, and i m. The knowledge set of i in },s is defined by: Ki,},s 5 hw u },sK wj. i Then, one easily verifies that 4 is equivalent to 226 W . van der Hoek et al. Mathematical Social Sciences 38 1999 215 –240 2 },sGw ⇒ Ki,},s w. 5 i 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 G, the principle of full communication does not m 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, • pq,x 5 pq,x9 5 1, pq,z 5 pq,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 is given by the solid lines in Fig. 1, and R is given by the dashed ones reflexive 1 2 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 • K1,},x 5 K2,},x, • },xGq, • q [ ⁄ K1,},x K2,},x 5 K1,},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 G, group knowledge does not satisfy the principle of full m communication.

5. Models in which group knowledge is distributed