Knowledge and group knowledge

216 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: c , c and c . 1 2 3 Assume that scientist 1 has proved c and therefore knows c . Analogously, for agents 2 1 1 and 3 regarding c and c 5 c ∧ c → w, respectively. If these computer scientists 2 3 1 2 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 conse- quence. 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- W . van der Hoek et al. Mathematical Social Sciences 38 1999 215 –240 217 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,K w,Gw [ + i m. i 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 K w is ‘agent i knows w ’ and Gw means ‘w is group i knowledge of the m agents.’ We assume the following ‘standard’ [the system below was originally introduced by Halpern and Moses 1985] inference system S5 G for the m multi-agent S5 logic extended with the operator G. Definition 2.2. The logic S5 G has the following axioms: m A1 any axiomatization for propositional logic A2 K w ∧ K w → c → K c i i i A3 K w → w i A4 K w → K K w i i i A5 ¬ K w → K ¬ K w i i i A6 K w → Gw i A7 Gw ∧ Gw → c → Gc A8 Gw → w A9 Gw → GGw A10 ¬ Gw → G ¬ Gw On top of that, we have the following derivation rules: R1 £ w,£w → c ⇒ £ c R2 £ w ⇒ £K w, for all i m. i 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 218 W . van der Hoek et al. Mathematical Social Sciences 38 1999 215 –240 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 £ , or £ for short, is defined in the usual way. That is, a S5 G m 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 c , c and c , each 1 2 3 known by one of the agents, let us consider the following derivation rule which is to be understood to hold for all w,c , i m: i R3£c ∧ ? ? ? ∧ c → w ⇒ £K c ∧ ? ? ? ∧ K c → Gw. 1 m 1 1 m m Theorem 2.3. Let the logic S5 G[R3 A6] be as S5 G, with axiom A6 replaced by m m the rule R3. Then , the two systems are equally strong : for all w, S5 G£w ⇔ S5 G[R3 A6]£w. m m 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 G we define the following two relations m 1 2 + 1 2 £ and £ 7 2 3 + : F £ c F £ c ⇔ w , . . . ,w with w 5 c, and such that for all 1 n n 1 i n: • either w [ F i • or there are j,k , i with w 5 w → w j k i • or w is an S5 G axiom i m • or w 5 K s where s is such that i k 1 s 5 w with j , i in the case of £ , j H 2 2 s in the case of £ . From a modal logic point of view, one establishes: 1 2 5£ w ⇔ 5£ w, 1 2 such that £w is unambiguously defined as 5£ w or 5£ w, and 2 1 £w ∧ . . . ∧ w → c ⇔ w , . . . ,w £ c ⇒ w , . . . ,w £ c. 1 n 1 n 1 n W . van der Hoek et al. Mathematical Social Sciences 38 1999 215 –240 219 2 1 In other words, the deduction theorem holds for £ , not for £ . Note that the relation 1 2 1 £ is more liberal than £ in the sense that £ allows for necessitation on premises, 2 1 where £ only applies necessitation to S5 G-theorems. Our definition of £ deviates m from standard terminology of premises: in classical logic, the formulas w , . . . ,w are 1 n called premises in a derivation of c if w ∧ . . . ∧ w → c is derivable. Thus, premises 1 n 1 in our sense in the context of £ 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. Theorem 2.5. Let hK , . . . ,K j be the set of sequences of epistemic operators. Then 1 m 2 1 hXw u X [ hK , . . . ,K j ,w [ F j£ c ⇔ F £ c. 1 m From Meyer and van der Hoek 1995a Lemma A1, for the one agent case without group knowledge i.e., S5 , one even obtains 1 2 1 hKw u w [ F j£ c ⇔ F £ c. 2 From an epistemic point of view, when using £ , 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 1 of Kaneko and Nagashima 1997], whereas in the case of £ , 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 1 mutual knowledge of the players as premises under the £ notion of derivability. Definition 2.6. A Kripke model } is a tuple } 5 kS,p,R , . . . ,R l where 1 m 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 7 S 3 S is an accessibility relation. For any s [ S and i m, i with R s we mean ht [ S u R stj. i i We refer to the class of Kripke models as _ or, when m is understood, as _. The class m of Kripke models in which the accessibility relations R are equivalence relations, is i denoted by 6 5 or by 6 5. m 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 ⇔ ps, p 5 1, },sw ∧ c ⇔ },sw and },sc, },s ¬ w ⇔ },s± u w, },sK w ⇔ },tw for all t with s,t [ R , i i },sGw ⇔ },tw for all t with s,t [ R . . . R . 1 m For a given w [ + and } [ 6 5, nwm 5 ht u },twj. For sets of formulae F, },sF is 220 W . van der Hoek et al. Mathematical Social Sciences 38 1999 215 –240 defined by: },sF ⇔ },sw 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 is reflexive; or, i 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 },sw for all s [ S; w is valid with respect to 65 iff } w for all } [ 6 5. The formula w is satisfiable in } iff },sw 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 1 2 of formulae and a formula: F w iff ;} [ } F ⇒ } w and F w iff ;} [ ;s},sF ⇒ },sw. If the class symbol is left out, we always mean 6 5 . m 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 2 2. F £ w ⇔ F w, 1 1 3. F £ w ⇔ F w. 3. Group knowledge is not always implicit Within the system S5 G, all agents are considered ‘equal’: if we make no additional m assumptions, they all know the same. Lemma 3.1. For all i, j m, we have : £K w ⇔ £K w. i j Remark 3.2. One should be sensitive for the comment that Lemma 3.1 is a meta statement about the system S5 G, and as such it should be distinguished from the m claim that one should be able to claim within the system S5 G that all agents know the m same : i.e., we do not have nor wish to have that £K w ↔ K w, if i ± j. Considering £w i j 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 G- theorems ’, since we have, for each i m: £w ⇔ m £K w i W . van der Hoek et al. Mathematical Social Sciences 38 1999 215 –240 221 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 hK ,K , . . . ,K ,Gj. Then: £Xw ⇔ £Yw. 1 2 m 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: a Definition 3.4. Given a notion of derivability £ where we will typically take a a [ h 1 , 2 j, we say that group knowledge is strongly implicit under £ if there is a a a formula w for which we have £ Gw, but for all i m Z K w. It is said to be weakly i a a a implicit under £ if there is a set of premises F for which F £ Gw, but F Z K w, for all i i m. Thus, Theorem 3.3 tells us that, using £ of S5 G, group knowledge is not strongly m implicit Although counter-intuitive at first, Theorem 3.3 also invites one to reconsider 1 the meaning of £w as opposed to a statement F £ 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. 1 1 • F £ Gw ⇔ F £ K w, i 2 2 • F £ K w ⇒ F £ Gw, i 2 2 • F £ Gw£F £ K w. i 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. 222 W . van der Hoek et al. Mathematical Social Sciences 38 1999 215 –240

4. Group knowledge is not always distributed