multiagent belief revision

Accepted Manuscript

Multiagent Belief Revision

Antoine Billot, Jean-Christophe Vergnaud, Bernard Walliser

PII: S0304-4068(15)00054-3DOI: http://dx.doi.org/10.1016/j.jmateco.2015.05.004Reference: MATECO 1973

To appear in: Journal of Mathematical Economics

Received date: 30 January 2014Revised date: 17 April 2015Accepted date: 5 May 2015

Please cite this article as: Billot, A., Vergnaud, J.-C., Walliser, B., Multiagent Belief Revision.Journal of Mathematical Economics (2015), http://dx.doi.org/10.1016/j.jmateco.2015.05.004

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http://dx.doi.org/10.1016/j.jmateco.2015.05.004

Multiagent Belief Revision∗

Antoine Billot, lemma-Paris 2 and iuf (Paris, France)†

Jean-Christophe Vergnaud, ces-cnrs-Paris 1 (Paris, France)‡

Bernard Walliser, pse (Paris, France)§

April 17, 2015

Abstract

An original epistemic framework is proposed for the modeling of beliefs and messageswithin a multiagent belief setting. This framework enables public, private and secret mes-sages as well, even when the latter contains errors. A revising rule– i.e. the product rule– is introduced in pure epistemic terms in order to be applied to all structures and message.Since any syntactic structure can be expressed through various semantic ones, an equiva-lence principle is given by use of the semantic notion of bisimilarity. Thereafter, a robustnessresult proves that, for a given prior structure, bisimilar messages yield bisimilar posteriorstructures (Theorem 1). In syntax, the beliefs revised thanks to the product rule are thenshown to be unique (Theorem 2). Finally, an equivalence theorem is established between theproduct rule and the Belief-Message Inference axiom (Theorem 3).Journal of Economic Literature Classification Number: C72, D82

1 Introduction

Belief revision has been intensively studied in epistemic logics. However, if epistemic models arewell-established in a multiagent setting, there is no convincing general modeling of the effect ofa message onto prior beliefs in this framework, in particular when this message is secret and/orwhen it contains false information on the world or, more subtly, on the information received byanother agent. Yet, a lot of economic and strategic situations involve erroneous information or‘secret communication’. The purpose of this paper is then to propose a framework allowing torevise consistently all prior beliefs by means of a generalized rule called the product rule (pr).This rule works as follows: first, a message is designed as a standard belief structure wherepossible worlds and possibility domains translate what is learned by agents and what is learnedabout what is learned; second, posterior worlds are built up by combining prior worlds withmessage worlds; third, at each posterior world, agents believe as possible all combinations ofprior and message worlds that are compatible.

∗We thank A. Baltag, G. Bonanno, P. Fleckinger, I. Gilboa, J. Lang, J.M. Tallon, J. van Benthem and threereferees for helpful remarks and comments on earlier version of this paper.†[email protected]‡[email protected]§[email protected]

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*ManuscriptClick here to view linked References

1.1 An example: the Rothschild ‘myth’

Nathan Mayer Rothschild is associated with one of the mythical cases of ‘insider trading’entail-ing erroneous information: he is said to have used his early knowledge of an English victory atWaterloo to fool people on the Stock Exchange and consequently make a vast fortune. A versionof this story can be found in Morton (1962): A Rothschild agent jumped into a boat at Ostend tobring a dispatch. When he received it, Nathan Rothschild let his eye fly over the lead paragraphs(claiming Wellington’s victory). Then he proceeded to the Stock Exchange. He did not invest.He sold. He dumped consols. Consols dropped still more. ‘Rothschild knows,’the whisper rippledthrough the Change. ‘Waterloo is lost.’ Nathan kept on selling and bought a giant parcel for asong. Let now consider an imaginary version of this story involving Nathan Mayer Rothschild(denoted R) and the British Prime Minister, Robert Banks Jenkinson (denoted J), which leavesout the strategic aspect to focus onto beliefs and revision. This version is developed along thetwo standard approaches in epistemics– namely, first in syntax and second in semantics.

For Rothschild as for Jenkinson, it does matter whether ‘Wellington wins’– a propositionformally denoted p– or ‘Wellington does not win’– i.e., ¬p. Before any message, they both haveno definite belief about the outcome of the battle and their prior beliefs (priors for brief, denotedBi) can then be formally expressed as follows: ¬Bip ∧ ¬Bi¬p with i ∈ {R, J}– meaning thatthey do not believe p and they do not believe ¬p neither– and it is common belief. Now, Roth-schild is secretly informed by a spy while Jenkinson is just informed through the offi cial channeland ignores that a spy came informing Rothschild. The content of the spy-message– ‘Wellingtonwins’– is captured by another proposition m while the content of the offi cial-message– ‘the out-come of the battle is still uncertain’– is captured by a proposition m′. (Notice that the languagein which the messages are represented differs from the language in which the facts of the matterare represented.) To formally describe the diffusion protocol of the message, the operator Biis introduced depicting what ‘agent i learns’. At a first level, the information displayed canbe captured by: BRm ∧ BJm′ ∧ ¬BJm. At a second level, Jenkinson erroneously learns thatRothschild has only learned the offi cial message: BJBRm′ ∧¬BJBRm, while Rothschild learnscorrectly that Jenkinson has only learned the offi cial message: BR

(BJm

′ ∧ ¬BJm). In addition,

Rothschild learns that Jenkinson learns erroneously that he has only learned the offi cial message:BRBJ

(BRm

′ ∧ ¬BRm). Posterior beliefs (posteriors for brief, denoted B∗i ) are then deduced

from a logical revision of priors according to what agents have learned. For Rothschild, learningm removes his initial doubt: hence, B∗Rp, while for Jenkinson, learningm

′ has no particular effecton his first-order posteriors: hence, ¬B∗Jp ∧ ¬B∗J¬p. Rothschild being able to simulate Jenkin-son’s reasoning, his second-order posteriors are then given by: B∗R (¬B∗Jp ∧ ¬B∗J¬p). In return,Jenkinson can also simulate Rothschild’s reasoning but, since he did not learn correctly whatRothschild has learned, he believes that Rothschild’s posteriors remain: B∗J (¬B∗Rp ∧ ¬B∗R¬p).Hence, Jenkison makes an obvious mistake when considering Rothschild’s final beliefs. The pos-teriors can be lengthened toward the higher level: since Rothschild has learned that Jenkinsonerroneously learned what he has learned himself, Rothschild believes that Jenkinson believesthat his posteriors remain: B∗RB

∗J (¬B∗Rp ∧ ¬B∗R¬p), etc.

Let now turn to a semantic interpretation of this story. For priors, a standard partitionalmodel (Aumann, 1976) with two possible worlds– one for p and one for ¬p– is suffi cient torepresent how Rothschild and Jenkinson both ignore the state of nature (the outcome of thebattle). At each world, they both regard these two worlds as possible:

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Insert Fig1

Similarly, the secret spy-message can be captured through a two-world structure: an actual worldwhere the correct message is m– ‘Wellington wins’– and a second world where no information isavailable– corresponding to m′. In the actual world, Rothschild believes this first world possiblewhile Jenkinson believes possible the second one at which, besides, both regard this world aspossible:

Insert Fig2

To deduce the semantical structure representing posteriors, pr works as follows. Firstly, thetrue actual world proceeds from the combination of the true prior world with the true world ofthe message– more generally, posterior worlds result from the combination of prior worlds withmessage worlds. In the true posterior world, Rothschild believes possible any combination ofthe prior worlds that he thought possible (i.e., w, w1) with compatible message worlds that hehas learned to be possible (i.e., w). Given the content of the message m, the two worlds w1and w are incompatible and, therefore, the only posterior world Rothschild believes possible is(w,w). The same process holds for Jenkinson but, since the world w1 that he has learned tobe possible is compatible with both w and w1, there are two posterior worlds that he believespossible: (w, w1) and (w1, w1).

Insert Fig3

Let us now display how pr is relevant to the revision process syntactically presented above.Recall that Jenkinson finally believes that it is common belief that Rothschild and he still doubtthe outcome of the battle. This corresponds to the two possible worlds (w, w1) and (w1, w1). Inreturn, Rothschild knows the outcome and Jenkinson’s beliefs as well. This corresponds to theactual world (w,w). Hence, three worlds are definitely necessary to describe posteriors in thiscase.

1.2 Background and motivation

In epistemic logics, Alchourron, Gärdenfors and Makinson (1985) can be seen as the first attemptto design and axiomatize belief revision. In their static set-up, a unique agent is endowedwith set-theoretic priors and she is assumed to receive a message defined as an event of thisspace. Messages are generally meant to clarify agents’beliefs but they could refute them as well(Fagin et al., 1995). More recently, in a dynamic set-up this time, Baltag, Moss and Solecki(1998) characterize a message by its content– i.e., its meaning– and its diffusion protocol– i.e.,a specification of the agents who receive the content of the message and the knowledge of allagents about its scattering. Messages are then semantically represented by a structure of possibleworlds. To explicit the revision process, two sorts of worlds need to be distinguished: prior worldsand message ones. A method is then given for mixing them up in order to obtain posterior worlds(see also Baltag and Moss, 2004; Aucher, 2009, 2011, and Baltag, van Ditmarsch and Moss,2008). Incidently, all these contributions focus on nonpartitional possibility correspondences.

In interactive epistemology, Aumann (1976) defines a framework where agents’beliefs arebased on two possible sources of information : probabilistic common priors and private informa-tion partitions over the same set of possible worlds. The existence of a Bayesian revision process

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is suggested but no precise method is made explicit about the way a prior partition evolves intoa posterior one. As for Geneakoplos and Polemarchakis (1982), they consider a dynamic processabout public communication connected with partition transformations. At each world, agentsrevise their prior partitions by intersection with the message partition. This corresponds to thestandard intersection rule (ir) where priors and message are both expressed over the same setof worlds. However, ir is not theoretically founded. Especially, there exists no current generalaxiomatization despite two interesting attempts we discuss now. First, Board (2004) considersinfinitely many possibility correspondences– each being associated with a proposition that be-haves as a message content– but does not account for a general structure to catch on the diffusionprotocol. Second, Bonanno (2005) suggests considering three possibility correspondences– onefor priors, one for messages and the latter for posteriors– but, in return, does only relate to asingle agent.

In the Rothschild-Jenkinson example, as shown above, it is impossible to apply ir becausethe number of posterior worlds is larger than that of prior worlds, even though, previously–that is for public or private messages– ir has been successfully applied. The diffi culty to useir in our case is apparently based upon the fact that a secret message can involve persistingerrors and these errors entail nonpartitional belief structures (to be considered each time theTruth axiom does no longer hold). However, Geneakoplos (1989) introduces a definition ofequivalence between information structures and shows that there always exists a partitionalstructure which is equivalent to a nonpartitional one. Therefore, the Rothschild-Jenkinsonsecret message example can be dealt with ir in a partitional framework. Yet, this equivalencerelation is founded in terms of decision– that is, two structures are equivalent if they lead tothe same decision according to the Expected Utility (eu) criterion. Here, we only considera pure epistemic framework where agents’ preferences are not specified. In our framework,the equivalence relation that associates two semantical structures corresponding to identicalbeliefs is given by a purely epistemic notion; i.e., bisimilarity. Furthermore, under bisimilarity,Geneakoplos’s conclusion may not hold: there does not always exist a partitional structure thatis bisimilar to a nonpartitional one– this is precisely the case for the posterior structure in theRothschild-Jenkinson example. In conclusion, ir is not an universal rule for revision, especiallyin the presence of a failure of the Truth axiom.

1.3 Overview

Section 2 is devoted to the presentation of a general semantical framework for beliefs and tothe definition of bisimilarity between belief structures or between worlds. Formally, beliefs aremodeled with semantic belief structures involving individual possibility correspondences be-tween worlds (see Perea, 2012). Each world describes a state of the material environment andcrossed beliefs about it. Messages are modeled with auxiliary belief structures which capturetheir content and their diffusion protocol and pr, the product rule, is semantically defined.Then, a robustness result proves that, for a given prior belief structure, bisimilar messages leadto bisimilar posterior belief structures (Theorem 1). Section 3 is devoted to the syntacticaltranslation of the semantical framework and also to a syntactical interpretation of pr. As it iswell-known (see, for instance, Hughes and Cresswell, 1996), a syntactical belief structure corre-sponds to a complete and consistent set of propositions written in a weak multi-modal versionof the standard kd logic. This correspondence holds for the prior structure, the message andthe posterior ones as well. Syntactical conditions are then designed in such a way that the belief

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revision axioms can be written in an extended language. A first axiom– called Belief-MessageInference– means that each agent’s posteriors can be directly obtained from the learning of amessage and from the priors. This can be viewed as a generalization of the ‘logical omniscience’condition. In order to ensure that posteriors define a consistent set of propositions, a secondaxiom– called Nonconflict– is also assumed. This implies that no agent can learn a propositionthat he believes to be a priori false. Formally, the revised beliefs are exhibited by finding outthe unique, complete and consistent set of propositions which satisfies the two axioms (Theorem2). Finally, an equivalence theorem (Theorem 3) clarifies the link between (semantical) pr andthe Belief-Message Inference axiom for belief revision and, in addition, pr is proved to be arobust revision rule by combining Theorems 2 and 3.

2 Semantics for belief revision

The epistemic framework is firstly introduced which is used for priors, message and posteriors.The rule pr is then formally defined. Moreover, bisimilarity between two structures is furtherconsidered. Finally, a first theorem is proved– setting up that pr is robust to substitution ofbisimilar messages.

2.1 Semantical structure

Agents are assumed to get crossed beliefs– of the type ‘agent i believes that agent j believesthat...’– about a common material environment. A semantic model à la Kripke is considered,based on a set of possible worlds which are said to be accessible by an agent in a given world.

A (generated) Pointed Epistemic Model (pem)1 is a list of components which satisfy a specificcondition– called Connectedness:

Definition 1 (PEM) A Pointed Epistemic Model (pem) for a set I of agents is defined as a5-uple H = (W, (Hi)i∈I , S,H0,w) where(i) W is a set of possible worlds denoted w,(ii) Hi is the agent i’s possibility correspondence defined as a correspondence from W toward2W \ {∅},(iii) S is a finite set of states of nature s,(iv) H0 is the possibility correspondence of nature defined as a function from W toward S,(v) w ∈W is the actual world,such that the following condition holds:(Connectedness) In the actual world w, for each world w, there exist a finite sequence of agentsi1, ..., in and a finite sequence of worlds w0, ..., wn such that w0 = w, wn = w and for allk ∈ [1, · · · , n], wk ∈ Hik (wk−1).

A possible world w ∈W is a full description of the material environment together with a fulldescription of agents’beliefs about this environment. The material part– called ‘nature’– is em-bodied within a state of nature denoted s. For all i ∈ I, the epistemic part– i.e., i’s beliefs aboutthe material environment– is embodied in an individual possibility correspondence denoted Hi.Each relation Hi brings together all possible worlds w which are considered ‘undiscernable’by

1A Pointed Epistemic Model is a usual term in epistemic logics with no precise equivalent in game theory(Blackburn et al., 2001).

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i when i faces the set W from a given world. We impose the possibility domains never to beempty since an empty possibility domain would reflect the absence of consistent beliefs at theconcerned world (see 3.2.). Besides, the possibility correspondence H0 assigns a unique state ofnature to any given world. Connectedness is a condition that ensures that the set W does notinclude any possible world which is inaccessible from w– the actual one. This condition is notrestrictive insofar as such inaccessible worlds have no influence on actual beliefs. Furthermore,in a belief revision process, posteriors are assumed to constitute a pem denoted H∗.

2.2 Message

A message is based on two components: a content and a diffusion protocol. The content cor-responds to the meaning of the message and the diffusion protocol specifies the agents whoreceive the message, as well as the knowledge of all agents about the scattering of the message.Since the content of a message relates to the material environment, and the diffusion protocolto crossed beliefs, a message can be modeled as a belief structure. Hence, a message can beformally defined as an (auxiliary) pem denoted H.

Definition 2 (message) Given a pem H = (W, (Hi)i∈I , S,H0,w), a message is an auxiliarypem H = (W,

(H i

)i∈I , S,H0,w) such that S is a finite subset of the prior set of events 2W \∅.2

The set W of worlds is made of elements w that describe possible instances of the message.The possibility correspondence H i characterizes the diffusion of the message for agent i in anymessage world w. Each state of nature s of S represents a possible content– that is an event ofW– and the possibility correspondence H0 associates an underlying message content with eachpossible world. Different contents may be sent in different situations. Practically, the simplestcase is as follows: a message whose content, denoted s0 or s1, is sent. By convention, we alwaysassume H0 (w) = s0. Four classical types of message can be considered according to their(different) diffusion protocols.

2.2.1 Public message

In a public message, denoted Hpub, each agent receives a message whose content is s0 ⊂ W .Moreover, everyone believes that everyone has received this message, hence the content anddiffusion of the message is common belief :{

W = {w} ; S = {s0} ; H0 (w) = s0,∀i ∈ I, H i (w) = {w} .

2.2.2 Private message

In a private message, denoted Hipriv, an agent– say i– receives a message whose content iseither s0 ⊂ W or its complement s1 = W\s0. Moreover, everyone believes that i was the onlyone to receive this message and i believes that everyone believes that he was the only one toreceive this message, hence the diffusion of the message is common belief:{

W = {w, w1} ; S = {s0, s1} ; H0 (w) = s0; H0 (w1) = s1,H i (w) = {w} ; H i (w1) = {w1} and ∀j 6= i, Hj (w) = Hj (w1) = {w, w1} .

2The finiteness condition on S is needed only to provide syntactic foundations. It plays no role in semantics.

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2.2.3 Secret message

In a secret message, denoted Hisec, agent i receives a message whose content is s0 ⊂ W , theother agents receiving nothing (i.e., s1 =W ). Then, everyone but i believes that no one receivesa message– consequently, different mistakes may occur:{

W = {w, w1} ; S = {s0, s1} ; H0 (w) = s0; H0 (w1) = s1,H i (w) = {w} ; H i (w1) = {w1} and ∀j 6= i, Hj (w) = Hj (w1) = {w1} .

2.2.4 Null message

A null message, denoted Hnul, is a public message whose content is not informative (i.e., s0 =W ): {

W = {w} ; S = {s0} ; H0 (w) = s0,∀i ∈ I, H i (w) = {w} .

2.3 Revision rule

The belief revision process determines the specification of posteriors, and more generally, the waythey are deduced from priors and the message. Formally, posteriors H∗ are assumed to resultfrom a ‘combination’of the prior pem H with the auxiliary one H by a product rule– denoted⊗:

H∗ = H⊗H.

Definition 3 (product rule– PR) Posteriors H∗ are derived from a product ⊗ of priors Hwith message H, in two steps:Step 1. Define H∗ =

(W ∗, (H∗i )i∈I , S

∗, H∗0 ,w∗)such that

(i) W ∗ =W ×W ,(ii) ∀ (w,w) ∈ W ∗, ∀i ∈ I, H∗i (w,w) =

⋃s∈S (Hi (w) ∩ s)× (H i (w) ∩H

−10 (s)),

(iii) S∗ = S,(iv) ∀ (w,w) ∈ W ∗, H∗0 (w,w) = H0 (w),(v) w∗ = (w,w) ∈ W ∗.Step 2. Define H∗ =

(W ∗, (H∗i )i∈I , S

∗, H∗0 ,w∗) as the connected part of H∗.

The set W ∗ is made of posterior worlds w∗, each w∗ being designed as the combination ofa prior world w with a message world w. For each posterior world, the possibility domain (foran agent) only keeps all pairs of prior worlds and message worlds which are consistent withthe basic content of the message. For each posterior world, the state of nature comes from thecorresponding prior world. Hence, there is no objective change taking place– a pure case ofbelief revision.

Remark 1 H∗ is not always a pem. Indeed, according to H and H, it could occur that anpossibility domain H∗i (w,w) is empty, even in a connected world (w,w).

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2.4 Bisimilarity

In epistemic logics, the semantical representation of a multiagent belief situation is not unique(see Hughes and Cresswell, 1996, or Blackburn, de Rijke and Venema, 2001). More precisely, sev-eral pems can be associated with the same beliefs. An equivalence relation between two pems isthen required. Consider two pems H = (W, (Hi)i∈I , S,H0,w) and H′ = (W ′, (H ′i)i∈I , S,H ′0,w′)and a binary relation denoted R, R ⊆W ×W ′, which links up worlds of the two structures thatshare relevant features. The relation R may satisfy the following properties:

Actual-world Equivalence (AE) – (w,w′) ∈ R.

Material Preservation (MP) – If (w,w′) ∈ R, then H ′0(w′) = H0(w).

Left Surjectivity (LS) – ∀i ∈ I, if (w,w′) ∈ R, then ∀w′ ∈ H ′i(w′), ∃w ∈ Hi(w) such that(w, w′) ∈ R.

Right Surjectivity (RS) – ∀i ∈ I, if (w,w′) ∈ R, then ∀w ∈ Hi(w), ∃w′ ∈ H ′i(w′) such that(w, w′) ∈ R.

AE means that the actual worlds w and w′ of two pems H and H′ are linked. MP meansthat, for two linked worlds w and w′, the material environment is the same. LS means that, fortwo linked worlds w and w′, any world of H′ which is accessible from w′ is linked with worlds ofH which are accessible from w. RS is the converse axiom: for two linked worlds w and w′, anyworld of H which is accessible from w is linked with worlds of H′ which are accessible from w′.

Definition 4 (bisimilar PEMs) Two pems H and H′ are bisimilar if there exists a binaryrelation R ⊆W ×W ′ which satisfies AE, MP, LS and RS.

As a classic result (Blackburn et al., 2001), beliefs represented by two bisimilar pems are syn-tactically identical, i.e., two bisimilar pems capture the same beliefs. This notion of bisimilaritycan be adapted to worlds belonging to two different pems: in two ‘bisimilar’worlds, beliefs shouldbe identical from the modeler’s point of view. To specify the link between these two notions ofbisimilarity, we need first to define what would be the pem H while the actual world w changes.Take any w ∈ W to be the new actual world. Denote H (w) = (W (w) , (Hw

i )i∈I , S,Hw0 , w) the

‘subpem’such that (1) W (w) contains w and all worlds w ∈ W which are connected with wand (2) (Hw

i )i∈I and Hw0 are respectively the restrictions of (Hi)i∈I and H0 to W (w). Now,

bisimilarity of worlds can be defined by bisimilarity of their corresponding pems.

Definition 5 (bisimilar worlds) Two worlds w and w′ belonging respectively to pems H andH′ are bisimilar if H (w) and H′ (w′) are bisimilar.

This definition holds naturally for two worlds belonging to the same pem: in a given pem,any world is bisimilar to itself. Besides, a subjective definition of bisimilarity, which capturesthe fact that some agent may have the same beliefs in two different pems, is also relevant. Anagent holds the same beliefs in two different pems if the worlds he considers possible in the twoactual worlds are bisimilar.

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Definition 6 (j-bisimilar PEMs) Two pems H and H′ are said to be j-bisimilar (or bisimilarfor agent j) if, for all w ∈ Hj(w), there exist w′ ∈ H ′j(w′) such that w and w′ are bisimilar,and conversely.

On the whole, it is easy to check that if two pems are j-bisimilar for all j ∈ I and if therespective actual worlds share the same states of nature, then the two pems are bisimilar.

2.5 Robustness

The following proposition establishes that, for given priors, the revision process is invariantunder bisimilarity of messages.

Theorem 1 Let H be a prior pem and let H and H′ two bisimilar messages. If H ⊗ H is apem, then H⊗H and H⊗H′ are two bisimilar pems.

Proof. Consider two bisimilar pems H and H′ and a relation R ⊆ W ×W ′ which satisfiesAE, MP, LS and RS.– (1) The following notations are used:{

H⊗H =(W ∗, (H∗i )i∈I , S

∗, H∗0 ,w∗) and

H⊗H′ =(W ∗′, (H∗′i )i∈I , S

∗′, H∗′0 ,w

∗′).

Define R∗ ⊆W ∗ ×W ∗′: for all ((w,w) , (w′, w′)) ∈W ∗ ×W ∗′, ((w,w) , (w′, w′)) ∈ R∗ iff w = w′

and (w,w′) ∈ R. By construction, R∗ satisfies AE and MP. Let ((w,w) , (w,w′)) ∈ R∗ and

i ∈ I. Consider(w, w

′) ∈ H∗′i ((w,w′)). Because w

′ ∈ H′i(w′), there exists– by LS– a world

w ∈ H i(w) s.t.(w, w

′) ∈ R. Since(w, w

′) ∈ H∗′i (w′), there exists sl s.t. w ∈ Hi (w)

and H′0

(w′)= sl. By MP, H0

(w)= sl. Therefore,

(w, w

)∈ H∗i ((w,w)) ⊆ W ∗. Thus,((

w, w),(w, w

′)) ∈ R∗ which proves that R∗ satisfies LS. (The proof for RS is similar.)– (2) We have to show now that if H⊗H is a pem, then H⊗H′ is also a pem. In a recursiveway based on the length of the path linking a state (w,w′) ∈W ∗′ with the actual world (w,w′),one can prove that for (w,w′) ∈ W ∗′, H∗′i ((w,w

′)) 6= ∅ and also that there exists (w,w) s.t.((w,w) , (w,w′)) ∈ R∗. Hence, since H∗i ((w,w)) 6= ∅, by AE and RS, there exists (w,w′) ∈H∗′i ((w,w

′)). Thus: H∗′i ((w,w′)) 6= ∅. Suppose that, for all (w,w′) ∈ W ∗′ which is linked with

the actual world (w,w′) by a path of length n, it is true that H∗′i ((w,w′)) 6= ∅ and also that

there exists (w,w) s.t. ((w,w) , (w,w′)) ∈ R∗. Then, let (w,w′) ∈W ∗′ be linked with the actualworld (w,w′) by a path of length n+1. Thus, there exist an agent j and a world

(w, w

′) ∈W ∗′s.t. (w,w′) ∈ H∗′j

((w, w

′)). Hence, there exists

(w, w

)∈ W s.t.

((w, w

),(w, w

′)) ∈ R∗.By LS, there exists (w,w) ∈ H∗i

((w, w

))s.t. ((w,w) , (w,w′)) ∈ R∗. Since H∗i ((w,w)) 6= ∅,

there exists(w, w

)∈ H∗i ((w,w)) and by RS, there also exists

(w, w

′) ∈ H∗′i ((w,w′)) s.t.((

w, w),(w, w

′)) ∈ R∗. Then, H∗′i ((w,w′)) 6= ∅.Remark that belief revision is entirely trivial for null messages– there is no modification of

agents’beliefs– and partly trivial for secret messages– there is no modification of agents’beliefsexcept for the one who receives the message.

9

Proposition 1 Let H be a pem.(1) H is bisimilar to H⊗Hnul.(2) For all j 6= i, H is j-bisimilar to H⊗Hisec.

Proof. Denote W ∗nul the set of possible worlds of H ⊗ Hnul and W ∗isec, the set of possibleworlds of H⊗Hisec.– (1) We know that W ∗nul = W × {w}. Define R ⊂ W ×W ∗nul as follows: for all {w, (w,w)} ∈W ×W ∗nul, {w, (w,w)} ∈ R iff w = w. It is then straightforward that R satisfies AE, MP, LSand RS.– (2) Consider j 6= i. Let w ∈ Hj(w) and consider H (w). Let w∗ = (w, w1) which belongsby definition to H∗j ((w,w)) and consider H∗ (w∗). Remark that W ∗isec (w

∗) = W (w) × {w1}(see Figure 1, for an example). Define R ⊂ W (w) × W ∗isec (w

∗) as: {w′, (w′′, w1)} ∈ R iffw′ = w′′. It is then straightforward that R satisfies AE,MP, LS and RS. It shows that w andw∗ are bisimilar. Conversely, any w∗ = (w, w1) ∈ H∗j ((w,w)) is bisimilar to w which belongsby definition to Hj(w). Therefore H is j-bisimilar to H⊗Hisecr.

2.6 Comparison of PR to IR

For the standard intersection rule ir, where priors and message are both expressed over thesame set of worlds, posteriors result from the intersection, at each world, of the prior possibilitydomain and the message. As already noticed in the introduction, ir is not always relevant:for instance, in Fig.3 (associated with the Rothschild-Jenkinson story), the prior pem containsonly two worlds, while three worlds, at least, are required for the representation of the posteriorpem. Yet, can we imagine to use ir in the Rothschild-Jenkinson example by working withbisimilar pems which include a suffi cient number of worlds? As claimed in the introduction, thebisimilarity equivalence precludes any possibility of finding a partitional pem that is bisimilarto the posterior pem in Fig.3. In addition, even though an abstract formal possibility exists,ir is not a feasible alternative to pr. Actually, finding out a convenient prior pem– that isconsistent with ir– needs previously, and thus paradoxically, knowing the posterior pem: now,the posterior pem is deduced from the belief revision rule and not the contrary!

3 Syntax for belief revision

The usual syntactical framework is firstly introduced for beliefs and messages. The transcrip-tion rules from possible world semantics to a propositional syntax are then defined with anotion of precondition for messages. Two belief revision axioms are expressed in an extendedlanguage– the first one clarifying the way revised formulas are obtained, and the other one en-suring consistency between priors and the message. The unique, complete and consistent set ofpropositions which satisfies the two axioms is consequently exhibited. Finally, after proposingtranscription rules for the extended language, an equivalence theorem specifies the link between(semantical) pr and the first axiom.

3.1 Syntactical structure

The material environment and agents’beliefs about it are translated through propositions whichbelong to an appropriate language.

10

Let I be a finite set of agents. A multiagent language L is defined according to threecomponents:(i) a finite nonempty set P of primitive propositions typically labeled p, q. . .(ii) propositional connectors, i.e., negation ¬, conjunction ∧ , disjunction ∨, material implication→,(iii) an epistemic operator for each agent i denoted Bi.

The well-formed formulas of this language are typically labeled ϕ or ψ... and satisfy thethree following requirements:

if p ∈ P, then p ∈ L,if ϕ,ψ ∈ L, then ¬ϕ, ¬ψ,ϕ ∧ ψ,ϕ→ ψ,ϕ ∨ ψ ∈ L,if ϕ ∈ L, then ∀i ∈ I, Biϕ ∈ L.

Primitive propositions describe the agents’common material environment. Besides, propo-sitional connectors allow to design formulas by combination of primitive propositions. Theformula Biϕ means: ‘agent i believes that ϕ is true’. Accordingly, formula BiBjϕ means:‘agent i believes that agent j believes that ϕ is true’.

Three axioms and two inference rules are required.

A1 – All tautologies of propositional calculus.

A2 Logical omniscience – Biϕ ∧Bi (ϕ→ ψ)→ Biψ.

A3 Weak consistency – ¬Bi⊥.3

MoPo Modus Ponens rule – From ϕ and ϕ→ ψ, infer ψ.

Nec Necessitation rule – From ϕ, infer Biϕ.

A SYntactic Structure (sys) is a list of formulas which are true.

Definition 7 (SYS) A SYntactic Structure (sys) is a maximally consistent set K such thatA1,A2, A3, MoPo and Nec hold.4

Coming back to belief revision, priors correspond to a sys denoted K and posteriors to asys denoted K∗. Both of them are expressed within two respective languages denoted L and L∗.The set of primitive propositions P is assumed to be the same in L and L∗. Moreover, epistemicoperators are respectively denoted Bi (for priors) and B∗i (for posteriors). A message is alsodefined as a sys and it is expressed within an auxiliary language, denoted L. This languageis based on a finite set M of primitive propositions labeled ml, which represent the primitivemessages. Each primitive message ml encapsulates a proposition ϕ of L through the followingprecondition mapping:

Definition 8 (precondition) The precondition, denoted d.c, is a mapping from M towardsL.

3⊥ represents any contradiction.4A set K of propositions is maximally consistent if, for all ϕ ∈ L, it contains either ϕ or ¬ϕ and not both.

11

This definition means that the content ml of a primitive message is a proposition ϕ ofthe initial language and allows to formally distinguish it from other messages. Note that themessage may deal with the material environment, for instance dmc = p, but can also deal withagents’ beliefs, for instance dmc = Bip. The content of any possible message m is definedas a combination of contents of primitive messages. By natural extension of the preconditionmapping (i.e, dm ∧m′c = dmc∧dm′c, d¬mc = ¬ dmc...), the content of m is linked to a formulaof the initial language L. We call L0 this finite set of messages m. The message epistemicoperator is denoted Bi– for each agent i– and it enables the characterization of the diffusionprotocol. The formula Bim means: ‘agent i learns a message whose content is dmc’and theformula BiBjm means: ‘agent i learns that agent j learns a message whose content is dmc’. Theformula Bi

(m→ Bjm

)means: ‘agent i learns that if the content is dmc then agent j learns

a message whose content is dmc’. For instance, the formula ¬Bim ∧Bi(m→ Bjm

)describes

how agent i learns that agent j receives a private message.

3.2 Transcription rules

Consider a pem H and define the interpretation function π as a mapping which associatesa truth assignment with the primitive propositions in each state s ∈ S– i.e., π (s) : p ∈ P →{true, false}. The truth of a formula ϕ in some particular world w within the pem H is written:(H, w) |= ϕ. The following rules express the principles of truth assignment:

(H, w) |= p iff π (H0 (w)) (p) = true,(H, w) |= ¬ϕ iff (H, w) 2 ϕ,(H, w) |= ϕ ∧ ψ iff (H, w) |= ϕ and (H, w) |= ψ,(H, w) |= Biϕ iff (H, w′) |= ϕ for all w′ such that w′ ∈ Hi (w) .

A pem H is said to be associated with a sys K by a bridge condition:

Definition 9 (bridge condition) ∀ϕ ∈ K, (H,w) |= ϕ.

The sys K gives all formulas which are true in the actual world w of the associated pem H,and only these formulas. The Kripke representation theorem establishes that it is possible toassociate several pems H with a single sys K (Blackburn, de Rijke and Venema, 2001), all ofthem being bisimilar. Without loss of generality, we assume:

State Distinguishability (SD) – For all s 6= s′, π (s) 6= π (s′).

Applying he same transcription rules, we can associate pems H and H∗ with syss K andK∗ in L and L∗. We also assume SD for the transcription rule between a message pem anda message sys. Nevertheless, an additional condition related to the precondition is needed toensure that a message pem H is associated with a message sys K:

Message Condition (MC) – ∀ml ∈ M, ∀s ∈ S, if π (s) (ml) = true (resp. false), then∀w ∈ s, (H, w) |= dmlc (resp. (H, w) 2 dmlc).

Note that this condition imposes a constraint on the precondition mapping d.c.

12

3.3 Belief revision axioms

To express the axioms that underlie the belief revision process, we need three distinct languages,denoted L, L and L∗, respectively related to priors, messages and posteriors, and a globallanguage L, denoted

(L,L,L∗

). It satisfies the three following requirements:

if ϕ ∈ L then ϕ ∈ L, if ϕ ∈ L then ϕ ∈ L, if ϕ ∈ L∗ then ϕ ∈ L,if ϕ,ψ ∈ L then ¬ϕ, ¬ψ and (ϕ ∧ ψ) ∈ L,if ϕ ∈ L then ∀i ∈ I, B∗iϕ ∈ L.

Within L, formulas resulting from priors, messages or posteriors as well as formulas resultingfrom posteriors about priors or messages can simultaneously be expressed. Then, L containspropositions using mixed operators such as B∗iBj or B

∗iBj but neither BiBj , BiB

∗j nor BiB

∗j .

Even if no explicit dynamics is involved in the revision process, propositions such as B∗iBjϕor B∗iBjϕ convey a kind of ‘memory’ effect: B∗iBjϕ can be interpreted as ‘agent i finally(ex post) believes that agent j initially (ex ante) believed that ϕ is true’ and B∗iBjϕ canbe interpreted as ‘agent i finally (ex-post) believes that agent j has learned that ϕ is true’.Conversely, propositions such as BiBjϕ, BiB∗jϕ or BiB∗jϕ involve the possibility to modelindividual expectations about beliefs (or messages)– a possibility which is definitely out of ourpurpose.

We now define a Multioperator sys (denoted Msys) by an extension of a sys.

Definition 10 A Multioperator sys (Msys) is a subset K of L such that:(i) K∩L is a sys in L,(ii) K∩L is a message sys in L,(iii) K∩L∗ is a sys in L∗,(iv) A1, A2, A3, MoPo and Nec hold wrt the posterior belief operators B∗i .

A Msys K captures the process of belief revision. It allows to make up each specific structure:K∩L is a sys in L corresponding to the priors K, K∩L is a sys in L corresponding to somemessage K and K∩L∗ is a sys in L∗ corresponding to posteriors K∗. In order to characterizeprecisely the belief revision process, two more axioms are introduced.

A4 Belief-message inference – For all ϕ in L and ψ in L,

∨m,m′,

m∗,m∗∈L0

( (Bi (m

∗) ∧Bi(m∗ → (dm ∨m′c)))

∨(Bi (m∗) ∧ (Bi(m∗ → (m ∨m′)))

)∧Bi (dmc → ϕ) ∧Bi (m′ → ψ)

→ B∗i (ϕ ∨ ψ) .

A4 describes how posteriors are inferred from priors and message. The message m ∨m′ isfirst obtained from deeper messages by combining Bi and Bi. Then, ϕ ∨ ψ is also obtainedby combining Bi and Bi.5 The following implications of A4 are easy to interpret, and werecommend that the more complex axiom–A4– be understood in terms of these implications(plus A5 which is explained below).

5A weaker axiom can be obtained when shortcuting the first step:

∨m,m′∈L0

( (Bi (m ∨m′) ∨Bi(dm ∨m′c)

)∧Bi (dmc → ϕ) ∧Bi (m

′ → ψ)

)←→ B∗i (ϕ ∨ ψ) .

13

A4a Belief inference – For any ϕ ∈ L, Bim ∧Bi (dmc → ϕ)→ B∗iϕ.

A4b Message inference – For any ψ ∈ L, Bi dmc ∧Bi (m→ ψ)→ B∗iψ.

These two subaxioms correspond to a sort of ‘logical omniscience’that combines operatorsBi with Bi. In words, ‘believing a priori that the content of a proposition m (as captured bythe precondition dmc) implies a proposition ϕ, and learning afterwards the proposition m itself’leads to the same posterior– i.e., B∗iϕ– than ‘believing a priori the content of m and learningafterwards that m implies ϕ’. Intuitively, belief revision is then viewed as a neutral processwith regard to inference and content: ceteris paribus, believing a priori an inference (resp. thecontent of a proposition) or learning it (resp. the proposition itself) afterwards is in some wayex-post analogous. Moreover, in case where dϕc = ϕ and dψc = ψ, the two axioms imply thatBi (ϕ)→ B∗i (ϕ) and Bi (ψ)→ B∗i (ψ), which means that priors are kept in memory– and whatis learned as well. Besides, A4 implies: B∗i (m→ dmc)6 which ensures that each agent interpretscorrectly the meaning of the message.

A5 Nonconflict – Bim→ ¬Bi¬ dmc.

A5 means that a message cannot contradict priors. This is a strong axiom: it prevents anyerroneous message to spread over true priors, but it also prevents any correct message to implyerroneous beliefs. When an agent faces a contradiction between what he learns and what hepreviously believes, everything works as if the belief revision process solved the contradiction:the agent disregards the message he learns or his priors or both. Belief conflict resolution isoutside the scope of this work (see on that subject, Board, 2004, for a partial treatment). A5is another implication of A4.

Proposition 2 If K is a M sys satisfying A4, then it also satisfies A5.

Proof. (Ad absurdum). Assume A4 and the negation of A5. For instance, according to thenegation of A5, agent i learns m (i.e., Bim) while he previously believed that the content ofm was false (i.e., Bi¬ dmc). From this second proposition, he also believes: dmc −→ ϕ for allproposition ϕ (i.e., Bi (dmc −→ ϕ)). Then, according toA4a, agent i finally believes propositionϕ (i.e., B∗iϕ). Since the same argument can be applied to ¬ϕ, he believes finally ϕ and ¬ϕ, thatis a contradiction.

3.4 Unicity of posteriors

Consider two MSYSs K and K′ satisfying A4 . If K∩L = K′∩L and K∩L = K′∩L, thenK∩L∗ = K′∩L∗. The following theorem establishes the unicity of the Msys K stemmed frompriors and message, and satisfying A4.

Theorem 2 Consider two M syss K and K′ satisfying A4. If K∩L = K′∩L and K∩L = K′∩L,then K∩L∗ = K′∩L∗.

6For instance, apply A4 with ϕ = dmc, ψ = m′ = ¬m and m∗ = m∗ = ⊥ to get B∗i (¬m ∨ dmc) =B∗i (m→ dmc).

14

Proof is in Appendix A.

Theorem 2 shows that, under A4, priors K = K∩L, when revised according to a messageK = K∩L, yield posteriors K∗ = K∩L∗ that are unique if K∗ is a sys. A4 is then proved to bea suffi cient axiom to characterize the revision process as unique.

3.5 General transcription rules

Let us now introduce the semantics of a Msys before extending the truth conditions. Define thesemantic structure H– call it a Mpem– as follows:

H =(H,H,H∗

),

where H =(W, (Hi)i∈I , S,H0,w

)is a pem, H =

(W,(Hi

)i∈I , S,H0,w

)an auxiliary pem

associated with H, and H∗ =(W ∗, (H∗i )i∈I , S,H

∗0 ,w

∗) a pem with W ∗ ⊂ W ×W and w∗ =(w,w). Truth value for any formula ϕ in Mpem H is defined in some particular world w∗ =(w,w) ∈ W ∗: (H, w∗) |= ϕ. The following valuation rules express the truth assignment in W ∗

for any formula:

if ϕ ∈ L, (H, (w,w)) |= ϕ iff (H, w) |= ϕ,if ϕ ∈ L, (H, (w,w)) |= ϕ iff

(H, w

)|= ϕ,

if ϕ ∈ L∗, (H, (w,w)) |= ϕ iff (H∗, (w,w)) |= ϕ,(H, (w,w)) |= ¬ϕ iff (H, (w,w)) 2 ϕ,(H, (w,w)) |= ϕ ∧ ψ iff (H, (w,w)) |= ϕ and (H, (w,w)) |= ψ,(H, (w,w)) |= B∗iϕ iff (H, w∗′) |= ϕ for all w∗′ such that w∗′ ∈ H∗i (w∗) .

A Msys K can be associated with a Mpem H if SD and MC hold.

3.6 Equivalence theorem

The following equivalence theorem– relevant for every kind of message, especially public, privateor secret– can be proved:

Theorem 3 Consider a pem H and an auxiliary pem H which are respectively associated witha sys K and an auxiliary sys K.(1) If H∗ = H⊗H is a pem, there exists a M sys K such that K = K∩L and K = K∩L, whichsatisfies A4 and which is associated with the Mpem H =

(H,H,H∗

).

Consider a M sys K which satisfies A4.(2) There exists a Mpem H =

(H,H,H∗

)which is associated with the M sys K and such that

H is associated with K = K∩L, H is associated with K = K∩L and H∗ = H ⊗H is associatedwith K∗ = K∩L∗.

Proof is in Appendix B.

Theorem 3 is like an existence theorem– it confirms that there exist Msys satisfying A4 and(i), (ii), (iii), (iv), at the same time. Besides, it justifies pr in a syntactical manner. It showsthat the belief revision process is unique and conforms to logical axioms that apply to all agents.Furthermore, in combining Theorem 2– which proves the unicity of posteriors– with Theorem3– which establishes pr as the semantical counterpart of the syntactic revision process– , pr isproved to be a robust revision rule.

15

4 References

Alchourron, C.E., Gardenfors, P. and Makinson, D. (1985): ‘On the logic of theorychange: partial meet contraction and revision functions,’Journal of Symbolic Logic 50, 510-530

Aucher, G. (2009): ‘BMS revisited,’Proceedings of the conference on Theoretical Aspectsof Rationality and Knowledge (TARK 2009)

Aucher, G. (2011): ‘DEL-sequents for progression,’Journal of Applied Non Classical Logic21, 289-321

Aumann, R.J. (1976): ‘Agreeing to disagree,’The Annals of Statistics 4, 1236-1239Aumann, R.J. (1999): ‘Interactive epistemology II: probability,’ International Journal of

Game Theory 28, 301-314Baltag, A. and Moss, L. (2004): ‘Logics for epistemic programs,’Synthese 139, 165-224Baltag, A., Moss, L. and Solecki, S (1998): ‘The logic of public announcements, com-

mon knowledge, and private suspicions,’Proceedings of the conference on Theoretical Aspects ofRationality and Knowledge (TARK 1998)

Baltag, A., van Ditmarsch H. and Moss, S (2008): ‘Epistemic logic and information up-date,’in P. Adriaans, and J. van Benthem, eds. Handbook on the Philosophy of Information,North-Holland, 361-456

Blackburn, P. de Rijke, M. and Venema, Y. (2001): Modal Logic. Cambridge Univer-sity Press

Board, O. (2004): ‘Dynamic interactive epistemology,’Games and Economic Behavior 49,49-80

Bonanno, G. (2005): ‘A simple modal logic for belief revision,’Synthese 147, 193-228Fagin, R., Halpern J.Y., Moses, Y. and Vardi, M.Y. (1995): Reasoning about knowl-

edge. MIT PressGeanakoplos, J. and Polemarchakis, H. (1982): ‘We cannot disagree for ever,’Journal

of Economic Theory 1, 192-200Hughes, G.E. and Cresswell, H.J. (1996): A New Introduction to Modal Logic. Rout-

ledgeMorton, F. (1962): The Rothschilds: A Family Portrait. Secker & WarburgPerea, A. (2012): Epistemic Game Theory. Cambridge University Press

5 Appendix A

Theorem 2 Consider two M syss K and K′ satisfying A4. If K∩L = K′∩L and K∩L = K′∩Lthen K ∩ L∗ = K′ ∩ L∗.

Proof. Define the depth of a formula ϕ in L, denoted d(ϕ), as the number of modal operatorsB∗i (and not Bi or Bi) which are hierarchically used. It is recursively defined by d(ϕ) = 0 ifϕ ∈ L ∪ L, d(¬ϕ) = d(ϕ) and d(ϕ ∧ ψ) = max(d(ϕ), d(ψ)) and d(B∗iϕ) = d(ϕ) + 1. Prove nowthat K = K′. (We proceed by induction on the depth of proposition in L.) Let Lα be the setof propositions of L whose depth is inferior or equal to α. First, consider propositions whosedepth is 0. Note that L0 is the closure of L ∪ L w.r.t. propositionnal connectors. Remarkthat K ∩

(L ∪ L

)= (K∩L) ∪

(K∩L

)= (K′∩L) ∪

(K′∩L

)= K′ ∩

(L ∪ L

). Since K and K′

satisfy MoPo, K ∩ L0 = K′ ∩ L0. Second, consider L1. For propositions such as B∗i (ϕ ∨ ψ),where ϕ ∈ L and ψ ∈ L, we have by A4, B∗i (ϕ ∨ ψ) equivalent to a proposition of L0 and

16

thus it follows that B∗i (ϕ ∨ ψ) ∈ K iff B∗i (ϕ ∨ ψ) ∈ K′. For propositions such as B∗i (ϕ), whereϕ ∈ L, we have B∗i (ϕ) ∈ K iff B∗i (ϕ ∨ >) ∈ K and thus B∗i (ϕ) ∈ K iff B∗i (ϕ) ∈ K′. Similarly,for propositions such as B∗i (ψ), where ψ ∈ L we have B∗i (ψ) ∈ K iff B∗i (ψ) ∈ K′. Finally, forpropositions B∗i (ϕ ∧ ψ), where ϕ ∈ L and ψ ∈ L, B∗i (ϕ ∧ ψ) ∈ K iffB∗i (ϕ) ∈ K and B∗i (ψ) ∈ K,and thus B∗i (ϕ ∧ ψ) ∈ K iff B∗i (ϕ ∧ ψ) ∈ K′.Therefore, K ∩ L1 = K′ ∩ L1. Suppose now thatK ∩ Lα = K′ ∩ Lα– for α ≥ 1. Consider a proposition ϕ whose depth is α+ 1. It contains a set{ψ1, ψ2...} of well-formed formulas whose depth is 1 and which begin with the modal operatorB∗i . By A4, each ψi is equivalent to a proposition ψ

′i of L0. Then, in proposition ϕ, if we

replace all ψi by ψ′i, we obtain a proposition ϕ

′ whose depth is α. By A4 and A2, ϕ ∈ K iffϕ′ ∈ K as well as ϕ ∈ K′ iff ϕ′ ∈ K′. Since ϕ′ ∈ K iff ϕ′ ∈ K′, then ϕ ∈ K iff ϕ ∈ K′ and, thus,K ∩ Lα+1 = K′ ∩ Lα+1.

6 Appendix B

Theorem 3(1) Consider a H and an auxiliary pem H which are respectively associated with a sys K andan auxiliary sys K. If H∗ = H⊗H is a pem, there exists a Msys K such that K = K∩L andK = K∩L, which satisfies A4 and which is associated with the Mpem H =

(H,H,H∗

).

(2) Consider a Msys K which satisfies A4. There exists a Mpem H =(H,H,H∗

)which is

associated with the Msys K and such that H is associated with K = K∩L, H is associated withK = K∩L and H∗ = H⊗H is associated with K∗ = K∩L∗.

Proof. For sake of simplicity, we will use the following equivalent expression of axiom A4:

∨m,m′,

m∗,m∗∈L0

Bi (dmc → ϕ)

∧Bi (m′ → ψ)

∧Bi(dm ∨m′ ∨m∗c∧¬ dm∗c

)∧Bi

((m ∨m′ ∨m∗)∧¬m∗

)

←→ B∗i (ϕ ∨ ψ) .

Consider H =(W, (Hi)i∈I , S,H0,w) and H =(W,(Hi

)i∈I , S,H0,w

)which are associated with

some sys K and K. Denote|ϕ|∗ = {w∗ ∈W ∗ s.t. (H, w∗) |= ϕ},|ϕ| = {w ∈W s.t. (H, w) |= ϕ},|ϕ| =

{w ∈W s.t.

(H, w

)|= ϕ

}.

– (1) Let us suppose that H∗ = H ⊗ H is a pem and consider the Mpem H =(H,H,H∗

).

Define K as: K = {ϕ s.t. (H,w∗) |= ϕ}. Since H is associated with K, theMC condition holdsand thus H is associated with K. We now prove that K is a a Msys which satisfies A4. Tocheck that K is a Msys is straightforward. To prove that K satisfies A4, we need to show that

17

for any (w,w) ∈W ∗,

(H, (w,w)) |=∨m,m′,

m∗,m∗∈L0

Bi (dmc → ϕ)

∧Bi (m′ → ψ)

∧Bi(dm ∨m′ ∨m∗c∧ dm∗c

)∧Bi

((m ∨m′ ∨m∗)∧¬m∗

)

←→ B∗i (ϕ ∨ ψ) .

(−→) We first prove that

(H, (w,w)) |=∨m,m′,

m∗,m∗∈L0

Bi (dmc → ϕ)

∧Bi (m′ → ψ)


)∧Bi

((m ∨m′ ∨m∗)∧¬m∗

)

−→ B∗i (ϕ ∨ ψ) .

Thus, we need to prove that, for any (w,w) ∈W ∗ such that

(H, (w,w)) |=∨m,m′,

m∗,m∗∈L0

Bi (dmc → ϕ)

∧Bi (m′ → ψ)

∧Bi(dm ∨m′ ∨m∗c∧¬ dm∗c

)∧Bi (m ∨m′ ∨m∗ ∧ ¬m∗)

(A)

we also have(H, (w,w)) |= B∗i (ϕ ∨ ψ) . (B)

If (A) holds, there exist m,m′,m∗,m∗ ∈ L0 such that

(H, (w,w)) |=

Bi (dmc → ϕ)

∧Bi (m′ → ψ)∧Bi (dm ∨m′ ∨m∗c ∧ ¬ dm∗c)

∧Bi((m ∨m′ ∨m∗)∧¬ dm∗c

) .

Step 1: Let us prove that if (A) holds, then for all (w′, w′) ∈ H∗i (w,w), (H, (w′, w′)) |=(dmc ∨ dm′c) ∧ (m ∨m′). Ad absurdum: let us suppose that there exists (w′, w′) ∈ H∗i (w,w)such that either w′ /∈ |dmc ∨ dm′c|∗ or w′ /∈ |m ∨m′|. Consider the case where w′ /∈ |dmc ∨ dm′c|∗which means: w′ ∈ |¬ dmc ∧ ¬ dm′c|. Since w′ ∈ Hi (w) and (H, w) |= Bi (dm ∨m′ ∨m∗c ∧ ¬ dm∗c),then w′ ∈ |dm∗c|. Now, pr implies that there exists s ∈ S such that w′ ∈ Hi (w) ∩ s,w′ ∈ H i (w) ∩ H

−10 (s). By MC, (H, w′) |= dm∗c implies that π (s) (m∗) = true and thus(

H, w′)|= m∗. But, then we face a contradiction with w′ ∈ H i (w) and

(H, w

)|= Bi (¬m∗). In

the case where (w′, w′) /∈ |m ∨m′|∗, we can get a contradiction by a similar proof.Step 2: Let us finally prove that (B) holds. Let consider (w′, w′) ∈ H∗i (w,w) and s ∈ S suchthat w′ ∈ Hi (w) ∩ s, w′ ∈ H i (w) ∩ H

−10 (s). By Step 1, we have that π (s) (m ∨m′) = true

and thus either π (s) (m) = true or π (s) (m′) = true. If π (s) (m) = true, then (H, w′) |= dmc.

18

Yet, (H, (w,w)) |= Bi (dmc → ϕ) implies that (H, w′) |= dmc → ϕ. Therefore, we also have(H, w′) |= ϕ and thus (H, (w′, w′)) |= ϕ. Similarly, if π (s) (m′) = true, then

(H, w′

)|= m′.

Hence, (H, (w,w)) |= Bi (m′ → ψ)) implies

(H, w′

)|= m′ → ψ. Thus

(H, w′

)|= ψ and

(H, (w′, w′)) |= ψ. Overall, for all (w′, w′) ∈ H∗i (w,w), (H, (w′, w′)) |= ϕ ∨ ψ which proves(B).(←−) We now prove that if (B) holds, then (A) holds. We know that (H, (w,w)) |= B∗i (ϕ ∨ ψ)is equivalent to H∗i (w,w) ⊂ |ϕ| ×W ∪W × |ψ|. Define

S ((w,w)) ={s ∈ S s.t. H∗i (w,w) ∩ s×H

−10 (s) 6= ∅

}and S (ϕ) =

{s ∈ S ((w,w)) s.t. H∗i (w,w) ∩ s×H

−10 (s) ⊂ |ϕ| ×W

},

S (ψ) ={s ∈ S ((w,w)) s.t. H∗i (w,w) ∩ s×H

−10 (s) ⊂W × |ψ|

}.

Step 1: Let us show that S (ϕ) ∪ S (ψ) = S ((w,w)). Ad absurdum: let suppose there exists

s ∈ S ((w,w)) \(S (ϕ) ∪ S (ψ)

)and thus, that there exist (w′, w′) ∈ H∗i (w,w) ∩ s × H

−10 (s)

such that w′ /∈ |ϕ| and (w′′, w′′) ∈ H∗i (w,w) ∩ s × H−10 (s) such that w

′′ /∈ |ψ|. But, by pr,(w′, w′′) ∈ H∗i (w,w), which is a contradiction with H∗i (w,w) ⊂ |ϕ| ×W ∪W × |ψ|. Define{

S (w) ={s ∈ S s.t. Hi (w) ∩ s 6= ∅

}\S ((w,w)) ,

S (w) ={s ∈ S s.t. H i (w) ∩H

−10 (s) 6= ∅

}\S ((w,w)) .

Step 2: Let us show that S (w) ∩ S (w) = ∅. Ad absurdum: let suppose there exist s ∈ S (w) ∩S (w), w′ ∈ Hi (w)∩ s and w′ ∈ H i (w)∩H

−10 (s). But, then pr implies (w

′, w′) ∈ H∗i (w,w) andthus s ∈ S ((w,w)) which is a contradiction.Step 3: We now exhibit m,m′,m∗,m∗ ∈ L0 such that

(H, (w,w)) |=

Bi (dmc → ϕ)

∧Bi (m′ → ψ)


)∧Bi

((m ∨m′ ∨m∗)∧¬m∗

)

.

For any s ∈ S, SD implies that there exists ms ∈ L0 such that π (s) (ms) = true andπ (s) (ms′) = false for any s 6= s′. Define m =

∨s∈S(ϕ)ms (m is the contradiction ⊥ if

S (ϕ) = ∅) m′ =∨s∈S(ψ)ms (m′ is the contradiction ⊥ if S (ϕ) = ∅), m# =

∨s∈S ¬ms,

m∗ = m# ∨(∨

s∈S(w)ms

)and m∗ =

∨s∈S(w)ms.

– Step 3.1: Let us show that (H, (w,w)) |= Bi (dmc → ϕ) . For all w′ ∈ Hi (w), either (a) thereexists s ∈ S (ϕ) such that w′ ∈ s or (b) w′ /∈

⋃s∈S(ϕ) s. In case (a), we have (H, w′) |= dmc ∧ ϕ

and in case (b), (H, w′) |= ¬ dmc since (H, w′) |= dmc iff w′ ∈⋃s∈S(ϕ) s. Therefore, for any

w′ ∈ Hi (w), (H, w′) |= dmc → ϕ and thus (H, (w,w)) |= Bi (dmc → ϕ).– Step 3.2: By a similar proof, we can show that (H, (w,w)) |= Bi (m′ → ψ).– Step 3.3: Let us show that (H, (w,w)) |= Bi (dm ∨m′ ∨m∗c ∧ ¬ dm∗c) .We need to provethat for any w′ ∈ Hi (w), (H, w′) |= dm ∨m′ ∨m∗c ∧ ¬ dm∗c. First, remark that for any

19

w′ ∈ Hi (w) either there exists s ∈ S such that w′ ∈ s and by Step 1, either s ∈ S (ϕ) and then(H, w′) |= dmc, or s ∈ S (ψ) and then (H, w′) |= dm′c, or s ∈ S (w) and then (H, w′) |= dm∗c oreither w′ /∈

⋃s∈S s and then (H, w′) |= dm∗c. Thus, in all cases, (H, w′) |= dm ∨m′ ∨m∗c. Sec-

ond, remark that by Step 2 that for any w′ ∈ Hi (w), w′ /∈⋃s∈S(w) s and thus (H, w′) |= ¬ dm∗c.

– Step 3.4: Let us show that (H, (w,w)) |= Bi (m ∨m′ ∨m∗ ∧ ¬m∗). We need to prove thatfor any w′ ∈ H i (w),

(H, w′

)|= m∨m′ ∨m∗ ∧¬m∗. For all w′ ∈ H i (w), there exists s ∈ S such

that w′ ∈ H−10 (s) and by Step 1, either s ∈ S (ϕ) and then(H, w′

)|= m, or s ∈ S (ψ) and then(

H, w′)|= m′ or s ∈ S (w) and then

(H, w′

)|= m∗. Furthermore, since w′ ∈

⋃s∈S H

−10 (s) by

MC(H, w′

)|= ¬m#. Finally, remark that by Step 2, w′ /∈

⋃s∈S(w)H

−10 (s) and thus by MC(

H, w′)|= ¬

(∨s∈S(w)ms

). Thus,

(H, w′

)|= ¬m∗.

– (2) Suppose now that K is a Msys which satisfies A4. Define H =(H,H,H∗

)such that

H is a pem associated with K = K∩L, H is an auxiliary pem associated with the messagesys K = K∩L, H∗ = H ⊗ H. Indeed, Kripke representation theorem ensures that there existpems associated with K and K. From a pem associated with K, it is straightforward to build apem associated with K that also satisfies SD and MC. Suppose (ad absurdum) that H is nota Mpem. This means that H∗ is not a pem and there exist at least a world (w,w) ∈ W ∗ andan agent i ∈ I s.t. H∗i (w,w) = ∅. By pr there exists a finite sequence of agents i1, ..., in and afinite sequence of worlds (w1, w1) , ..., (wn, wn) s.t. in = i, (w1, w1) = (w,w), (wn, wn) = (w,w)and, for all k ∈ [1, n− 1], (wk+1, wk+1) ∈ H∗ik (wk, wk). Note then that for all k ∈ [1, n− 1],wk+1 ∈ Hik (wk) and wk+1 ∈ H ik (wk). Define S (w) =

{s ∈ S s.t. H i (w) ∩H

−10 (s) 6= ∅

}and

note m∗ =∨s∈S(w)ms.

– Consider n = 1 (i.e (w,w) = (w,w)) and let us show that Bi¬ dm∗c ∧Bim∗ ∈ K, which isa violation of A5. Since by Proposition 4 we know that if a Msys satisfies A4 it also satisfiesA5, this would demonstrate a contradiction. Thus (w,w) = (w,w) and by pr it follows thatfor any s ∈ S: if H i (w)∩H

−10 (s) 6= ∅, then Hi (w)∩s = ∅. Since H i (w) ⊂

⋃s∈S(w)H

−10 (s) and

Hi (w) ∩⋃s∈S(w) s = ∅, then

(H,w

)|= Bim∗ and (H,w) |= Bi¬m∗. This means Bi¬m∗ ∈ K

and Bim∗ ∈ K and therefore Bi¬ dm∗c ∧Bim∗ ∈ K.– Consider n = 2 and let us show that ¬B∗i1

(Bim∗ → ¬Bi¬ dm∗c

)∈ K which is a violation of

A5 and thus a contradiction with the hypothesis that K is a Msys which satisfies A4. Supposethat B∗i1


)= B∗i1

(¬Bim∗ ∨ ¬Bi¬ dm∗c

)∈ K, and thus by A4, there exist

m, m′, m∗, m∗ ∈ L0 such thatBi1 (dmc → ¬Bi¬ dm∗c)∧Bi1

(m′ → ¬Bim∗

)∧Bi1 (dm ∨ m′ ∨ m∗c ∧ ¬ dm∗c)∧Bi1 ((m ∨ m′ ∨m∗) ∧ ¬m∗)

∈ K.Thus w ∈ |dm ∨ m′ ∨ m∗c ∧ ¬ dm∗c| and w ∈ |(m ∨ m′ ∨ m∗) ∧ ¬m∗|. By pr there exists s suchthat w ∈ Hi1 (w) ∩ s and w ∈ H i1 (w) ∩H

−10 (s). Then by MC, s ⊂ |dm ∨ m′ ∨ m∗c ∧ ¬ dm∗c|

and H−10 (s) ⊂ |(m ∨ m′ ∨ m∗) ∧ ¬m∗|. By MC again, that implies that s ⊂ |dm ∨ m′c| and

H−10 (s) ⊂ |m ∨ m′|. Suppose first that s ⊂ |dmc| and thus (H, w) |= dmc. Since (H,w) |=

Bi1 (dmc → ¬Bi¬ dm∗c) = Bi1 (¬ dmc ∨ ¬Bi¬ dm∗c), we also have (H, w) |= ¬ dmc∨¬Bi¬ dm∗cand thus (H, w) |= ¬Bi¬ dm∗c which is a contradiction. Second, in the case where s ⊂ |dm′c|,

20

we have H−10 (s) ⊂ |m′| and thus

(H, w

)|= m′. Since

(H,w

)|= Bi1

(m′ → ¬Bim∗

)=

Bi1(¬m′ ∨ ¬Bim∗

), we also have

(H, w

)|= ¬m′ ∨ ¬Bim∗ and thus

(H, w

)|= ¬Bim∗ which is

a contradiction.– For n > 2, let us show that ¬B∗i1

(B∗i2 ...B

∗in−1


))∈ K which is a viola-

tion of A5 and thus a contradiction with the hypothesis that K is a Msys which satisfies A4.Suppose on the contrary that

B∗i1

(B∗i2 ...B

∗in−1


))= B∗i1

(B∗i2 ...B

∗in−1

(¬Bim∗ ∨ ¬Bi¬ dm∗c

))∈ K,

and thus by A4, there exist mn−1, m′n−1, m∗n−1, m∗n−1 ∈ L0 such that

B∗i1 ...B∗in−2

Bin−1 (dmn−1c → ¬Bi¬ dm∗c)∧Bin−1

(m′n−1 → ¬Bm∗

)∧Bin−1

( ⌈mn−1 ∨ m′n−1 ∨ m∗n−1

⌋∧¬ dm∗n−1c

)∧Bin−1

( (mn−1 ∨ m′n−1 ∨ m∗n−1

)∧¬m∗n−1

)

∈ K.

Then, by A4a and A4a, there exist mn−2, m′n−2 ∈ L0 such that

B∗i1 ...B∗in−3

Bin−2

(dmn−2c → ϕn−2

)∧Bin−2 (mn−2)∧Bin−2

(⌈m′n−2

⌋)∧Bin−2

(m′n−2 → ψn−2

) ∈ K,

where ϕn−2 =

Bin−1 (dmn−1c → ¬Bi¬ dm∗c)

∧Bin−1( ⌈

mn−1 ∨ m′n−1 ∨ m∗n−1⌋

∧¬ dm∗n−1c

) ,ψn−2 =

Bin−1(m′n−1 → ¬Bim∗

)∧Bin−1

( (mn−1 ∨ m′n−1 ∨ m∗n−1

)∧¬m∗n−1

) .Recursively, for k = 1 to n− 3, there exist mk, m

′k ∈ L0 such that

Bi1 (dm1c → ϕ1)

∧Bi1 (m1)∧Bi1 (dm′1c)∧Bi1 (m′1 → ψ1)

∈ Kwith

ϕk =

(Bik+1

(dmk+1c → ϕk+1

)∧Bik+1

(⌈m′k+1

⌋) ),

ψk =

(Bik+1 (mk+1)

∧Bik+1(m′k+1 → ψk+1

) ) .By pr, for all k = 1 to n − 1, there exist sk such that wk+1 ∈ Hik (wk) ∩ sk and wk+1 ∈H ik (wk) ∩H

−10 (sk). Let us show by recursivity that for all k = 1 to n − 2, sk ⊂ |dmk ∧ m′kc|,

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