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Communication between agents in dynamic epistemic logic

This manuscript studies actions of communication between epistemic logic agents. It starts by looking into actions through which all/some agents share all their information, defining the model operation that transforms the model, discussing its properties, introducing a modality for describing it and providing an axiom system for the latter. The main part of the manuscript focuses on an action through which some agents share part of their information: they share all that they know about a topic defined by a given formula. Once again, the manuscript defines the model operation that transforms the model, discusses its properties, introduces a modality for describing it and provides an axiom system for the latter.


Introduction
Epistemic logic (EL; Hintikka 1962) is a logical system for reasoning about the knowledge a set of agents might have. On the syntactic side, its language extends propositional logic with a modality K i for every agent i, with formulas of the form K i ϕ read as "agent i knows that ϕ is the case". On the semantic side, it typically relies on relational 'Kripke' models, assigning to each agent an indistinguishability relation among epistemic possibilities. 1 The crucial idea is that knowledge is defined in terms of uncertainty: agent i knows that ϕ is the case when ϕ holds in all situations she considers possible. 2 Despite its simplicity (or maybe because of it), EL has become a widespread tool, contributing to the formal study of complex multi-agent epistemic notions in philosophy (Hendricks 2006), computer science (Fagin et al. 1995, Meyer andvan der Hoek 1995) and economics (de Bruin 2010, Perea 2012).
One of the reasons for the success of EL and its variations is that it allows a natural representation of actions that affect the agents' information (e.g., *comm-2022-10-10-arXiv.tex, compiled 11th October 2022 To appear in Revista Mexicana de Lógica 1(1). 1 There are other alternatives; see Footnote 5.
knowledge and beliefs). The two paradigmatic examples are public announcements (Plaza 1989, Gerbrandy andGroeneveld 1997), representing the effect of agents receiving truthful information, and belief revision (van Ditmarsch 2005, van Benthem 2007, Baltag and Smets 2008, representing actions of agents receiving information that is reliable and yet potentially fallible. These two frameworks are part of what is known as dynamic epistemic logic (DEL; , van Benthem 2011, a field whose main feature is that actions are semantically represented not as relations (as done, e.g., in propositional dynamic logic, Harel et al. 2000), but rather as operations that transform the underlying semantic model.
The mentioned DEL frameworks have been used for representing communication between agents (e.g., Ågotnes et al. 2010, van Ditmarsch 2014, Baltag and Smets 2013, Galimullin and Alechina 2017). Yet, they were originally designed to represent the effect of external communication, with the information's source being some entity that is not part of the system. This can be observed by noticing that, in these settings, the incoming information χ does not need to be known/believed by any of the involved agents.
This manuscript studies epistemic actions in which the information that is being shared is information some of the agents already have. In this sense, the actions studied here are true actions of inter-agent communication. For this, the crucial notion is that of distributed knowledge (Hilpinen 1977, Halpern and Moses 1984, 1985, 1990, representing what a group of agents would know by putting all their information together. Distributed knowledge thus 'pre-encodes' the information a group of agents would have if they were to share their individual pieces. Then, the actions studied here can be seen as (variations of) actions that fulfil this promise, doing so by defining the model that is obtained after communication takes place.
In defining these communication actions, it is important to emphasise that, under relational 'Kripke' models, epistemic logic defines knowledge in terms of uncertainty. This is because these models only represent the epistemic uncertainty of the agent, without 'explaining' why some uncertainty (i.e., epistemic possibility) has been discarded and why some other remains. This has two important consequences.
• First, as discussed in van der Hoek et al. (1999), distributed knowledge does not satisfy the "principle of full communication": there are situations in which a group knows distributively a formula ϕ, and yet ϕ does not follow from the individual knowledge of the groups' members. Thus, under relational models, distributed knowledge is better understood as what a group of agents would know (in the "information as range" sense) if they indicated to one another which epistemic possibilities they have already discarded.
• Second, recall that an agent's uncertainty is represented by her indistinguishability relation. Thus, although changes in uncertainty can be represented by changing what each epistemic possibility describes (technically, by changing the model's atomic valuation), they are more naturally represented by changes in the relation itself. 3 This text is organised as follows. Section 2 recalls the basics of EL, including the semantic model representing the agents' uncertainty, the formal language used for describing them and an axiom system characterising validities. Then, while Section 3 discusses communication actions through which all/some agents share all their information with everybody (comparing them with proposals in the literature), Section 4 discusses a novel action through which some agents share part of their information with everybody. Section 5 is a brief discussion of the issues arising when only some agents receive the shared information. Finally, Section 6 summarises the work, discussing also further research lines. While the proofs of propositions are found within the text, the proofs of theorems can be found in the appendix. The semantic interpretation of D G ϕ deserves some comments. Recall: R D G is the intersection of the relations of agents in G. Thus, R D G wu holds if and only if R i wu holds for every i in G, that is, if and only if every agent in G considers u possible when at w or, equivalently, if and only if no agent in G can discard u when at w. Using the notation · , the semantic interpretation of D G ϕ is equivalently stated as Note also that the abbreviation K i ϕ behaves as expected: so agent i knows ϕ at (M, w) if and only if every world she cannot distinguish from w is a ϕ-world. At (M 1 , w 0 ) all atoms are true; yet, no agent knows this. First, agent a knows that p holds, but knows the truth value of neither q nor r: (M 1 , w 0 ) K a p ∧ (¬ K a q ∧ ¬ K a ¬q) ∧ (¬ K a r ∧ ¬ K a ¬r).
Then, b knows that q holds, but knows the truth value of neither p nor r: Finally, c knows that r holds, but knows the truth value of neither p nor q: Still, each agent knows that a knows p's truth-value, that b knows q's truthvalue, and that c knows r's truth-value: Finally, agents would benefit from sharing their individual information. In particular, if they all shared, they would know which the real situation is: (ii) Let A and P be as in Item (i); consider the pointed model depicted below.
Again, all atoms are true in the real situation; yet, no agent knows this. On the one hand, a knows p ∨ q without knowing the truth-value of p or q, On the other hand, b knows q ∨ r without knowing the truth-value of q or r: Agent c has slightly more information, as she knows that q is true but still ignores the truth-value of p and r: This time, while communicating would help a and b, it would not help c.
In fact, collectively, the agents do not have enough information to find out which the real situation is: (iii) Take A = {a, b} and P = p, q ; consider (M 3 , w 0 ) depicted below.
In the pointed model, both p and q are true. Both agents have partial information about this: while agent a knows p but does not know whether q, agent b does not know whether p but knows q: However, both agents have misleading information about what the other knows: a thinks b knows p without knowing whether q, and b thinks a does not know whether p but knows q: If they were to share their (partially misleading) information, they would believe inconsistencies: (M 3 , w 0 ) D {a,b} ⊥. ◭

Axiom system
As the reader can imagine, there are other alternatives for defining a logical framework describing the individual and distributed knowledge a set of agents might have. The epistemic logic framework recalled in this section is based on some concrete choices (e.g., relying on a model that represents uncertainty [via relations], and then defining knowledge in terms of it), and these choices define the properties of the notions of knowledge that arise. Still, some properties might not be easy to identify just from the model and the language's semantic interpretation. In such cases, it is useful to look for an axiom system characterising the formulas in L D that are valid in pointed (multi-agent relational) models. By doing so, the axiom system provides a list of the essential laws governing the behaviour and interaction of individual and distributed knowledge in this setting.
It is well-known (e.g., Halpern and Moses 1985, Fagin et al. 1995, Baltag and Smets 2020) that the axiom system L D , whose axioms and rules are shown on Table 1, characterise the formulas in L D that are valid in pointed M A -models. While PR and MP characterise the behaviour of Boolean operators, axioms K D , M D and rule G D characterise distributed knowledge: D G contains all validities (rule G D ), it is closed under modus ponens (axiom K D ) and it is monotone on the set of agents (axiom M D : if ϕ is distributively known by the agents in G, then it is also distributively known by any larger group G ′ ).

Sharing everything with everybody
As discussed before, the concept of distributed knowledge relies on the idea of agents communicating their individual information. Indeed, the fact that a set of agents G has distributed knowledge of ϕ (D G ϕ) encodes the intuition that, if they all would share their information, then afterwards each one of them would know that ϕ is the case. This 'encoding' can be made explicit by following the DEL approach: define an operation that, when receiving an initial model representing the agents' individual information, returns the model that results from the agents sharing their information. The current section explores two variations of this idea: an operation representing an action through which all agents share all her information with everybody (Subsection 3.1), and an operation representing an action through which some agents share all her information with everybody (Subsection 3.2).

Everybody shares everything with everybody
The simplest form of communication among EL agents is one through which every agent shares everything with everybody. This will be called an act of ∀∀∀-communication.

Operation and modality
Recall that a multi-agent relational model represents not the knowledge of each agent, but rather her uncertainty: the worlds she considers possible from a given one. Thus, in this setting, an action through which every agent shares everything she knows with everybody corresponds, formally, to a model operation through which every agent discards every possibility that has been already discarded by any agent. Such a operation is straightforward: the indistinguishability relation of each agent in the new model is defined as the intersection of the relations of all agents in the original model. 4 Recall that D ∅ is not a modality in L D . If it were, then L D would need further axioms and rules. In fact, since R D ∅ (w) = D(M) for any w ∈ D(M) and any M ∈ M A , it follows that (M, w) D ∅ ϕ if and only if (M, u) ϕ for all u ∈ D(M). In other words, D ∅ is nothing but the global universal modality (Goranko and Passy 1992). Because of this, an axiom system for D ∅ requires not only extending K D , G D and M D to allow G = ∅, but also the use of three additional axioms:

Definition 3.1 (∀∀∀-communication)
Thus, after the operation an agent i cannot distinguish u from w (that is, R ! i wu) if and only if, before the operation, no agent in A could distinguish u from w (that is, R j wu for all j ∈ A).
It should be emphasised that, despite the given intuition, the just defined operation is not one through which the agents share the information that has allowed them to discard certain possibility. As discussed in the introduction, relational models only represent the epistemic uncertainty of the agents (the edges). Thus, an agent's communication amounts to sharing the epistemic possibilities she has already discarded (by indicating which edges are not in her indistinguishability relation), so others can discard them too. 5 Here are some small yet useful observations.
• Each new relation R ! i can be equivalently defined as for any world w in any model M, it follows that the action of ∀∀∀-communication can only reduce the uncertainty of each agent.
• The operation preserves universal relational properties: if all relations in R j | j ∈ A are, e.g., reflexive/transitive/symmetric/Euclidean, then so is each resulting R ! i . Thus, in particular, the operation preserves equivalence relations.
Here is a modality for describing the effects of ∀∀∀-communication. p, q, r w 0 As shown in the diagram above on the right, the ∀∀∀ operation removes those edges that were not in the indistinguishability relation of every agent (i.e., it preserves only A-edges). Thus, (ii) Analogously, recall M 2 from Example 2.1.(ii), shown below on the left.

Properties
Here are some properties describing the effects of the ∀∀∀ operation. First, intuitively, the action turns distributive knowledge of the whole group into individual knowledge. The following proposition shows that this is true only up to a certain extent. (i.e., when applying "!" to M does not reduce ϕ's truth-set). However, Proof. By semantic interpretation, (M, w) D A ϕ holds for a given (M, w) Hence, by the as- In general, there is not guarantee that every world making ϕ true in a given M (a world in ϕ M ) is also a world making ϕ true in M ! (a world in ϕ M ! ).
The fact that D A ϕ → [!] K i ϕ is not valid should not be taken to mean that ∀∀∀ is flawed. First, this formula one intuitively expects to be valid is so in those situations one intuitively considers: those in which ϕ describes ontic (i.e., propositional) facts. 6 The formula is in fact valid for a larger class of formulas in L D including, e.g., those describing knowledge about propositional formulas K j γ | γ is propositional 7 and more. Second, the formula is not valid for an arbitrary ϕ ∈ L D because L D can express not only an agent's (distributed) knowledge but also her ignorance, which might be reduced by the operation. This is used in the counterexample in Proposition 3.1, as ¬ K b p states that "agent b does not know that p holds".
Here are two further results.  The second observation states that each relation (R ! ) ! i in the model after two ∀∀∀ acts is exactly as R ! i , its matching relation in the model after a single ∀∀∀ act. Since M ! and (M ! ) ! have the same domain and atomic valuation, this implies that the ∀∀∀ operation is idempotent: M ! = (M ! ) ! for any M in M A . Thus,6 The ∀∀∀-act does not affect atomic valuations, so γ M = γ M ! for every M and every propositional γ. Thus, by Proposition 3.1, But, from Proposition 3.2.(i) and semantic interpretation, the latter is equivalent to In particular, the first item of Proposition 3.3 provides a characterisation of the knowledge any agent i has after an act of ∀∀∀-communication: Axiom system To axiomatise a modality whose semantic interpretation relies on a model operation, a common DEL strategy is to provide recursion axioms: valid formulas and validity-preserving rules defining a translation that takes a formula with model-changing modalities (a formula in the 'dynamic' language) and returns one without them (a formula in the initial 'static' language). To prove soundness within this strategy, it is enough to show that the new axioms and rules are valid and preserve validity; this also shows that a formula and its translation are semantically equivalent. To prove completeness, notice that the recursion axioms make a formula and its translation provably equivalent, and thus one can rely on the completeness of the axiom system for the 'static' language. The reader is referred to van Ditmarsch et al. (2008, Chapter 7) and Wang and Cao (2013) for an extensive explanation of this technique.
For the case at hand, the recursion axioms appear on states that replacing logical equivalents within the scope of [!] preserves logical equivalence. As detailed in the proof of Theorem 2 (Subsection A.1), these axioms and rule define an "inside-out' translation (see Plaza 1989 andGerbrandy 1999, Section 4.4) that, when dealing with nested [!], works first with the deepest occurrence before dealing with the rest.

Somebody shares everything with everybody
A simple generalisation of the act of ∀∀∀ communication is one where only some agents share everything they know with everybody. This will be called an act of ∃∀∀-communication.

Operation and modality
The operation representing ∃∀∀ communication is a simple variation of the ∀∀∀-case. First, worlds and valuation are preserved, as before. Then, the indistinguishability relation of each agent in the new model is defined as the intersection of her original relation with those of the agents that share their information (the senders).

Definition 3.3 (∃∀∀-communication)
Thus, after the operation, an agent i cannot distinguish u from w (that is, R S! i wu) if and only if, before the operation, neither she nor any agent in S could distinguish u from w (that is, R i wu and R D S wu). Here are some small and yet useful observations.
. Hence, just as ∀∀∀, the action of ∃∀∀-communication can only reduce the uncertainty of each agent.
• As one might expect, an ∃∀∀-communication with A the communicating agents is exactly an ∀∀∀-communication, as • If R i and the relations in R j | j ∈ S are all reflexive (resp., transitive, symmetric, Euclidean), then so is the resulting R ! i . In particular, the ∃∀∀ operation preserves equivalence relations.
Here is the associated modality.

Example 3.2
Here are examples of this operation at work.
The action {a, b}! produces the model depicted by the diagram above on the right. The new uncertainty of each sharing agent is given by the old uncertainty of the sharers (thus, e.g., R {a,b}! a = R D {a,b} ), and the new uncertainty of each non-sharing agent is given by the old uncertainty of the sharers and herself (thus, e.g., In words, while a and b know p ∧ q distributively, c knows r. Then, after a and b share all their information to everyone, they both get to know p ∧ q but still do not know whether r. However, c gets to know what the real situation is. Analogous situations result if the communicating agents are {a, c} or {b, c}. (ii) Now recall M 2 from Example 2.1.(ii) (diagram below on the left).
, which results from a and b sharing all their information to everyone, appears above on the right. Note that the same model results if c is the only communicating agent (i.e., M {a,b}! The reason for this is that what agents in {a, b} know distributively is exactly what c knows individually (i.e., R D {a,b} = R c ). ◭ Properties What is the effect of a ∃∀∀ operation? Intuitively, the action turns distributive knowledge of a group into individual knowledge of the group's members. Just as in the ∀∀∀ case, this is true only up to a certain extent.
Proof. As that of Proposition 3.1, here using Again, there is not guarantee that every world making ϕ true in a given M (a world in ϕ M ) is also a world making ϕ true in M S! (a world in ϕ M S! ).
As in the ∀∀∀ case, should not be taken as a drawback for the ∃∀∀ operation: the formula is valid for the situations one intuitively considers (i.e., when ϕ is propositional) and also for further fragments of L D (see the discussion after Proposition 3.1).
Here are two further results.
Analogous to Proposition 3.2 for the ∀∀∀-case, Proposition 3.5 states two facts. First, it states that the relation for interpreting D G in the new model, (R S! ) D G , is the same as the relation for interpreting D S∪G in the original model, R D S∪G . As in the ∀∀∀ case, this provides a validity (Proposition 3.6.(i)) that is crucial for an axiomatisation by translation.
The second fact is relative to how two consecutive ∃∀∀ acts can be 'comprised' into a single one. More precisely, it indicates how a '∃∀∀-conversation' in which group S 1 shares first and group S 2 shares second can be replaced by a single ∃∀∀-act in which the sharing agents are those in S 1 ∪ S 2 . Given that (M S1! ) S2! and M (S 1 ∪S 2 )! have the same domain and atomic valuation, this implies But, from Proposition 3.5.(i) and semantic interpretation, the latter is equivalent to Thanks to these validities and previous observations, one can find further validities describing properties of an act of ∃∀∀ communication. First, as it was mentioned, the ∀∀∀ operation of the previous subsection is the particular case of the ∃∀∀ operation in which all agents share. Thus, From Proposition 3.6.(i), it follows that the knowledge any agent i has after an act of ∃∀∀-communication is what was distributively known among her and the sharing agents about the effects of the action: From Proposition 3.6.(ii) it follows that an act of ∃∀∀-communication by the same group is idempotent: Moreover, it also follows that, when two groups communicate, the order in which they do so is irrelevant: Thus, repeated ∃∀∀-sharing by the same group does not provide anything new, regardless of whether it is immediate or after some other groups have shared: Axiom system The axiom system for the extended language L D, [S!] relies again on the DEL reduction axioms technique, with the axioms and rule for the case at hand being those on Table 3. Axioms A p S! , A ¬ S! , A ∧ S! and rule RE S! are as in the ∀∀∀ case, indicating respectively that [S!] does not affect atomic propositions, commutes with negation, distributes over conjunction and 'preserves' logical equivalence. Axiom A D S! is the one that distinguishes ∃∀∀ from ∀∀∀, indicating that what a group G knows distributively after [S!] is exactly what the group S ∪ G knew distributively about [S!]'s effects. Together, these axioms and rule define an "inside-out' translation from L D, [S!] to L D such that a formula and its translation are both semantically and provably equivalent (see the proof of Theorem 3 on Subsection A.2).

Other operations for communication in the literature
The operations proposed in this section are not the first ones representing actions of agents sharing their individual information. The action of "tell Then there is the action for 'resolving the distributed knowledge of a group' studied in Ågotnes and Wáng (2017). Through it, agents in a group G share all their information only within G itself. Thus, while the new indistinguishability relation of every agent not in G remains exactly as before, that of each agent in G is defined as R D G . 8 This action can be also seen as an instance of the act of ∃∀∀-communication (Subsection 3.2) (only agents in G share), but note that only agents in G receive the information. In the terminology of this manuscript, this action is better described as an act of '∃∀∃-communication' (Section 5 will discuss briefly ∃∃∃-communication). Finally, there is the more general action of 'semi-public reading events' of Baltag and Smets (2020), where each agent i gets a set α(i) ⊆ A satisfying i ∈ α(i). Intuitively, α(i) contains those agents whose information i will receive when communication occurs. Thus, for each agent i, the operation defines her new indistinguishability relation as R D α(i) . Note again that this form of communication is semi-public: even though each agent i only 'hears' what agents in α(i) 'say', the definition implies that i still learns that every agent j receives the information provided by j's sources (agents in α(j)). This can be seen as a generalisation of a '∃∀∀'-communication: while some agents (those in α(i) for some agent i) share all their information, every agent receives information from potentially different agents (each agent i only 'listen to' those agents in α(i)).

Sharing something with everybody
Section 3 discussed model operations for actions through which all/some agents share all the information with everybody; thus, the actions represent acts of full communication (from a subset of agents). As discussed, the operations are small variations of proposals already present in the literature.
This section, which constitutes the core of this contribution, discusses a variation of the ∃∀∀ case: one in which the sharing agents communicate only part of their information. This action makes the process of communication between agents a more realistic one. Indeed, there are natural restrictions on the 'amount' of information an agent can communicate at once, and operations with such restrictions can represent more realistic 'conversations'. In defining this operation, probably the most important question is the following: what defines what each agent will communicate? There are indeed several possibilities (see the discussion in Section 6), but a natural one is to assume that the 'conversation' is relative to a given subject/topic. This manuscripts uses this idea, assuming that this subject/topic is defined by a formula χ. Following the previous notation, this action through which some agents share some of their information (that relative to the given subject χ) with everybody will be called an act of ∃∃∀-communication.
Operation and modality For the definitions, the following will be useful.
In order to understand the intuition behind the definition, note how the set R j ∩ M χ can be seen as the epistemic contribution of agent j about formula χ in model M, as it contains those pairs in the M-uncertainty about χ (the set M χ ) that j has already discarded (the set R j ). With this in mind, the definition states that agent i's uncertainty after agents in S share all their information on subject χ (i.e., her uncertainty in the new model, R S χ ! i ) is given by her previous uncertainty (i.e., R i ) minus the sum ( ) of the epistemic contribution (i.e., R j ∩ M χ ) of each agent j in S. 9 Note also how, through the operation, agents in S communicate all they know about χ: intuitively, they share all the information that has allowed them to discard any uncertainty between χ-and ¬χ-worlds, and thus only edges in M χ can be eliminated. This emphasises the fact that χ is taken to be the subject/topic of the conversation, with agents intuitively sharing what has allowed them to discard edges between worlds disagreeing on χ's truth-value. 10 The relations in the model resulting from ∃∃∀-communication can be described in a simpler way.
Then, from the definitions of both set subtraction and ∼ M χ , the right-hand side becomes . Before discussing the ∃∃∀ operation further, here are some examples.  The definition might be easier to grasp by taking a 'knowledge' perspective. Intuitively, the knowledge of the agent after a conversation in which agents in S share what they know about χ is her initial knowledge plus the knowledge any agent in S has about χ. The provided definition describes exactly the same idea, stating it in terms of the agent's uncertainty. 10 There is a natural alternative in which χ is rather taken to be the content of the conversation. In this asymmetric version, agents intuitively share only what has allowed them to discard ¬χ as a possibility, and thus only edges pointing to ¬χ-worlds can be eliminated. new model, the operation looks at the original model, focussing only on edges between worlds disagreeing on the topic's truth-value (edges across partition cells in the diagrams above) and leaving the rest as they are. For example, for topic p, the operation focuss on edges between worlds in {w 0 , w 1 , w 2 } and worlds in {w 3 }, disregarding (i.e., not affecting) the rest. When all agent share, as in this case, the operation simply removes the edges under consideration (i.e., across partition cells) that are not in R D A .
p, q, r w 0 As the diagrams show, the operation behaves as expected. For example, a conversation among all agents about p (leftmost model) benefits b and c (they get to know p's truth-value) but does not benefit the only agent who knew p's truth-value before, namely a. 11  In all these cases, only agents in S = {a, b} communicate. The operation again looks only at edges across partition cells, with each agent i ∈ S keeping only those in R D S and each agent i S keeping only those in R D S∪{i} . For example, after {a, b} exchange (recall: publicly) information about p ∧ q (leftmost model), every agent will know the truth-value of both p and q (since c additionally knew r's, now she knows which the real situation is). But if a and b's conversation is rather about q ∧ r (rightmost model), then less information is shared (neither of them knew whether r, and a did not get the chance to talk about what she knew: p). In fact, afterwards no agent knows the truth-value of all atoms, and only two agents (a and c) know the truth-value of two of them (q and, respectively, p and r). ◭ After the examples, here are some observations.
• An ∃∃∀ act is not a generalisation of ∃∀∀: there is no formula χ such that, for every model M = W, R, V and every set of agents S, an 'Sconversation about χ' (producing R S χ ! i = R i ∩ (R D S ∪ ∼ M χ )) has the same effects as an 'S-conversation' about everything (producing R S! i = R i ∩R D S ). For that, one would need a χ such that R i ∩(R D S ∪∼ M χ ) = R i ∩R D S (and thus, ∼ M χ ⊆ R D S ) for every M and every S. 12 Still, one can find specific situations (a model M, a set of agents S and a formula χ) an 'S-conversation about χ' has the same effects as an 'S-conversation about everything'.
• If R i and the relations in R j | j ∈ S are all reflexive/symmetric, then so is the resulting R ! i . 13 However, the operation does not preserve transitivity and neither 'Euclideanity' (see, e.g., Example 4.1.(ii), in particular the effects of the action {a, b} p∧r ! on the relation for agent a). 12 There is no χ satisfying ∼ M χ ⊆ R D S for every M and S: there are M and S with R D S = ∅, and yet the (reflexive) relation ∼ M χ is never empty (domains are non-empty). But even when working with equivalence relations, no χ satisfies the requirement for every M and every S. 13 For reflexivity, take any w ∈ W. By the assumption, R i ww and R D S ww; moreover, w∼ M χ w.

Properties
Here are some observations about the ∃∃∀ operation, starting again with a caveat. Proof. Since (M, w) D S ϕ, then R D S (w) ⊆ ϕ M and thus, by the assumption, The first has been obtained. For the second, take any u ∈ (R i ∩ ∼ M ϕ )(w), so w and u coincide in ϕ's truth-value in M. By reflexivity, w ∈ R D S (w), so w ∈ ϕ M (the antecedent D S ϕ) and hence u ∈ ϕ M . Therefore, by the assumption, u ∈ ϕ M Sϕ! , as required.
Both requirements are essential. In Fact 4.1 it will be shown that reflexivity The following proposition provides the two useful observations. Proposition 4.3 Let M = W, R, V be in M A ; take S, S 1 , S 2 ⊆ A and let χ, χ 1 , χ 2 be formulas. Let M S χ ! = W, R S χ ! , V and (M S 1 χ 1 ! ) S 2 χ 2 ! = W, (R S 1 χ 1 ! ) S 2 χ 2 ! , V be as indicated in Definition 4.2. Then, Proof.
(ii) By algebraic manipulations starting from On the one hand, the second result of the previous proposition shows the indistinguishability relation of an agent i after two ∃∃∀ operations in a row. By comparing it with the same relation after a single ∃∃∀ operation, , one understands why, different from the two previous cases, two successive ∃∃∀ actions cannot be compressed into a single one.
On the other hand, the first result indicates that the relation for interpreting provides the following validity, which will be useful for the axiomatisation.

Proposition 4.4 [S
But, from Proposition 4.3.(i) and semantic interpretation, the latter is equivalent , whose conjuncts are equivalent, respectively, to w ∈ D S∪G [S χ !] ϕ M and w ∈ D χ G [S χ !] ϕ M . This validity yields immediately the following one, characterising the knowledge of an agent after ∃∃∀ communication: Here is another useful validity, a consequence of an earlier observation: In the actions of Section 3, each relation in the new model is defined in terms of relations in the initial one. However, the ∃∃∀ operation defines each 'new' relation in terms of 'old' ones and the relation of only knowing χ fully, ∼ χ (with χ the communication's topic). This and the fact that model operations can change the truth-set of a formula is what makes ∃∃∀ behaves less similar to the ∀∀∀and ∃∀∀-actions, and more similar to public announcements. 14 Indeed, by definition, ∼ M Proof. In the model below (with equivalence relations), each world indicates the truth-value of atoms m a , m b , m c (in that order) by using "•" (the atom holds) or "•" (the atom fails). 15 M 1 : Note how, at worlds in {w 0 , w 2 , w 3 , w 6 }, every agent i has uncertainty about whether m i holds (i.e., χ i fails for every i, and thus so does χ ∨ ). However at worlds in {w 1 , w 4 , w 5 , w 7 }, at least one agent i knows whether m i (so χ ∨ is the case). 16 (i) Since χ ∨ M 1 = {w 0 , w 2 , w 3 , w 6 }, applying A χ ∨ ! to M 1 (note the generated partition) yields M 2 below. Then, since χ ∨ M 2 = {w 2 }, a further application of A χ ∨ ! yields M 3 .
even when the model is reflexive (cf. the discussion in the proof of Proposition 4.2).
(ii) On the one hand, χ a On the other hand, (iv) As it can be seen from Item (ii), (M 1 , w 3 ) [A χ a !] [A χ c !] χ b (the model that results from A χ a ! is also M ′ 2 , and a further A χ c ! yields M ′ 3 too). However, χ a ∧ χ c M 1 = {w 5 }, so applying A (χ a ∧χ c ) ! on M 1 has no effect, and thus The provided counterexamples show that the given formulas are not valid, even under equivalence relations.
Axiom system Once again, the axiom system for L D, [S χ !] relies on reduction axioms (Table 4). Axioms A p S χ ! , A ¬ S χ ! , A ∧ S χ ! and rule RE S χ ! are similar to their previous matching cases. Axiom A D S χ ! is the important one, indicating that a group G knows ϕ distributively after the action ([S χ !] D G ϕ) if and only if the group S∪G knew, distributively, that ϕ would hold after the action (D S∪G [S χ !] ϕ) and the group G knew, distributively and relative to similarity on χ, that ϕ would hold after the action (D χ

Coda: sharing with somebody
Section 3 introduced two forms of inter-agent communication: ∀∀∀, through which all agents share all their information with everybody, and ∃∀∀, through which some agents share all their information with everybody. Then, Section 5 introduced ∃∀∀, a form of communication through which some agents share some of their information with everybody. The next natural step is defining a form of communication through which some agents share some of their information with somebody. This operation, which could be denoted by ∃∃∃, will not be studied in this manuscript. Instead, the text will only discuss some of the modelling options that arise.
First, when defining the ∃∃∃ operation, one might decide that the shared information is received only by agents in a given group R ⊆ A. This would represent a scenario similar to a round table with a potentially distracted audience: the people on 'the table' get to talk about a given topic, with some members of the audience listening and the distracted ones missing the information. However, one could also think of a different scenario, one in which every sharing agent i has a specific set of listeners R i . This is closer to what happens in some online social networks, where only 'friends' or 'followers' can receive what a given agent sends (cf. with Baltag and Smets 2020, discussed in Subsection 3.3). In fact, the scenario with a single set R can be seen as the special case of the scenario with {R i } i∈A in which all sets R i are the same.
A further possibility is to take the social networks idea seriously, and work with models that represent explicitly the social connections each agent has. These structures have been used by the logic community  is one of the earliest proposals), either for studying information flow in social networks (peer pressure: Zhen and Seligman 2011; reflexive influence: Christoff et al. 2016;diffusion and prediction: Baltag et al. 2019) or for studying social network formation (as a side effect of peer pressure: Pedersen and Slavkovik 2017; by similarity: Smets and Velázquez-Quesada 2018. A second decision to make when defining an ∃∃∃ operation is the following: what information, if any, obtain the agents that do not 'hear' the communication? On one extreme, these agents might be oblivious, not only to the content of the communication, but also to the fact that a communication took place. On the technical side, representing this form of 'private' communication requires an operation that changes not only the model's indistinguishability relations, but also its domain. Indeed, while the 'real' part of the model would change as the communication takes place (the receiving agents might learn something), for non-receiving agents one should keep 'a copy' of the original model, to indicate that they see no change at all. 17 On the other extreme, non-receiving agents might not hear the content of the communication, but they might notice that the communication takes place (the semi-public form of communication from Ågotnes and Wáng 2017 and Baltag and Smets 2020 discussed in Subsection 3.3). Even more, they might know the topic of the conversation. Combining this with some previous knowledge about what the communicating agents know, the non-receivers might get to know part of what is being shared, and thus they might get to know (part of) the new epistemic state of the communicating agents.

Summary and further research lines
This paper has discussed three communication actions. Different from most epistemic acts present in the literature, they truly describe processes of interagent communication: those in which the information that is being shared is information some of the agents already have. Through the first two actions, ∀∀∀ and ∃∀∀, all/some agents share all their information with everybody. Through the third, ∃∃∀, the core of this contribution, some agents share some of their information with everybody. The text has presented examples of these actions at work, discussing some of their basic properties and providing, in all cases, a sound and complete axiom system for modalities describing their effects.
There are several research lines that arise from the current proposal. Here are some of them.
• Distributed knowledge has played a crucial role in this manuscript. Yet, the equally important notion of common knowledge (referring to the infinite iteration including everybody knows, everybody knows that everybody knows, and so on; Lewis 1969, Vanderschraaf and Sillari 2014) has been absent. Common knowledge is an important piece when discussing communication between agents, as one would like to know not only what each individual agent gets from the action (their individual knowledge), but also what a group as a whole learns from it (the group's common knowledge). Adding the common knowledge operator to the studied languages involves the use of additional tools, in particular for the axiomatisation.
• In a ∃∃∀ action, what each sharing agent communicates is given by a formula χ, understood as the subject/topic of the conversation. This is the reason why the operation only considers for elimination edges between worlds differing in χ's truth-value. However, as mentioned, there are other alternatives. One can understand χ as the content of the conversation instead; in such case, the operation should consider for elimination only edges pointing to ¬χ-worlds.
One could also take a set of formulas. If this represents the set of topics then, under some reasonable restrictions on the class of models, such variation could turn out to be a generalisation of one of the other actions: an ∃∃∀communication on all topics could be equivalent to an ∃∀∀ communication.
In any of these alternatives, one can also look for a less uniform treatment, assigning to each sharing agent (and even to sets of of them) a potentially different (set of) formula(s) (cf. van Benthem and Minica 2012, Baltag et al. 2018). All these options deserve a proper exploration.
• In the communication actions studied in this text, each agent i communicates by sharing (some of) the possibilities she has already discarded (those in R i ). Thus, it has been implicitly assumed that, while agents might not share 'everything that they know', what they share is 'something that they do know' (in other words, they 'communicate' a set S ⊆ R i ). But one can also find agents that share 'more than what they know' (they communicate a set S ⊃ R i ), and even only 'things they do not know' (they communicate a set S with S ∩ R i = ∅). These alternatives can be used for studying acts of lying, which so far have been modelled only by adapting tools from public announcement logic (van Ditmarsch 2014).
• The success of public announcement logic comes from the fact that it provides formal tools for studying different epistemic change phenomena. Among them, one can find studies on arbitrary announcements. Indeed, some proposals (e.g., Balbiani et al. 2008, Galimullin andÅgotnes 2021) have worked with modalities of the form ⋆! ϕ, read as "there is a formula that can be truthfully announced, and after doing so ϕ will be the case". The setting presented here offers a further alternative for quantifying over information change: one can define an operation through which a given agent (and eventually a set of them) shares an arbitrary set S ⊆ R i . Thus, one can study what a set of agents can get to know (i.e., a form of knowability) by sharing their information.

Definition A.1 (Translation)
The translation τ is given by

This translation works with formulas of the form [!]
[!] ϕ in an "inside-out' fashion, dealing first with the deepest occurrence of [!] (i.e., translating [!] ϕ) before dealing with the rest (i.e., before translating [!] τ([!] ϕ)). Because of this, the strategy proving strong completeness is as follows (cf. Plaza 1989 andGerbrandy 1999, Section 4.4): (i) show that τ is a proper recursive translation that returns formulas in L D (Proposition A.1), (ii) show that a formula and its translation are both provably and semantically equivalent (Proposition A.2), and (iii) use τ and the completeness of L D to show that, if Ψ ∪ ϕ ⊆ L D, [!] , then from Ψ ϕ it follows that ϕ is derivable from Ψ in L D, [!] .
Relying on the complexity ordering c, the following proposition states that τ is a proper recursive translation from L D, [!] to L D .
Proof. The proof is by structural induction on ϕ. Here are the cases.
Now, for the actual proposition, the proof proceeds by induction on c(ϕ), the complexity of ϕ, with both Item (τ1) and Item (τ2) proved simultaneously.
• Base case (ϕ such that c(ϕ) is minimum). A ϕ with the minimum c should be minimum at both ndc (i.e., ndc(ϕ) = 0, so any ϕ without observation operators) and nsc (i.e., nsc(ϕ) = 1). The only such formula is p. Proving Item (τ1) is immediate, as the definition of τ(p) does not use τ; proving Item (τ2) is also immediate, as τ(p) = p is a formula in L D .

Thus, c([!] D
• Using the ordering c, the proposition below shows that a formula ϕ ∈ L D, [!] and its translation τ(ϕ) ∈ L D are both provably and semantically equivalent.
Proof. By induction on c(ϕ). The following rule will be useful.

(τ1)
Here are the cases.
Finally, the argument for strong completeness, which has three steps.

A.2 Proof of Theorem 3 (System L D, [S!] )
Soundness Again, the soundness of axioms and rules in L D is well-known. For those in Table 3, the soundness of A p S! , A ¬ S! , A ∧ S! and RE S! is as in the ∀∀∀ case (for the latter, recall that M S! is a model in M A ), and A D S! has been proved valid already (Proposition 3.6).

Completeness
The argument relies on the following translation. This translation works again in an "inside-out' fashion, dealing first with the deepest occurrence of [S!] before dealing with the rest. The strategy for proving strong completeness is as in the ∀∀∀ case: (i) show that τ is a proper recursive translation that returns formulas in L D (Proposition A.3), (ii) show that a formula and its translation are both provably and semantically equivalent (Proposition A.4), and (iii) use τ and the completeness of L D to show that, if Ψ ∪ ϕ ⊆ L D, [S!] , then from Ψ ϕ it follows that ϕ is derivable from Ψ in L D, [S!] .
The proof of the proposition is by induction on c(ϕ) relying on Lemma A.3, just as in the ∀∀∀ case. Again, Item (τ1) and Item (τ2) are proved simultaneously. Given that nsc, ndc, c and τ are defined as for the ∀∀∀ instance, all cases here are exactly as in their ∀∀∀ counterpart (Proposition A.1).
Then, a formula ϕ ∈ L D, [S!] and its translation τ(ϕ) ∈ L D are both provably and semantically equivalent.  Finally, the argument for strong completeness is as the ∀∀∀ case (page 32), relying on Proposition A.3 and Proposition A.4 instead.

A.3 Proof of Theorem 4 (System L D, [S χ !] )
Soundness The soundness of axioms and rules in L D is well-known. For those in Table 4, soundness of A p S χ ! , A ¬ S χ ! , A ∧ S χ ! and RE S χ ! is as in the previous cases (for the latter, recall that M S χ ! is a model in M A ), and A D S χ ! has been proved valid already (Proposition 4.4).
Proof. The following lemma will be useful. Proof. The proof is by structural induction on ϕ. For atoms, Boolean operators and the D G modality, the functions nsc, ndc and c are all defined as for the ∀∀∀ instance; thus, the cases for p, ¬ϕ, ϕ 1 ∧ϕ 2 and D G ϕ are exactly as in Lemma A.1.
The proof of the proposition is by induction on c(ϕ) relying on Lemma A.4, analogous to the ∀∀∀ case. Again, Item (τ1) and Item (τ2) are proved simultaneously. For atoms, Boolean operators and formulas of the form D G ϕ, the functions nsc, ndc, c and the translation τ are defined as for the ∀∀∀ instance; thus, cases p, ¬ϕ, ϕ 1 ∧ ϕ 2 and D G ϕ are exactly as in Proposition A.1. Here are the remaining cases. Finally, the argument for strong completeness is as the ∀∀∀ case (page 32), relying on Proposition A.5 and Proposition A.6 instead.