The Pleasure of Gossip

Maduka Attamah, Hans van Ditmarsch∗ †, Davide Grossi, Wiebe van der Hoek

January 28, 2015

Abstract

Rohit Parikh has written on levels of knowledge [PK92]. Levels of knowledgeare relevant for the analysis of gossip protocols. Gossip protocols describe the dis-semination of information over a network. We present some examples of epistemicgossip protocols, wherein the agents or processes communicate with each other bypeer-to-peer contact (telephone calls), as in the usal gossip protocols, but whereinthe decision to contact another agent is based on the calling agent’s information only.This is, as far as we know, unusual in gossip protocols. In this we wish to honourRohit Parikh’s long career and many contributions to logic and computer science.

∗Hans van Ditmarsch is corresponding author. We thank a reviewer for insightful comments. Hans isalso affiliated to IMSc, Chennai, as research associate, and he acknowledges support from ERC projectEPS 313360.†To appear in: C. Baskent, L. Moss and R. Ramanujam (Eds.), Rohit Parikh on Logic, Language and

Society, Springer 2015

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1 Gossip Protocols

Six friends each know a secret. They can call each other. In each call theyexchange all the secrets they know. How many calls are needed for everyone toknow all secrets?

We solve this for n friends, and let us work our way upwards bottom-up. For one friend,no calls need to be made, and for two friends, a single call between them is sufficient. Forthree friends, any two need to call each other first, then the friend who did not make a callyet needs to call either of the two who just called, and finally either of those who madethe second call, call the friend not involved in that call. That makes three calls.

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Let there now be four friends Amal, Bharat, Chandra, and Devi (a, b, c, d) who hold,respectively, secrets A,B,C,D. First, some terminological and modelling assumptions.We see a secret as a propositional variable such that ‘knowing the secret’ means knowingthe truth value of that variable. ‘Amal knows secret A’ means that Amal knows whetherA, i.e., Amal knows that A is false or Amal knows that A is true. We represent by ab acall from a to b. The informative consequences of a call (i.e., which secrets are exchanged)are independent from who initiates a call, so in that sense a call ab is the same as a callba. We will define protocols wherein ab is allowed but not ba, but in our examples we tendto respect lexicographic order for convenience of presentation (for example, when both aband ba are possible). Given the setting of our contribution we prefer to talk about agentsinstead of friends (just consider ‘multi-friend epistemic logic’). However, we will continueto talk about secrets and not about propositions, propositional variables, or facts. Knowinga proposition or a fact tends to mean that you know that it is true, whereas knowing asecret tends to mean that you know whether it is true. That is what we want.

Back now to the four agents Amal, Bharat, Chandra, and Devi (a, b, c, d). The fourcalls ab; cd; ac; bd distribute all secrets. The underlying protocol, of which this call sequenceis an execution, is as follows.

Protocol 1 (Four Agents): Any two agents make the first call; the second call is then be-tween the remaining two agents; the third call is then between an agent who made the firstcall and an agent who made the second call; and the fourth call is between the two whowere not chosen in the third call.

The distribution of secrets given the four calls is as follows. The rows list the distributionof secrets after a particular call took place.

a b c dA B C D

ab AB AB C Dcd AB AB CD CDac ABCD AB ABCD CDbd ABCD ABCD ABCD ABCD

No other protocol of sequential calls solves this in four calls, and that less than four callsis insufficient to distribute all secrets. We can easily show this:

In an execution of any other protocol one of the first callers will also make the secondcall. So, it has to start like this.

a b c dA B C D

ab AB AB C Dac ABC AB ABC D

. . .

How will this continue? For the third call, let us distinguish between the case that Devi(d) is not involved and the case that she is involved. If Devi is not involved, then another

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call ac does not result in more information, and the cases ab and bc are symmetric. Takethe first, then we get

a b c dA B C D

ab AB AB C Dac ABC AB ABC Dab ABC ABC ABC D

. . .

We now have to make three more calls in order for all agents to know all secrets: forexample, one by one, Devi now calls Amal, Bharat, and Chandra. That makes six callsaltogether.

But if the third call involves Devi, there also will always remain two agents who do notknow d yet. Again, two or three further calls are needed, so that we need at least five callsaltogether.

This also demonstrates that less than four calls is insufficient to distribute all secrets,because any execution starts with either ab; ac (at least five calls to termination) or ab; cd(at least four calls to termination), modulo a permutation of agents.

For n = 4, 2n − 4 calls are sufficient to distribute all the secrets. Let there now ben > 4 agents. Then 2n− 4 is also sufficient. Suppose the agents are a, b, c, d, e, f, . . . .

Protocol 2 (Fixed Schedule): First, a makes a call to all agents e, f, . . . except b, c, d.Then, the calls ab, cd, ac, bd are made (this is an execution of Protocol 1). Finally a makes,again, a call to all agents e, f, . . . except b, c, d.

This adds up to (n − 4) + 4 + (n − 4) = 2n − 4 calls. It will be clear that all secrets arethen distributed over all agents.

Let us do this for n = 6, such that we get 2n− 4 = 8 calls. Given are six agents Amal,Bharat, Chandra, Devi, Ekram, and Falguni (a, b, c, d, e, f) who hold secretsA,B,C,D,E, F .Amal starts by calling Ekram and then Falguni, etc.

a b c d e fA B C D E F

ae AE B C D AE Faf AEF B C D AE AEFab ABEF ABEF C D AE AEFcd ABEF ABEF CD CD AE AEFac ABCDEF ABEF ABCDEF CD AE AEFbd ABCDEF ABCDEF ABCDEF ABCDEF AE AEFae ABCDEF ABCDEF ABCDEF ABCDEF ABCDEF AEFaf ABCDEF ABCDEF ABCDEF ABCDEF ABCDEF ABCDEF

Less than 2n − 4 calls are insufficient to distribute all secrets. This has been shown in[Tij71], as also related in [HHL88, Hur00].

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This is not the only protocol to distribute the secrets in 2n− 4 calls. For example, inProtocol 2 some calls are made more than once. For the depicted n = 6 execution, theseare ae and af . The following also achieves distribution of all secrets over all agents but inall different calls.

a b c d e fA B C D E F

ab AB AB C D E Fcd AB AB CD CD E Fef AB AB CD CD EF EFac ABCD AB ABCD CD EF EFde ABCD AB ABCD CDEF CDEF EFaf ABCDEF AB ABCD CDEF CDEF ABCDEFbd ABCDEF ABCDEF ABCD ABCDEF CDEF ABCDEFce ABCDEF ABCDEF ABCDEF ABCDEF ABCDEF ABCDEF

Not all sequences of eight different calls distribute the secrets over all agents. Forexample, when we change the sixth call from af into bf , Amal will only know the secretsA,B,C,D after those eight calls.

Maximum number of calls If gossip is the goal, prolonging gossip is better! As longas two agents who call each other still exchange all the secrets that they know and at leastone of them learns something new from the call, what is the maximum number of calls todistribute all secrets?

The maximum number of calls to distribute all secrets is(n2

)= n·(n−1)

2. This is also the

maximum number of different calls between n agents/agents. For six agents a, b, c, d, e, fthe following calls can be made such that in every call at least one agent learns one secret—for convenience we generate the execution sequence in lexicographic order again.

ab; ac; ad; ae; af ; bc; bd; be; bf ; cd; ce; cf ; de; df ; ef

For four agents we getab; ac; ad; bc; bd; cd

Let us be explicit and give the detailed distribution of secrets for four agents:

a b c dA B C D

ab AB AB C Dac ABC AB ABC CDad ABCD AB ABC ABCDbc ABCD ABC ABC ABCDbd ABCD ABCD ABC ABCDcd ABCD ABCD ABCD ABCD

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Rounds of calls Instead of consecutive telephone calls wherein all secrets are exchangedbetween both parties, several calls between two agents might as well take place at the sametime, to speed up the exchange of information. A number of simultaneous telephone callsis called a round. What is the minimum number of rounds to communicate all secretsbetween n agents? This is analyzed in an elegant one-page journal contribution [Kno75].The answer is: dlog2ne for n even, and dlog2ne+ 1 for n odd. We will not show the proof,except for the obvious case, but instead give some examples.

First consider the case that n = 2m. (This is the obvious case.) Then we need exactlym rounds. In this case, let the n agents be named 1, . . . , n. We count modulo 2m. The firstround consists of 2m−1 parallel calls between two agents: for i = 1 to i = 2m−1, all agents2i (simultaneously) call their neighbour 2i + 1 (i.e., for future convenience, 2i + 21 − 1).The second round also consists of 2m−1 parallel calls but now between agents that were notpaired in the first round. A way to implement this, is for all agents 2i (simultaneously) tocall agents 2i+ 3, i.e., 2i+ 22 − 1. (And nobody will find the line engaged!) We continueto do so m times altogether, namely until in the mth round all 2i (simultaneously) call2i + 2m − 1. For example, for eight agents a, b, c, d, e, f, g, h (i.e., 1, 2, . . . , 8) the threerounds are {ab, cd, ef, gh}; {ac, bd, eg, fh}; {ae, bf, cg, dh}. Let us be explicit again.

a b c d e f g hA B C D E F G H

i AB AB CD CD EF EF GH GHii ABCD ABCD ABCD ABCD EFGH EFGH EFGH EFGHiii ABCDEFGH . . . . . . . . . . . . . . . . . . . . .

When the number of agents is not a power of 2 this requires some more work, e.g., forn = 5 the minimal number of rounds is dlog25e + 1 = 4. Such a four round parallel callsequence is: {ab, cd}; {ac, be}; {ae, bc}; {ad}. Isn’t it strange that the case n = 5 needsmore calls than the case n = 8? One can easily verify from the table below that less thanfour is indeed impossible.

a b c d eA B C D E

{ab, cd} AB AB CD CD E{ac, be} ABCD ABE ABCD CD ABE{ae, bc} ABCDE ABCDE ABCDE CD ABCDE{ad} ABCDE ABCDE ABCDE ABCDE ABCDE

Another configuration for the first two rounds starts with {ab, cd}; {ac, bd}; . . . . But thenwe even need three more rounds, and therefore five in total. A minimal completion of thatis {ab, cd}; {ac, bd}; {ae}; {ab, ce}; {de}. Note that in the third round, there is nothing elseto do but to make the single call between e and another agent.

Knowledge-Based Gossip Protocols So far, we assumed that the agents can coordi-nate their actions before making any calls. In the Fixed Schedule protocol the individual

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called Amal has been assigned to make the first call, and to make this call to the individ-ual named Bharat. Then, the individual named Chandra calls the individual named Devi.And so on. A better way to define a protocol seems that, e.g., any two agents can makethe first call, and then any two other agents can make the second call. So it doesn’t have tobe between these exact four individuals. The question then becomes if the protocol can berephrased such that these scheduling decisions can be made by the agents themselves, atexecution time, and based on what they know. There are many realistic settings whereinwe cannot assume the existence of a global scheduler to assign optimal behaviour to allagents, but wherein agents (or nodes) have to figure this out by themselves: by way ofwhat they know. Consequently, such protocols may not be those with the shortest possibleexecutions, but merely those with the best under such more restricted circumstances.

All agents only know their own secret initially. They cannot distinguish between theother agents by their knowledge. So for the first call we can choose any two. One caneasily justify that the first two callers are selected non-deterministically. One of them issimply the agent getting through before the others, in making a call, and the recipient ofthat call can be any other agent. But for the second call we have a problem. Now there aretwo agents who know two secrets, and the remaining agents only know one secret. In otherwords, they have different knowledge. We may pick any agent who only knows one secretto initiate a call, this choice is knowledge-based (and anyone fulfilling the condition canbe chosen), and this rules out those who made the call in the first round. In our attemptto generalize the Fixed Schedule protocol, this agent now has to call another agent whoonly knows one secret. But the agent initiating that second call cannot choose such a one-secret-only agent based on its knowledge. If Chandra initiates the second call, she has noreason to prefer Devi over any other agent, if she were ignorant about who made the firstcall. It seems not unreasonable to assume that she only knows that she was not involvedherself in that first call. That means that, from Chandra’s point of view, the first call couldhave been between Amal and Bharat, or between Amal and Devi, or between Bharat andDevi. (And from each pair, either agent initiating the call.) She does not know which onereally happened! We can also say that Chandra cannot distinguish different histories ofcalls, as in [PR85, PR03].

Learn New Secrets protocol Let us now consider an epistemic protocol wherein anagent makes a decision about whom to call based on its knowledge only, and such that anyagent fulfilling the knowledge condition is chosen non-deterministically. This knowledgemay be about the secrets that it knows, it may be about the protocol that it is engagedin, and about its call history. In the protocols we present, we only use knowledge of se-crets. The agent names remain Amal, Bharat, . . . (a, b, . . . ), and the corresponding secretsA,B, . . . ; but agent variables will be x, y, . . . and propositional variables will be px, py, . . . ;where px is the secret of agent x, and so on.

Protocol 3 (Learn New Secrets): Until all agents know all secrets: choose an agent x whodoes not know all secrets, let x choose an agent y whose secret it does not know, and let xcall y.

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It is easy to see that this protocol will achieve the epistemic goal that everybody knowsevery secret. No call sequence obtained from the Fixed Schedule protocol can be obtainedby the Learn New Secrets protocol, because in the final two calls (ae; af) from the FixedSchedule protocol agent a contacts agents of which she already knows the secret. Butthe same information transitions can be achieved by an execution sequence of Learn NewSecrets: instead of final calls ae; ef , make final calls from Ekram and Falguni to Bharat (orto Chandra, or to Devi): eb; fb. This is legal, as Ekram and Falguni do not know Bharat’ssecret at the time of that call. The Learn New Secrets protocol also allows for longerexecution sequences than the Fixed Schedule protocol. The longest possible execution oflength n · (n − 1)/2 already mentioned above is a possible execution sequence of LearnNew Secrets. For example, for n = 4: ab; ac; ad; bc; bd; cd. One can easily show that anylength of call sequence between the minimum of 2n− 4 and the maximum of n · (n− 1)/2can be realized by the Learn New Secrets protocol. (On the assumption that any non-deterministic choice between available callers is uniformly random, what would then be theexpected length of an execution sequence? Some examples are given in [AvDGvdH14].)

Expected and Known Information Growth protocols There are many such know-ledge-based protocols. In the (not knowledge-based) Fixed Schedule protocol, Amal callsthe same agents at the end as at the beginning, because she has learnt something newin the intervening calls that she wishes to tell them. This feature can also be used in aknowledge-based protocol.

The reason for Amal to call Ekram again is that Ekram will learn something new inthat call, because Amal has learnt new secrets in the intervening calls. When Amal firstcalled Ekram, Amal learnt E and Ekram learnt A. When Amal calls Ekram again, Ekramlearns B,C,D, and F from Amal. Amal learns nothing from Ekram in the second call.Instead of Amal calling Ekram, the fifth call could also have resulted from Ekram callingAmal. Let us define ‘pz is learnt in the call xy’ as ‘before the call, x (exclusive) or y didnot know whether pz, but after the call x and y know whether pz’. Now consider theseprotocols. (Below, there is no probabilistic meaning associated to ‘consider x possible’. Itmerely means that x is in the collection of objects that cannot be ruled out.)

Protocol 4 (Expected Information Growth): Until all agents know all secrets: choose anagent x, let x choose an agent y such that x considers it possible that there is a secret pzthat would be learnt in the call xy, and let x call y.

Protocol 5 (Known Information Growth): Until all agents know all secrets: choose an agentx, let x choose an agent y such that x knows that there is a secret pz that would be learntin the call xy, and let x call y.

The Expected Information Growth protocol may loop and therefore termination is notguaranteed! For example, for four agents, the following is an infinite execution sequence:ab; cd; ab; cd; ab; . . . (i.e., (ab; cd)∗ is an initial call sequence for all finite iterations). In thethird call, a considers it possible that b has learnt something new, namely if the second

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call had been one of bc, bd, cb, db. Therefore, after ab; cd, call ab can be chosen accordingto the protocol. We could also say that at the moment of the third call Amal is unable todistinguish the call sequences / histories ab; cd, ab; dc, ab; bc, ab; cb, ab, bd, ab, db.

At first sight, the Expected Information Growth protocol seems to have an advantageover the Known Information Growth protocol. Maybe there are situations wherein after acertain number of calls, due to the uncertainty about who called who, no agent knows forcertain that calling any other agent will result in information growth. That would cause adeadlock in the Known Information Growth protocol, from which an agent can still escapewhen using the Expected Information Growth protocol.

But on second sight, such a situation cannot occur. Consider any situation wherein itis not yet the case that all agents know all secrets. Then there is an agent x who does notknow some secret py. That agent knows that when it calls agent y, it will learn py. So theknowledge condition that x knows that there is a pz that is learnt in the call xy, is fulfilled,namely for pz = py.

A crucial moment is when exactly the secret is chosen that the agent who initiatesthe call will learn. Consider the following ‘de re’ variation of the Known InformationGrowth protocol—that for the purpose of making it different should be considered de ‘dedicto’ Known Information Growth. A similar variation exists for the Expected InformationGrowth protocol. ‘De re’ and ‘De dicto’ knowledge are considered in [JvdH04, vB01,AvD11].

Protocol 6 (Known Information Growth (de re)): Until all agents know all secrets: choosean agent x, let x choose an agent y and choose a secret pz such that x knows that pz islearnt in the call xy, and let x call y.

Consider the sequenceab; cd; ac; ab

and let us assume that the system is synchronous: calls are made at regular intervals.Clearly, this is not a possible execution of the Learn New Secrets protocol, because twoagents will never call each other twice in that protocol. But it is a possible execution ofthe Known Information Growth (de re) protocol. Consider the point of view of agent b,Bharat. After the initial call ab, Bharat is not involved in the two subsequent calls (as callsare at regular intervals, he knows that he missed two calls), and is then called again. Thetwo intervening calls must therefore have been between a, c and d. The following historiesare then possible—here, we only take into account the informational consequences andidentify calls xy with yx:

ab; ac; ad; ab (i)ab; ac; cd; ab (ii)ab; ad; ac; ab (iii)ab; ad; cd; ab (iv)ab; cd; ac; ab (v)ab; cd; ad; ab (vi)

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In all of those cases b will learn new secrets from agent a in the fourth call: namely C,Din (i), C in (ii), C,D in (iii), D in (iv), C,D in (v), and C,D in (vi). Therefore, Bharatknows that there is a secret (namely C or D) that he will learn by calling Amal again.Therefore, the Known Information Growth (de dicto) protocol applies. However, Bharatdoes not know that he will learn secret C from calling Amal and Bharat also does not knowthat he will learn secret D from calling Amal. So, it is not the case that there is a secretsuch that Bharat knows that he will learn that secret from calling Amal. Therefore, theKnown Information Growth (de re) protocol does not apply . . . to Bharat. But, it appliesto Amal initiating a call at that point and making the call to Bharat. Because, for Amalonly the actual call sequence is considered possible, given the sequence ab; cd; ac; ab (in thethird call ac, Amal learns that the second call was cd). Therefore, there is a secret thatAmal knows that Bharat will learn from the fourth call, namely C (or D).

Many more variations are possible, e.g., (i) knowledge-based gossip protocols with un-certainty over the number of calls that have taken place (asynchronous systems), such thatin sequence ab; cd; ac; ab Bharat learns that two intervening calls must have taken placewhen he is called again; (ii) knowledge-based gossip protocols for rounds of parallel calls,wherein, for five agents a, b, c, d, e, Ekram learns that two calls must be taking place in agiven round when he finds every other agent engaged (so he is unable to distinguish rounds{ab, cd}, {ac, bd}, {ad, bc}), and (iii) knowledge-and-history-based gossip protocols, e.g.,when Amal, after she has called someone, is not allowed to initiate the next call (this isnot a knowledge condition).

Variations such as between rounds of calls and sequential calls are standard in the gossipprotocol community, where we based ourselves on the combinatorial survey [HHL88]. Theabsence of a global scheduler is common in this community (related to network theory andsignal processing), where the method to make up for this absence is to assume randomscheduling instead [KDG03, BGPS06]. We do not know of approaches where one makesup for the absence of a scheduler by using the knowledge of the processes and let themschedule instead, i.e., knowledge-based protocols, that are based on information available inindividual network nodes. It would be interesting to find out how the speed of disseminationin knowledge-based scheduling compares to that in random scheduling. It may well be thatknowledge-based protocols are only computationally feasible in small networks. The costof determining whom to call, and only then to make a call, may exceed that of just callinganyone such that eventually you learn something—doesn’t this sound like real life!

2 Knowledge and Gossip

What sort of knowledge do the agents obtain in these protocols? This becomes interestingif we do not only consider what agents know about the secrets but also consider whatthey know about each other. We systematically overview knowledge in the initial state ofinformation (wherein every agent only knows its own secret), the change of knowledge due

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to a call between two agents, and the knowledge conditions after termination of a protocolconsisting of such calls.

We will informally use ‘agent x knows something’ (Kxϕ), ‘the agents in B have generalknowledge of ϕ’ (EBϕ), and ‘the agents in B have common knowledge of ϕ’ (CBϕ). Butnot imprecisely! We will merely use the English paraphrases. The formal language, theKripke models, and the semantics of calls will only be defined in the next section.

Initial state of information We can represent the uncertainty of the agents abouttheir secrets in a multi-agent Kripke model. In that case we consider Amal’s secret A asa proposition A of which the value is initially only known by Amal (a). For four agents anice depiction would be already a four-dimensional model, so let us depict the one for threeagents. Below, a node like 011 stands for ‘A is false and B is true and C is true’ (in theorder a, b, c the digits 0 and 1 stand for the value of the propositions A,B,C, respectively).

000

001

010

011

100

101

110

111

bc

bc

bc

ac

ac ac

ab

ab

ab

abbc

ac

This is a most standard kind of situation. The secret of an agent is its local state,and every agent only knows its local state (and this is common knowledge). (So, it is aninterpreted system [FHMV95].) For example, in state 011 we have that Amal knows thatA is false (because A is false in 011 and 111, the two states considered possible by a), andthat Bharat knows that B is true and that Chandra knows that C is true.

The distributions of secrets over agents that we already considered in the previoussection correspond in a precise way to such a Kripke model. We represent the distributionof secrets over agents as a list (or, if you want, as a function from agents to subsets ofthe set of all secrets), as we already have done in the previous section. The one above issuccinctly represented by A.B.C. The situation AB.AB.C is represented by

000

001

010

011

100

101

110

111

c

c

c

c

c c

ab

ab

ab

abc

c

11

As we are only interested in whether agents know secrets, and as any agent knows thesame number of secrets in any state of such a model, it is sufficient to stick to this modelperspective in the logic. For example ‘Amal knows whether A and whether B’ is a modelvalidity of the above model. A list like A.B.C and AB.AB.C we call a gossip state. In agossip state, the agents have common knowledge of the distribution of secrets, i.e., eachagent knows for all agents how many secrets those agents know, and it knows its ownsecrets.

Executing a phone call Let us now execute a telephone call in this setting. We getfrom A.B.C to AB.AB.C by executing the call ab. What sort of dynamics is a telephonecall? A telephone call is a very different form of communication than an announcement inthe presence of other agents. An announcement is public. This means that, after Amal says‘The old name of Chennai is Madras’ in the presence of Bharat and Chandra, then Bharatknows that the old name of Chennai is Madras, but Chandra also knows that Bharatknows that, and Amal knows that Chandra knows that Bharat knows that, and so on.The information that the old name of Chennai is Madras, is common knowledge betweenthe three agents. But if first Amal calls Bharat to tell him that, and then Bharat callsChandra, all three know that the old name of Chennai is Madras, but it is not commonknowledge. It is even impossible that this becomes common knowledge if nothing is knownabout the timing of the phone calls.

From the left-most information transition below it is clear that a telephone call is notpublic announcement. No worlds are eliminated and no links are completely cut. (Wecould still conceive a public announcement of ‘the truth about A’ as a non-deterministicannouncement !A∪!¬A, resulting in cutting all bc labelled links of the initial cube.) Insteadof the mere transition for the call ab we list those for the sequence ab; ac; bc.

000

001

010

011

100

101

110

111

bc

bc

bc

ac

ac ac

ab

ab

ab

abbc

ac

ab→ 000

001

010

011

100

101

110

111

c

c

c

c

c c

ab

ab

ab

abc

c

ac→

12

000

001

010

011

100

101

110

111

b

b

b

b

bc→ 000

001

010

011

100

101

110

111

The corresponding transitions between the gossip states (the list of who knows what secrets)are as follows.1

A.B.Cab→ AB.AB.C

ac→ ABC.AB.ABCbc→ ABC.ABC.ABC

Now here is an obvious but surprising observation. Having first explained that calls do notcreate common knowledge of the secrets, after all, at the end, there is common knowledgethat all three agents know all the secrets. We can understand this as follows: the agentshave common knowledge what protocol is being carried out. In this case, this could be aFixed Schedule protocol, but also an execution sequence of the Learn New Secrets protocol.On the assumption of synchronization, if there are three agents and the second call is ac,then Bharat knows that the call ac—or ca—is taking place, because that is the only callthat he is not involved in. The agents know that after three steps all agents know allsecrets. In each step there is some change in common knowledge, that finally results incommon knowledge of all secrets.

For any Fixed Schedule Protocol this remains the case for more than four agents, underconditions of synchronicity. We then assume that there is common knowledge which callsequence is executed, and with what time interval between calls. We could imagine theagents sitting around a table and making the calls from there, in view of each other, butwhispering, so that any other person only notices that a call is made, but not what is said.

Executing an epistemic protocol Now consider four agents, and a call between Amaland Bharat such that Chandra and Devi consider any other call possible that does not

involve them. Although the real transition is A.B.C.Dab→ AB.AB.C.D, Chandra considers

it possible that the transition was A.B.C.Dab→ AD.B.C.AD, or A.B.C.D

ab→ A.BD.C.BD.As we model the information change resulting from a call—which secrets are learnt—weabstract from who initiates the call and who receives it, so ab and ba are treated on a par.We now get the transition resulting in the following model.

1We will later define a restricted logical language which justifies to use only that succinct representation,there is no need for the more complex representation.

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A.B.C.Dab→

AB.AB.C.D

A.BD.C.BD AC.B.AC.D

AD.B.C.AD A.BC.BC.D

A.B.CD.CD

dc d

c

bd

b

c

b

a

a

a

We call this a gossip model. In fact, the unit of interpretation is the combination ofthe gossip model and a designated gossip state: what really happened, namely the resultAB.AB.C.D. Such a gossip model also represents a multi-agent Kripke model, namelyone wherein we replace each gossip state by the corresponding Kripke model, take theirdisjoint union, and add accessibility links appropriately (two worlds in gossip states areindistinguishable for an agent, even across gossip states, if it knows the same secrets inboth).

Postconditions of protocol execution Clearly the agents have general knowledge ofall secrets after the execution of such a protocol, i.e., every agent x knows the valueof all secrets A,B, . . .—wherein we consider such secrets as propositional variables withvalue true or false. Do they know more than that? This depends somewhat on furtherassumptions. If they do not know the protocol, not much more can be said. If we supposethat a Fixed Schedule protocol is known to the agents, the last agents to call, now knowthat everybody knows all secrets. This is already a bit stronger. Under such conditions wecan also obtain higher order knowledge: let subsequently to the last call, an agent involvedin that call, call all other agents. For example, following ae; af ; ab; cd; ac; bd; ae; af let anow call everybody else once more, in five calls ab, ac, ad, ae, af . We have now obtainedgeneral knowledge of general knowledge of all secrets. (And a and f have general knowledgeof that.) And so on.

So, instead of 2n − 4 calls to achieve general knowledge of the secrets, we need 2n −4 + (n− 1) = 3n− 5 calls to achieve general knowledge of general knowledge of the secrets.And so on... Still, it remains out of reach to make this common knowledge.

Let us additionally assume that calls take place at regular intervals, like one every tenminutes, and that this is also known to the agents. In other words, we assume synchro-nization. Then the secrets are common knowledge after termination of a Fixed Scheduleprotocol! For example, for six agents, this will take only one hour and forty minutes.

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But for epistemic protocols it is harder to achieve common knowledge. Take the LearnNew Secrets protocol, and four agents. We have seen that the executions consists of betweenfour and six calls (between 2n− 4 and n · (n− 1)/2). What if general knowledge is alreadyobtained after four calls? The two agents not involved in that call do not necessarily knowthat. In this case, ‘stuffing’ the protocol with a number of a skip actions will still achievecommon knowledge, after six time steps: one hour.

3 Outline of Gossip Logic

Let a finite set of n agents (agents) A = {a, b, . . . } and a corresponding set of secrets(propositional variables) P = {A,B, . . . } be given (i.e.: A is the secret of a, B of b, etc.).We let x, y, z, . . . be variables over A, and px, py, pz . . . the corresponding variables for theirsecrets (where px is the secret of x, etc.).

Definition 1 (Language): The language L is defined as

L 3 ϕ ::= Kwxpy | ¬ϕ | (ϕ ∧ ϕ) | Kwxϕ | [π]ϕπ ::= ?ϕ | xy | (π ; π) | (π ∪ π) | π∗

where x, y ∈ A, x 6= y, and py ∈ P.

Disjunction and implication are defined as usual. We read Kwxϕ as ‘agent x knows whetherϕ’. ‘Agent x knows ϕ’ is defined by abbreviation as Kxϕ ::= ϕ ∧ Kwxϕ, and the dual ofKxϕ is written Kxϕ. (As we also have that Kwxϕ is definable as Kxϕ ∨ Kx¬ϕ, this ismerely basic modal logic but with atomic formulas Kwxpy instead of py.) For any Q ⊆ P,KwxQ ::=

∧py∈Q Kwxpx, which means that agent x knows the value of all secrets in Q.

Formula [xy]ϕ stands for ‘after a call from agent x to y, ϕ (is true)’. For (?¬ϕ ; π)∗ ; ?ϕwe may write ‘until ϕ do π’. For (?ϕ ; π) ∪ (?¬ϕ ; π′) we may write ‘if ϕ then π else π′’.Epistemic protocols will be defined as such programs π but with additional constraints.Informally, a protocol is a program that intends to get all agents to know all secrets.

The extension of a program π is the set of its execution sequences of calls.

Definition 2 (Extension of a program): Defined by induction of the structure of programs.Σ(xy) = {xy}, Σ(?ϕ) = ∅, Σ(π ; π′) = {σ ; σ′ | σ ∈ Σ(π) and σ′ ∈ Σ(π′)}, Σ(π ∪ π′) =Σ(π) ∪ Σ(π′), Σ(π∗) = {σ∗ | σ ∈ Σ(π)}.

A gossip state assigns a subset of all propositional variables to each agent. But not inany way, it has to be done properly. We can also see a gossip state as a partition on theset {0, 1}P of valuations of propositional variables, which defines a Kripke model.

Definition 3 (Gossip state and gossip model): We define gossip states by induction. Thefunction g : A → P with g(x) = px is a gossip state. Let g be a gossip state and x, y ∈A. Consider h : A → P defined as: h(x) = h(y) = g(x) ∪ g(y), and for all z 6= x, y:h(z) = g(z). Then h is a gossip state. We also write g{x,y} for h. For each agent, knowing

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the same number of secrets induces an equivalence relation on the set of gossip states: forg(x) = g′(x) we write g ≈x g

′. A gossip model is a pair G = (S,≈), where S is a set ofgossip states and for each agent x, ≈x is such an equivalence relation. A pointed gossipmodel is a pair (G, g), where g ∈ S.

The initial gossip state is the gossip state g with g(x) = px for all agents. The initialgossip model is the singleton gossip model (({g},≈), g) for the initial gossip state g, where≈x = {(g, g)} for all agents.

Given a set of agents and propositional variables, a multi-agent Kripke model is atriple (S,R, V ) (with R a function assigning to each agent a binary relation on S, and Va function assigning to each state in S a valuation of propositional variables; for R(x) wewrite Rx and for V (p), for such a propositional variable, we write Vp). If all accessibilityrelations Rx are equivalence relations ∼x we call this an epistemic model. A gossip stateand a gossip model correspond to an epistemic model for the set of agents A and the setof propositional variables P. This is how to construct these epistemic models.

Every gossip state g : A → P(P) induces an epistemic model M(g) = (S,∼, V ) suchthat S = {0, 1}P, where for every x ∈ A, s ∼x t iff for all py ∈ g(x), s(py) = t(py), andwhere valuation V : S → P → {0, 1} is such that Vs = s. (As the domain consists of theset of all valuations it is superfluous to list the valuation function separately, as we do.)

Every gossip model G = (S,≈) induces a multi-pointed epistemic model M(G) =(S ′,∼′, V ′) such that: S ′ = {sg | g ∈ S and s ∈ M(g)}; sg ∼′x th iff g ≈x h and, in g,s ∼x t; and V ′sg = s (observe that s and t are valuations and that the domain of anygossip state consists of all valuations). To a pointed gossip model (G, g) corresponds amulti-pointed epistemic model (M(G),M(g)).

Definition 4 (Semantics on gossip models): Let G = (S,≈) be a gossip model. We induc-tively define the interpretation of a formula ϕ ∈ L on a gossip state g ∈ S.

G, g |= Kwxpy iff py ∈ g(x)G, g |= ¬ϕ iff G, g 6|= ϕG, g |= ϕ ∧ ψ iff G, g |= ϕ and G, g |= ψG, g |= Kwxϕ iff for every h ≈x g : G, h |= ϕ iff G, g |= ϕG, g |= [π]ψ iff for all (G′, g′) such that (G, g)[[π]](G′, g′) : G′, g′ |= ψ

where[[xy]] = {((G, g), (Gcall, g{x,y}))}[[?ϕ]] = {((G, g), (G, g))} whenever G, g |= ϕ[[π; π′]] = [[π]] · [[π′]][[π ∪ π′]] = [[π]] ∪ [[π′]][[π∗]] = [[π]]∗

and where Gcall = (S ′,≈′) such that S ′ = {g{z,w} | g ∈ S and z 6= w ∈ A}, and for anyg′, h′ ∈ S: g′ ≈z h

′ iff g′(z) = h′(z).

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The advantage of a logical language wherein Kwxpy is an atomic proposition instead of theusual propositional variable py, now appears from the semantics: this less expressive logicallows us to have gossip models as primitive semantic objects instead of (more complex)epistemic models. Anything we do in gossip models can still be done at the level of epistemicmodels (but not vice versa, for example, we cannot evaluate py in a gossip model).

Instead of the semantics of Kwxpy as given, we could alternatively have defined it moreclosely resembling the usual semantics of ‘knowing whether’, on epistemic models. LetM(g) = (S,∼, V ), then

M(G),M(g) |= Kwxpy iff for all s ∈ S, for all t ∼x s : Vs(py) = Vt(py).

Instead of the semantics of Kwxϕ as given, we could alternatively have defined knowingthat ϕ, or have stipulated Kxϕ as a primitive in the language. They are interdefinable:Kwxϕ↔ (Kxϕ∨Kx¬ϕ), and Kxϕ↔ (ϕ∧Kwxϕ). With Kwxpy as an atomic propositionit seemed more elegant to have Kwxϕ as a primitive construct than Kxϕ. For Kxϕ we get:

G, g |= Kxϕ iff for every h ≈x g : G, h |= ϕ.

When two agents call each other in a given gossip state, then instead of taking the unionof the secrets they know, we can alternatively take the intersection of the equivalencerelations for those agents in the induced epistemic models: given a gossip state g withinduced epistemic model M(g) = (S,∼, V ), we have that M(g{x,y}) = (S,∼′, V ) with∼′x = ∼′y = ∼x ∩ ∼y and for all z 6= x, y, ∼′z = ∼z.

The informative effect of x calling y is the same as the informative effect of y callingx. In other words, [[xy]] = [[yx]]. For that reason we write g{x,y} and not gxy for the gossipstate resulting from a call between x and y in g.

Although a gossip state induces an epistemic model consisting of different states (worlds),in our logic we never need to evaluate formulas in a given state. This is because the basicformulas are expressions Kwxpy that are either valid on such an epistemic model, or theirnegation is valid.

We now define epistemic gossip protocols as programs π ∈ L satisfying some additionalconstraints. If a formula ψ contains a subformula Kwxϕ or Kw ypx we say that x occurs inψ. We write ψ(x) to denote all (possibly 0) occurrences of x in ψ.

Definition 5 (Epistemic gossip protocol): An epistemic gossip protocol is program π ofform

until∧x∈A

KwxP do⋃

x,y∈A

(?Kxψ(y) ; xy),

where ψ(y) ∈ L.

An assumption that the agents know the protocol needs to be hardwired in the semanticsof a call, by restricting the construction of the gossip model Gcall resulting from a call xy.Instead of the Gcall = (S ′,≈′) with S ′ = {g{z,w} | g ∈ S and z 6= w ∈ A} we then need aS ′ = {g{z,w} | g ∈ S and zw ∈ π(g)}, where π(g) is the set of calls possible in g accordingto π (defined in the obvious way). Details are omitted.

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These are some epistemic gossip protocols, as they have already featured informally inprior sections. In their description we implicitly use the validities ¬Kwxϕ ↔ Kx¬Kwxϕand Kxϕ↔ KxKxϕ: these are really knowledge conditions. In all protocols below there isa single occurrence of the agent variable y in the knowledge condition ?Kxψ(y).

Definition 6 (Knowledge-based Protocols in L): Below, ∇ denotes exclusive disjunction.

• until∧

x∈A KwxP do⋃

x,y∈A(?¬Kwxpy ; xy)Learn New Secrets (Protocol 3)

• until∧

x∈A KwxP do⋃

x,y∈A(?Kx

∨pz∈P(Kwxpz ∇ Kw ypz) ; xy)

Expected Information Growth (Protocol 4)

• until∧

x∈A KwxP do⋃

x,y∈A(?Kx

∨pz∈P(Kwxpz ∇ Kw ypz) ; xy)

Known Information Growth (de dicto) (Protocol 5)

• until∧

x∈A KwxP do⋃

x,y,z∈A(?Kx(Kwxpz ∇ Kw ypz) ; xy)Known Information Growth (de re) (Protocol 6)

Proposition 7 (Protocol 3 ⊂ Protocol 5 ⊂ Protocol 4): Every execution (call sequence) ofthe Learn New Secrets protocol (Protocol 3) is an execution of the Known InformationGrowth protocol (Protocol 5), and every execution of the Known Information Growth pro-tocol (Protocol 5) is an execution of the Expected Information Growth protocol (Protocol4). The inclusion is strict.

We recall that the Expected Information Growth protocol (Protocol 4) has executions ofinfinite length.

Proposition 8 (Protocol 6 ⊂ Protocol 5): Every execution (call sequence) of the (de dicto)Known Information Growth protocol (Protocol 5) is an execution of the de re Known In-formation Growth protocol (Protocol 6).

Not surprisingly, in this logic of knowledge and change of knowledge, the dynamicsof calls in gossip logic can be described in other, more general, dynamic epistemic logics.Gossip calls have been modelled in [vD00, Section 6.6] in a dialect of dynamic epistemiclogic, they have been modelled in a logic of messages proposed in [Sie12], and a modellingin action model logic is found in [AvDGvdH14].

We recall the difference between the extension Σ(π) of a protocol according to Definition1, and the meaning [[π]] of a protocol, according to Definition 4. The extension is the numberof its executions, the number of branches of the execution tree. The meaning is the numberof different information transitions, i.e., the number of branches of the tree induced by theinitial gossip model and π.2 For example, call sequences ab; ac; bc and ab; ac; ab are different

2In other words, the meaning is the set of pointed gossip model sequences, where the points are theactual gossip states. Note that this is different from a set of gossip state sequences. Different pointedgossip model sequences may have the same induced gossip state sequence; i.e., if you just look at theirpoints, they are the same.

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in the extension, but have the same meaning: in both cases they induce the transitionsA.B.C → AB.AB.C → ABC.AB.ABC → ABC.ABC.ABC.

For example, for the Known Information Growth protocol, for three agents, the ex-tension consists of 96 different call sequences, and the meaning consists of 6 differenttransformation chains of gossip models. For the Learn New Secrets protocol, these figuresfor three agents are 24, and 6. The extension of a protocol allows to determine its averageexecution length (as a function of the number of agents), and that would allow us to com-pare different epistemic gossip protocols. Surely, we are interested in a minimum expectedexecution length.

It is interesting for all agents to know all secrets. But for a given agent, it may evenbe more interesting to be the first to know all secrets. Epistemic gossip protocols may alsoplay a role in ‘gossip games’, say, in a setting similar to some of Rohit Parikh’s work, suchas [PTW13, CPP04].

References

[AvD11] T. Agotnes and H. van Ditmarsch. What will they say? - Public announce-ment games. Synthese, 179(S.1):57–85, 2011.

[AvDGvdH14] M. Attamah, H. van Ditmarsch, D. Grossi, and W. van der Hoek. Knowl-edge and gossip. In Proc. of 21st ECAI, pages 21–26. IOS Press, 2014.

[BGPS06] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah. Randomized gossip algo-rithms. IEEE/ACM Trans. Netw., 14(SI):2508–2530, 2006.

[CPP04] S. Chopra, E. Pacuit, and R. Parikh. Knowledge-theoretic properties ofstrategic voting. In Proc. of 9th JELIA, LNCS 3229, pages 18–30, 2004.

[FHMV95] R. Fagin, J.Y. Halpern, Y. Moses, and M.Y. Vardi. Reasoning about Knowl-edge. MIT Press, Cambridge MA, 1995.

[HHL88] S.M. Hedetniemi, S.T. Hedetniemi, and A.L. Liestman. A survey of gossip-ing and broadcasting in communication networks. Networks, 18:319–349,1988.

[Hur00] C.A.J. Hurkens. Spreading gossip efficiently. Nieuw Archief voor Wiskunde,5/1(2):208–210, 2000.

[JvdH04] W. Jamroga and W. van der Hoek. Agents that know how to play. Funda-menta Informaticae, 63:185–219, 2004.

[KDG03] D. Kempe, A. Dobra, and J. Gehrke. Gossip-based computation of aggre-gate information. In Proceedings of the 44th FOCS, pages 482–491. IEEEComputer Society, 2003.

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[Kno75] W. Knodel. New gossips and telephones. Discrete Mathematics, 13:95,1975.

[PK92] R. Parikh and P. Krasucki. Levels of knowledge in distributed systems.Sadhana, 17(1):167–191, 1992.

[PR85] R. Parikh and R. Ramanujam. Distributed processing and the logic ofknowledge. In Logic of Programs, LNCS 193, pages 256–268. Springer,1985. Similar to JoLLI 12: 453–467, 2003.

[PR03] R. Parikh and R. Ramanujam. A knowledge based semantics of messages.Journal of Logic, Language and Information, 12:453–467, 2003.

[PTW13] R. Parikh, C. Tasdemir, and A. Witzel. The power of knowledge in games.IGTR, 15(4), 2013.

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