Ámbito familiar

In this chapter, we proposed epistemic gossip protocols, where an agent will call another agent based on its current knowledge. We described various such protocols, we gave some of their logical properties, and we modelled them in dynamic epistemic logic.

We considered the case where only one pairwise call is staged in a given round of gossiping. There are other alternatives however, namely,parallel callsin which more than one pairwise call can be staged simultaneously. One strategy for allowing the execution of parallel calls in an epistemic gossip protocols could be to allow each of all the agents to try to make a call with one of the agents with whom the calling condition is satisfied for the given protocol and for the given network topology. Another strategy for making parallel calls could be throughk-party calls, in which an agent makes a conference call

with all the agents with whom it can call in a given situation. For example, if the calling condition is satisfied for an agent a to call agent b and agent c, then we can set up a conference call among agents a, band c in which all three agents exchange their secrets among themselves.

Furthermore, in this thesis we assumed that in each pairwise call the calling pair exchange all the secrets they know. This assumption can be relaxed so that one can also consider one-way calls, that is, calls in which only one of the callers sends its secrets to the other calling partner. This case is analogous totext messaging or electronic mail

between the communicating pair. And to take this line still further, one can consider a case where one of the agents broadcasts its secrets to a group of agents rather than to one other agent, similar to the popular manner in which information in shared in social networks among a group of friends. Calls such as one-way and broadcast calls can be used to describe the spread of a disease, a news item, or a commercial within a population.

Epistemic gossip protocols that are based on parallel calls, one-way calls, broadcast calls, or any combination of various such types of calls could also be described and analysed.

Another interesting line of future work is to consider strategic issues. Suppose the agents are allowed to choose from a set of protocols, or from a set of possible calls due to a protocol, can an agent ensure, for example, that it is the first to know all secrets, or, for that matter not the last?

A Framework for Epistemic Gossip

Protocols

4.1

Introduction

In this chapter we present Epistemic Gossip Protocol (EGP), which is a tool to anal- yse epistemic gossip protocols. Particularly we introduce Epistemic Gossip Protocol Language (EGPL), which is a high-level programming language for epistemic gossip pro- tocols. Then, we describe the details of an interpreter for the EGPL. The tool EGP outputs key dynamic properties of an epistemic gossip protocol. In the next chapter we apply this tool to the epistemic gossip protocols introduced in Chapter 3, and then, we present empirical results.

The initial setting of the gossip scenarios we consider is as follows. There are a finite number of agents, and each agent knows a unique piece of information called a secret. Only pairwise communications between the agents are allowed. These communications are known ascalls, and only one call is allowed in a round. In each call, the calling pair exchange all the secrets they know. The goal of such communications is to reach a state where all the agents know all the secrets in the scenario.

Each epistemic gossip protocol can be considered as a rule with some epistemic con- dition which has to be satisfied for one agent to call another agent. We call such rule a

calling condition∗. In each round, a pair of agents is chosen non-deterministically from the set of pairs for which the calling condition is satisfied, and allowed to make a call. This call can be made in one of several modes. For example, the calling pair can make the call publicly such that every other agent knows who is calling who in any round. This mode is referred to as thepublic synchronous mode (while there is no uncertainty

∗

Note that a calling condition is not limited only to an epistemic (calling) condition (epistemic calling condition is defined in Section 3.6), but can also include conditions about the underlying communication graph structure. For example, a pair of agents cannot engage each other in a pairwise call if they are not neighbours on the underlying network graph. Furthermore, the reader should assume a complete

communication graph wheresoever in this thesis we have not made explicit the structure of the underlying communication graph. Note that our discussions so far in this thesis have assumed a complete graph as the underlying network graph, and as such the calling condition for a protocol have so far been tantamount to an epistemic calling condition.

about who is calling whom in each round, the contents of the calls, namely the secrets, are not observed). Another mode is that in which the calls are made in private such that, apart from the pair involved in the call, the other agents may not be sure which pair of agents is making the call, but all the agents are sure a call is made in each round. This mode is referred to as the private synchronous mode. The private asynchronous mode is like the private synchronous mode except that the agents consider it possible that no call is made in a round even though there is some pair for which the calling condition is satisfied.

In this chapter, we assume that the protocols are based on the private synchronous call mode. Therefore the agents would have to reason about possible situations which are due to all the possible calls in the previous round. For example: at the initial situation of a gossip scenario comprising of four agents, no other situation is considered possible. But after one round of calls, there could be up to twelve new and different possible situations due to possible calls at the initial situation. Note that we distinguish between the call aiaj and the call ajai, where aiaj denotes the call from agent ai to another agent, aj. Hence after a maximal series of rounds we can think of a tree structure in which each path is an execution sequence of calls of a given protocol. We refer to this tree as the

call tree or gossip tree of the given protocol, and the set of all the paths of this tree is theextension of the protocol.

The gossip tree offers a platform to protocol designers for the evaluation and compari- son of epistemic gossip protocols. In the gossip protocol literature it is typical to measure the performance of a protocol by considering the length of its execution sequence, that is, the number of calls in the execution sequence [30]. Correspondingly we measure the performance of epistemic gossip protocols by considering the average length of the exe- cution sequences in the protocol’s extension, together with the size of the extension of the given protocol. Whereas the size of the extension gives an idea of the computational memory required by an agent to reason about possible situations, the average execution length gives an idea of how fast it will take for all agents to know all secrets under the given protocol. We also make use of the following definitions.

Definition 4.1 (Initial State and Goal State). Given a setAg={a1, . . . , an}of agents and a setP ={A1, . . . , An} of secrets. Let secret Ai be the unique secret of ai, and let

Si be the set of secrets known byai whereai∈Ag, and where initiallySi ={Ai}. Then, agossip situation is an-tuplehS1, . . . , Sni, the initial stateish{Ai}, . . . ,{An}i, and the

goal state ishP, . . . , Pi.

Definition 4.2 (Terminating and Non-terminating Execution Sequences). An execu- tion sequence is terminating if and only if it is finite. An execution sequence is non- terminating if and only if it is infinite.

Definition 4.3 (Terminating and Non-terminating Protocol). An epistemic gossip pro- tocol is terminating if all its execution sequences are finite. Otherwise the protocol is

The following protocols were described in Chapter 3, reproduced informally here. 1. Learn New Secrets. An agent ai can call another agent aj if ai does not know

the secret of aj.

2. Known Information Growth de Dicto. An agent ai can call another agent aj if ai knows that there issome secret Ak that would be learnt in the callaiaj. 3. Known Information Growth de Re. An agent ai can call another agentaj if

there issome secret Aksuch that ai knows that it would be learnt in the callaiaj. 4. Possible Information Growth de Dicto. An agent ai can call another agent aj if ai considers it possible that there is some secret Ak that would be learnt in the call aiaj.

5. Possible Information Growth de Re. An agentai can call another agentaj if there issome secret Ak such thatai considers it possible that it would be learnt in the call aiaj.

Whereas Protocols 1, 2 and 3 are terminating, Protocols 4 and 5 are non-terminating. Take a scenario with four agents a, b, c, dand consider the following execution sequence of Protocol 4: ab;cd;ab;cd;. . .. After the first two calls, agent a considers it possible that agentb learnt some new secret in the second round, therefore acallsbin the third round, which turns out to be redundant. Likewise in the fourth round agentc considers it possible that agent d learnt some new secret in the third round, so c calls d in the fourth round. In this way the loopab;cd;. . . could go on infinitely. The same example works for Protocol 5.

In a given round of gossiping, an agent may be uncertain about the actual execution sequence that occurred. However the agent in question may consider some execution sequences possible based on the fact that it cannot distinguish between these execution sequences. Under Known Information Growth de Dicto and Known Information Growth de Re, for agent ai to call agent aj at a gossip situation, it is required that at every execution sequence that agentai considers possible at the given gossip situation, there is some secret that only one of agent ai and aj knows. As such, if the callaiaj is possible at the given gossip situation, then one of agentai andaj is sure to learn some new secret from the other.

The difference between the Known Information Growth and Possible Information Growth protocols is as follows. Whereas in Known Information Growth de Re agent ai is certain of the particular secret that will be learnt in such aiaj call, in Known Information Growth de Dictoai may remain uncertain about the particular secret that will be learnt in theaiaj call. For example, consider again the gossip scenario with four agents a, b, c, d. After the execution sequence ab;bc;cd, agent a is certain that it will learn some new secret by calling agent b in the fourth round (since agent b must have been involved in some call in either the second round or the third round), however, agent

ais uncertain about the secret it will learn from agentbin the fourth round (since agent bmay have called with either agentcor agentdin the second or third round). Therefore the epistemic calling condition for Known Information Growth de Dicto holds for agent ato call b in the fourth round, but not for Known Information Growth de Re. But, for the same scenario and for the same execution sequence, in the fourth round, agenta is certain that it will learn the secret of agent c in anac call. Thus the epistemic calling condition for Known Information Growth de Re is satisfied. For a broader discussion of the De Re / De Dicto distinction, see [34]. The difference between Possible Information Growth de Dicto and Possible Information Growth de Re protocols is obvious from the foregoing discussion.