Two Generals' Problem

In computing, the Two Generals' Problem is a thought experiment meant to illustrate the pitfalls and design challenges of attempting to coordinate an action by communicating over an unreliable link. It is related to the more general Byzantine Generals' Problem (though published long before that later generalization) and appears often in introductory classes about computer networking (particularly with regard to the Transmission Control Protocol), though it can also apply to other types of communication. A key concept in epistemic logic, this problem highlights the importance of common knowledge. Some authors also refer to this as the Two Armies Problem or the Coordinated Attack Problem.^[1][2]

Definition

Two armies, each led by a general, are preparing to attack a fortified city. The armies are encamped near the city, each on its own hill. A valley separates the two hills, and the only way for the two generals to communicate is by sending messengers through the valley. Unfortunately, the valley is occupied by the city's defenders and there's a chance that any given messenger sent through the valley will be captured.

Positions of the armies. Armies A1 and A2 need to communicate but their messengers may be captured by army B.

While the two generals have agreed that they will attack, they haven't agreed upon a time for attack. It is required that the two generals have their armies attack the city at the same time in order to succeed, else the lone attacker army will die trying. They must thus communicate with each other to decide on a time to attack and to agree to attack at that time, and each general must know that the other general knows that they have agreed to the attack plan. Because acknowledgement of message receipt can be lost as easily as the original message, a potentially infinite series of messages are required to come to consensus.

The thought experiment involves considering how they might go about coming to consensus. In its simplest form one general is known to be the leader, decides on the time of attack, and must communicate this time to the other general. The problem is to come up with algorithms that the generals can use, including sending messages and processing received messages, that can allow them to correctly conclude:

Yes, we will both attack at the agreed-upon time.

Allowing that it is quite simple for the generals to come to an agreement on the time to attack (i.e. one successful message with a successful acknowledgement), the subtlety of the Two Generals' Problem is in the impossibility of designing algorithms for the generals to use to safely agree to the above statement.

Illustrating the problem

The first general may start by sending a message "Attack at 0900 on August 4." However, once dispatched, the first general has no idea whether or not the messenger got through. This uncertainty may lead the first general to hesitate to attack due to the risk of being the sole attacker.

To be sure, the second general may send a confirmation back to the first: "I received your message and will attack at 0900 on August 4." However, the messenger carrying the confirmation could face capture and the second general may hesitate, knowing that the first might hold back without the confirmation.

Further confirmations may seem like a solution - let the first general send a second confirmation: "I received your confirmation of the planned attack at 0900 on August 4." However, this new messenger from the first general is liable to be captured too. Thus it quickly becomes evident that no matter how many rounds of confirmation are made, there is no way to guarantee the second requirement that each general be sure the other has agreed to the attack plan. Whichever general sends the final messenger will always be left wondering whether the messenger got through.

Proof

For deterministic protocols with a fixed number of messages

Suppose there is any fixed-length sequence of messages, some successfully delivered and some not, that suffice to meet the requirement of shared certainty for both generals to attack. In that case there must be some minimal non-empty subset of the successfully delivered messages that suffices (at least one message with the time/plan must be delivered). Consider the last such message that was successfully delivered in such a minimal sequence. If that last message had not been successfully delivered then the requirement wouldn't have been met, and one general at least (presumably the receiver) would decide not to attack. From the viewpoint of the sender of that last message, however, the sequence of messages sent and delivered is exactly the same as it would have been, had that message been delivered. Therefore the general sending that last message will still decide to attack (since the protocol is deterministic). We've now constructed a circumstance where the purported protocol leads one general to attack and the other not to attack - contradicting the assumption that the protocol was a solution to the problem.

For nondeterministic and variable-length protocols

Such a protocol can be modeled as a labeled finite forest, where each node represents a run of the protocol up to a specified point. The roots are labeled with the possible starting messages, and the children of a node N are labeled with the possible next messages after N. Leaf nodes represent runs in which the protocol terminates after sending the message the node is labeled with. The empty forest represents the protocol that terminates before sending any message.

Let P be a protocol that solves the Two Generals' problem. Then, by a similar argument to the one used for fixed-length protocols above, P' must also solve the Two Generals' problem, where P' is obtained from P by removing all leaf nodes. Since P is finite, it follows that the protocol represented by the empty forest solves the Two Generals' problem. But clearly it does not, contradicting the existence of P.

Engineering approaches

A pragmatic approach to dealing with the Two Generals' Problem is to use schemes that accept the uncertainty of the communications channel and not attempt to eliminate it, but rather mitigate it to an acceptable degree. For example, the first general could send 100 messengers, anticipating that the probability of all being captured is low. With this approach the first general will attack no matter what, and the second general will attack if any message is received. Alternatively the first general could send a stream of messages and the second general could send acknowledgments to each, with each general feeling more comfortable with every message received. As seen in the proof, however, neither can be certain that the attack will be coordinated. There's no algorithm that they can use (e.g. attack if more than four messages are received) which will be certain to prevent one from attacking without the other. Also, the first general can send a marking on each message saying it is message 1, 2, 3 ... of n. This method will allow the second general to know how reliable the channel is and send an appropriate number of messages back to ensure a high probability of at least one message being received. If the channel can be made to be reliable, then one message will suffice and additional messages do not help. The last is as likely to get lost as the first.

Assuming that the generals must sacrifice lives every time a messenger is sent and intercepted, an algorithm can be designed to minimize the number of messengers required to achieve the maximum amount of confidence the attack is coordinated. To save them from sacrificing hundreds of lives to achieve a very high confidence in coordination, the generals could agree to use the absence of messengers as an indication that the general who began the transaction has received at least one confirmation, and has promised to attack. Suppose it takes a messenger 1 minute to cross the danger zone, allowing 200 minutes of silence to occur after confirmations have been received will allow us to achieve extremely high confidence while not sacrificing messenger lives. In this case messengers are used only in the case where a party has not received the attack time. At the end of 200 minutes, each general can reason: "I have not received an additional message for 200 minutes; either 200 messengers failed to cross the danger zone, or it means the other general has confirmed and committed to the attack and has faith I will too".

History

The Two Generals' Problem and its impossibility proof was first published by E. A. Akkoyunlu, K. Ekanadham, and R. V. Huber in 1975 in "Some Constraints and Trade-offs in the Design of Network Communications",^[3] where it is described starting on page 73 in the context of communication between two groups of gangsters.

This problem was given the name the Two Generals Paradox by Jim Gray^[4] in 1978 in "Notes on Data Base Operating Systems"^[5] starting on page 465. This reference is widely given as a source for the definition of the problem and the impossibility proof, though both were published previously as above.

References