(image unavailable: Two Generals' Problem diagram)

Positions of the armies. Armies A1 and A2 cannot see one another directly, so need to communicate
by messengers, but their messengers may be captured by army B.

In computing, the Two Generals' Problem (or Chinese Generals Problem ¹) is a thought
experiment meant to
illustrate the pitfalls and design challenges of attempting to coordinate an action by
communicating over an unreliable link. In the experiment, two generals are only able to communicate
with one another by sending a messenger through enemy territory. The experiment asks how they might
reach an agreement on the time to launch an attack, while knowing that any messenger they send
could be captured.

The Two Generals' Problem appears often as an introduction to the more general Byzantine
Generals problem in
introductory classes about computer networking (particularly with regard to the Transmission Control
Protocol, where it shows that TCP cannot guarantee state consistency between endpoints and why
this is the case), though it applies to any type of two-party communication where failures of
communication are possible. A key concept in epistemic
logic, this problem highlights
the importance of common knowledge. Some authors also refer to this as the Two Generals' Paradox, the
Two Armies Problem, or the Coordinated Attack Problem.^{2 3} The Two Generals' Problem
was the first computer communication problem to be proven to be unsolvable.⁴ An important
consequence of this proof is that generalizations such as the Byzantine Generals problem are also
unsolvable in the face of arbitrary communication failures, thus providing a base of realistic
expectations for any distributed consistency protocols.

Definition

Two armies, each led by a different
general, are preparing to attack a fortified
city. The armies are encamped near the city, each in its own valley. A third 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 is a chance that any given
messenger sent through the valley will be captured.⁵

While the two generals have agreed that they will attack, they haven't agreed upon a time for an
attack. It is required that the two generals have their armies attack the city simultaneously to
succeed, lest the lone attacker army 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 is required to come to
consensus.⁶

The thought experiment involves considering how they might go about coming to a consensus. In its
simplest form, one general is known to be the leader, decides on the time of the 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 is sure the other has agreed to the attack plan. Both generals will
always be left wondering whether their last messenger got through.⁷

Proof

Because this protocol is deterministic, suppose there is a sequence of a fixed number of messages, one or more
successfully delivered and one or more not. The assumption is that there should be a shared
certainty for both generals to attack. Consider the last such message that was successfully
delivered. If that last message had not been successfully delivered, then 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. Since the protocol is deterministic, the general sending
that last message will still decide to attack. We've now created a situation where the suggested
protocol leads one general to attack and the other not to attack—contradicting the assumption
that the protocol was a solution to the problem.

A non-deterministic protocol with a potentially variable message count can be compared to an
edge-labeled finite tree, where each node in the tree represents an explored example up to a specified point. A
protocol that terminates before sending any messages is represented by a tree containing only a
root node. The edges from a node to each child are labeled with the messages sent in order to reach
the child state. Leaf nodes represent points at which the protocol terminates. Suppose there exists
a non-deterministic protocol P which solves the Two Generals' Problem. Then, by a similar
argument to the one used for fixed-length deterministic protocols above, P' must also solve the
Two Generals' Problem, where the tree representing P' is obtained from that for P by removing
all leaf nodes and the edges leading to them. Since P is finite, it then follows that the
protocol that terminates before sending any messages would solve the problem. But clearly, it does
not. Therefore, a non-deterministic protocol that solves the problem cannot exist.

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 is no algorithm that they can use (e.g. attack if more than
four messages are received) that 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 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 confidence 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",⁸ 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 ⁹ in 1978 in "Notes on Data Base Operating Systems" ¹⁰ 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 mentioned above.

References