In the previous post, we show that State Machine Replication for any f<n failures is possible in the synchronous model when the adversary can only cause parties to crash. In this post, we show that omission failures are more challenging. It requires f<n/2.

Theorem: It is impossible to implement State Machine Replication with two replicas and an adversary that can cause omission failures to one replica (and any number of clients) even in a lock-step model.

As in our previous lower bounds, we assume a solution that is safe and live exists and reach a contradiction. For the contradiction, we define multiple worlds and use an indistinguishability argument. In each world, there are two clients called client $1$ and client $2$ and two servers called server $1$ and server $2$.

### World A:

In this world, client $1$ wants to send command $C1$, and the adversary blocks all communication to and from server $2$. Since the protocol is safe and live, any correct solution must notify client $1$ that command $C1$ is the only committed command.

### World B:

In this world, client $2$ wants to send command $C2$, and the adversary blocks all communication to and from server $1$. Since the protocol is safe and live, any correct solution must notify client $2$ that command $C2$ is the only committed command.

### World C:

In this world, client $1$ wants to send command $C1$, and client $2$ wants to send command $C2$. The adversary causes both clients to fail by omission as follows: it blocks all communication between client $1$ and server $2$ and all communication between client $2$ and server $1$. Finally, without loss of generality, the adversary also causes server 2 to have omission failures by blocking all communication between server $1$ and server $2$.

Observe that the view of server 1 in world A and world C is indistinguishable. Since in worlds A and C, client $1$ only communicates with server $1$, it also has indistinguishable views.

Similarly, the view of server 2 in world B and world C is indistinguishable. Since in worlds B and C, client $2$ only communicates with server $2$, it also has indistinguishable views.

So in world C, the two clients will see conflicting states and this is a violation of safety.

Notes:

1. The proof heavily uses the fact that the clients are prone to omission failures. If clients can just crash (or are non-faulty) then the clients can implement SMR using the replicas as relays.
2. This lower bound can be generalized to $n$ replicas and $f$ omission failures for any $n\leq 2f$.
3. This lower bound holds even if there is a setup and a PKI.