In this second post, we show the fundamental lower bound on the number of rounds for consensus protocols in the synchronous model.
Theorem 1: Any protocol solving consensus in the synchronous model that is resilient to $t$ crash failures must have an execution with at least $t+1$ rounds.
 Bad news: Deterministic synchronous consensus is slow. Even for crash failures, it requires $O(n)$ rounds.
 Good news: With randomization, synchronous consensus is possible in constant expected time. See this post for a survey. Note that randomization does not circumvent the existence of an execution that takes $t+1$ rounds, it just (exponentially) reduces the probability of this event.
We use the proof approach of Aguilera and Toueg that is based on uncommitted (bivalent) configurations.
This post assumes you are familiar with the definitions in the previous post and with Lemma 1 that we proved in the previous post:
Lemma 1: (Lemma 2 of FLP85): $\mathcal{P}$ has an initial uncommitted configuration.
To prove the theorem we will prove the following two lemmas:
Lemma 2: Round $(t1)$ Uncommitted (Lemma 3 of AT99): If $\mathcal{P}$ is tolerant to an adversary that can crash at most one party every round, then there must exist an uncommitted configuration at the end of round $t1$.
Recall the proof pattern for showing the existence of an uncommitted configuration:
 Proof by contradiction: assume all configurations are either 1committed or 0committed.
 Define some notion of adjacency. Find two adjacent configurations $C$ and $C’$ such that $C$ is 1committed and $C’$ is 0committed.
 Reach a contradiction due to an indistinguishability argument between the two adjacent configuration, $C$ and $C’$. The adjacency allows the adversary to cause indistinguishability via crashing of just one party.
Proof of the round $(t1)$ Uncommitted Lemma: By induction, each time showing an uncommitted configuration in round $0 \leq k \leq t1$. The base case of $k=0$ follows from the Lemma 1. For the induction step, assume we are at an uncommitted round $k$ configuration $C_k$ and show that there must exist a round $k+1$ uncommitted configuration $C_{k1}$. Naturally we will use the recurring proof pattern:
 Assume all round $k+1$ configurations $C_k \rightarrow C$ are either 1committed or 0committed.

Define two round$k+1$ configuration $C,C’$ as $j,i$adjacent if the only difference is that in $C$ party $j$ crashes right before sending to noncrashed party $i$ and in $C’$ party $j$ crashes right after sending to party $i$ (and $j$ is the only new crash happening in round $k+1$).
2.1. Claim: there must exist two $j,i$adjacent configurations at the end of round $k+1$ such that $C$ is 1committed and $C’$ is 0committed.
2.2. Proof: assume with no crashes in round $k+1$, the system is 1committed at the end of round $k+1$. There must be some party $j$ whose crash in $k+1$ before sending to $i$ changes the configuration to be 0committed. In particular, it cannot be that $j$ crashes after sending all its messages because that will be indistinguishable from $j$ crashing only in round $k+2$.
 Now unsurprisingly, consider in both words $C$ and $C’$ the case where party $i$ crashes at the beginning of round k+2. Clearly, these two worlds are indistinguishable and must decide the same value. This is a contradiction.
This concludes the proof of lemma 2.
So we reached round $t1$ with an uncommitted configuration. Can we decide in one round and terminate in round $t$? The following Lemma shows we cannot, and completes the theorem:
Lemma 3: Round $t$ Cannot Always Decide (Lemma 1 of AT99): If $\mathcal{P}$ has an execution that crashes at most one party every round and leads to an uncommitted configuration at the end of round $t1$, then in that execution $\mathcal{P}$ cannot always commit by the end of round $t$.
Proof of Lemma 3 is a trivial variation of the proof pattern that looks at deciding configurations instead of committed configurations.

Assume that $\mathcal{P}$ always decides after $t$ rounds. So all $C_{t1} \rightarrow C$ are either 0deciding or 1deciding.

Define two round$t$ configurations $C,C’$ as $j,i$adjacent if the only difference is that in $C$ party $j$ crashes right before sending to noncrashed party $i$ and in $C’$ party $j$ crashes right after sending to party $i$ (and $j$ is the only new crash happening in round $k+1$). So there must exist two $j,i$adjacent configurations in the end of round $t$ such that $C$ is 1committed and $C’$ is 0committed.

Given $n>t+1$ there are at least two noncrashed parties in round $t$, so at least one of them is not $i$ or $j$. However, this honest has no way to distinguish between the two deciding configurations $C$ and $C’$. This is a contradiction.
This concludes the proof of Lemma 3, which concludes the proof of Theorem 1.
Discussion. The proof started with an initial uncommitted configuration (Lemma 1) and then showed that we could continue for $t1$ rounds to reach an uncommitted configuration (Lemma 2) at round $t1$. Finally, we used one more crash to show that we cannot always decide in $t$ rounds (Lemma 3). Note that the theorem is nonconstructive, and all it shows is that a $t+1$ round execution must exist (but does not say it probability measure of this event).
A fascinating observation from this lower bound is that in order to create a long execution the adversary must corrupt just one party every round. This fact is critical for some upper bounds and will be discussed in later posts.
In the next post, we will show that the same proof approach can be used to show that in the asynchronous model, there must exist an infinite execution even for protocols that are resilient to just one crash failure.
Acknowledgment. We would like to thank Kartik for helpful feedback on this post.
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