In this third post, we conclude with the celebrated Fischer, Lynch, and Paterson impossibility result from 1985. It is the fundamental lower bound for consensus in the asynchronous model.

Theorem 1 (FLP85): Any protocol $\mathcal{P}$ solving consensus in the asynchronous model that is resilient to even just one crash failure must have an infinite execution.

• Bad news: Deterministic asynchronous consensus is impossible.
• Good news: With randomization, asynchronous consensus is possible in constant expected time. See this paper for a recent result. Note that randomization does not circumvent the existence of a non-terminating execution, it just reduces the probability measure of this event to have measure zero.

This post assumes you are familiar with the definitions of the first post and with Lemma 1 that we proved in the first post:

Lemma 1: (Lemma 2 of FLP85): $\mathcal{P}$ has an initial uncommitted configuration.

Recall that given a configuration $C$ there is a set $M$ of pending messages. These are messages that have been sent but not delivered yet. For $e \in M$ we write $e=(p,m)$ to denote that party $p$ has been sent a message $m$. Also, recall that an uncommitted configuration is a configuration where no party can decide because the adversary can still change the decision value.

Given an initial uncommitted configuration, our goal will be to build an infinite execution such that:

1. The sequence is uncommitted: every configuration of the infinite execution is uncommitted. If a configuration is uncommitted, then no party can decide.
2. The sequence is fair: every message sent is eventually delivered. This is the essence of asynchrony, the adversary can delay messages, but only by a finite amount.

To prove the theorem we will prove the following technical Lemma:

Lemma 2: Uncommitted Configurations Can Always be Extended (Lemma 3 of FLP85): If $C$ is an uncommitted configuration and $e=(p,m)$ is any pending message of $C$, then there exists some $C \stackrel{\pi}{\rightsquigarrow} C’ \xrightarrow{e=(p,m)} C’’$ such that $e \notin \pi$ and $C’’$ is an uncommitted configuration.

#### Proof of Theorem 1 from Lemma 1 and Lemma 2:

Start with Lemma 1, to begin with, an uncommitted configuration. Repeat Lemma 2 infinitely often; each time apply it to the pending messages in a FIFO order. Clearly, from Lemma 2, the sequence is uncommitted. For fairness, due to FIFO, a message $e$ that has $|M|$ pending messages before it will be derived after at most $|M|+1$ applications of Lemma 2.

#### Proof of Lemma 2:

Recall the proof pattern for showing the existence of an uncommitted configuration:

1. Proof by contradiction: assume all configurations are either 1-committed or 0-committed.
2. Find a local structure: two adjacent configurations $X$ and $X’$ such that $X$ is 1-committed and $X’$ is 0-committed.
3. Reach a contradiction due to an indistinguishability argument between the two adjacent configurations, $X$ and $X’$ using the adversary’s ability to crash one party.

Proof of Lemma 2 follows this pattern exactly: The contradiction of the statement of Lemma 2 is that: there exists a configuration $C$ and a message $e=(p,m)$ such that for all $\pi$ with $e \notin \pi$, let $C \stackrel{\pi}{\rightsquigarrow} C’$, let $C’ \xrightarrow{e=(p,m)} C’’$, then either $C’’$ is 1-committed or $C’’$ is 0-committed ($C’’$ is not uncommitted).

Define two configurations $X,X’$ as adjacent if $X \xrightarrow{e’=(p’,m’)} X’$ and $e’$ is a pending message in $X$.

Claim: there must exist two adjacent configurations $Y \xrightarrow{e’} Y’$ and a pending message $e’=(p’,m’)$ in $Y$ such that: 1. $C \rightsquigarrow Y \xrightarrow{e} Z$ and Z is 1-committed. 2. $C \rightsquigarrow Y \xrightarrow{e’} Y’ \xrightarrow{e} Z’$ and $Z’$ is 0-committed.

Proof of claim: Since $C$ is an uncommitted configuration, there must exist two sequences $\tau_0$ and $\tau_1$ such that $C \stackrel{\tau_0}{\rightsquigarrow} D_0$ and $C \stackrel{\tau_1}{\rightsquigarrow} D_1$, where $D_0$ is 0-committed and $D_1$ is 1-committed.

For each $i \in {0,1}$, let $\pi_i$ be the longest prefix of $\tau_i$ that does not contain $e$. Let $C \stackrel{\pi_0}{\rightsquigarrow} C_0 \xrightarrow{e} C’_0$ and $C \stackrel{\pi_1}{\rightsquigarrow} C_1 \xrightarrow{e} C’_1$. It follows from the assumption that $C’_0,C’_1$ must be committed and from $\tau_0,\tau_1$ that $C’_0$ is 0-committed and $C’_1$ is 1-committed.

Since $\pi_0,\pi_1$ start from the same configuration $C$, let $G$ be the least common ansenstor configuration of $\pi_0,\pi_1$ and assume without loss of generality that $G\xrightarrow{e} G’$ is such that $G’$ is 1-committed.

Now examine the sub-sequence $G=Y_1,\dots,Y_k=C_0$ of $\pi_0$ from $G$ to $C_0$. Let $G’=Z_1,\dots,Z_k=C’_0$ be such that $Y_i \xrightarrow{e} Z_i$. Since $Z_1=G’$ is 1-committed and $Z_k=C’_0$ is 0-commiteed then clearly there must exist two adjacent configurations $Y \xrightarrow{e’} Y’$ (in the path $Y_1,\dots,Y_k$) such that that $Z$ is 1-committed and $Z’$ is 0-committed where $Y \xrightarrow{e} Z$ and $Y’ \xrightarrow{e} Z’$. Note that this follows from the discrete version of the intermediate value theorem.

Proof of Lemma 2 given the claim

Let $Y,Y’$ be these two adjacent configurations. There are two cases to consider about $e=(p,m)$ and $e’=(p’,m’)$:

1. Case 1 (trivial case): $p \neq p’$. This implies that processing $e$ and then $e’$ will lead to a different outcome than processing $e’$ and only then $e$. But since $e,e’$ reach different parties there is no way to distinguish these two worlds.

Formally $Y \xrightarrow{e=(p,m)} Z$ is 1-committed and so $Y \xrightarrow{e=(p,m)} Z \xrightarrow{e’=(p’,m’)} Z’’$ is 1-committed. But $Y \xrightarrow{e’=(p’,m’)} Y’ \xrightarrow{e=(p,m)} Z’$ is 0-committed. This is a contradiction because $Z’’$ and $Z’$ have exactly the same configuration and pending messages.

2. Case 2: $p=p’$. This implies that the committed value must change between the world where $p$ receives $m$ before it receives $m’$ relative to the world where $p$ receives $m’$ before it receives $m$! But what if $p$ crashes? These two worlds will be indistinguishable to the rest of the parties! Moreover, $p$ does not need to crash; it can just be slow!

Formally, consider some execution where party $p$ crashes at $Y$. So there must be some $Y \stackrel{\sigma}{\rightsquigarrow} D$ where $D$ is a deciding configuration and $\sigma$ does not contain party $p$. But if party $p$ was just slow then $Y \stackrel{\sigma}{\rightsquigarrow} D \xrightarrow{e} D’$ must be a configuration where all parties other than $p$ have decided.

Since $Y \xrightarrow{e} Z$ is 1-committed then $Y \xrightarrow{e} Z \stackrel{\sigma}{\rightsquigarrow} D’$ must be 1-committed.

Since $Y \xrightarrow{e’} Y’ \xrightarrow{e} Z’$ is 0-committed then $Y \xrightarrow{e’} Y’ \xrightarrow{e} Z’ \stackrel{\sigma}{\rightsquigarrow} D’’$ must be 0-committed.

For parties other than $p$, $D’$ and $D’’$ are indistinguishable. This is a contradiction.

This completes the proof of Lemma 2, and that completes the proof of the FLP Theorem.

Discussion. We started from an uncommitted configuration (Lemma 1) and then showed that we could extend this to another uncommitted configuration infinitely many times and do this while eventually delivering every pending message (Lemma 2).

This proof is non-constructive; it shows that an infinite execution must exist. Using randomization, there are protocols that are almost surely terminating (their probability measure of terminating is one). There exist asynchronous consensus protocols that terminate in an expected constant number of rounds. More on that in later posts.

Acknowledgment. We would like to thank Nancy Lynch, Kartik Nayak, Ling Ren, Nibesh Shrestha, Sravya Yandamuri for helpful feedback on this post.