Comment by FabHK

The basic results of SMR theory are as follows, where "sync", "async" and "partially sync" refer to specific network models; PKI is public key infrastructure (that is, each node knows all the other nodes and has their public keys); "f" is the number of failed/dishonest/byzantine nodes (out of n total nodes); and only deterministic protocols are considered.

1) Permissioned, Sync, PKI: SMR possible, any f (!), Dolev-Strong (1983, [-5])

2) Permissioned, Sync, no PKI: SMR impossible if f >= n/3, PSL (1980), FLM (1985) (the hexagon proof, [-4])

3) Permissioned, Async: SMR impossible even with f=1 (!), FLP (1985) ("endless bivalent", [-3])

4) Permissioned, partially sync: SMR with "eventual availability" impossible if f >= n/3 [-2], possible otherwise (eg Tendermint [-1], Byzantine Paxos, PBFT)

In setting 4), PBFT-type protocols such as Tendermint guarantee consistency (among the "honest" nodes following the protocol as intended - you can't make any guarantees wrt to faulty or byzantine nodes) and eventual availability (that is, all requests sent by clients will "sooner or later" be dealt with) once network functionality is resumed.

That is consensus, for all intents and purposes, given that more consensus isn't really possible due to 2), 3). And arguably better consensus than Nakamoto consensus, which improves the boundary in 4) to n/2 (without selfish mining) at the cost of being stochastic, not deterministic, but replaces "consistency always, availability eventually" with "consistency eventually, availability always", arguably the wrong choice for financial applications.

[-5] https://timroughgarden.github.io/fob21/l/l2.pdf

[-4] https://timroughgarden.github.io/fob21/l/l3.pdf

[-3] https://www.youtube.com/watch?v=vJhm9uhd34E&list=PLEGCF-WLh2...

[-2] https://timroughgarden.github.io/fob21/l/l6.pdf

[-1] https://timroughgarden.github.io/fob21/l/l7.pdf