First let us realize that the weaknesses of those approaches is they must use some centralization to prevent Sybil attacks:
Still another approach to consensus is Byzantine agreement [Pease et al. 1980; Lam-
port et al. 1982], the best known variant of which is PBFT [Castro and Liskov 1999].
Byzantine agreement ensures consensus despite arbitrary (including non-rational) be-
havior on the part of some fraction of participants. This approach has two appealing
properties. First, consensus can be fast and efficient. Second, trust is entirely decou-
pled from resource ownership, which makes it possible for a small non-profit to help
keep more powerful organizations, such as banks or CAs, honest. Complicating mat-
ters, however, all parties must agree on the the exact list of participants. Moreover,
attackers must be prevented from joining multiple times and exceeding the systems
failure tolerance, a so-called Sybil attack [Douceur 2002]. BFT-CUP [Alchieri et al.
2008] accommodates unknown participants, but still presupposes a Sybil-proof cen-
tralized admission-control mechanism.
Generally, membership in Byzantine agreement systems is set by a central authority
or closed negotiation. Prior attempts to decentralize admission have given up some of
the benefits.
The new Stellar SCP protocol/algorithm (above white paper) morphs the Sybil attack problem from one of divergence
to one of perpetual preemption (unless of course centralization of trust is used by participants to thus remove the Sybil attack). It also provides asymptotic security that Satoshi's PoW doesn't have.
Note that Bitcoin does not have asymptotic security, meaning if ever someone with greater hashrate could come along in the future, they could rewrite the block chain. Iota has an interesting point about the
insecurity of PoW hashes in the context of quantum computing. However, I argue that the community will enforce checkpoints, because our transaction history is valuable to us.
you can't solve byzantine generals problem with a probabilistic model unless you've first solved sybil with a probabilistic model and Bitcoin doesn't do that
because there's no way of telling if all pools are owned by the same person, then it's not collusion or 51% attack, it's a sybil attack
since the essence of the byzantine generals problem is sybil attack, dealing with sybil comes first in the hierarchy before byzantine generals is discussed at all
I made this same point in either 2013 or 2014.
Afaics, the only solution is
unprofitable PoW which is
the design I am now pursuing.
Bitcoin solves the byzantine generals problem within the bounds of the assumptions in the model. If one entity controls a majority of hashing power, that is outside of the bounds.
Circular logic. Bitcoin didn't solve the Sybil attack problem when pools control 51% and no one can know whether they do and reroute their PoW shares.
The stated problem bounds do not include being able to tell whether someone controls >50% of the hash rate. That isn't in the paper at all. The wording of the paper is "As long as a majority of CPU power is controlled by nodes that are not cooperating to attack the network". It doesn't matter whether they cooperate via pools or otherwise, either way it is outside the bounds.
Without considering the Sybil attack, then one isn't solving the Byzantine fault issue, i.e. isn't solving the Byzantine Generals problem (which is the correct title of this thread). Just because Satoshi failed to mention that he hadn't solved what he was implying to have solved, doesn't make that just having a majority of the hashrate is the only consideration in a PoW solution to the Byzantine Generals problem.
Even if we remove the economics which drives hashrate to concentrate into mining farms such as my suggestion to make mining unprofitable (and an ASIC resistant PoW protocol such as a memory hard hash would help improve the ratio of PoW shares from the marginal mines which are the payers required to make mining unprofitable for the lowest-cost miners which are the mining farms), we still have the problem that if payers are not full nodes and thus have to choose another server to do verification and select transactions for each block, the Sybil attack problem remains in that one can't know if many servers are owned/controlled by the same entity. And in fact, I have shown that verification MUST due to economics be centralized because those full nodes which have higher hashrate (even if hidden behind a Sybil attack from the public's perspective) thus earn more block reward and/or transaction fees per verification than those who control less hashrate, thus pools/full nodes are forced to be centralized (and hide it from the public with a Sybil attack because we all are delusional and expect Satoshi's design to remain decentralized when it can't).
But let's consider what damage the Sybil attack on full nodes can do, and how it can be detected and mitigated. In Satoshi's design, the Sybil attacking full node has lower costs for verification (and maybe can also potentially do a selfish mining attack but that isn't required to make my point) and thus will eventually drive the other full nodes bankrupt as a result. Thus Satoshi's design centralizes because of the inviolable and insoluble economic reality.
The other bad things centralization can do is censor some transactions and execute long-con double-spend attacks.
The solution is to centralize only the verification, but keep the control of the PoW computation decentralized, and make it such that the blame for censoring transactions and long-con double-spending is not ambiguous as it is in Satoshi's design.
That is exactly what my design accomplishes, while also enabling instant transactions that are sound. White paper and implementation forthcoming.