Satoshi Nakamoto did not understand the "Byzantine Generals Problem," as is evidenced by the following: https://satoshi.nakamotoinstitute.org/emails/cryptography/11/

"**The proof-of-work chain is a solution to the Byzantine Generals' Problem.** I'll try to rephrase it in that context."

Let us first make it quite clear that, for 2n+1+m nodes (which vertices are simultaneously message-radiating "generals", message-reflecting or, in the case of faulty/malignant elements, transmitting or absorbing witnesses, and message-absorbing lieutenants), no "do gooder consensus", in the case interesting us to be understood as meaning that truth is what "good guys" *decide* it to be, can be ascertained unless the traitors/fomenters' contingent m isn't above n (the problem isn't a problem if n vanishes, because for evil to overcome good, it has to exist.)

Instead of a reductio ad absurdum we shall succinctly analyze i.e., slaughter the beast at hand and excise from within its innards, that is in the very question, its general solution.

We shall as well discover that substituting reason for faith, computation for trust, complicates, slows down, hampers, squanders, pollutes i.e., is, in a word, inefficient.

Imagine a multitude of nodes and let one of them, x, ask a question to some other, say y. Now if x were endowed with simple and despised faith it would accept y's answer confining annoying solicitations to a singleton, but this age is a haughty and voluble one, we ask for proofs even of axioms, blind as we are to the fact that we still are giving our credence to authorities or, as is the case for Bitcoin, to the hardware, to the software, to the network supporting it, if only they belittle faith. Such a x won't be satisfied with y's answer but will have to beg all the other nodes for what is it that y told them in response to x's question. Far from being done with it, taking a majority of the reports, it will have to, for the same reason, ask, for any z, all the remaining nodes what is it that z told them that x said. As you can judge, we have an infinite regress in the making, "fortunately" enough for infidels, they are to be held in the chains of finitude and therefore these sequences can be terminated. How deep such a pit need be? If it were to heap n "good" nodes t0, t1, ... tn-1 where t0 would be y, then if there still remained, in addition to x, taken as "good" for argument's sake, n other "good" nodes, and at most n "bad" ones, then lo, x would find itself in the pleasant position of agreeing with any of the n other remaining "good" nodes concerning tn-1's witness, this because tn-1 being "good" will send the same message to all the askers, and so will "reflect" x, mirror as it were, or the n remaining "do gooders." Hence when x will ask these n "good" nodes "what is it that tn-1 said that ... t1 said that t0 vomited?," it will find itself sprinkled with the same droplets of noisome diarrhea that tn-1 itself benefitted it with, and therefore whatever the no more than n "false witnesses" told x in regard to t0, ... tn-1, it will have a majority of n+1 equal votes vs. less than or equal to n diverging ones. Now that, if you take some time to think instead of indulging in prolefeed such as that ridiculous but nonetheless pernicious "Lucifer" TV show or some pr0n, belongs to the species of worse scenario and by that we mean that the maximum number of "goodies" is chained for n badies, beyond which disagreement among the nice vertices is not impossible. Now picture yourself tn-1 as "bad", shall the unchained n+2 "good" nodes agree on t0, ... tn-1? Not necessarily, as a counterexample shows: tn-1, being a contrarian, isn't bound to send the same datum to all nodes, let it report i to x, j to a "good" w and any random heterogeneous, all different from i and j, garbage to all the others, where i and j are different. Now x and w will ask all the remaining nodes, whether "good" or "bad" they have no inkling of, what tn-1 told that ... t0 said. The evil nodes may very well bombard x with i all the while assuring w of j. For x this state of affairs will result in his assigning i to t0, ... tn-1 seeing that it has a majority of n occurrences of i against n+1 votes, no n of which are equal but w will conclude j for the same t0, ... tn-1. And yet, although the "good" nodes will differ on t0, ... tn-1 with an evil tn-1 appendage, they still will agree on t0, ... tn-2, because they will value the well-ending t0, ... tn-1 equally which is, for each "goodie", n+1 identical decisions against n on bad-ending t0, ... tn-1. The cases where t0, ... tn-1 is mongrelous i.e., with both "good" and "evil" ti, i < n-1 are easier because there are fewer free fomenters. This procedure ensures hivemindeness on the "good" nodes' part: they'll decide on the same value for t0, ... tn-2, for t0, ... tn-3 and so on until they come to consensus on t0 that is y.

The belabored *digression* that follows S. Nakamoto's d'entrée de jeu conclusion, for such it proves to be as far as the object of the exchange is concerned, and which is purported to cast the Blockchain into the BG's framework, is not even coherent viz.:

"It has been decided that anyone who feels like it will announce a time, and whatever time is heard first will be the official attack time.  **The problem is that the network is not instantaneous, and if two generals announce different attack times at close to the same time, some may hear one first and others hear the other first.**" continued with "**They use a proof-of-work chain to solve the problem.**"

How? "Once each general **receives whatever attack time he hears first**, he sets his computer to **solve an extremely difficult proof-of-work problem** that includes the attack time in its hash."

This to my mind looks like a rather silly petitio principii.

"The proof-of-work is so difficult, it's **expected to take 10 minutes of them all working at once before one of them finds a solution.**"

So, for them to solve the problem of agreeing on a concerted-attack time against the King's WiFi password, they find no better than duplicating that same problem, for except they work *all at once* on the "extremely difficult problem" (a stupid exhaustive search), they'll hardly "solve" it within 10 minutes.

"Once one of the generals finds a proof-of-work, he broadcasts it to the network, **and everyone changes their current proof-of-work computation to include that proof-of-work in the hash they're working on.** If anyone was working on a different attack time, they switch to this one, because its proof-of-work chain is now longer."

He who wrote the above doesn't have a clear idea of what a Blockchain is, let alone the BGP: why should the less lucky, or as is often the case in practice not as powerful as the de facto general, lieutenants decide to prolong said general's already hashed and received block thereby having to restart hashing their own, now altered blocks because of their overwriting the parent block's "address" i.e., hash field? That certainly doesn't make much sense if their blocks have the same payload as is suggested by Nakamoto-san's answer. (I believe him to be a vertical, beyond our surfacic senses or in this case imaginary, vertical cross-product of the orthogonal and horizontal pair D. Kleiman, C. Wright.) Why should the would be generals not compete against one another instead, growing parallel block branches? After all it is significantly a matter of luck for one to bump into an adequate block-hash. The BGP's solution doesn't define "good" as "most powerful", but demands that the good nodes be more than twice as numerous as the bad ones.

The Blockchain is an interesting *antisolution* to the BGP, a minority of nodes (i.e., violating the indispensable to any solution minimum ratio of 2n+1:n of "goodies" to "badies") can very well "speak" a longest branch of the Blockchain tree (for it is a *tree*, not a sequence as it appears to be presenting itself) and thereby impose their truth, their history, their law.

It is not difficult in the least to graft and grow a longest "bad future" on the Timechain: https://www.wingclips.com/movie-clips/back-to-the-future-2/alternative-future