user84232: So from what I understand, Bitcoin's PoW is prone to 51% attack, but as a distributed system it is also prone to BFT's 1/3 attack right? I think it's mathematically proven that in a distributed system, if you have more than 1/3 bad nodes collaborate together, then you cannot safely reach a correct consensus no matter what?
So for Bitcoin, there are two types of possible attack scenarios, one is if a miner node consistently has more than 51% computing power of the whole network then it can double-spend indefinitely and basically has unlimited cash to spend. 
The other is if more than 1/3 of the nodes in the whole network are bad guys working together with modified malicious node code then they can potentially stop the correct block being accepted and make it impossible for others to know which is the correct longest chain, thus prevent valid transactions being processed and recorded.
So are my understandings correct?

user84114: PoW and BFT solve different problems. A couple of differences:
BFT assumes the nodes (aka miners) are known in advance. PoW / Nakamoto Consensus allows anyone to become a node (miner).
BFT assumes nodes do not change over time. PoW / Nakamoto Consensus does not make this assumption.
BFT does not allow forks. PoW/NC allows temporary forks under normal situations (no one suddenly accumulates a lot of hash power). PoW/NC even allows someone to rewrite the entire history of Bitcoin back to block 1 if they could amass enough hash power. 
It makes it hard to compare the two algorithms - it's like comparing an algorithm to find a number in a sorted list versus an algorithm for an unsorted list. 

user52021: 
So from what I understand, Bitcoin's PoW is prone to 51% attack, but as a distributed system it is also prone to BFT's 1/3 attack right?

No. Bitcoin is not a "consensus system" by any of the traditional definitions: It never reaches a state from which it can't, at least, theoretically roll back. In theory today's blocks could get undone years from now, but in Bitcoin the computationally difficulty of that increases over time.
As a result the impossibility results for consensus don't generally apply.

The other is if more than 1/3 of the nodes in the whole network are bad guys working together with modified malicious node code then they can potentially stop the correct block being accepted.

Nah. So long as an attacker does not manage to partition the graph of honest nodes, they could control (say) 99% of the nodes and the network would continue to run just fine. There isn't some magic threshold number of nodes where partitioning happens,  you need only have a single honest peer to be completely functional.

84232: <p>So from what I understand, Bitcoin's PoW is prone to 51% attack, but as a distributed system it is also prone to BFT's 1/3 attack right? I think it's mathematically proven that in a distributed system, if you have more than 1/3 bad nodes collaborate together, then you cannot safely reach a correct consensus no matter what?</p>

<p>So for Bitcoin, there are two types of possible attack scenarios, one is if a miner node consistently has more than 51% computing power of the whole network then it can double-spend indefinitely and basically has unlimited cash to spend. </p>

<p>The other is if more than 1/3 of the nodes in the whole network are bad guys working together with modified malicious node code then they can potentially stop the correct block being accepted and make it impossible for others to know which is the correct longest chain, thus prevent valid transactions being processed and recorded.</p>

<p>So are my understandings correct?</p>
 84114: <p>PoW and BFT solve different problems. A couple of differences:</p>

<p>BFT assumes the nodes (aka miners) are known in advance. PoW / Nakamoto Consensus allows anyone to become a node (miner).</p>

<p>BFT assumes nodes do not change over time. PoW / Nakamoto Consensus does not make this assumption.</p>

<p>BFT does not allow forks. PoW/NC allows temporary forks under normal situations (no one suddenly accumulates a lot of hash power). PoW/NC even allows someone to rewrite the entire history of Bitcoin back to block 1 if they could amass enough hash power. </p>

<p>It makes it hard to compare the two algorithms - it's like comparing an algorithm to find a number in a sorted list versus an algorithm for an unsorted list. </p>
 52021: <blockquote>
  <p>So from what I understand, Bitcoin's PoW is prone to 51% attack, but as a distributed system it is also prone to BFT's 1/3 attack right?</p>
</blockquote>

<p>No. Bitcoin is not a "consensus system" by any of the traditional definitions: It never reaches a state from which it can't, at least, theoretically roll back. In theory today's blocks could get undone years from now, but in Bitcoin the computationally difficulty of that increases over time.</p>

<p>As a result the impossibility results for consensus don't generally apply.</p>

<blockquote>
  <p>The other is if more than 1/3 of the nodes in the whole network are bad guys working together with modified malicious node code then they can potentially stop the correct block being accepted.</p>
</blockquote>

<p>Nah. So long as an attacker does not manage to partition the graph of honest nodes, they could control (say) 99% of the nodes and the network would continue to run just fine. There isn't some magic threshold number of nodes where partitioning happens,  you need only have a single honest peer to be completely functional.</p>
