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18:55 < andytoshi> amiller: i've been thinking a bit about compact SPV over the last couple weeks and i think i finally have a simple understanding of what you were saying back when we were writing the sidechains whitepaper
18:55 < andytoshi> i'm going to summarize it here and maybe you can tell me if i'm off-base
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18:55 < andytoshi> let's assume constant difficulty and blocktime "1" so that we have something tractable
18:57 < andytoshi> if i hand you a chain with a million blocks in it, the time it took to generate that (assuming constant hashrate etc) will follow the distribution \sum_1^{1000000} Exp(1) = Erlang(1000000, 1)
18:57 < andytoshi> compare this to the extreme case of a "compact SPV" chain where it's just one block with a million times the PoW on it. this has generation time distribution Exp(1000000)
18:59 < andytoshi> the erlang one has variance proportional to a million; the exponential proportional to a million squared ... with the result that you've gotta spend ~ 1000000 ± 2000 blocktimes to have a nonnegligible chance of producing the whole chain. basically it really is a proof that a million blocktimes' worth of work was done
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19:01 < andytoshi> contrast the exponential distribution which is so flat that there's almost the same probability that it took 1 blocktime to generate this super-work block as there is that it took a million. (more precisely what i mean is that the CDF climbs from 0 to nearly one smoothly as the time taken goes from 0 to a couple million)
19:01 < andytoshi> so even though both these "chains" (a million 1-diff blocks) and (one million-diff blocks) have same expected work, the resulting chains prove drastically different things
19:06 < andytoshi> an actual compact SPV chain with log-many blocks is going to be weirdly weighted, you expect it'll have one block with a ton of work, one with half that, one with half that, and so on .. but we can imagine for a minute that actually the blocks are all equal-difficulty, there are just log-many of them, and this will give some intuition as to the tradeoff between blocksize length and variance. (this is
19:07 < andytoshi> a really bad approximation, but you can sum exponential distributions of equal characteristic time to get erlang distributions .... as soon as you start having variable targets you have to do things numerically and it's a mess)
19:08 < andytoshi> the variance we get doing this is n_blocks * block_weight^2  .... lower variance is good, the lower the variance the closer the expected time is to how much work is actually "proven". we can see the two extremes i described, variance a million (a million diff-one blocks) and variance million^2 (one diff-million block)
19:08 < andytoshi> lemme stop saying "million" sorry, i'm talking asymptotics here, i'll just say N
19:09 < andytoshi> so we have N for N blocks, N^2 for 1 block ... for log(N) blocks of difficulty N/log(N) (same total expected work) our equation becomes log(N)*(N/log(N))^2 = N^2/log(N)
19:11 < andytoshi> you can replace log with any function in the above. you see if you try to play difficulty games to trim the blockchain length down from N to f(N), the variance of your time distribution goes from N to N^2/f(N)
19:13 < andytoshi> so here's a stats/real analysis question .. how is P(chain generated in less than expected time * "negligible") affected by the shape of f?
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19:55 < amiller> that looks right to me
19:58 < andytoshi> :s cool.
19:58 < amiller> imo it is enough to use chernoff bounds and there's no good reason to do anything else
19:58 < andytoshi> oh nice, i didn't remember chernoff bounds
20:00 < amiller> chernoff bounds let you say that there's negligible probablity that a random variable (in this case the amount of work done) differs from its expected value (the expected work based on just adding up the difficulty) by more than some fraction
20:01 < andytoshi> oh, cool, this makes it very clear. i'd never heard of this bound i think
20:01 < amiller> it's negligible like meaning the probability is exp(-k), where k is the number of samples that make up the variable
20:01 < andytoshi> i sorta bumped into erlang distributions by accident which gave me a special case..
20:01 < andytoshi> yep understood https://en.wikipedia.org/wiki/Chernoff_bound is pretty clear
20:02 < amiller> yeah erlang distribution is a good model for this
20:06 < andytoshi> :/ i don't have our conversation logs from the time we were writing the paper, but i think i remember you mentioning chernoff's bound and saying "this is pretty inherent, you can't drop blocks and prove a statement of the same strength"
20:06 < andytoshi> but here we are like 2 years later and it's just clicking now
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