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Author Topic: using Shannon's information to measure proof-of-work  (Read 3400 times)
grondilu
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June 25, 2011, 08:31:31 PM
 #1

Guys, I am so excited about this new idea of mine that I can't wait until tomorrow when I get my usal internet access to tell it to you.  So I'm writing this from a cybercafé.  I hope you'll enjoy.

I was working on my catalaxia project and I was thinking very hard about how I could avoid using Satoshi's compicated algorith for adjusting difficulty.  Somehow I beleived there should be a more elegant, mathematically pure way to do this.

I was troubled by the idea of mesuring the "strengh" of a block chain, and I must confess I never managed to understand exactly how Satoshi does it.  I wanted something much more simple.

Then I remembered my old lessons in physical statistics and information theory.   Then I thought about entropy and Shannon's information concept.

Basically shannon's information is a way to measure unlikelyness of an event.  A bit like probability, but with additive properties.

In probabilities, when two events e1 and e2 occurs in the same time, the probability of this to happen is the product of the probability:

P(e1 & e2) = P(e1) * P(e2)

Shannon wanted to have the same thing, but with additive property:

I(e1 & e2) = I(e1) + I(e2)

Obviously the solution was to use the logarithm function, so he came up with this formula:

I(e) = - ln ( P(e) )

And he called that "the information".  It's measured in bits if you divide by the natural logarithm of two, and it's quite a fascinating concept.

Ok, enough for the theory.

The idea now is to use this as a measure for the "weight" of a block.  Basically all we have to do is to divide 2^256 by the block hash, and take the natural logarithm.  The total information of the block chain is nothing but the sum of each information for each block.

The good news is that we won't have to start the block chain from scratch or whatever, as the longuest block chain currently almost certainly has the highest Shannon's information.

I think there are many advantages on this, such as simplicity and robustness.  There are also many interesting implications such as the possibility to make the information being a inside secondary currency inside the block chain, which could be used to trade between miners in order to avoid chain forks.  I'll explain later.

There are many things to say about why I think this is a good idea, but right now I just wanted to tell it to you before someone else does, so I can claim the glory of being the first to have dropped it, in case it appears as great an idea as I think it is.
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dinox
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June 25, 2011, 10:21:05 PM
 #2

I can assure you that satoshi's method is a lot simpler.

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June 26, 2011, 12:11:18 PM
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Then I remembered my old lessons in physical statistics and information theory.   Then I thought about entropy and Shannon's information concept.

That Del Shannon guy sure was an underrated hero wasn't he?

"Del Shannon"

Quote
Basically shannon's information is a way to measure unlikelyness of an event.  A bit like probability, but with additive properties.

OH! That Shannon. Now we are talking heroes of another order of magnitude (no offence Del, wherever you are  Cheesy)  Here he is:

"Claude Ellwood Shannon"

How many people realize that in 'coining' the term (pardon the pun) 'bitcoin' Satoshi was actually acknowledging (perhaps inadvertantly) the father of information theory? The 'bit' being the unit of information that halves prior uncertainty of an unknown value. It's generally recognized that Shannon essentially defined the 'bit' if he didn't literally coin the term (again, pardon the pun). I don't understand programming all that well nor mathematics either for that matter, but I do love information theory, so Shannon is a legend in my eyes, in that he devised the mens to quantify redundancy. I don't know how Satoshi's algorithm does the weight calculation, but I assume it's some sort of Bayesian approach, heavy on recursive looping too I expect.

Quote
In probabilities, when two events e1 and e2 occurs in the same time, the probability of this to happen is the product of the probability:

P(e1 & e2) = P(e1) * P(e2)

Shannon wanted to have the same thing, but with additive property:

I(e1 & e2) = I(e1) + I(e2)

Obviously the solution was to use the logarithm function, so he came up with this formula:

I(e) = - ln ( P(e) )

And he called that "the information".  It's measured in bits if you divide by the natural logarithm of two, and it's quite a fascinating concept.

Fascinating indeed. Particularly for it's implications for entropy, but that's another story. BTW, even that explanation helped my mathematics understanding. I can see now, how the cumulative 'surprise value' of each bit, is derived from Shannon's Log function. Thanks for that.
 
Quote
The idea now is to use this as a measure for the "weight" of a block.  Basically all we have to do is to divide 2^256 by the block hash, and take the natural logarithm.  The total information of the block chain is nothing but the sum of each information for each block.

I can't wait till you write a verbose non-mathematical description for us non-mathematical types. It sounds like a very grass roots 'bottom up' approach, to avoid the need to calculate the weight by measuring it directly. But again, I don't know how it's ordinarily done in Satoshi's implementation. 

Quote
The good news is that we won't have to start the block chain from scratch or whatever, as the longuest block chain currently almost certainly has the highest Shannon's information.

I think there are many advantages on this, such as simplicity and robustness.  There are also many interesting implications such as the possibility to make the information being a inside secondary currency inside the block chain, which could be used to trade between miners in order to avoid chain forks.  I'll explain later.

I'll be looking forward to it.  Wink

Quote
There are many things to say about why I think this is a good idea, but right now I just wanted to tell it to you before someone else does, so I can claim the glory of being the first to have dropped it, in case it appears as great an idea as I think it is.

Well done ol' chap. I hope somebody will take note of this. It sounds like an elegant alternative. I don't suppose you ever read Richard Dawkins' article The Information Challenge, in which he honors Shannon and provides a great layman's explanation of information as an economic commodity and the relevance of redundancy. I particularly like the parallels it draws (in my mind at least), with energy and entropy. He comes tantalizingly close to venturing into this realm, but unfortunately forestalls it. Anyhow, I wish you well with this promising refinement and intuitively, I honestly do think you have a good development refinement to offer. Please, keep us informed.

Pieter Wuille
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June 26, 2011, 12:36:00 PM
 #4

Bitcoin counts blocks are 'worth' as much as the statically expected number of hashes that have been performed to generate it. As far as I can see, that's the theoretically best criterion possible.

aka sipa, core dev team

Tips and donations: 1KwDYMJMS4xq3ZEWYfdBRwYG2fHwhZsipa
grondilu
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June 27, 2011, 07:23:17 PM
 #5

I worked hard during two days but I finaly wrote a minimalist distributed timestamp server in bash!!!

Unfortunately I am once again in a cybercafé now so I can't publish it right now.   I'll do it tomorow, I promise.


Stay tuned as I think this is gonna be an MAJOR BREAKTHROUGH in crypto anarchism!
bji
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June 27, 2011, 07:55:55 PM
 #6

I was working on my catalaxia project and I was thinking very hard about how I could avoid using Satoshi's compicated algorith for adjusting difficulty.  Somehow I beleived there should be a more elegant, mathematically pure way to do this.

I didn't read deeply enough to understand every aspect of Satoshi's algorithm, but I thought the basic idea was to average the block generation time for the previous two weeks and then compute a difficulty which, given the same expected computational power, would have resulted in the target 10 minute average interval.  Basically, adjust the difficulty so that if it had been the difficulty over the prior two weeks, then everything else being equal, a block would have been produced on average once every 10 minutes.

It seems pretty straightforward and "elegant" to me already ...

Quote
I was troubled by the idea of mesuring the "strengh" of a block chain, and I must confess I never managed to understand exactly how Satoshi does it.  I wanted something much more simple.

I don't understand why you think the block difficulty measures the "strength" of the block chain, or even what "strength" means in this context.   Unless I am wrong (and I often am!), the only measure is how many blocks were produced over the past two weeks.  Individual blocks are not considered.  If it took 1 minute to generate 2015 blocks and then the next 1 block took 20159 minutes, there would be no adjustment to difficulty over the next two week period because exactly 2016 blocks had been generated in exactly 20160 minutes, which gives the perfect 10 minute average interval.  The contents of the blocks and the actual time taken to produce the hash for any given block don't matter, so from the difficulty adjustment algorithm's point of view, all blocks are the same, the only metric is the average over 2016 blocks of the time taken to generate the hash.

Quote
The idea now is to use this as a measure for the "weight" of a block.  Basically all we have to do is to divide 2^256 by the block hash, and take the natural logarithm.  The total information of the block chain is nothing but the sum of each information for each block.

The good news is that we won't have to start the block chain from scratch or whatever, as the longuest block chain currently almost certainly has the highest Shannon's information.

What would you use this "weight" for?  If it's for adjusting difficulty, I don't see how it's useful.

Quote
I think there are many advantages on this, such as simplicity and robustness.  There are also many interesting implications such as the possibility to make the information being a inside secondary currency inside the block chain, which could be used to trade between miners in order to avoid chain forks.  I'll explain later.

Please do, as this paragraph confuses me utterly.
patvarilly
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June 28, 2011, 08:08:59 AM
 #7

From main.h, here's how a block's work gets calculated:

    CBigNum GetBlockWork() const
    {
        CBigNum bnTarget;
        bnTarget.SetCompact(nBits);
        if (bnTarget <= 0)
            return 0;
        return (CBigNum(1)<<256) / (bnTarget+1);
    }


The total work done up to a certain block is the sum of the work of all previous blocks, down to the genesis block

Basically, the "work" the highest possible hash value divided by the maximum acceptable hash value at the block's difficulty.  As Pieter Wuille remarked, this is just the expected number of hashes you'd have to calculate before solving a block.  What you're suggesting calculating is -ln( 1 / "work" ) = ln( "work" ) and adding these up.  Pieter is right to point out that the original definition of total work estimates how much computational power has been invested to date into securing the block chain.  Your definition of total work would be appropriate only if it took a constant amount of calculation to get one an extra bit at the beginning of the hash to be zero.  But it doesn't take a constant amount of work, it takes twice as much work.

Here's a vivid example of what would go wrong.  Say I have 16 block chained together, A1->...->A16, each of which has "work" of 2, and additionally, three blocks X->Y->Z, each of "work" 16.  Under the original definition of aggregate work, the total work in A1->A16 is 32, while the total work in X->Z is 48. These two aggregate works are proportional to the amount of time it would take you to secure each block chain with a constant-speed miner.  Under your definition of work, A1->A16 would have aggregate work 16*ln(2) = 11.1, while X->Y->Z would have aggregate work 8.3, even though X->Y->Z took much longer to secure.
grondilu
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June 28, 2011, 11:11:16 AM
 #8

From main.h, here's how a block's work gets calculated:

    CBigNum GetBlockWork() const
    {
        CBigNum bnTarget;
        bnTarget.SetCompact(nBits);
        if (bnTarget <= 0)
            return 0;
        return (CBigNum(1)<<256) / (bnTarget+1);
    }


The total work done up to a certain block is the sum of the work of all previous blocks, down to the genesis block

Basically, the "work" the highest possible hash value divided by the maximum acceptable hash value at the block's difficulty.  As Pieter Wuille remarked, this is just the expected number of hashes you'd have to calculate before solving a block.  What you're suggesting calculating is -ln( 1 / "work" ) = ln( "work" ) and adding these up.  Pieter is right to point out that the original definition of total work estimates how much computational power has been invested to date into securing the block chain.  Your definition of total work would be appropriate only if it took a constant amount of calculation to get one an extra bit at the beginning of the hash to be zero.  But it doesn't take a constant amount of work, it takes twice as much work.

Here's a vivid example of what would go wrong.  Say I have 16 block chained together, A1->...->A16, each of which has "work" of 2, and additionally, three blocks X->Y->Z, each of "work" 16.  Under the original definition of aggregate work, the total work in A1->A16 is 32, while the total work in X->Z is 48. These two aggregate works are proportional to the amount of time it would take you to secure each block chain with a constant-speed miner.  Under your definition of work, A1->A16 would have aggregate work 16*ln(2) = 11.1, while X->Y->Z would have aggregate work 8.3, even though X->Y->Z took much longer to secure.

Thanks for your comment.  I'll read that later but it's too late to discourage me, as I have just finished writing a prototype:

http://s0.barwen.ch/~grondilu/cgi-bin/timestamp

grondilu
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June 28, 2011, 11:22:21 AM
 #9

Well done ol' chap. I hope somebody will take note of this. It sounds like an elegant alternative. I don't suppose you ever read Richard Dawkins' article The Information Challenge, in which he honors Shannon and provides a great layman's explanation of information as an economic commodity and the relevance of redundancy. I particularly like the parallels it draws (in my mind at least), with energy and entropy. He comes tantalizingly close to venturing into this realm, but unfortunately forestalls it. Anyhow, I wish you well with this promising refinement and intuitively, I honestly do think you have a good development refinement to offer. Please, keep us informed.

Thanks for this reference.  I am a huge fan of Richard Dawkins and I will read the link you provided.  Your description of the subject makes it extremely relevant to this thread, as I realised that the cryptographic timestamp server I have implemented could be used as a currency.  I'll describe how later.  Anyway this would turn information, or time, into a currency.

Crazy stuff, really.

PS.  I'll put the code here as well, just in case:

$ xz < timestamp |xxd -p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June 28, 2011, 03:12:20 PM
 #10

Where's the distributed timestamp server in bash?  I want to see that.  And you should add it to the effort of the guys that are doing bitcoin cli tools:

http://forum.bitcoin.org/index.php?topic=2461.0
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June 29, 2011, 07:08:20 AM
 #11

I don't suppose you ever read Richard Dawkins' article The Information Challenge, in which he honors Shannon and provides a great layman's explanation of information as an economic commodity and the relevance of redundancy.

I don't understand this conclusion:
Quote
Shannon’s unit of information is the bit, short for “binary digit”. One bit is defined as the amount of information needed to halve the receiver’s prior uncertainty...
Why is bit defined to remove only 50% of receiver's prior uncertainty? A bit should remove 100% of receiver's prior uncertainty! It is either '1' or '0'. There is no 1/2 binary digit?!

“It rained in Oxford every day this week”  or “It rained in the Sahara desert every day this week” - both remove 100% of receiver's prior uncertainty. What do I miss?

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June 29, 2011, 10:28:06 AM
 #12

I don't suppose you ever read Richard Dawkins' article The Information Challenge, in which he honors Shannon and provides a great layman's explanation of information as an economic commodity and the relevance of redundancy.

I don't understand this conclusion:
Quote
Shannon’s unit of information is the bit, short for “binary digit”. One bit is defined as the amount of information needed to halve the receiver’s prior uncertainty...
Why is bit defined to remove only 50% of receiver's prior uncertainty? A bit should remove 100% of receiver's prior uncertainty! It is either '1' or '0'. There is no 1/2 binary digit?!

In Shannon's meaning, one bit is an amount, it is not '1' or '0'.  It is one bit.  It halves the uncertainty because if you add one bit, you divide the probability by 2, by definition.
 
Quote
“It rained in Oxford every day this week”  or “It rained in the Sahara desert every day this week” - both remove 100% of receiver's prior uncertainty. What do I miss?

No, because the amount of information is not the same for both sentences, as rain in Sahara is much more of an unlikely event as rain in Oxford.
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June 29, 2011, 12:23:41 PM
 #13

Quote
“It rained in Oxford every day this week”  or “It rained in the Sahara desert every day this week” - both remove 100% of receiver's prior uncertainty. What do I miss?

No, because the amount of information is not the same for both sentences, as rain in Sahara is much more of an unlikely event as rain in Oxford.
I'm not discussing the amount of information, but if a bit of information is removing 50% or 100% of receiver's uncertainty. If you've learned about the rain in Sahara that doesn't make you less uncertain about the rain in Oxford!

Same confusion goes here:

Quote
He can’t see any details, so a nurse has agreed to hold up a pink card if it is a girl, blue for a boy. How much information is conveyed when, say, the nurse flourishes the pink card to the delighted father? The answer is one bit -- the prior uncertainty is halved.
My answer is different. My answer is one bit -- the prior uncertainty is removed entirely, because pink card means girl. Unless, I don't understand what does 'flourish' mean in this context?

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June 29, 2011, 01:57:18 PM
 #14

Okay, how about re-phrasing it by saying that the prior certainty is doubled?

Before the nurse flashed the card, you were 50% certain it was whichever it turned out later to be.

After the nurse flashed the card, you were 2 * 50 = 100 % certain it was whichever it turned out to be.

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June 29, 2011, 03:03:01 PM
 #15

Okay, how about re-phrasing it by saying that the prior certainty is doubled?

Before the nurse flashed the card, you were 50% certain it was whichever it turned out later to be.

After the nurse flashed the card, you were 2 * 50 = 100 % certain it was whichever it turned out to be.

-MarkM-

As a logical block above is okay, but then the entire article should be rewritten as some assumptions won't be correct. You can't double the value of something by multiplying with 1, because 1 bit is, by definition, 1 binary digit - '1' or '0'. Unless, you can prove that 1 bit = 2. But then, what is 2?

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June 29, 2011, 04:14:52 PM
 #16

It is not really one bit, it is one additional bit.

It was conceptually easy to transpose halving uncertainty into doubling certainty, but trying to then re-write the sensible-sounding story of how your certainty doubled as somehow being a case of your uncertainty having halved was beyond me. It'd seem to imply you were 100% uncertain until the card was flashed then became 50% uncertain, which doesn't sound right at all. So the whole idea that halving uncertainty implies doubling certainty seems dubious. Boolean logic might work better somehow maybe as it smells of one of those "from x implies y we cannot deduce that y implies x" type logic thingies.

Sounded good for a moment though, didn't it? Wink

We were 50% uncertain that it was the one it turned out later not to be, until the card was flashed, then we were 100% uncertain it was the one it turned out not to be... Oops, uncertainty doubled, not halved. Smiley

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June 29, 2011, 08:23:31 PM
 #17

From this:
http://www.noanswersingenesis.org.au/dawkinschallenge.htm

one can see (in the 3rd and 4th paragraphs of the 'Information' section) that it is the *number of possibilities* (not precisely 'uncertainty') which is halved by the transmission of a single bit.  In your examples, the two possibilities are:

1 - it has rained X number of continuous days in Y
2 - it has not rained X number of continuous days in Y

If instead we had an example where there were 8 possible outcomes, as each additional bit comes in, the number of remaining possibilities is reduced by 1/2.  8 > 4 > 2 > 1.  After receiving 3 bits, we have precisely identified which of the 8 possibilities occurred.

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June 30, 2011, 02:07:53 PM
 #18

... I realised that the cryptographic timestamp server I have implemented could be used as a currency.  I'll describe how later.  Anyway this would turn information, or time, into a currency.
Crazy stuff, really.

I'm burning with curiosity: did you just devise a new form of freecoin or you're just suggesting a major upgrade to the existing bitcoin protocol?
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July 01, 2011, 11:41:55 AM
 #19

I'm burning with curiosity: did you just devise a new form of freecoin or you're just suggesting a major upgrade to the existing bitcoin protocol?

Didn't know about this freecoin thing.  I'll have look someday if I have time.

I had several other ideas, such as using the chain as a data storage, and having nodes pay for each bit of information they store in the chain.

Also, I'll use general bitcoin addresses (with any version number from 0x00 to 0xFF).

Here is the current state of my code (it doesn't work yet but it will give you an idea of where I am):

$ xz < timestamp.sh | base64
/Td6WFoAAATm1rRGAgAhARYAAAB0L+Wj4EZuGttdABGIQkY99Bhqpmevep9vaWgy9uJUdLDJuwGD
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grondilu
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July 01, 2011, 08:37:52 PM
 #20


Guys, this stuff is totally blowing my mind.  As I keep modifying my design again and again, I get to think more and more about the whole thing, and the more I do that, the more I realize how deep it is.

I think it goes way beyond currencies and information technology.  I'm starting to get convinced that it is related to fundamental physical concepts such as time, energy, uncertainty principle and Everett's interpretation of quantum mechanics.   I'm really starting to think that I'm up to something big.

The idea of using chained proofs of work to measure time was something that I immediately blew me up when I got to know about bitcoin.  Now I realize why.  This idea is full of profound consequences.

If one agrees to define time as "a causal sequence of events", then it does make sense to use probabilities of these events to measure time.  And if you want to have additive properties for this thing you measure and that you call time, then it's not exactly probabilities you should use, but rather the Shannon information of these probabilities.  Now, instead of saying you are measuring, what if you say that you are actually defining it?

Finally, if you remember that quantum mechanics is all based about probability waves, you're beginning to realize how cool this idea is.
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