3 ms·

Dumb question: what does this finding means for practical applications?

by znpy 3y ago

ColinWright 3y ago
Someone always asks this, and it's been answered many times before. In short: this specific result probably has no practical applications just now. But there are several points to consider: 0: The techniques developed to answer the question might be useful for other applications; 1: It can be fun to chase down problems like this; 2: Sometimes results like this end up being useful decades, even centuries later; 3: The skills the researchers develop might be able to be used on other problems; 4: Not everything has to be useful. Consider music, art, hobbies. As examples of some of these: a: Elliptic curves were initially studied purely out of curiosity. Now they form the basis of modern cryptosystems; b: Factorising Integers was regarded as the pursuit of cranks, until it became an important tool in breaking cryptosystems; c: Latin Squares and Lattices were studied well before they were suggested as a possibility for post-quantum cryptography; d: Graph Theory was chased purely because it was interesting, and now forms the foundation of a gazillion algorithms; e: Eigenvectors were studied purely out of curiosity, then suddenly they became the basis of the Page Rank algorithm; f: Non-Euclidean Geometry was created purely to show that classic geometry was probably right, then showed it was actually "wrong", then turned out to be useful in relativity. And so on. There are more, and thousands, possibly millions of words have been written on the subject of how pure research, pursued simply because it was interesting, has turned out (in some cases!) to be useful. The 9th Dedekind number is unlikely to be useful, but its pursuit may be valuable in ways we can't imagine.
- znpy 3y ago
  Thanks for the wall of text, I don’t doubt basic research is valuable and never wrote anything like that, so you could have saved that typing. > The 9th Dedekind number is unlikely to be useful, but its pursuit may be valuable in ways we can't imagine. The fact that you can’t imagine it doesn’t mean somebody else can’t imagine it. If you can’t imagine it, save the typing and let somebody that can imagine it reply, next time. Btw your reply was useless.
  - ColinWright 3y ago
    To paraphrase you, the fact that you found it useless doesn't mean others found it useless. It's had a significant number of upvotes, so ... shrug But thanks for the feedback. I'll keep it in mind next time I have the urge to try to be helpful.
booleandilemma 3y ago
I have a new example to point to the next time someone asks me for an impractical result in mathematics.
nullc 3y ago
I spent some time searching for an algebraic expression or straight forward recursive formula for dedekind numbers in the past because such a formula would make it easy to construct an efficient encoding function that maps a monotone boolean function to an integer optimally. Why was I interested in optimally encoding monotone boolean functions? Because they're exactly equivalent to the set of credible multi-party signing policies. So, for example you have some N parties that can participate in signing to control some Bitcoin-- which combinations of them should be allowed? People commonly use thresholds, M of N, but sometimes not all participants are equal. The full set of policies you could possibly have is the monotone functions (since it doesn't make sense to have a policy that says a party CANT participate, since a party could always refrain to satisfy the policy). Knowing the next dedekind number, even if not through a an efficient expression provides extra information that could help understand the structure of the problem. For my application though having an optimal encoding isn't strictly necessary (and in any case may not be possible, given the amount of thought that has gone into solving the problem finding the dedekind numbers)-- though it would be neat. The naive encoding is to just express it as a boolean function, which takes 2^n bits, but already at 9 inputs an optimal encoding of monotone functions would be 26.8% of the size and the gap will increase as the number of inputs increases. More interesting to me would be an encoding that efficiently (if not optimally) expressed the monotone functions in terms of how many bits were allowed to be false, because I'd come up with a commitment scheme that allows the policy setter to set the policy as a single hash (which has a hidden bound on he number of missing parties), while the signers can collaboratively produce a signature of size proportional to the number of missing parties. In the case that the policy is a simple threshold, this construct alone is enough, but if it isn't additional information for which missing-sets aren't allowed is needed and to maximize policy privacy it would be best if he information was hierarchically encoded. E.g. if none are missing then no information is needed, if 1 is missing which parties are it not allowed to be, if 2 are missing which pairs, and so on. (this structure also has more promise for an efficient encoding, since its easy to optimally encode an n choose m selection). For these more generalized encoding questions, particularly if they don't seek to encode optimally the next dedekind number is less useful, but exploration of it could still yield helpful insights towards more efficient, if not optimal encodings or somewhat efficient encodings with special properties like the hierarchical one I wanted above.

Dumb question: what does this finding means for practical applications?

by znpy 3y ago

Dumb question: what does this finding means for practical applications?

Someone always asks this, and it's been answered many times before. In short: this specific result probably has no practical applications just now. But there are several points to consider: 0: The techniques developed to answer the question might be useful for other applications; 1: It can be fun to chase down problems like this; 2: Sometimes results like this end up being useful decades, even centuries later; 3: The skills the researchers develop might be able to be used on other problems; 4: Not everything has to be useful. Consider music, art, hobbies. As examples of some of these: a: Elliptic curves were initially studied purely out of curiosity. Now they form the basis of modern cryptosystems; b: Factorising Integers was regarded as the pursuit of cranks, until it became an important tool in breaking cryptosystems; c: Latin Squares and Lattices were studied well before they were suggested as a possibility for post-quantum cryptography; d: Graph Theory was chased purely because it was interesting, and now forms the foundation of a gazillion algorithms; e: Eigenvectors were studied purely out of curiosity, then suddenly they became the basis of the Page Rank algorithm; f: Non-Euclidean Geometry was created purely to show that classic geometry was probably right, then showed it was actually "wrong", then turned out to be useful in relativity. And so on. There are more, and thousands, possibly millions of words have been written on the subject of how pure research, pursued simply because it was interesting, has turned out (in some cases!) to be useful. The 9th Dedekind number is unlikely to be useful, but its pursuit may be valuable in ways we can't imagine.

znpy 3y ago
Thanks for the wall of text, I don’t doubt basic research is valuable and never wrote anything like that, so you could have saved that typing. > The 9th Dedekind number is unlikely to be useful, but its pursuit may be valuable in ways we can't imagine. The fact that you can’t imagine it doesn’t mean somebody else can’t imagine it. If you can’t imagine it, save the typing and let somebody that can imagine it reply, next time. Btw your reply was useless.
- ColinWright 3y ago
  To paraphrase you, the fact that you found it useless doesn't mean others found it useless. It's had a significant number of upvotes, so ... shrug But thanks for the feedback. I'll keep it in mind next time I have the urge to try to be helpful.

booleandilemma 3y ago

I have a new example to point to the next time someone asks me for an impractical result in mathematics.

nullc 3y ago

I spent some time searching for an algebraic expression or straight forward recursive formula for dedekind numbers in the past because such a formula would make it easy to construct an efficient encoding function that maps a monotone boolean function to an integer optimally. Why was I interested in optimally encoding monotone boolean functions? Because they're exactly equivalent to the set of credible multi-party signing policies. So, for example you have some N parties that can participate in signing to control some Bitcoin-- which combinations of them should be allowed? People commonly use thresholds, M of N, but sometimes not all participants are equal. The full set of policies you could possibly have is the monotone functions (since it doesn't make sense to have a policy that says a party CANT participate, since a party could always refrain to satisfy the policy). Knowing the next dedekind number, even if not through a an efficient expression provides extra information that could help understand the structure of the problem. For my application though having an optimal encoding isn't strictly necessary (and in any case may not be possible, given the amount of thought that has gone into solving the problem finding the dedekind numbers)-- though it would be neat. The naive encoding is to just express it as a boolean function, which takes 2^n bits, but already at 9 inputs an optimal encoding of monotone functions would be 26.8% of the size and the gap will increase as the number of inputs increases. More interesting to me would be an encoding that efficiently (if not optimally) expressed the monotone functions in terms of how many bits were allowed to be false, because I'd come up with a commitment scheme that allows the policy setter to set the policy as a single hash (which has a hidden bound on he number of missing parties), while the signers can collaboratively produce a signature of size proportional to the number of missing parties. In the case that the policy is a simple threshold, this construct alone is enough, but if it isn't additional information for which missing-sets aren't allowed is needed and to maximize policy privacy it would be best if he information was hierarchically encoded. E.g. if none are missing then no information is needed, if 1 is missing which parties are it not allowed to be, if 2 are missing which pairs, and so on. (this structure also has more promise for an efficient encoding, since its easy to optimally encode an n choose m selection). For these more generalized encoding questions, particularly if they don't seek to encode optimally the next dedekind number is less useful, but exploration of it could still yield helpful insights towards more efficient, if not optimal encodings or somewhat efficient encodings with special properties like the hierarchical one I wanted above.