What would you do with a solution to 3-SAT?
I'll tell you what I'd do, man: two chicks at the same time, man.
Anyway, my actual answer would have been about the same as jimrandomh's, but, assuming I'm the only one who has the polynomial-time solution to 3-SAT, in the absence of sufficiently specific knowledge about how to create an FAI, I would use it to make huge amounts of money (either by using it to get a significant advantage in prediction and using that to play the stock market (etc.) or just by making super-useful thitherto-impossible products or services using it) and then use that to support said universe-optimization efforts.
There's a nice paper about that by Scott Aaronson (pdf)
If such a procedure existed, then we could quickly find the smallest Boolean circuits that output (say) a table of historical stock market data, or the human genome, or the complete works of Shakespeare. It seems entirely conceivable that, by analyzing these circuits, we could make an easy fortune on Wall Street, or retrace evolution, or even generate Shakespeare’s 38th play. For broadly speaking, that which we can compress we can understand, and that which we can understand we can predict.
He...
The perpetual motion machines you refer are only that in a very metaphorical sense -- they don't allow an infinite extraction of some energy-like metric.
Glider guns produce an endless stream of gliders to give the simplest example.
That is a bit poetic. In the Fibonacci case, we know that there is a simple explanation/formula. For the stock market, genome, or Shakespeare, it is not obvious that the smallest circuit will provide any significant understanding. On the other hand, if there's any regularity at all in the stock market, the shortest efficient description will take advantage of this regularity for compression. And, therefore, you could use this automatically discovered regularity for prediction as well.
On the other hand, if several traders get their hands on efficient NP computers at once, it's safe to bet that historical regularities will go out the window.
OBVIOUSLY the answer to this question is:
I would assemble a documentary crew, and make a movie about me visiting the town and city halls of every town and city in America. I would travel the minimum distance possible, and do nothing particularly interesting at any location. I would release my video into the Creative Commons, and open up a website to go with the video. There would be a google map-like toy that invites users to plot their own tour of all the towns and cities in Europe. This will be my way of testing initiates into the Better Bayesian Conspir...
What would you do with a solution to 3-SAT?
Any answer other than "create a superintelligent friendly AI to optimize the universe" would be a waste of this particular genie, but there are some steps in between that and 3-SAT which I can't specify yet.
Do you agree that the FAI problem has to be solved sooner or later?
...And right now, thinking about possible replies to your comment, I finally switched to agreeing with that. Thanks.
Oh hell. This changes a lot. I need to think.
I would settle all famous conjectures in mathematics. Or at least the ones with short enough proofs/refutations.
Daniel's statement:
"Given a statement S of ZFC and a number n, is there a proof of S that is shorter than n?"
Is trivially in NP.
Their existence is a trivial implication of multiple states mapping onto the same state. They may not have found specific GoE patters, but they surely had the concept (if not by that name).
I'm not entirely sure it is a trivial implication:
In a sense, you're right, in that on any finite life-field run on a computer, which has only a finite number of possible states, the existence of convergent patterns does trivially imply Garden of Eden patterns. However, most life-theorists aren't interested in finite fields, and it was considered possible that Garden of Eden patterns might only work by exploiting weird but uninteresting things that only occur on the boundary.
In an infinite field, you have an uncountable infinity of states, and uncountable sets can have functions defined from them to themselves that are surjective but not injective, so the trivial implication does not work.
On the other hand, if you only look at a finite subset of the infinite field, then you find that knowing the exact contents of a n by n box in one generation only tells you the exact contents of an (n-2) by (n-2) box in the next generation. You have 2^(n^2) patterns mapping to 2^((n-2)^2) patterns, the former is 16^(n-1) times as large as the latter. This makes the existence of convergent patterns trivial, and the existence of Garden of Eden patterns quite surprising.
Another way to look at this is to see that the smallest known Garden of Eden pattern is a lot larger than the smallest pair of convergent patterns.
On the other hand, if you only look at a finite subset of the infinite field, then you find that knowing the exact contents of a n by n box in one generation only tells you the exact contents of an (n-2) by (n-2) box in the next generation. You have 2^(n^2) patterns mapping to 2^((n-2)^2) patterns, the former is 16^(n-1) times as large as the latter. This makes the existence of convergent patterns trivial, and the existence of Garden of Eden patterns quite surprising.
I agree with the GoE part, but does this really single-handedly imply convergent patter...
Many experts suspect that there is no polynomial-time solution to the so-called NP-complete problems, though no-one has yet been able to rigorously prove this and there remains the possibility that a polynomial-time algorithm will one day emerge. However unlikely this is, today I would like to invite LW to play a game I played with with some colleagues called what-would-you-do-with-a-polynomial-time-solution-to-3SAT? 3SAT is, of course, one of the most famous of the NP-complete problems and a solution to 3SAT would also constitute a solution to *all* the problems in NP. This includes lots of fun planning problems (e.g. travelling salesman) as well as the problem of performing exact inference in (general) Bayesian networks. What's the most fun you could have?