The Infinity Machine

[simont]

Probably most of my friends have heard me waffling on about the Infinity Machine at one time or another. If anyone reading this has managed to miss it so far, you can find my article introducing the concept at http://www.chiark.greenend.org.uk/~sgtatham/infinity.html. Today I have a question about it to ask geek-inclined readers.

The Infinity Machine has been a fun thought experiment for most of my life, but in one respect it's slightly frustrating. Its ability to search a (countably) infinite space in finite time would enable it to solve quite a few problems that are famously unsolved in the real world; but quite a few of those problems would simultaneously be rendered pointless to solve anyway by the presence of Infinity Machines in the world. For example, you could use the Infinity Machine to search all possible computer programs to find the one which was fastest at factorising large integers – but you wouldn't want to, because if we had Infinity Machines then a perfectly naïve factorisation algorithm would be just as efficient in practice and far easier to get working correctly.

It occurred to me this week that there is a scenario in which that slight frustration might be resolved. Suppose you were suddenly taken away from your normal life and sat down in front of an Infinity Machine for a few days, or a week, or a month. Suppose you were free to write programs and run them on it, and free to write finite quantities of the output to real-world storage media to take back with you, but when your time expired the Infinity Machine would vanish and nobody would ever get their hands on one again. In this situation, asking the Machine for efficient finite algorithms would be entirely sensible, in principle.

What would you get it to compute, in this situation?

(Ground rules: to help you write your programs you can have access to a large library, and perhaps archives of reference websites such as Wikipedia if you want them. But you don't get unfettered Internet access while you're using the Infinity Machine: I don't want you doing things like factorising every key on the PGP keyservers, or all the root CA keys, because that's against the spirit of what I'm interested in asking.)

Some thoughts on my own answer:

There are three categories of things you might try for. You might try for global altruism: find out things that improve the sum of human knowledge or the global quality of life. Or you might try for personal profit, e.g. finding algorithms you can sell. Or you might simply try to satisfy your own curiosity. I would certainly like to think I'd mostly try for the former, but then, who knows what I'd really do…

The trouble is that quite a few of the ‘does there exist an efficient algorithm’ questions I can think of are things where I want the answer to be no. If I found out there was an integer factorisation algorithm so efficient that RSA keys had to be impractically big to defeat it, I'm not sure I'd want to publish it to the world anyway. Finding out there wasn't would be more pleasant, but less dramatic.

It might be more interesting to set the Machine searching for unbreakable crypto primitives. Instead of obsoleting RSA (or perhaps as well), I could tell the Machine to work on finding me a block cipher, a hash function and an asymmetric key scheme which have no computationally feasible attacks against them. But the trouble with that is, what would I do with them when I got back to the real world? I'm not a published cryptographer; how would I get people to believe my algorithms were better than those of any other random crank?

It would certainly be interesting to set up some mathematical formal systems and search through all possible derivations within them for proofs of unsolved problems. Goldbach's conjecture would make a good warm-up exercise. The Riemann hypothesis would probably be the one I was most interested in tackling in this way, although I'm not sure what kind of formal system it would take to even express the question coherently, let alone have a good chance of containing a proof. And let's not forget Fermat's Last Theorem: just because it's been solved doesn't mean it wouldn't still be interesting to search through the formal system of Peano arithmetic to see if it contained an elementary proof short enough to have nearly fitted into Fermat's margin. Again, you have the problem of distinguishing yourself from a crank on your return to the real world; but the nice thing about formal systems is that they're independently checkable. I could publish the description of my formal system, a formal proof written in that system, and my derivation-checking program; then people could verify the code of my own checker, and/or easily write their own derivation-checkers, to independently test that the proof was formally valid.

Back to algorithms, it occurred to me that finding the best possible optimisation and code generation algorithms for a compiler (within reasonable running-time limits) might be a worthwhile thing. One could certainly sell that if one were of a mind, without suffering from the crank problem – cryptographic algorithms require careful review to determine their security, but the output of a compiler speaks for itself and the market would figure it out eventually. Alternatively one could publish it as free software, according to taste; either way, I'm pretty confident it would be genuinely useful to people in helping them get stuff done.

So, what do other people think?

Perhaps I should ask some subquestions as well:

• What could you do that would have the biggest effect on improving the world?
• What would you do in order to make yourself the biggest profit?
• What would you be most curious to know the answers to, even if nobody would ever believe you?
• And which of those would you prioritise most highly: what would be the thing you'd actually do if given the chance?

Comments

[simont]

The trouble with having the IM search for an English-speaking program is that the search basically has to be done by iterating over all programs and running some acceptance criterion over each of them. So if you wanted it to find an English-speaking program, you'd have to have already written an acceptance tester capable of recognising a genuinely English-speaking program, as opposed to one which was merely cheating in some way. In particular, you couldn't define an English-speaking program as one which would give specific answers to some finite set of test questions, because almost certainly the shortest program capable of giving those answers to those questions would be one that simply had them hardwired in in some fashion. So I fear that one might be infeasible.

Chess shouldn't be a problem, though. You could certainly find out which player won in perfectly played chess, and you might also try searching for the shortest program capable of producing perfect play within reasonable execution time, to see if it turned out that there was a relatively simple way to make the optimal strategy practically implementable. (As there is in nim, for instance. Presumably for chess it would be far more complex than that, but it might still turn out to be tractable by a finite computer.)

I think I'd be more interested in asking about go than chess, but that's just me.

[naath.livejournal.com]

Ah, but I don't know the rules for Go; maybe with wikipedia I could work out what they were though.

I'd be tempted to assume that an IM would by its very nature be an AI - but I'd probably be wrong.

And yes, the search is hard... but what you can do is make a giant neural net and train it with all the available data (rather than whatever subset you have time for), and you could train as many neural nets as you could invent that might work (obviously you generate these somehow, not just think about them yourself). So I think you'd probably have something much better than what we currently have at the end of it.

[simont]

An Infinity Machine contains no AI, unless you write it. It isn't qualitatively different from a normal finite computer, except that it has unlimited storage space and you get to ask questions of the form "will this program ever produce an answer, if I run it forever?". It does ordinary mechanistic computation, just very very fast.

Large-scale neural nets are an interesting idea, though, and certainly they neatly dodge my basic methodological objection. I think I wouldn't be confident enough of getting decent results to spend my limited time on that particular approach, but good luck with it if you ever find yourself in this scenario :-)