Scott_Aaronson

The Weighted Majority Algorithm

*I'm not making any nonstandard claims about computational complexity, only siding with the minority "derandomize things" crowd*

Note that I also enthusiastically belong to a "derandomize things" crowd! The difference is, I think derandomizing is *hard work* (sometimes possible and sometimes not), since I'm unwilling to treat the randomness of the problems the world throws at me on the same footing as randomness I generate myself in the course of solving those problems. (For those watching at home tonight, I hope the differences are now reasonably clear...)

Above-Average AI Scientists

*When Scott Aaronson was 12 years old, he: "set myself the modest goal of writing a BASIC program that would pass the Turing Test by learning from experience and following Asimov's Three Laws of Robotics..."*

As I think back on that episode, I realize that even at the time, I didn't really expect to succeed; I just wanted to see how far I could get and what would happen if I tried. And it's not clear to me in retrospect that it wasn't worth a day's work: at the least, I learned something about how to write tokenizers and user interfaces! Certainly I've spent many, many days less usefully. For similar reasons, it's probably worth it for budding computer scientists to spend a few days on the P vs. NP question, even if their success probability is essentially zero: it's the only way to get a gut, intuitive feel for why the problem is hard.

Is it likewise possible that some of the AGI researchers you've met (not the creationist, but the other ones) aren't *quite* as stupid as they seemed? That even if they don't succeed at their stated goal (as I assume they won't), the fact that they're actually building systems and playing around with them makes it halfway plausible that they'll succeed at *something*?

How An Algorithm Feels From Inside

While "reifying the internal nodes" must indeed be counted as one of the great design flaws of the human brain, I think the *recognition* of this flaw and the attempt to fight it are as old as history. How many jokes, folk sayings, literary quotations, etc. are based around this one flaw? "in name only," "looks like a duck, quacks like a duck," "by their fruits shall ye know them," "a rose by any other name"... Of course, there wouldn't *be* all these sayings if people didn't keep confusing labels with observable attributes in the first place -- but don't the sayings suggest that recognizing this bug in oneself or others doesn't require any neural-level understanding of cognition?

Rationality Quotes 5

Are there any witticisms that *don't* count as 'rationality quotes'?

Natural Selection's Speed Limit and Complexity Bound

*So here's my question: Can you actually do asymptotically better than natural natural selection by applying an error-correcting code that doesn't hamper beneficial mutations?*

In principle, yes. In a given generation, all we want is a mutation rate that's nonzero, but *below* the rate that natural selection can correct. That way we can maintain a steady state indefinitely (if we're indeed at a local optimum), but still give beneficial mutations a chance to take over.

Now with DNA, the mutation rate is fixed at ~10^-8. Since we need to be able to weed out bad mutations, this imposes an upper bound of ~10^8 on the number of functional base pairs. But there's nothing special *mathematically* about the constant 10^-8 -- that (unless I'm mistaken) is just an unwelcome intruder from physics and chemistry. So by using an error-correcting code, could we make the "effective mutation rate" nonzero, but as far below 10^-8 as we wanted?

Indeed we could! Here's my redesigned, biology-beating DNA that achieves this. Suppose we want to simulate a mutation rate ε<also stick in "parity-check pairs" from a good error-correcting code. These parity-check pairs let us correct as many mutations as we want, with only a tiny probability of failure.

Next we let the physics and chemistry of DNA do their work, and corrupt a 10^-8 fraction of the base pairs. And then, using exotic cellular machinery whose existence we get to assume, we read the error syndrome off the parity-check pairs, and use it to *undo all but one mutation* in the unencoded, functional pairs. But how do we decide which mutation gets left around for evolution's sake? We just pick it at random! (If we need random bits, we can just extract them from the error syndrome -- the cosmic rays or whatever it is that cause the physical mutations kindly provide us with a source of entropy.)

Natural Selection's Speed Limit and Complexity Bound

Wiseman, let M be the number of "functional" base pairs that get mutated per generation, and let C be the number of those base pairs that natural selection can affect per generation. Then if M>>C, the problem is that the base pairs will become mostly (though not entirely) random junk, *regardless* of what natural selection does. This is a point about random walks that has nothing to do with biology.

To illustrate, suppose we have an n-bit string. At every time step, we can change one of the bits to anything we want, but then *two* bits get chosen at random and inverted. Question: in "steady state," how many of the bits can we ensure are 0?

I claim the answer is only 3n/4. For suppose pn of the bits are 1, and that we always pick a '1' bit and change it to 0. Then the expected change in the number of bits after a single time step is

D(p) = (1/4) [p^2 (-3) + 2p(1-p) (-1) + (1-p)^2 (1)].

Setting D(p)=0 and solving yields p=1/4.

So we get an either/or behavior: *either* the mutation rate is small enough that we can keep the functional DNA pretty much exactly as it is, *or else* a huge fraction of the "functional" DNA becomes randomized. In other words, there's no intermediate regime where the functional DNA keeps mutating around within some "interesting" region of configuration space, without spilling out into a *huge* region of configuration space.

Note that for the above argument, I've assumed that the "interesting" regions of DNA configuration space are necessarily small -- and in particular, that they can't correspond to Hamming balls of size c*n where c is a constant. This assumption is basically a restatement of our earlier observation that natural selection hasn't discovered error-correcting codes. As such, it seems to me to be pretty secure biologically. But if this assumption fails then so does my argument.

Natural Selection's Speed Limit and Complexity Bound

Wiseman, if it's true that (1) copying DNA inherently incurs a 10^-8 probability of error per base pair, and that (2) evolution hasn't invented any von-Neumann-type error-correction mechanism, then all the objections raised by you and others (and by me, earlier!) are irrelevant.

In particular, it doesn't matter how capable a species is of coping with a few detrimental mutations. For if the mutation rate is higher than what natural selection can correct, the species will just keep on mutating, from one generation to the next, until the mutations finally *do* become detrimental.

Also, your idea that sections of DNA can become more robust once they've "proven themselves" violates the 10^-8 assumption -- which I'd imagine (someone correct me if I'm wrong) comes from physics and chemistry, not because it's favored by natural selection.

Natural Selection's Speed Limit and Complexity Bound

Eliezer, sorry for spamming, but I think I finally understand what you were getting at.

Von Neumann showed in the 50's that there's no in-principle limit to how big a computer one can build: even if some fraction p of the bits get corrupted at every time step, as long as p is smaller than some threshold one can use a hierarchical error-correcting code to correct the errors faster than they happen. Today we know that the same is true even for quantum computers.

What you're saying -- correct me if I'm wrong -- is that biological evolution never discovered this fact.

If true, this is a beautiful argument for one of two conclusions: either that (1) digital computers shouldn't have as hard a time as one might think surpassing billions of years of evolution, or (2) 25MB is enough for pretty much anything!

Natural Selection's Speed Limit and Complexity Bound

OK, I posted the following update to my blog entry:

Rereading the last few paragraphs of Eliezer's post, I see that he actually argues for his central claim -- that the human genome can’t contain more than 25MB of "meaningful DNA" -- on different (and much stronger) grounds than I thought! My apologies for not reading more carefully.

In particular, the argument has nothing to do with the number of generations since the dawn of time, and instead deals with the maximum number of DNA bases that can be simultaneously protected, *in steady state*, against copying errors. According to Eliezer, copying DNA sequence involves a ~10^-8 probability of error per base pair, which — because only O(1) errors per generation can be corrected by natural selection — yields an upper bound of ~10^8 on the number of "meaningful" base pairs in a given genome.

However, while this argument is much better than my straw-man based on the number of generations, there's still an interesting loophole. Even with a 10^-8 chance of copying errors, one could imagine a genome reliably encoding far more than 10^8 bits (in fact, arbitrarily many bits) by using an error-correcting code. I'm not talking about the "local" error-correction mechanisms that we know DNA has, but about something more global -- by which, say, copying errors in any small set of genes could be completely compensated by other genes. The interesting question is whether natural selection could read the syndrome of such a code, and then correct it, using O(1) randomly-chosen insertions, deletions, transpositions, and reversals. I admit that this seems unlikely, and that even if it's possible in principle, it's probably irrelevant to real biology. For apparently there are examples where changing even a single base pair leads to horrible mutations. And on top of that, we can't have the error-correcting code be *too* good, since otherwise we'll suppress beneficial mutations!

Incidentally, Eliezer's argument makes the falsifiable prediction that we shouldn't find *any* organism, anywhere in nature, with more than 25MB of functional DNA. Does anyone know of a candidate counterexample? (I know there are organisms with far more than humans' 3 billion base pairs, but I have no idea how many of the base pairs are functional.)

Lastly, in spite of everything above, I'd still like a solution to my "pseudorandom DNA sequence" problem. For if the answer were negative -- if given any DNA sequence, one could efficiently reconstruct a nearly-optimal sequence of insertions, transpositions, etc. producing it -- then even my original straw-man misconstrual of Eliezer's argument could put up a decent fight! :-)

once you know what your probability distribution is...I'd merely stress that that's an enormous "once." When you're writing a program (which, yes, I used to do), normally you have only the foggiest idea of what a typical input is going to be, yet you want the program to work anyway. This is not just a hypothetical worry, or something limited to cryptography: people have actually run into strange problems using pseudorandom generators for Monte Carlo simulations and hashing (see here for example, or Knuth vol 2).

Even so, intuition suggests it should be possible to design PRG's that defeat anything the world is likely to throw at them. I share that intuition; it's the basis for the (yet-unproved) P=BPP conjecture.

"Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin." --von Neumann