Can you give an example of the kind of argument that you are trying to refute here?

The argument:

"a (real-world) agent such as a human or AI cannot self improve because the K-complexity of a closed system is constant"

seems obviously silly. A human or AI is not a closed system, it receives sensory inputs, and no-one would argue that self-improvement has to happen without any sensory inputs.

Aside: I would caution, echoing Eliezer, that theoretical results about what is computable and what isn't, and about Turing machines with infinite memory and how... (read more)

Roko: here.

Henry: Kolmogorov complexity.

what exactly do you mean by the phrase "the 'complexity' of that entire universe... would be 500 bits."

He means 500 bits plus log2 the size of the initial conditions, plus log2 the number of instants elapsed since then - which is the number of bits you would need to specify a current multiverse state under the specified assumptions.

You're right about what you said about the instants. However, we don't know how big the initial conditions were - and I'm not clear on what it would mean for them not to exist. Maybe you are talking about eternal return? Even then there's a specification of which path you are on.

@Nick Tarleton:

Thanks, Nick. I can see I'm going to have to spend an exciting evening or 10 alone with the SL4 archives...

Perhaps one could consider Eliezer's OB posts as an "edited and sanitized summary of the morass of SL4"?

Show us simple laws which predict how many galaxies worth of atoms there would be in a universe (why not one?), and I will take 500 bits for generating the universe somewhat seriously...

Suppose, for example, that you had a computer program that defined a synthetic biology and a synthetic gene system. And this computer program ran through an artificial evolution that simulated 10^44 organisms (which is one estimate for the number of creatures who have ever lived on Earth), and subjected them to some kind of competition. And finally, after some predefined number of generations, it selected the highest-scoring organism, and printed it out. In an intuitive sense, you would (expect to) say that the best organisms on each round were getting ... (read more)

The "500 bits" only works if you take a hidden variable or Bohmian position on quantum mechanics. If (as the current consensus would say) non-linear dynamics can amplify quantum noise then enormous amounts of new information are being "produced" locally everywhere all the time. The current state of the universe incorporates much or all of that information. (Someone who understands the debates about black holes and the holographic principle should chime in with more precise analysis.)

I couldn't follow the whole argument so I'm not sure how this affects it, but given that Eliezer keeps referring to this claim I guess it is important.

Tim: The initial wavefunction may be completely determined by the laws, and under MWI that's all you need.

Roko: I actually don't notice much duplication between OB and SL4. It's definitely worth reading.

Will: Why don't you take it seriously? You know the Mandelbrot set is generated from very few bits, and I don't imagine you expect there to be a simple derivation of its structures. Taken literally, your request is highly unreasonable because it involves so many levels of poorly understood processes - inflation, baryogenesis, galaxy formation... and the ans... (read more)

Will Pearson: Show us simple laws which predict how many galaxies worth of atoms there would be in a universe (why not one?), and I will take 500 bits for generating the universe somewhat seriously...

Been done. It didn't work out, and Eddington's theory is now only history, but the example shows that it is not out of the question that there be such laws.

It might be worthwhile to note that cogent critiques of the proposition that a machine intelligence might very suddenly "become a singleton Power" do not deny the inefficacies of the human cognitive architecture offering improvement via recursive introspection and recoding, nor do they deny the improvements easily available via hardware substitution and expansion of more capable hardware and I/O.

The do, however, highlight the distinction between a vastly powerful machine madly exploring vast reaches of a much vaster "up-arrow" space of ... (read more)

I would drop dead of shock if there were a 500-state Turing machine that solved the halting problem for all the Turing machines up to 50 states.

Here's the algorithm: given as input a Turing machine M with less than 400 states,

1. Compute a number k greater than BB(400).
2. Simulate M for k steps.
3. If it halts by then, then output "M halts". If it doesn't, then output "M doesn't halt".

Correctness proof: If it halts in less than k steps, then obviously it halts. If it doesn't, then by the definition of BB(400), it never will.

Analysis of number of states: This is possible with 400+C states because you use about 400 states for step 1, and a constant number of overhead states for simulating Turing machines. Because there exist universal Turing machines with less than 25 states, you can arrange for C to b... (read more)

I thought you had some reason for giving us extra detail, why the number 500. Yeah it could be short as 500 bits, but it could equally be very long.

I will like shorter explanation for the universe if I am presented with one compared to a longer one, but there is no reason to expect one of a certain length. Since we don't have any that actually manage to do so, why attach any number to it?

I don't think there is a Theory of Everything for the universe. Just as that there isn't one for Mathematics.

What if the universe is not continuous, the simulation does lazy evaluation, and the laws of nature can only be described by an infinite number of bit? How would this change your reasoning?

drops dead of shock

Oh... well, at least there's no reasonable way for a superintelligence that can perform any finite computation, to obtain that particular 500-bit Turing machine, through any amount of internal thinking or induction from sensory experience, without using a reasoning method guaranteed at some point to make a mistake.

I don't understand. Am I too dumb or is this gibberish?

@pk I don't understand. Am I too dumb or is this gibberish?

It's not so complicated; it's just that we're so formal...

Kolmogorov complexity of universe is more than just a description of its physical laws.

First of all, besides the physical laws, K should include the initial state of the universe. It may be large or even infinite. And it should be properly expressed in qubits.

Second, you'd need to pick the "present state", whatever that means, our of all possible future and past states. In a classical universe it may be only a 100bits or so, in a quantum multiverse, it's at least 1 bit per every branch.

Third, it is not at all obvious that the laws of universe are... (read more)

In fact, it's just bloody hard to fundamentally increase your ability to solve math problems in a way that "no closed system can do" just by opening the system. So far as I can tell, it basically requires that the environment be magic and that you be born with faith in this fact.

Eliezer, you're making an important error here. I don’t think it affects the main argument you're making in this article (that considerations of "complexity" doesn't rule out self-improving minds), but this error may have grave consequences elsewhere. The error... (read more)

Wei, essentially the black box has to be such that it seems more likely for it to be a halting oracle than something only slightly smarter than you are - since when it says "doesn't halt", and you don't know if it halts or not, it's not like you can easily check.

Suppose, though, that the black box has the apparent qualities of a closed timelike curve in a universe that permits CTCs to be constructed with perfection (even if our own universe permitted CTCs they would break down well before up-arrow time). Then it would seem pretty plausible, afte... (read more)

Good point, but when the box says "doesn't halt", how do I know it's correct?

If humanity unfolded into a future civilization of infinite space and infinite time, creating descendants and hyperdescendants of unlimitedly growing size, what would be the largest Busy Beaver number ever agreed upon?

Suppose they run a BB evaluator for all of time. They would, indeed, have no way at any point of being certain that the current champion 100-bit program is the actual champion that produces BB(100). However, if they decide to anthropically reason that "for any time t, I am probably alive after time t, even though I have no direct eviden... (read more)

Boris: There's a small amount of subtlety in actually doing step 1.

Isn't it simply impossible? That doesn't interfere with your claim that such a Turing machine exists, but step 1 claims that it's computable.

Nick wrote: Good point, but when the box says "doesn't halt", how do I know it's correct?

A halting-problem oracle can be used for all kinds of things besides just checking whether an individual Turing machine will halt or not. For example you can use it to answer various mathematical questions and produce proofs of the answers, and then verify the proofs yourself. You should be able to obtain enough proofs to convince yourself that the black box is not just giving random answers or just being slightly smarter than you are.

If P!=NP, you should be ... (read more)

+1 to Anatoly Vorobey. Using K-complexity to capture the human notion of complexity seems to be even worse than using game-theoretic rationality to capture human rationality - something that's been attacked to death already.

Boris: There's a small amount of subtlety in actually doing step 1.

Douglas: Isn't it simply impossible? That doesn't interfere with your claim that such a Turing machine exists, but step 1 claims that it's computable.

It's impossible to bound BB(n) computably in n, but that leaves open the following question, on which Boris's argument depends.

Let BBB(n), the Busy Beaver Bound function, be the size of the smallest Turing machine that, starting from a blank tape, prints out an upper bound for BB(n). Boris's step (1) claims that BBB(n) is bounded by n+c for some constant c. It is not obvious to me either that this is true or that this is false.

Richard: Boris's step (1) claims that BBB(n) is bounded by n+c for some constant c. It is not obvious to me either that this is true or that this is false. c is the size of a universal Turing machine which, in addition to simulating its argument, keeps track of how many steps it simulates. BB(n) is the maximum number of steps that any halting TM of size n performs, therefore there is some particular TM of size n that performs BB(n) steps. Give that TM as input the the aforementioned UTM, and it will compute BB(n). The uncomputability of BB prevents you from finding the right TM, and from proving it correct if it's handed to you by a magic light in the sky. But the TM still exists even if you'll never know which it is.

Eliezer, I'm not quite sure why you are obsessed with the ability to find mathematical truths. You can different sets of mathematical truths if you take the parallel postulate true or not. Which ones are applicable in the real world depends where you are.

Unless you know where you are in the universe can not know what is friendly for you to do (or even useful).

In fact, it's just bloody hard to fundamentally increase your ability to solve math problems in a way that "no closed system can do" just by opening the system. So far as I can tell, it basically requires that the environment be magic and that you be born with faith in this fact.

As Wei mentioned, P≠NP is basically the conjecture that this isn't true: i.e., that you can exponentially increase your ability to solve math problems by your environment being magic and your not being born with faith in that fact. So for example, if your environment im... (read more)

I would drop dead of shock

Eliezer, just as it was interesting to ask what probability estimate 'Nuts!' amounted to, I think it would be very useful for the forum of Overcoming Bias to ask what your implicit probability estimate for a 500 state TM being able to solve the halting problem for all TMs of up to 50 states.

I imagine that 'I would drop dead of shock' was intended to convey a probability of less than 1 in 10,000, or maybe 1 in 1,000,000?

@Toby: Why, yes, I was feeling rather grateful at that point that I hadn't quantified the probability. It's fair to assume that it would have been low enough that I couldn't plausibly recover from the calibration hit, like 1% or something.

@Scott: This entire discussion assumes unlimited finite internal computing power, so P and NP cut no dice here. Otherwise, of course a larger environment can outsmart you mathematically.

@Will: Mathematical truths are about which axioms imply which theorems.

@Wei: A halting oracle is usually said to output 1s or 0s, n... (read more)

Let me expand my point. Assume you run your hyper civilisation for 1 googol years, you ask it, "Do the angles in a three straight sided shape add up to 180?"

This can be answered yes or no, depending upon what axiom scheme you adopt, cartesian, hyperbolic or something more obscure. Having a system that has thought of all mathematical truths doesn't get you a Powerful system, unless it knows which apply to solving the problems you ask.

Beautiful post, Eli. Seriously, one of the better --- and more useful, at least pedagogically --- ones. Should be required reading for, well, just about everyone.

Oops, I mis-stated the result. If we fix a universal Turing machine (equivalently, a programming language), then there exists a constant C so that deciding the halting problem for programs of length less than N can be done by a program of length less than N+C. Pengvado's solution works here.

Unfortunately, as Richard observed, you can't do this with Turing machines, essentially because Turing machines can not inspect their own finite control, unlike the stored program on a von Neumann machine. In fact, it appears that my claim---in Richard's notation, that BBB(N) is N+O(1)---is open. It is known that BBB(N) is O(N), and moreover, a 2002 paper "Improved Bounds for Functions Related to Busy Beavers" by Ben-Amran and Petersen showed that BBB(N) is N+o(N). I haven't found better results, but on the off-chance that I do, I'll post another comment here.

Now, because my claim is open, we still have to check if Eliezer's original intuition (that 500-state Turing machines can't solve the halting problem for 50-state ones) holds up. Luckily for me, it doesn't. It's known that BBB(N) is less than 3N+6 (that's the O(N) result above), so at worst, there exists a 500-state Turing ... (read more)

Otherwise, of course a larger environment can outsmart you mathematically.

No, not of course. For example, suppose P were equal to PSPACE. Then even though a larger environment could fundamentally outsmart you mathematically (say by solving the halting problem), it couldn't prove to you that it was doing so. In other words, the situation with polynomial-time computation would be more-or-less the same as it is with unlimited computation: superintelligent machines could only prove their superintelligence to other superintelligent machines.

That the situatio... (read more)

Scott, while I don't mean to deflate in any sense the shattering import of your wisdom...

...there's a pretty large segment of the planetary population upon whom the distinction between "polynomially more computing power" and "exponentially more computing power" would be lost.

These unenlightened fools, if they had computing power N, would be just as impressed by a superintelligence that had N^googol computing power as a superintelligence that had 2^N computing power. They'd just lump both of those together as "the environment is smarter than me, and can prove things I can't and then exhibit the proof". Poor bastards.

Um, except that we also don't know whether there are computations that can be checked in N time but only performed in Ngoogol time. The situation is qualitatively the same as for N versus 2N.

(the superscripts didn't show up: that was N^googol and 2^N)

The initial wavefunction may be completely determined by the laws, and under MWI that's all you need.

Bundling the initial state into the laws of physics seems like a confused way of looking at things to me. The state and the laws are different things - if you muddle them together conceptually, your accounting is likely to get equally muddled. Also, most of the rationale for the size of the "laws of physics" being small evaporates. The initial state might be enormous, for all anyone knows. The laws of physics could be infinite too - but there the hope that they are small and finite seems slightly more reasonable.

Henry: "Both of those concepts seem completely apt for describing perfectly deterministic systems. But, in describing the "complexity" of the universe even in something as simple as the 'pattern of stars that exists' one would still have to take into account potential non-deterministic factors such as human behavior. [...] [A]re you saying that you are a strict determinist?"

I'll take this one. Yes, we're presuming determinism here, although the determinism of the Many Worlds Interpretation is a little different from the single-world det... (read more)

POSTSCRIPT-- Strike that stray does in the penultimate sentence. And re compatibilism, the short version is that no, you don't have free will, but you shouldn't feel bad about it.

A halting oracle is usually said to output 1s or 0s, not proofs or halting times, right?

It's easy to use such an oracle to produce proofs and halting times. The following assumes that the oracle outputs 1 if the input TM halts, and 0 otherwise.

For proofs: Write a program p which on inputs x and i, enumerates all proofs. If it finds a proof for x, and the i-th bit of that proof is 1, then it halts, otherwise it loops forever. Now query the oracle with (p,x,0), (p,x,1), ..., and you get a proof for x if it has a proof.

Halting times: Write a program p which o... (read more)

I don't recall if I've mentioned this before, but Solomonoff induction in the mixture form makes no mention of the truth of its models. It just says that any computable probability distribution is in the mixture somewhere, so you can do as well as any computable form of cognitive uncertainty up to a constant. Eliezer, if what you say is true, then it shouldn't be possible for anyone, using just a Turing machine, to beat Solomonoff Induction at a pure prediction game (by more than a constant), even if the input sequence is uncomputable. But here is a count

Good question, Eliezer. If the human mathematician is computable, why isn't it already incorporated into Solomonoff Induction? It seems to me that the human mathematician does not behave like a Bayesian. Let H be the hypothesis that the input sequence is the unary encodings of Busy Beaver numbers. The mathematician will try to estimate, as best as he can, P(next symbol is 0|H). But when the next symbol turns out to be 1, he doesn't do a Bayesian update and decrease P(H), but instead says "Ok, so I was wrong. The next Busy Beaver number is bigger than ... (read more)

@ Wei. Thanks for the response. I will look at the refs but haven't yet done so. I'm skeptical about whether they'll change my mind on the subject, but I'll take a look.

It seems to me that the belief in pure determinism is axiomatic (I'm not even sure what it means to be a Bayesian in a purely deterministic system!), so most of this discussion appears to be pure conjecture. I'm not sure that it has any meaningful application.

@Wei: p(n) will approach arbitrarily close to 0 as you increase n.

This doesn't seem right. A sequence that requires knowledge of BB(k), has O(2^-k) probability according to our Solomonoff Inductor. If the inductor compares a BB(k)-based model with a BB(k+1)-based model, then BB(k+1) will on average be about half as probable as BB(k).

In other words, P(a particular model of K-complexity k is correct) goes to 0 as k goes to infinity, but the conditional probability, P(a particular model of K-complexity k is correct | a sub-model of that particular model with K-complexity k-1 is correct), does not go to 0 as k goes to infinity.

After further thought, I need to retract my last comment. Consider P(next symbol is 0|H) again, and suppose you've seen 100 0's so far, so essentially you're trying to predict BB(101). The human mathematician knows that any non-zero number he writes down for this probability would be way too big, unless he resorts to non-constructive notation like 1/BB(101). If you force him to answer "over and over, what their probability of the next symbol being 0 is" and don't allow him to use notation like 1/BB(101) then he'd be forced to write down an inconsistent probability distribution. But in fact the distribution he has in mind is not computable, and that explains how he can beat Solomonoff Induction.

Rolf, I was implicitly assuming that even knowing BB(k), it still takes O(k) bits to learn BB(k+1). But if this assumption is incorrect, then I need to change the setup of my prediction game so that the input sequence consists of the unary encodings of BB(1), BB(2), BB(4), BB(8), …, instead. This shouldn’t affect my overall point, I think.

"produced by inductive natural selection;"

Natural selection is not an inductive process.

In practice, natural selection produces intelligent agents, who can do induction, and then they make selective choices that affect who lives, who dies and who reproduces. What that means is that any idea that evolution (or natural selection) in the real world does not involve induction is not correct.

Nice try Tyler. What individuals "do" does not define what natural selection does.

One could also say: "In practice, natural selection produces intelligent agents, who can predict, and then they make selective choices that affect who lives, who dies and who reproduces."

That does NOT mean that natural selection operates via a predictive process. Ask any good biologist and they will tell you than natural selection is not predictive. That's why species go extinct all the time.

Natural selection is a non-inductive proces that can pro... (read more)

That does NOT mean that natural selection operates via a predictive process. [...]

It most certainly does - if we are talking about natural selection in evolution and nature.

Examples of failure to predict do not prove evolution does not predict - just that it does not predict perfectly.

The process of natural selection is "Trial and error" which is quite literally analogous to Poppers theory of falsification.

Heh. Not even science works via Popper's theory of falsification.

it is not at all clear the intelligent agent operates primarily using

You cite the same rhetoric by Dawkins that I do. Dawkins is one of those whom I am arguing against. The issue is simple enough not to require an appeal to authority to resolve. Natural selection does not involve prediction if it acts on a system which does not make predictions. It does involve prediction when it acts on a system which does make predictions. If the selecting agent is intelligent, and thinks about the future, you cannot fully model the resulting process without modelling that predictive process. Tree leaves do not make predictions - and I n... (read more)

Tim,

I think you are confusing an emergent property created by a system with how the system operates on its own. That architects draft blueprints that result in public housing projects that leads to forced living relationships that makes it hard to evict drug dealers doesn’t mean that the discipline of architecture runs on drug dealing. Even if that drug dealing impacts the architects.

You seem to have written some simulations of natural selection. In writing those algorithms did you have to code in the ability to predict? Of course not... (read more)

Since the dolphin has evolved flippers would you therefore say that “natural selection operates on flippers?”

Yes - provided we were talking about a process that included dolphin evolution. The process of natural selection in general does not necessarily involve prediction - but from the beginning I have been discussing natural selection in evolution - which does involve prediction.

Regarding Popper, I won't say much - since I would be preaching to the choir on this blog. I recommend you start by going and reading the bit beginning: "Previously,... (read more)

"Since the dolphin has evolved flippers would you therefore say that “natural selection operates on flippers?"

"Yes - provided we were talking about a process that included dolphin evolution."

I'm flabbergasted by this response. There is nothing inherent about natural selection that is going to give you flippers. That's dependent on the environment, the mutations, and other factors. The process itself has no "flippers". It's a process that works fine without flippers, yet you insist that natural selection "opera... (read more)

Re: the philosophy of science - you don't seem to get it. Oh well, I tried.

Re: evolution and natural selection - yes there is "the class of all things which involve inheritance, variation and selection". However, there is also the instance of evolution via natural selection running on our planet. You have consistently talked about the former. I have consistently talked about the latter. I figure that once you stop attempting to apply my comments to the former notion - which they simply do not apply to - our disagreement here will evaporate.

T... (read more)

Tim,

The problem is that there is no mechanism in the process of natural selection for stuffing that foresight generated by brains back into the genome. Learn all you want but it isn't passed onto you kids via your genes. That's the rub. That's why natural selection is blind to the future. The idea that natural selection is blind is perfectly accurate.

That's also true for the small minority of organisms on the planet that actually have brains that predict the future. So I am talking about the instance of natural selection operating on this planet. Y... (read more)

The problem is that there is no mechanism in the process of natural selection for stuffing that foresight generated by brains back into the genome. Learn all you want but it isn't passed onto you kids via your genes. That's the rub.

There most certainly are such mechanisms. Most obviously sexual selection and the Baldwin effect - but also via vanilla natural selection. I explain this issue clearly enough in my essay.

No, sexual selection does not determine which mutations occur. It's merely a reinforcing feedback loop that is actually considered an example of how evolution can run off the rails, so to speak.

Sure females might by accident happen to pick a feature in males that might prove adaptive. Unfortunately for your argument it is not based on prediction, but is happenstance. Even were the "correct" feature choosen initially there is then the tendency of sexual selection to over select for the feature merely because other females find the feature att... (read more)

To me, this is all as silly as the Chinese Room. Naive intuition says that the Chinese Room doesn't understand chinese in the way we do... because it doesn't, running as it does on exponentially slower timescales. Similarly, I can close the box on my turing machine and it can, over mega-exponential timescales and using mega-exponential amounts of tape, get just as much smarter as it could by looking around and talking to people, but there are still several reasons why the latter strategy is superior.

Of course, I may just not understand what is supposed to ... (read more)

I hardly know where to begin - and I'm not sure I should bother, since we are not really very on topic, and this has gone on long enough already. Very briefly:

1. Nobody ever claimed that sexual selection determines which mutations occur. Sexual selection provides examples of selection by intelligent agents affecting the course of evolution;

2. Sexual selection is not "happenstance". Females tend to pick traits that - among other things - reflect male health, fitness, and resistance to parasite infestations;

3. The cases where sexual selection drive... (read more)

Associating Kolmogorov complexity with intelligence causes brain damage: Example.