The Power of Intelligence

It is often implicitly assumed that the power of a superintelligence will be practically unbounded. There seems like there could be “ample headroom” above humans, i.e. that a superintelligence will be able to vastly outperform us across virtually all domains.

By “superintelligence,” I mean something which has arbitrarily high cognitive ability, or an arbitrarily large amount of compute, memory, bandwidth, etc., but which is bound by the physical laws of our universe.[1] There are other notions of “superintelligence” which are weaker than this. Limitations of the abilities of this superintelligence would also apply to anything less intelligent.

There are some reasons to believe this assumption. For one, it seems a bit suspicious to assume that humans have close to the maximal possible intelligence. Secondly, AI systems already outperform us in some tasks,[2] so why not suspect that they will be able to outperform us in almost all of them? Finally, there is a more fundamental notion about the predictability of the world, described most famously by Laplace in 1814:

Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it – an intelligence sufficiently vast to submit this data to analysis – it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present in its eyes.[3]

We are very far from completely understanding, and being able to manipulate, everything we care about. But if the world is as predictable as Laplace suggests, then we should expect that a sufficiently intelligent agent would be able to take advantage of that regularity and use it to excel at any domain.

This investigation questions that assumption. Is it actually the case that a superintelligence has practically unbounded intelligence, or are there “ceilings” on what intelligence is capable of? To foreshadow a bit, there are ceilings in some domains that we care about, for instance, in predictions about the behavior of the human brain. Even unbounded cognitive ability does not imply unbounded skill when interacting with the world. For this investigation, I focus on cognitive skills, especially predicting the future. This seems like a realm where a superintelligence would have an unusually large advantage (compared to e.g. skills requiring dexterity), so restrictions on its skill here are more surprising.

There are two ways for there to be only a small amount of headroom above human intelligence. The first is that the task is so easy that humans can do it almost perfectly, like playing tic-tac-toe. The second is that the task is so hard that there is a “low ceiling”: even a superintelligence is incapable of being very good at it. This investigation focuses on the second.

There are undoubtedly many tasks where there is still ample headroom above humans. But there are also some tasks for which we can prove that there is a low ceiling. These tasks provide some limitations on what is possible, even with arbitrarily high intelligence.

Chaos Theory

The main tool used in this investigation is chaos theory. Chaotic systems are things for which uncertainty grows exponentially in time. Most of the information measured initially is lost after a finite amount of time, so reliable predictions about its future behavior are impossible.

A classic example of chaos is the weather. Weather is fairly predictable for a few days. Large simulations of the atmosphere have gotten consistently better for these short-time predictions.[4]

After about 10 days, these simulations become useless. The predictions from the simulations are worse than guessing what the weather might be using historical climate data from that location.

Chaos theory provides a response to Laplace. Even if it were possible to exactly predict the future given exact initial conditions and equations of motion,[5] chaos makes it impossible to approximately predict the future using approximate initial conditions and equations of motion. Reliable predictions can only be made for a short period of time, but not once the uncertainty has grown large enough.

There is always some small uncertainty. Normally, we do not care: approximations are good enough. But when there is chaos, the small uncertainties matter. There are many ways small uncertainties can arise: Every measuring device has a finite precision.[6] Every theory should only be trusted in the regimes where it has been tested. Every algorithm for evaluating the solution has some numerical error. There are external forces you are not considering that the system is not fully isolated from. At small enough scales, thermal noise and quantum effects provide their own uncertainties. Some of this uncertainty could be reduced, allowing reliable predictions to be made for a bit longer.[7] Other sources of this uncertainty cannot be reduced. Once these microscopic uncertainties have grown to a macroscopic scale, the motion of the chaos is inherently unpredictable.

Completely eliminating the uncertainty would require making measurements with perfect precision, which does not seem to be possible in our universe. We can prove that fundamental sources of uncertainty make it impossible to know important things about the future, even with arbitrarily high intelligence. Atomic scale uncertainty, which is guaranteed to exist by Heisenberg’s Uncertainty Principle, can make macroscopic motion unpredictable in a surprisingly short amount of time. Superintelligence is not omniscience.

Chaos theory thus allows us to rigorously show that there are ceilings on some particular abilities. If we can prove that a system is chaotic, then we can conclude that the system offers diminishing returns to intelligence. Most predictions of the future of a chaotic system are impossible to make reliably. Without the ability to make better predictions, and plan on the basis of these predictions, intelligence becomes much less useful.

This does not mean that intelligence becomes useless, or that there is nothing about chaos which can be reliably predicted. 

For relatively simple chaotic systems, even when what in particular will happen is unpredictable, it is possible to reliably predict the statistics of the motion.[8] We have learned sophisticated ways of predicting the statistics of chaotic motion,[9] and a superintelligence could be better at this than we are. It is also relatively easy to sample from this distribution to emulate behavior which is qualitatively similar to the motion of the original chaotic system.

But chaos can also be more complicated than this. The chaos might be non-stationary, which means that the statistical distribution and qualitative description of the motion themselves change unpredictably in time. The chaos might be multistable, which means that it can do statistically and qualitatively different things depending on how it starts. In these cases, it is also impossible to reliably predict the statistics of the motion, or to emulate a typical example of a distribution which is itself changing chaotically. Even in these cases, there are sometimes still patterns in the chaos which allow a few predictions to be made, like the energy spectra of fluids.[10] These patterns are hard to find, and it is possible that a superintelligence could find patterns that we have missed. But it is not possible for the superintelligence to recover the vast amount of information rendered unpredictable by the chaos.

This Investigation

This blog post is the introduction to an investigation which explores these points in more detail. I will describe what chaos is, how humanity has learned to deal with chaos, and where chaos appears in things we care about – including in the human brain itself. Links to the other pages, blog posts, and report that constitute this investigation can be found below.

Most of the systems we care about are considerably messier than the simple examples we use to explain chaos. It is more difficult to prove claims about the inherent unpredictability of these systems, although it is still possible to make some arguments about how chaos affects them.

For example, I will show that individual neurons, small networks of neurons, and in vivo neurons in sense organs can behave chaotically.[11] Each of these can also behave non-chaotically in other circumstances. But we are more interested in the human brain as a whole. Is the brain mostly chaotic or mostly non-chaotic? Does the chaos in the brain amplify uncertainty all the way from the atomic scale to the macroscopic, or is the chain of amplifying uncertainty broken at some non-chaotic mesoscale? How does chaos in the brain actually impact human behavior? Are there some things that brains do for which chaos is essential?

These are hard questions to answer, and they are, at least in part, currently unsolved. They are worth investigating nevertheless. For instance, it seems likely to me that the chaos in the brain does render some important aspects of human behavior inherently unpredictable and plausible that chaotic amplification of atomic-level uncertainty is essential for some of the things humans are capable of doing.

This has implications for how humans might interact with a superintelligence and for how difficult it might be to build artificial general intelligence.

If some aspects of human behavior are inherently unpredictable, that might make it harder for a superintelligence to manipulate us. Manipulation is easier if it is possible to predict how a human will respond to anything you show or say to them. If even a superintelligence cannot predict how a human will respond in some circumstances, then it is harder for the superintelligence to hack the human and gain precise, long-term control over them.

So far, I have been considering the possibility that a superintelligence will exist and asking what limitations there are on its abilities.[12] But chaos theory might also change our estimates of the difficulty of making artificial general intelligence (AGI) that leads to superintelligence. Chaos in the brain makes whole brain emulation on a classical computer wildly more difficult – or perhaps even impossible.

When making a model of a brain, you want to coarse-grain it at some scale, perhaps at the scale of individual neurons. The coarse-grained model of a neuron should be much simpler than a real neuron, involving only a few variables, while still being good enough to capture the behavior relevant for the larger scale motion. If a neuron is behaving chaotically itself, especially if it is non-stationary or multistable, then no good enough coarse-grained model will exist. The neuron needs to be resolved at a finer scale, perhaps at the scale of proteins. If a protein itself amplifies smaller uncertainties, then you would have to resolve it at a finer scale, which might require a quantum mechanical calculation of atomic behavior. 

Whole brain emulation provides an upper bound on the difficulty of AGI. If this upper bound ends up being farther away than you expected, then that suggests that there should be more probability mass associated with AGI being extremely hard.

I will explore these arguments, and others, in the remainder of this investigation. Currently, this investigation consists of one report, two Wiki pages, and three blog posts.


  • Chaos and Intrinsic Unpredictability. Background reading for the investigation. An explanation of what chaos is, some other ways something can be intrinsically unpredictable, different varieties of chaos, and how humanity has learned to deal with chaos.

Wiki Pages:

  • Chaos in Humans. Some of the most interesting things to try to predict are other humans. I discuss whether humans are chaotic, from the scale of a single neuron to society as a whole.
  • AI Safety Arguments Affected by Chaos. A list of the arguments I have seen within the AI safety community which our understanding of chaos might affect.

Blog Posts:

Other Resources

If you want to learn more about chaos theory in general, outside of this investigation, here are some sources that I endorse:

  • Undergraduate Level Textbook:
    S. Strogatz. Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, and Engineering. (CRC Press, 2000).
  • Graduate Level Textbook:
    P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner and G. Vattay, Chaos: Classical and Quantum. (Niels Bohr Institute, Copenhagen 2020).
  • Wikipedia has a good introductory article on chaos. Scholarpedia also has multiple good articles, although no one obvious place to start.
  • What is Chaos? sequence of blog posts by The Chaostician.


  1. ^

    In this post, “we” refers to humanity, while “I” refers to the authors: Jeffrey Heninger and Aysja Johnson.

  2. ^
  3. ^

    The quote continues: “The human mind offers, in the perfection which it has been able to give to astronomy, a feeble idea of this intelligence. Its discoveries in mechanics and geometry, added to that of universal gravity, have enabled it to comprehend in the same analytic expressions the past and future states of the system of the world. Applying the same method to some other objects of its knowledge, it has succeeded in referring to general laws observed phenomena and in foreseeing those which given circumstances ought to produce. All these efforts in the search for truth tend to lead it back continually to the vast intelligence which we have just mentioned, but from which it will always remain infinitely removed. This tendency, peculiar to the human race, is that which renders it superior to animals; and their progress in this respect distinguishes nations and ages and constitutes their true glory.”

    Laplace. Philosophical Essay on Probabilities. (1814) p. 4.

  4. ^

    Interestingly, the trend appears linear. My guess is that the linear trend is a combination of exponentially more compute being used and the problem getting exponentially harder.

    Nate Silver. The Signal and the Noise. (2012) p. 126-132.

  5. ^

    Whether or not this statement of determinism is true is a perennial debate among scholars. I will not go into it here.

  6. ^

    The most precise measurement ever is of the magnetic moment of the electron, with 9 significant digits.

    NIST Reference on Constants, Units, and Uncertainty.

  7. ^

    Because the uncertainty grows exponentially with time, if you try to make longer-term predictions by reducing the initial uncertainty, you will only get logarithmic returns.

  8. ^

    If the statistics are predictable, this can allow us to make a coarse-grained model for the behavior at a larger scale which is not affected by the uncertainties amplified by the chaos.

  9. ^

    Described in the report Chaos and Intrinsic Unpredictability.

  10. ^
  11. ^

    The evidence for this can be found in Chaos in Humans.

  12. ^

    This possibility probably takes up too much of our thinking, even prior to these arguments.

    Wulfson. The tyranny of the god scenario. AI Impacts. (2018)

New Comment
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I don’t see anything in this to put a useful limit on what a superintelligence can do. We humans also have to deal with chaotic systems such as the weather. We respond not simply by trying to predict the weather better and better (although that helps so far as it goes) but by developing ways of handling whatever weather happens. In your own example of pinball, the expert player tries to keep the machine in a region of state space where the player can take actions to keep it in that space, avoiding the more chaotic regions.

Intelligence is not about being a brain in a vat and predicting things. It is about doing things to funnel the probability mass of the universe in intended directions.

It seems like your comment is saying something like:

These restrictions are more relevant to an Oracle than to other kinds of AI.

Even an Oracle can act by answering questions in whatever way will get people to further its intentions.

If the AI is deliberately making things happen in the world, then I would say it’s not an Oracle, it’s an Agent whose I/O channel happens to involve answering questions. (Maybe the programmers intended to make an Oracle, but evidently they failed!)

My response to @Jeffrey Heninger would have been instead:

“If you have an aligned Oracle, then you wouldn’t ask it to predict unpredictable things. Instead you would ask it to print out plans to solve problems—and then it would come up with plans that do not rely on predicting unpredictable things.”

The title is “superintelligence is not omniscience”. Then the first paragraph says that we’re talking about the assumption “There is “ample headroom” above humans.” But these are two different things, right? I think there is ample headroom above humans, and I think that superintelligence is not omniscience. I think it’s unhelpful to merge those together into one blog post. I think it’s fine to write a post about “Things that even superintelligent AI can’t do” and it’s fine to write a post “Comparing capabilities between superintelligent AIs & humans / groups-of-humans”, but to me, those seem like they should be two different posts.

For example, as I discussed here, it will eventually be possible to make an AI that can imitate the input-output behavior of 10 trillion unusually smart and conscientious humans, each running at 100× human speed and working together (and in possession of trillions of teleoperable robot bodies spread around the world). That AI will not be omniscient, but it would certainly illustrate that there’s ample headroom.

I will show that individual neurons, small networks of neurons, and in vivo neurons in sense organs can behave chaotically. Each of these can also behave non-chaotically in other circumstances. But we are more interested in the human brain as a whole. Is the brain mostly chaotic or mostly non-chaotic? Does the chaos in the brain amplify uncertainty all the way from the atomic scale to the macroscopic, or is the chain of amplifying uncertainty broken at some non-chaotic mesoscale? How does chaos in the brain actually impact human behavior? Are there some things that brains do for which chaos is essential?

See here. The human brain does certain a-priori-specifiable and a-priori-improbable things, like allowing people to travel to the moon and survive in Antarctica. There has to be some legible reason that they can do those things—presumably it has certain learning algorithms that can systematically pick up on environmental regularities, blah blah. Whatever that legible reason is, I claim we can write computer code that operates on the same principles. I don’t think these algorithms can “rely on chaos” in a sense that can’t be replaced by an RNG, because the whole point of chaos is that it won’t have any predictable useful consequences (beyond the various useful things you can do with an RNG). So if you’re going to make an argument about the difficulty of AGI on this basis, I’m skeptical. (If you’re going to make an argument that you can’t forecast a very specific human’s exact thoughts and behaviors hours and days into the future, then sure, chaos is relevant; but that would be a crazy thing to expect anyway.)

Another way to describe chaotic systems is steerable systems. The fact that they have sensitive dependence on initial conditions means that if you know the dynamics and current state of the system, you can steer them into future knowable states with arbitrarily weak influence.


If you knew the precise dynamics and state of a classically chaotic system, you could predict it. If it's unpredictable in practice, you don't know those things.

To clarify further: Without any steering, any finite level of precision in a chaotic system means that you have a corresponding finite horizon beyond which you have essentially zero information about the state of the system.

If you can influence the system even a tiny bit, there exists a finite precision of measurement and modelling that allows you to not just predict, but largely control the states as far into the future as you like.

It's helpful to avoid second-person in statements like this.  It matters a whole lot WHO is doing the predicting, and at least some visions of "superintelligent" include a relative advantage in collecting and processing dynamics and state information about systems.

Just because YOU can't predict it at all doesn't mean SOMETHING can't predict it a bit.


I don't use "you" to mean."me".

The main point is that "you" is the same in both cases it might a well be "X".

There's no free ability of an agent to control beyond that agent's ability to predict.

That's my confusion - why is "you" necessarily the same in both cases?  Actually, what are the two cases again?  In any case, the capabilities of a superintelligence with respect to comprehending and modeling/calculating extremely complex (chaotic) systems is exactly the sort of thing that is hard to know in advance.

There are LOTS of free lunches, from the perspective of currently-inefficient human-modeled activities. Tons of outcomes that machines can influence with more subtlety than humans can handle, toward outcomes that humans can define and measure just fine.


That’s my confusion—why is “you” necessarily the same in both cases?

Because those are the cases I am talking about.

intelligence with respect to comprehending and modeling/calculating extremely complex (chaotic) systems is exactly the sort of thing that is hard to know in advance.

I didn't say anything about superintellences.

yeah coming back to this again, something seems very wrong with this to me. if you know a lot about the system you can make a big ripple but if there are active controllers with tighter feedback loops they can compensate for your impact with much less intelligence unless your impact can reliably disable them. if they can make themselves reliably unpredictable to you eg by basing decisions on high quality randomness that they can trust you can't influence (eg in a deterministic universe this might be the low bits of an isolated highly chaotic system), then they can make it extremely hard for your small intervention to accumulate into an impact that affects them - it can be made nearly impossible to interfere with another agent unless you manage to yourself inject an agent glider into the chaotic system, ie induce self repairing behavior that can implement closed loop control towards the outcomes you initially intended to achieve. certainly you don't need to vary that many dimensions in order to get a fluid simulator to end up hitting a complicated target, but it gets less tractable fast if you aren't allowed to keep checking back in and interfering again.

agreed, but only with ongoing intervention. if a system is chaotic, losing connection with it means it will stop doing what you said to.

It's not obvious that any system is chaotic at a physical level, to all theoretically possible measurement and prediction capabilities.  It's possible there's quantum uncertainty and deterministic causality only, and "chaos" as determined-but-incalculable behavior is a description of the observer's relationship to a phenomenon, not the phenomenon itself.  

The question is whether a given superintelligence is powerful enough to comprehend and predict some important systems which are chaotic to current human capabilities.

chaos is not randomness. a deterministic universe still has sensitive dependence on initial conditions, the key trait of chaos. fluid dynamics is chaotic, so even arbitrarily superintelligent reasoners can't get far ahead of physics before the sensitive dependence makes your prediction mismatch reality. this is true even if your mechanistic understanding is perfect and the universe isn't random, so long as the system is in a chaotic regime and you don't have perfect knowledge of its starting state.

I was sort of with you until 

If some aspects of human behavior are inherently unpredictable, that might make it harder for a superintelligence to manipulate us.

Humans are very hackable and easy to manipulate. We can see the proof of it all the time and in almost every person. The manipulator does not need to be superintelligent, a lot of humans are already very good at it. So, yes, it is possible that there is a predictability ceiling of some sort in the universe. No, that would no save us from sufficiently smart, fast and determined non-human agents.

Well it's mildly reassuring, at least. if a weakly superintelligent system is not able to craft interventions that guide a human into an attractor durably, and instead needs ongoing interventions due to chaos in the human's mind (a big if, given even merely human level intelligence), then a nature retreat is enough to clear their mind of the influence.

this is most relevant for considering what it's reasonable to assume an ai can be aimed at, not to constrain the much wider space of things an ai might do if destructive.

I feel like this is making a claim that sounds reasonable denotation wise in order to imply something misleading connotation wise. Sure, I can't exercise my intelligence to get unbounded outcomes over others, but that's often because I expect to be stopped by those close to me in intelligence. Consider a sociopathic Einstein running a ward full of people with down syndrome with zero other oversight. Sure, his influence isn't literally unbounded, but he doesn't find it difficult to shape their behavior or kill them if he wants, regardless of their actions against him. Having wildly divergent physical capabilities that can't be understood eg 'has infrared cameras and can thus see in the dark' can be used to extend the metaphor.