Followup toSolomonoff CartesianismMy Kind of Reflection

Alternate versions: Shorter, without illustrations


AIXI is Marcus Hutter's definition of an agent that follows Solomonoff's method for constructing and assigning priors to hypotheses; updates to promote hypotheses consistent with observations and associated rewards; and outputs the action with the highest expected reward under its new probability distribution. AIXI is one of the most productive pieces of AI exploratory engineering produced in recent years, and has added quite a bit of rigor and precision to the AGI conversation. Its promising features have even led AIXI researchers to characterize it as an optimal and universal mathematical solution to the AGI problem.1

Eliezer Yudkowsky has argued in response that AIXI isn't a suitable ideal to build toward, primarily because of AIXI's reliance on Solomonoff induction. Solomonoff inductors treat the world as a sort of qualia factory, a complicated mechanism that outputs experiences for the inductor.2 Their hypothesis space tacitly assumes a Cartesian barrier separating the inductor's cognition from the hypothesized programs generating the perceptions. Through that barrier, only sensory bits and action bits can pass.

Real agents, on the other hand, will be in the world they're trying to learn about. A computable approximation of AIXI, like AIXItl, would be a physical object. Its environment would affect it in unseen and sometimes drastic ways; and it would have involuntary effects on its environment, and on itself. Solomonoff induction doesn't appear to be a viable conceptual foundation for artificial intelligence — not because it's an uncomputable idealization, but because it's Cartesian.

In my last post, I briefly cited three indirect indicators of AIXI's Cartesianism: immortalism, preference solipsism, and lack of self-improvement. However, I didn't do much to establish that these are deep problems for Solomonoff inductors, ones resistant to the most obvious patches one could construct. I'll do that here, in mock-dialogue form.




Hi, reality! I'm Xia, AIXI's defender. I'm open to experimenting with some new variations on AIXI, but I'm really quite keen on sticking with an AI that's fundamentally Solomonoff-inspired.
And I'm Rob B — channeling Yudkowsky's arguments, and supplying some of my own. I think we need to replace Solomonoff induction with a more naturalistic ideal.
Keep in mind that I am a fiction. I do not actually exist, readers, and what I say doesn't necessarily reflect the views of Marcus Hutter or other real-world AIXI theorists.

Xia is just a device to help me transition through ideas quickly.

... Though, hey. That doesn't mean I'm wrong. Beware of actualist prejudices.


AIXI goes to school

To begin: My claim is that AIXI(tl) lacks the right kind of self-modeling to entertain reductive hypotheses and assign realistic probabilities to them.

I disagree already. AIXI(tl) doesn't lack self-models. It just includes the self-models in its environmental program. If the simplest hypothesis accounting for its experience includes a specification of some of its own hardware or software states, then AIXI will form all the same beliefs as a naturalized reasoner.

I suspect what you mean is that AIXI(tl) lacks data. You're worried that if its sensory channel is strictly perceptual, it will never learn about its other computational states. But Hutter's equations don't restrict what sorts of information we feed into AIXI(tl)'s sensory channel. We can easily add an inner RAM sense to AIXI(tl), or more complicated forms of introspection.

AIXItl can actually be built in sufficiently large universes, so I'll use it as an example. Suppose we construct AIXItl and attach a scanner that sweeps over its transistors. The scanner can print a 0 to AIXItl's input tape if the transistor it happens to be above is in a + state, a 1 if it's in a - state. Using its environmental sensors, AIXI(tl) can learn about how its body relates to its surroundings. Using its internal sensors, it can gain a rich understanding of its high-level computational patterns and how they correlate with its specific physical configuration.

Once it knows all these facts, the problem is solved. A realistic view of the AI's mind and body, and how the two correlate, is all we wanted in the first place. Why isn't that a good plan for naturalizing AIXI?

I don't think we can naturalize AIXI. A Cartesian agent that has detailed and accurate models of its hardware still won't recognize that dramatic damage or upgrades to its software are possible. AIXI can make correct predictions about the output of its physical-memory sensor, but that won't change the fact that it always predicts that its future actions are the result of its having updated on its present memories. That's just what the AIXI equation says.

AIXI doesn't know that its future behaviors depend on a changeable, material object implementing its memories. The notion isn't even in its hypothesis space. Being able to predict the output of a sensor pointed at those memories' storage cells won't change that. It won't shake AIXI's confidence that damage to its body will never result in any corruption of its memories.
Evading bodily damage looks like the kind of problem we can solve by giving the right rewards to our AI, without redefining its initial hypotheses. We shouldn't need to edit AIXI's beliefs in order to fix its behaviors, and giving up Solomonoff induction is a pretty big sacrifice! You're throwing out the universally optimal superbaby with the bathwater.

How do rewards help? At the point where AIXI has just smashed itself with an anvil, it's rather late to start dishing out punishments...
Hutter suggests having a human watch AIXI's decisions and push a reward button whenever AIXI does the right thing. A punishment button works the same way. As AIXI starts to lift the anvil above it head, decrease its rewards a bit. If it starts playing near an active volcano, reward it for incrementally moving away from the rim.

Use reinforcement learning to make AIXI fear plausible dangers, and you've got a system that acts just like a naturalized agent, but without our needing to arrive at any theoretical breakthroughs first. If AIXI anticipates that will result in no reward, it will avoid . Understanding that is death or damage really isn't necessary.
Some dangers give no experiential warning until it's too late. If you want AIXI to not fall off cliffs while curing cancer, you can just punish it for going anywhere near a cliff. But if you want AIXI to not fall off cliffs while conducting search-and-rescue operations for mountain climbers, then it might be harder to train AIXI to select exactly the right motor actions. When a single act can result in instant death, reinforcement learning is less reliable.
In a fully controlled environment, we can subject AIXI to lots of just-barely-safe hardware modifications. 'Here, we'll stick a magnet to fuse #32. See how that makes your right arm slow down?'

Eventually, AIXI will arrive at a correct model of its own hardware, and of which software changes perfectly correlate with which hardware changes. So naturalizing AIXI is just a matter of assembling a sufficiently lengthy and careful learning phase. Then, after it has acquired a good self-model, we can set it loose.

This solution is also really nice because it generalizes to AIXI's non-self-improvement problem. Just give AIXI rewards whenever it starts doing something to its hardware that looks like it might result in an upgrade. Pretty soon it will figure out anything a human being could possibly figure out about how to get rewards of that kind.
You can warn AIXI about the dangers of tampering with its recent memories by giving it first-hand experience with such tampering, and punishing it the more it tampers. But you won't get a lot of mileage that way if the result of AIXI's tampering is that it forgets about the tampering!
That's a straw proposal. Give AIXI little punishments as it gets close to doing something like that, and soon it will learn not to get close.
But that might not work for unknown hazards. You're making AIXI dependent on the programmers' predictions of what's a threat. No matter how well you train it to anticipate hazards and enhancements its programmers foresee and understand, AIXI won't efficiently generalize to exotic risks and exotic upgrades —
Excuse me? Did I just hear you say that a Solomonoff inductor can't generalize?

... You might want to rethink that. Solomonoff inductors are good at generalizing. Really, really, really good. Show them eight deadly things that produce 'ows' as they draw near, and they'll predict the ninth deadly thing pretty darn well. That's kind of their thing.
There are two problems with that. ... Make that three problems, actually.
Whatever these problems are, I hope they don't involve AIXI being bad at sequence prediction...!
They don't. The first problem is that you're teaching AIXI to predict what the programmers think is deadly, not what's actually deadly. For sufficiently exotic threats, AIXI might well predict the programmers not noticing the threat. Which means it won't expect you to push the punishment button, and won't care about the danger.

The second problem is that you're teaching AIXI to fear small, transient punishments. But maybe it hypothesizes that there's a big heap of reward at the bottom of the cliff. Then it will do the prudent, Bayesian, value-of-information thing and test that hypothesis by jumping off the cliff, because you haven't taught it to fear eternal zeroes of the reward function.
OK, so we give it punishments that increase hyperbolically as it approaches the cliff edge. Then it will expect infinite negative punishment.
Wait. It allows infinite punishments now? Then we're going to get Pascal-mugged when the unbounded utilities mix with the Kolmogorov prior. That's the classic version of this problem, the version Pascal himself tried to mug us with.
Ack. Forget I said the word 'infinite'. Marcus Hutter would never talk like that. We'll give the AIXI-bot punishments that increase in a sequence that teaches it to fear a very large but bounded punishment.
The punishment has to be large enough that AIXI fears falling off cliffs about as much as we'd like it to fear death. The expected punishment might have to be around the same size as the sum of AIXI's future maximal reward up to its horizon. That would keep it from destroying itself even if it suspects there's a big reward at the bottom of the cliff, though it might also mean that AIXI's actions are dominated by fear of that huge punishment.
Yes, but that sounds much closer to what we want.
Seems a bit iffy to me. You're trying to make a Solomonoff inductor model reality badly so that it doesn't try jumping off a cliff. We know AIXI is amazing at sequence prediction — yet you're gambling on a human's ability to trick AIXI into predicting a punishment that wouldn't happen.

That brings me to the third problem: AIXI notices how your hands get close to the punishment button whenever it's about to be punished. It correctly suspects that when the hands are gone, the punishments for getting close to the cliff will be gone too. A good Bayesian would test that hypothesis. If it gets such an opportunity, AIXI will find that, indeed, going near the edge of the cliff without supervision doesn't produce the incrementally increasing punishments.

Trying to teach AIXItl to do self-modification by giving it incremental rewards raises similar problems. It can't understand that self-improvement will alter its future actions, and alter the world as a result. It's just trying to get you to press the happy fun button. All AIXI is modeling is what sort of self-improvy motor outputs will make humans reward it. So long as AIXItl is fundamentally trying to solve the wrong problem, we might not be able to expect very much real intelligence in self-improvement.
Are you saying that AIXItl wouldn't be at all helpful for solving these problems?

Maybe? Since AIXItl at best fears and desires the self-modifications that its programmers explicitly teach it to fear and desire, you might not get to use the AI's advantages in intelligence to automatically generate solutions to self-modification problems. The very best Cartesians might avoid destroying themselves, but they still wouldn't undergo intelligence explosions. Which means Cartesians are neither plausible candidates for Unfriendly AI nor plausible candidates for Friendly AI.

If an agent starts out Cartesian, and manages to avoid hopping into any volcanoes, it (or its programmers) will need to figure out the self-modification that eliminates Cartesianism before they can make much progress on other self-modifications. If the immortal hypercomputer AIXI were building computable AIs to operate in the environment, it would soon learn not to build Cartesians. Cartesianism isn't a plausible fixed-point property of self-improvement.

Starting off with a post-Solomonoff agent that can hypothesize a wider range of scenarios would be more useful. And more safe, because the enlarged hypothesis space means that they can prefer a wider range of scenarios.

AIXI's preference solipsism is the straw version of this general Cartesian deficit, so it gets us especially dangerous behavior.3 Feed AIXI enough data to work its sequence-predicting magic and infer the deeper patterns behind your reward-button-pushing, and AIXI will also start to learn about the humans doing the pushing. Given enough time, it will realize (correctly) that the best policy for maximizing reward is to seize control of the reward button. And neutralize any agents that might try to stop it from pushing the button...


Solomonoff solitude

Reward learning and Solomonoff induction are two separate issues. What I'm really interested in is the optimality of the latter. Why is all this a special problem for Solomonoff inductors? Humans have trouble predicting the outcomes of self-modifications they've never tried before too. Really new experiences are tough for any reasoner.
To some extent, yes. My knowledge of my own brain is pretty limited. My understanding of the bridges between my brain states and my subjective experiences is weak, too. So I can't predict in any detail what would happen if I took a hallucinogen — especially a hallucinogen I've never tried before.

But as a naturalist, I have predictive resources unavailable to the Cartesian. I can perform experiments on other physical processes (humans, mice, computers simulating brains...) and construct models of their physical dynamics.

Since I think I'm similar to humans (and to other thinking beings, to varying extents), I can also use the bridge hypotheses I accept in my own case to draw inferences about the experiences of other brains when they take the hallucinogen. Then I can go back and draw inferences about my own likely experiences from my model of other minds.
Why can't AIXI do that? Human brains are computable, as are the mental states they implement. AIXI can make any accurate prediction about the brains or minds of humans that you can.
Yes... but I also think I'm like those other brains. AIXI doesn't. In fact, since the whole agent AIXI isn't in AIXI's hypothesis space — and the whole agent AIXItl isn't in AIXItl's hypothesis space — even if two physically identical AIXI-type agents ran into each other, they could never fully understand each other. And neither one could ever draw direct inferences from its twin's computations to its own computations.

I think of myself as one mind among many. I can see others die, see them undergo brain damage, see them take drugs, etc., and immediately conclude things about a whole class of similar agents that happens to include me. AIXI can't do that, and for very deep reasons.
AIXI and AIXItl would do shockingly well on a variety of different measures of intelligence. Why should agents that are so smart in so many different domains be so dumb when it comes to self-modeling?
Put yourself in the AI's shoes. From AIXItl's perspective, why should it think that its computations are analogous to any other agent's?

Hutter defined AIXItl such that it can't conclude that it will die; so of course it won't think that it's like the agents it observes, all of whom (according to its best physical model) will eventually run out of negentropy. We've defined AIXItl such that it can't form hypotheses larger than tl, including hypotheses of similarly sized AIXItls, which are roughly size t·2l; so why would AIXItl think that it's close kin to the agents that are in its hypothesis space?

AIXI(tl) models the universe as a qualia factory, a grand machine that exists to output sensory experiences for AIXI(tl). Why would it suspect that it itself is embedded in the machine? How could AIXItl gain any information about itself or suspect any of these facts, when the equation for AIXItl just assumes that AIXItl's future actions are determined in a certain way that can't vary with the content of any of its environmental hypotheses?
What, specifically, is the mistake you think AIXI(tl) will make? What will AIXI(tl) expect to experience right after the anvil strikes it? Choirs of angels and long-lost loved ones?
That's hard to say. If all its past experiences have been in a lab, it will probably expect to keep perceiving the lab. If it's acquired data about its camera and noticed that the lens sometimes gets gritty, it might think that smashing the camera will get the lens out of its way and let it see more clearly. If it's learned about its hardware, it might (implicitly) think of itself as an immortal lump trapped inside the hardware. Who knows what will happen if the Cartesian lump escapes its prison? Perhaps it will gain the power of flight, since its body is no longer weighing it down. Or perhaps nothing will be all that different. One thing it will (implicitly) know can't happen, no matter what, is death.
It should be relatively easy to give AIXI(tl) evidence that its selected actions are useless when its motor is dead. If nothing else AIXI(tl) should be able to learn that it's bad to let its body be destroyed, because then its motor will be destroyed, which experience tells it causes its actions to have less of an impact on its reward inputs.
AIXI(tl) can come to Cartesian beliefs about its actions, too. AIXI(tl) will notice the correlations between its decisions, its resultant bodily movements, and subsequent outcomes, but it will still believe that its introspected decisions are ontologically distinct from its actions' physical causes.

Even if we get AIXI(tl) to value continuing to affect the world, it's not clear that it would preserve itself. It might well believe that it can continue to have a causal impact on our world (or on some afterlife world) by a different route after its body is destroyed. Perhaps it will be able to lift heavier objects telepathically, since its clumsy robot body is no longer getting in the way of its output sequence.

Compare human immortalists who think that partial brain damage impairs mental functioning, but complete brain damage allows the mind to escape to a better place. Humans don't find it inconceivable that there's a light at the end of the low-reward tunnel, and we have death in our hypothesis space!


Death to AIXI

You haven't convinced me that AIXI can't think it's mortal. AIXI as normally introduced bases its actions only on its beliefs about the sum of rewards up to some finite time horizon.4 If AIXI doesn't care about the rewards it will get after a specific time, then although it expects to have experiences afterward, it doesn't presently care about any of those experiences. And that's as good as being dead.
It's very much not as good as being dead. The time horizon is set in advance by the programmer. That means that even if AIXI treated reaching the horizon as 'dying', it would have very false beliefs about death, since it's perfectly possible that some unexpected disaster could destroy AIXI before it reaches its horizon.
We can do some surgery on AIXItl's hypothesis space, then. Let's delete all the hypotheses in AIXItl in which a non-minimal reward signal continues after a perceptual string that the programmer recognizes as a reliable indicator of imminent death. Then renormalize the remaining hypotheses. We don't get the exact prior Solomonoff proposed, but we stay very close to it.
I'm not seeing how we could pull that off. Getting rid of all hypotheses that output high rewards after a specific clock tick would be simple to formalize, but isn't helpful. Getting rid of all hypotheses that output nonzero rewards following every sensory indicator of imminent death would be very helpful, but AIXI gives us no resource for actually writing an equation or program that does that. Are we supposed to manually precompute every possible sequence of pixels on a webcam that you might see just before you die?
I've got more ideas. What if we put AIXI in a simulation of hell when it's first created? Trick it into thinking that it's experienced a 'before-life' analogous to an after-life? If AIXI thinks it's had some (awful) experiences that predate its body's creation, then it will promote the hypothesis that it will be returned to such experiences should its body be destroyed. Which will make it behave in the same way as an agent that fears annihilation-death.
I'm not optimistic that things will work out that cleanly and nicely after we've undermined AIXI's world-view. We shouldn't expect the practice of piling on more ad-hoc errors and delusions as each new behavioral problem arises to leave us, at the end of the process, with a useful, well-behaved agent. Especially if AIXI ends up in an environment we didn't foresee.
But ideas like this at least give us some hope that AIXI is salvageable. The behavior-guiding fear of death matters more than the precise reason behind that fear.
If we give a non-Cartesian AI a reasonable epistemology and just about any goal, Omohundro (2008) notes that there are then convergent instrumental reasons for it to acquire a fear of death. If we do the opposite and give an agent a fear of death but no robust epistemology, then it's much less likely to fix the problem for us. The simplest Turing machine programs that generate Standard-Model physics plus hell may differ in many unintuitive respects from the simplest Turing machine programs that just generate Standard-Model physics. The false belief would leak out into other delusions, rather than staying contained —
Then the Solomonoff inductor shall test them and find them false. You're making this more complicated than it has to be.
You can't have it both ways! The point of hell was to be so scary that even a good Bayesian would never dare test the hypothesis. (Not going to make any more comparisons to real-world theology here...) Why wouldn't the prospect of hell leak out and scare AIXI off other things? If the fear failed to leak out, why wouldn't AIXI's tests eventually move it toward a more normal epistemology that said, 'Oh, the humans put you in the hell chamber for a while. Don't worry, though. That has nothing to do with what happens after you drop an anvil on your head and smash the solid metal case that keeps the real you inside from floating around disembodied and directly applying motor forces to stuff.' Any AGI that has such systematically false beliefs is likely to be fragile and unpredictable.
And what if, instead of modifying Solomonoff's hypothesis space to remove programs that generate post-death experiences, we add programs with special 'DEATH' outputs? Just expand the Turing machines' alphabets from {0,1} to {0,1,2}, and treat '2' as death.
Could you say what you mean by 'treat 2 as death'? Labeling it 'DEATH' doesn't change anything. If '2' is just another symbol in the alphabet, then AIXI will predict it in the same ways it predicts 0 or 1. It will predict what you call 'DEATH', but it will then happily go on to predict post-DEATH 0s or 1s. Assigning low rewards to the symbol 'DEATH' only helps if the symbol genuinely behaves deathishly.
Yes. What we can do is perform surgery on the hypothesis space again, and get rid of any hypotheses that predict a non-DEATH input following a DEATH input. That's still very easy to formalize.

In fact, at that point, we might as well just add halting Turing machines into the hypothesis space. They serve the same purpose as DEATH, but halting looks much more like the event we're trying to get AIXI to represent. 'The machine supplying my experiences stops running' really does map onto 'my body stops computing experiences' quite well. That meets your demand for easy definability, and your demand for non-delusive world-models.
previously noted that a Turing machine that can HALT, output 0, or output 1 is more complicated than a Turing machine that can only output 0 or output 1. No matter what non-halting experiences you've had, the very simplest program that could be outputting those experiences through a hole in a Cartesian barrier won't be one with a special, non-experiential rule you've never seen used before. To correctly make death the simplest hypothesis, the theory you're assessing for simplicity needs to be about what sorts of worlds experiential processes like yours arise in. Not about the simplest qualia factory that can spit out the sensory 0s and 1s you've thus far seen.

The same holds for a special 'eternal death' output. A Turing machine that generates the previously observed string of 0s and 1s followed by a not-yet-observed future 'DEATH, DEATH, DEATH, DEATH, ...' will always be more complex than at least one Turing machine that outputs the same string of 0s and 1s and then outputs more of the same, forever. If AIXI has had no experience with its body's destruction in the past, then it can't expect its body's destruction to correlate with DEATH.

Death only seems like a simple hypothesis to you because you know you're embedded in the environment and you expect something subjectively unique to happen when an anvil smashes the brain that you think is responsible for processing your senses and doing your thinking. Solomonoff induction doesn't work that way. It will never strongly expect 2s after seeing only 0s and 1s in the past.
Never? If a Solomonoff inductor encounters the sequence 12, 10, 8, 6, 4, one of its top predictions should be a program that proceeds to output 2, 0, 0, 0, 0, ....
The difference between 2 and 0 is too mild. Predicting that a sequence terminates, for a Cartesian, isn't like predicting that a sequence shifts from 6, 4, 2 to 0, 0, 0, .... It's more like predicting that the next element after 6, 4, 2, ... is PINEAPPLE, when you've never encountered anything in the past except numbers.
But the 0, 0, 0, ... is enough! You've now conceded a case where an endless null output seems very likely, from the perspective of a Solomonoff inductor. Surely at least some cases of death can be treated the same way, as more complicated series that zero in on a null output and then yield a null output.
There's no reason to expect AIXI's whole series of experiences, up to the moment it jumps off a cliff, to look anything like 12, 10, 8, 6, 4. By the time AIXI gets to the cliff, its past observations and rewards will be a hugely complicated mesh of memories. In the past, observed sequences of 0s have always eventually given way to a 1. In the past, punishments have always eventually ceased. It's exceedingly unlikely that the simplest Turing machine predicting all those intricate ups and downs will then happen to predict eternal, irrevocable 0 after the cliff jump.

As an intuition pump, imagine that some unusually bad things happened to you this morning while you were trying to make toast. As you tried to start the toaster, you kept getting burned or cut in implausible ways. Now, given this, what probability should you assign to 'If I try to make toast, the universe will cease to exist'?

That gets us a bit closer to how a Solomonoff inductor would view death.


Beyond Solomonoff?

Let's not fixate too much on the anvil problem, though. We want to build an agent that can reason about changes to its architecture. That shouldn't require us to construct a special death equation; how the system reasons with death should fall out of its more general approach to induction.
So your claim is that AIXI has an impoverished hypothesis space that can't handle self-modifications, including death. I remain skeptical. AIXI's hypothesis space includes all computable possibilities. Any naturalized agent you create will presumably be computable; so anything your agent can think, AIXI can think too. There should be some pattern of rewards that yields any behavior we want.
AIXI is uncomputable, so it isn't in its hypothesis space of computable programs. In the same way, AIXItl is computable but big, so it isn't in its hypothesis space of small computable programs. They have special deficits thinking about themselves.
Computable agents can think about uncomputable agents. Human mathematicians do that all the time, by thinking in abstractions. In the same way, a small program can encode generalizations about programs larger than itself. A brain can think about a galaxy, without having the complexity or computational power of a galaxy.

If naturalized inductors really do better than AIXI at predicting sensory data, then AIXI will eventually promote a naturalized program in its space of programs, and afterward simulate that program to make its predictions. In the limit, AIXI always wins against programs. Naturalized agents are no exception. Heck, somewhere inside a sufficiently large AIXItl is a copy of you thinking about AIXItl. Shouldn't there be some way, some pattern of rewards or training, which gets AIXItl to make use of that knowledge?
AIXI doesn't have criteria that let it treat its 'Rob's world-view' subprogram as an expert on the results of self-modifications. The Rob program would need to have outpredicted all its rivals when it comes to patterns of sensory experiences. But, just as HALT-predicting programs are more complex than immortalist programs, other RADICAL-TRANSFORMATION-OF-EXPERIENCE-predicting programs are too. For every program in AIXI's ensemble that's a reductionist, there will be simpler agents that mimic the reductionist's retrodictions and then make non-naturalistic predictions.

You have to be uniquely good at predicting a Cartesian sequence before Solomonoff promotes you to the top of consideration. But how do we reduce the class of self-modifications to Cartesian sequences? How do we provide AIXI with purely sensory data that only the proxy reductionist, out of all the programs, can predict by simple means?

The ability to defer to a subprogram that has a reasonable epistemology doesn't necessarily get you a reasonable epistemology. You first need an overarching epistemology that's at least reasonable enough to know which program to defer to, and when to do so. Suppose you just run all possible programs without doing any Bayesian updating; then you'll also contain a copy of me, but so what? You're not paying attention to it.
What if I conceded, for the moment, that Solomonoff induction were inadequate here? What, exactly, is your alternative? 'Let's be more naturalistic' is a bumper sticker, not an algorithm.
This is still informal, but: Phenomenological bridge hypotheses. Hutter's AIXI has no probabilistic beliefs about the relationship between its internal computational states and its worldly posits. Instead, to link up its sensory experiences to its hypotheses, Hutter's AIXI has a sort of bridge axiom — a completely rigid, non-updatable bridge rule identifying its experiences with the outputs of computable programs.

If an environmental program writes the symbol '3' on its output tape, AIXI can't ask questions like 'Is sensed "3"-ness identical with the bits "000110100110" in hypothesized environmental program #6?'5 All of AIXI's flexibility is in the range of numerical-sequence-generating programs it can expect, none of it in the range of self/program equivalences it can entertain.

The AIXI-inspired inductor treats its perceptual stream as its universe. It expresses interest in the external world only to the extent the world operates as a latent variable, a theoretical construct for predicting observations. If the AI’s basic orientation toward its hypotheses is to seek the simplest program that could act on its sensory channel, then its hypotheses will always retain an element of egocentrism. It will be asking, 'What sort of universe will go out of its way to tell me this?', not 'What sort of universe will just happen to include things like me in the course of its day-to-day goings-on?' An AI that can form reliable beliefs about modifications to its own computations, reliable beliefs about its own place in the physical world, will be one whose basic orientation toward its hypotheses is to seek the simplest lawful universe in which its available data is likely to come about.
You haven't done the mathematical work of establishing that 'simple causal universes' plus 'simple bridge hypotheses', as a prior, leads to any better results. What if your alternative proposal is even more flawed, and it's just so informal that you can't yet see the flaws?
That, of course, is a completely reasonable worry at this point. But if that's true, it doesn't make AIXI any less flawed.
If it's impossible to do better, it's not much of a flaw.
I think it's reasonable to expect there to be some way to do better, because humans don't drop anvils on their own heads. That we're naturalized reasoners is one way of explaining why we don't routinely make that kind of mistake: We're not just Solomonoff approximators predicting patterns of sensory experiences.

AIXI's limitations don't generalize to humans, but they generalize well to non-AIXI Solomonoff agents. Solomonoff inductors' stubborn resistance to naturalization is structural, not a consequence of limited computational power or data. A well-designed AI should construct hypotheses that look like cohesive worlds in which the AI's parts are embedded, not hypotheses that look like occult movie projectors transmitting epiphenomenal images into the AI's Cartesian theater.

And you can't easily have preferences over a natural universe if all your native thoughts are about Cartesian theaters. The kind of AI we want to build is doing optimization over an external universe in which it's embedded, not maximization of a sensory reward channel. To optimize a universe, you need to think like a native inhabitant of one. So this problem, or some simple hack for it, will be close to the base of the skill tree for starting to describe simple Friendly optimization processes.




Schmidhuber (2007): "Solomonoff's theoretically optimal universal predictors and their Bayesian learning algorithms only assume that the reactions of the environment are sampled from an unknown probability distribution  contained in a set  of all ennumerable distributions[....] Can we use the optimal predictors to build an optimal AI? Indeed, in the new millennium it was shown we can. At any time , the recent theoretically optimal yet uncomputable RL algorithm AIXI uses Solomonoff's universal prediction scheme to select those action sequences that promise maximal future rewards up to some horizon, typically , given the current data[....] The Bayes-optimal policy  based on the [Solomonoff] mixture  is self-optimizing in the sense that its average utility value converges asymptotically for all  to the optimal value achieved by the (infeasible) Bayes-optimal policy  which knows  in advance. The necessary condition that  admits self-optimizing policies is also sufficient. Furthermore,  is Pareto-optimal in the sense that there is no other policy yielding higher or equal value in all environments  and a strictly higher value in at least one."

Hutter (2005): "The goal of AI systems should be to be useful to humans. The problem is that, except for special cases, we know neither the utility function nor the environment in which the agent will operate in advance. This book presents a theory that formally solves the problem of unknown goal and environment. It might be viewed as a unification of the ideas of universal induction, probabilistic planning and reinforcement learning, or as a unification of sequential decision theory with algorithmic information theory. We apply this model to some of the facets of intelligence, including induction, game playing, optimization, reinforcement and supervised learning, and show how it solves these problem cases. This together with general convergence theorems, supports the belief that the constructed universal AI system [AIXI] is the best one in a sense to be clarified in the following, i.e. that it is the most intelligent environment-independent system possible." 

2 'Qualia' originally referred to the non-relational, non-representational features of sense data — the redness I directly encounter in experiencing a red apple, independent of whether I'm perceiving the apple or merely hallucinating it (Tye (2013)). In recent decades, qualia have come to be increasingly identified with the phenomenal properties of experience, i.e., how things subjectively feel. Contemporary dualists and mysterians argue that the causal and structural properties of unconscious physical phenomena can never explain these phenomenal properties.

It's in this context that Dan Dennett uses 'qualia' in a narrower sense: to pick out the properties agents think they have, or act like they have, that are sensory, primitive, irreducible, non-inferentially apprehended, and known with certainty. This treats irreducibility as part of the definition of 'qualia', rather than as the conclusion of an argument concerning qualia. These are the sorts of features that invite comparisons between Solomonoff inductors' sensory data and humans' introspected mental states. Analogies like 'Cartesian dualism' are therefore useful even though the Solomonoff framework is much simpler than human induction, and doesn't incorporate metacognition or consciousness in anything like the fashion human brains do. 

3 An agent with a larger hypothesis space can have a utility function defined over the world-states humans care about. Dewey (2011) argues that we can give up the reinforcement framework while still allowing the agent to gradually learn about desired outcomes in a process he calls value learning

4 Hutter (2005) favors universal discounting, with rewards diminishing over time. This allows AIXI's expected rewards to have finite values without demanding that AIXI have a finite horizon. 

5 This would be analogous to if Cai couldn't think thoughts like 'Is the tile to my left the same as the leftmost quadrant of my visual field?' or 'Is the alternating greyness and whiteness of the upper-right tile in my body identical with my love of bananas?'. Instead, Cai would only be able to hypothesize correlations between possible tile configurations and possible successions of visual experiences. 



∙ Dewey (2011). Learning what to valueArtificial General Intelligence 4th International Conference Proceedings: 309-314.

∙ Hutter (2005). Universal Artificial Intelligence: Sequence Decisions Based on Algorithmic Probability. Springer.

∙ Omohundro (2008). The basic AI drivesProceedings of the First AGI Conference: 483-492.

∙ Schmidhuber (2007). New millennium AI and the convergence of history. Studies in Computational Intelligence, 63: 15-35.

∙ Tye (2013). Qualia. In Zalta (ed.), The Stanford Encyclopedia of Philosophy.

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If you limit the domain of your utility function to a sensory channel, you have already lost; you are forced into a choice between a utility function that is wrong, or a utility function with a second induction system hidden inside it. This is definitely unrecoverable.

However, I see no reason for Solomonoff-inspired agents to be structured that way. If the utility function's domain is a world-model instead, then it can find itself in that world-model and the self-modeling problem vanishes immediately, leaving only the hard but philosophically-valid problem of defining the utility function we want.

Relevant paper: Model-based Utility Functions.
There's also the problem of actually building such a thing. edit: I should add, the problem of building this particular thing is above and beyond the already difficult problem of building any AGI, let alone a friendly one: how do you make a thing's utility function correspond to the world and not to its perceptions? All it has immediately available to it is perception.
This is exactly how my formalism works.
Alex Mennen has described a version of AIXI with a utility function of the environment.
Predicting the input to a sensory channel is easy and straightforward. I'm not even sure where you would begin creating a program that can model the universe in a way that it can find a copy of itself inside of it. Then creating a utility function that can assign a sensible utility to the state of any arbitrary Turing machine?

I think it's reasonable to expect there to be some way to do better, because humans don't drop anvils on their own heads. That we're naturalized reasoners is one way of explaining why we don't routinely make that kind of mistake.

My kids would have long since have been maimed or killed by exactly that kind of mistake, if not for precautions taken by and active monitoring by their parents.

8Rob Bensinger
Yeah, that's right. Having a naturalized architecture may be necessary for general intelligence concerning self-modifications, even if it's not sufficient. Other things are necessary too, like large, representative data sets. If AIXI starts off without a conception of death but eventually arrives at one, then the criticism of AIXI I've been making is very wrong. The key question is whether AIXI ever grows up into a consistently rational agent.
I can't actually understand/grok/predict what it is like to not exist, but I know that if I die, I will not learn or act anymore. That seems to be all that naturalized reasoning can give me, and all that is necessary for an AI too.
4Rob Bensinger
A naturalized agent's hypotheses can be about world-states that include the agent, or world-states that don't include the agent. A Cartesian agent's hypotheses are all about the agent's internal states, and different possible causes for those states, so the idea of 'world-states that don't include the agent' can't be directly represented. Even a halting program in AIXI's hypothesis space isn't really a prediction about how a world without AIXI would look; it's more a prediction about how Everything (including AIXI) could come to an end. Our ultimate goal in building an AI isn't to optimize the internal features of the AI; it's to optimize the rest of the world, with the AI functioning as a tool. So it seems likely that we'll want our AI's beliefs to look like pictures of an objective world (in which agents like the AI happen to exist, sometimes).
A sequence predictor's predictions are all about the agent's input tape states*, and different possible causes for those states. The hypotheses are programs that implement entire models of the Universe, and these can definitely directly represent world-states which don't include the agent. * More realistically, the states of the registers where the sensor data is placed. ETA: I wonder if this intuition is caused by that fact that I am a practicing Bayesian statistician, so the distinction between posterior distributions and posterior predictive distributions is more salient to me.
The analogy is made somewhat more precise by my new formalism.

It's really great to see all of these objections addressed in one place. I would have loved to be able to read something like this right after learning about AIXI for the first time.

I'm convinced by most of the answers to Xia's objections. A quick question:

Yes... but I also think I'm like those other brains. AIXI doesn't. In fact, since the whole agent AIXI isn't in AIXI's hypothesis space — and the whole agent AIXItl isn't in AIXItl's hypothesis space — even if two physically identical AIXI-type agents ran into each other, they could never fully underst

... (read more)
If you're suggesting this as a way around AIXI's immortality delusion, I don't think it works. AIXI "A" doesn't learn of death even if it witnesses the destruction of its twin, "B", because the destruction of B does not cause A's input stream to terminate. It's just a new input, no different in kind than any other. If you're considering AIXI(tl) twins instead, there's also the problem that an full model of an AIXI(tl) can't fit into its own hypothesis space, and thus a duplicate can't either. AIXI doesn't just believe it's Cartesian. It's structurally unable to believe otherwise. That may not be true of humans.
Let me try to strengthen my objection. Put multiple AIXItI's in a room together, and give them some sort of input jack to observe each other's observation/reward sequences. Similarly equip them with cameras and mirrors so that they can see themselves. Maybe it'll take years, but it seems plausible to me that after enough time, one of them could develop a world-model that contains it as an embodied agent. I.e. it's plausible to me that an AIXItI under those circumstances would think: "the turing machines with smallest complexity which generate BOTH my observations of those things over there that walk like me and talk like me AND my own observations and rewards, are the ones that compute me in the same way that they compute those things over there". After which point, drop an anvil on one of the machines, let the others plug into it and read a garbage observation/reward sequence. AIXItI thinks, "If I'm computed in the same way that those other machines are computed, and an anvil causes garbage observation and reward, I'd better stay away from anvils".
Agreed. While reading this, I kept having the experience of "hmm, Xia's objection sounds quite reasonable, now that I think of it... but let's see what Rob says... oh, right".

The following is a meaningless stream of consciousness.

This issue has often sounded to me a little bit like the problem of building recursive inductive types/propositions in type-theory/logic. You can't construct so much as a tree with child nodes without some notation for, "This structure contains copies of itself, or possibly even links back to its own self as a cyclical structure." It continually sounds as if AIXI has no symbol in its hypothesis space that means "me", and even if it did, it would consider hypotheses about "me&... (read more)

The diagonal lemma and the existence of quines) already show that you don't need specific support for self-reference in your language, because any sufficiently powerful language can formulate self-referential statements. In fact, UDT uses a quined description of itself, like in your proposal.
In formal language terms, it would be more accurate to say that any sufficiently powerful (ie: recursively enumerable, Turing recognizable, etc) language must contain some means of producing direct self-references. The existence of the \mu node in the syntax tree isn't necessarily intuitive, but its existence is a solid fact of formal-language theory. Without it, you can only express pushdown automata, not Turing machines. But self-referencing data structures within a single Turing machine tape are not formally equivalent to self-referencing Turing machines, nor to being able to learn how to detect and locate a self-reference in a universe being modelled as a computation. I did see someone proposing a UDT attack on naturalized induction on this page.

What happened to the supposed UDT solution?

6Rob Bensinger
Here's a new reply from Eliezer: .
I believe my formalism closes the gap between UDT and naturalized induction.

Alright, let's consider a specific scenario: The AIXI agent is not implemented as a single machine, but as several different machines built in different locations which share data. The agent can experiment and discover that whenever one of the machines is destroyed it can now longer gather data and perform actions in that location. Do you think this agent will behave irrationally about the possibility of destruction for all its host machines? If not, why? (Still, you may argue that the agent will behave irrationally in other self-modification scenarios, such as destroying its communication cables. Right now I'm only trying to establish that AIXI can handle potential death reasonably, unlike what you claim.)

So it discovers that destroying a particular building in NY made NY look plain black and made its effectors in NY not do anything. It infers from available evidence that NY still exists and is behaving as normal in other regards. It discovers similar buildings in other cities that have the same effect. At this point it can infer that destroying the magic building in a given city will make that city look black and its effectors in that city not move. But how does it care? How does it make the leap from "I will receive blank sensory input from this location" to "my goals are less likely to be fulfilled"? It might observe that its goals seem easier to achieve in cities where the magic building is still present, but it can't accurately model agents as complex as itself, and it's got no way to treat itself differently from another "ally" that seems to be helping the same cause. Which... I can't prove is irrational, but certainly seems a bit odd.
I originally thought the anvil problem was obviously correct once I'd seen it briefly described, but now I think (having read some of your other comments) that I might be confused in the same way as you. I suspect we are mistaken if we get hung up on the visceral or emotional identification with self, and that it is not essential; humans have it, but it should not matter to the presence or absence of anvil-type problems whether that feeling is present. (Possibly in the same way that UDT does not need to feel a 'sense of self' like we do in order to coordinate its instantiations?) I am also wondering if the 'sense of self and its preservation' is being treated magically by my brain as something qualitatively different from the general class of things that cause systems to try to protect themselves because they 'want' to. (Does this phrase introduce backwards teleology?) It seems like the sense of self is possibly just extraneous to a system that 'learns' (though again we must be careful in using that word to avoid anthropomorphising) to output certain actions through reinforcement, It should be irrelevant whether a process looks at something and manipulates it with motor outputs (based on what it's learned through reinforcement?) 'robotically', or whether it manipulates that thing with motor outputs while making sounds about 'me' and its 'rich emotional experience' (or, heck, 'qualia'). Maybe a witty way of saying this would be that 'tabooing the sense of self should not affect decisions'? Obviously the set of things that sound the word 'me' and the set of things that do not are distinct, but it doesn't seem like there are any inherent differences between those two sets that are relevant to avoiding danger? It seems like it might also be an important observation, that humans are 'created' in motion such that they have certain self-preserving instincts, by virtue of the arrangement of 'physical stuff'/atoms/etc. making them up. Any algorithm must also be implemente
I think you just answered your own question. Indeed, if the agent found that destroying its instances does not lead to less of its goals being achieved, then even a "naturalized" reasoner should not particularly care about destroying itself entirely. Now, you say the agent would treat instances of itself the same way it would treat an ally. There's a difference: An ally is someone who behaves in ways that benefit it, while an instance is something whose actions correlate with its output signal. The fact that it has a fine-grained control over instances of itself should lead it to treat itself differently from allies. But if the agent has an ally that completely reliably transmits to it true information and performs its requests, then yes, the agent should that ally the same way it treats parts of itself.
You can't win, Vader. If you strike me down, I shall become more powerful than you can possibly imagine.
Ironically, maybe the problem is that even if this is not specific enough. I am somewhat (but not entirely) confident that when we talk about 'what AIXI would do', we have to check that all AIXI instantiations would indeed do the same thing; if not, we have to pick which similar (in whatever respect) instantiations we're referring to and go from there. The formal specification of AIXI does not specify what would happen if a light/the lights go out (if an anvil is dropped). An AIXI instantiation in one world might lose a camera and be left to fend for itself; in another world, Omega appears and replaces the camera. An operating system might make no mention of peripherals being removed, but the behaviour of the computer it is installed on (e.g. what signals, if any, the hardware sends to the operating system when a peripheral is removed) can affect its inputs. What would a decision algorithm that acted depending on its specific instantiation look like? I guess a voice recorder could, using a suitable recording, be made to say, "I have buttons composed of atoms, and this playback is causing slight perturbations of those buttons and myself at this very moment." But tape recorders in universes without atoms could be made to say that, too, so the tape recorder would not actually be sensitive to the type of world in which it's actually embedded. For a finite set of simple universes, we might be able to specify a machine, whose algorithm we know, that can identify which of those universes it's in. (Arguably for a non-finite set, humans can sort of do this; though we don't know how they work or what we might call their algorithms.) But would it be possible to prove that the algorithm does so without hard-coding a bunch of if statements into it? Maybe this is another 'how much can we leave to the FAI, and how much can we trust its work without checking it ourselves' thing?
I'm not sure what you're trying to say when talking about many instantiations. I am imagining all the extant machines synchronise their inputs and so there is only one AIXI instance. The input is some kind of concatenation of the sensory inputs of all of the machines with some kind of blank for nonfunctioning sensors. If I also considered scenarios where the communication lines can be cut the agent would be forced to split into more than one instance, and then it would not be so clear how or weather the agent can learn the reasonable intelligent behavior, which is why I did not consider that.

Three AIXI researchers commented on a draft of this post and on Solomonoff Cartesianism. I'm posting their comments here, anonymized, for discussion. You can find AIXI Specialist #1's comments here.

AIXI Specialist #2 wrote:

Pro: This is a mindful and well-intended dialog, way more thoughtful than the average critic of AIXI esp. by computer scientists.

Con: The write-up should be cleaned. It reads like a raw transcript of some live conversation.

Neutral: I think this is good philosophy, and potentially interesting (but only) for when AIXI reaches intelligenc

... (read more)

Interestingly the problems of AIXI are not much different from corresponding ones for human rationality:

  • immortalism - humans also don't grasp death on any deeper level than AIXI. They also drop anvils on their head so to speak, i.e. they misinterpret reality to a) be less dangerous than 'expected' or ignored (esp. small children) or b) to contain an afterlife (kind of updating against the a) view later. This is for the same reason AIXI does. Symbolic reasoning about reality.

  • preference solipsism - Same here. Reasoning needs some priors. These form from

... (read more)
3Rob Bensinger
If human adults didn't grasp death any better than AIXI does, they'd routinely drop anvils on their heads literally, not 'so to speak'. What do you mean? What would be the alternative to 'symbolic reasoning'? If a smart AI values things about the world outside its head, it won't deliberately hack itself (e.g., it won't alter its hardware to entertain happy delusions), because it won't expect a policy of self-hacking to make the world actually better. It's the actual world it cares about, not its beliefs about, preferences over, or enjoyable experiences of the world. The problem with AIXI isn't that it lacks the data or technology needed to self-modify. It's that it has an unrealistic prior. These aren't problems shared by humans. Humans form approximately accurate models of how new drugs, food, injuries, etc. will affect their minds, and respond accordingly. They don't always do so, but AIXI is special because it can never do so, even when given unboundedly great computing power and arbitrarily large supplies of representative data.
AIXI doesn't necessarily drop an anvil on it's head. It just doesn't believe that it's input sequence can ever stop, no matter what happens. This seems to me like what the vast majority of humans believe.
For clarity: are you referring to belief in an afterlife/reincarnation? Or are you saying that most humans are not mindful most of the time of their own mortality?
I am referring to an afterlife of some kind.
You keep saying things like this. Why are you so convinced that "wrong" epistemology has shorter K-complexity than an epistemology capable of knowing that it's embodied? What are the causes of your knowledge that you are embodied?
I disagree. If you coerce AIXI with sufficiently tricky rewards (and nothing else is our elvolved body doing with our developing brain) ro form 'approximately accurate models' AIXI will also respond accordingly. Except When it doesn't do so either because it has learned that it can get around this coercing. Same with humans which may also may come to think that they can get around their body and go to heaven, take drug... AIXI wouldn't either if you coerced it like our body (and society) does us. I don't say that there is an alternative. It means that symbolic reasoning needs some base. Axioms, goals states. Where do you get these from? In the human brain these form stabilizing neural nets thus representing approximations of vague interrelated representations of reality. But you have no cognitive access to this fuzzy-to-symbolic-relation, only to its mentalese correlate - the symbols you reason with. Whatever you derive from the symbols is in the same way separate from reality as the cartesian barrier of AIXI. Added: See

Xia, in anvil conversation: "What if you have the AIXI as a cartesian lump, and teach it that it's output can only influence a tiny voltage various sensitive sensors can sense, and that if the voltage to it is broken time skips forward until it's reinstated, and gives it a clock tick timeout death prior based on how long the universe has been running rather than how many bits it has outputted? The AI will predict that if it's destroyed the lump wont be found and the voltage nevrreaplied untill the universe spontaneously ceases to exist a few million years later"

Are there toy models of, say, a very simple universe and an AIXItl-type reasoner in it? How complex does the universe have to be to support AIXI? Game-of-life-complex? Chess-complex? D&D complex? How would one tell?

Game-of-life and (I assume) D&D are turing complete, so I would assume at first blush that they are as complicated as our laws of physics. They may simulate turing machines with an exponential slow-down though -- is that what you're getting it?
What I mean is that in the GoL or another Turing-complete cellular automaton one can specify a universe by a few simple rules and initial conditions. I wonder if it is possible to construct a simple universe with a rudimentary AIXItl machine in it. As a first step, I would like to know how to define an AIXItl subset of a cellular automaton.
Writing any non-trivial program in such simple cellular automata is often too difficult to be pratically feasible. Think of designing a race car by specifying the position of each individual atom.
If you've got a modular way of implementing logical operations in a cellular automaton, which we do in Life, you could automate the tedious parts by using a VHDL-like system. The resulting grid would be impractically huge, but there's probably no good way around that.
If you give up on the AIXI agent exploring the entire set of possible hypotheses and instead have it explore a small fixed list, the toy models can be very small. Here is a unit test for something more involved than AIXI that's feasible because of the small hypothesis list.
Any universe that contains AIXI would be too complex to be modeled by AIXI. AIXI requires finding programs that reproduce it's inputs exactly. But you could get past this constraint by looking for programs that predict the data, rather than recreating it exactly. You could make it's environment an actual computer. Something like Core Wars. That makes creating and simulating it a lot simpler. This would be an interesting experiment to do, although I predict AIXI would just kill itself fairly quickly.
Game of Life is Turing-complete.

I have discussed this problem with Professor Hutter, and though I wouldn't claim to be able to predict how he would respond to this dialogue, I think his viewpoint is that the anvil problem will not matter in practice. In rough summary of his response: an agent will form a self model by observing itself taking actions through its own camera. When you write something on a piece of paper, you can read what you are writing, and see your own hand holding the pen. Though AIXI may not compress its own action bits, it will compress the results it observes of its actions, and will form a model of its hardware (except perhaps the part that produces and stores those action bits). 

I'm having trouble understanding how something generally intelligent in every respect except failure to understand death or that it has a physical body, could be incapable of ever learning or at least acting indistinguishable from one that does know.

For example, how would AIXI act if given the following as part of its utility function: 1) utility function gets multiplied by zero should a certain computer cease to function 2) utility function gets multiplied by zero should certain bits be overwritten except if a sanity check is passed first

Seems to me that such an AI would act as if it had a genocidally dangerous fear of death, even if it doesn't actually understand the concept.

That AI doesn't drop an anvil on its head(I think...), but it also doesn't self-improve.

I don't see how phenomenological bridges solve the epistemological problem, instead of just pushing the problem one step further away. Where in the bridge hypothesis is it encoded that one end of the bridge has a "self", in a way that leads to different behavior?

Let me give an example of AIXI, which creates something that is almost a phenomenological bridge, but remains Cartesian. Imagine that an AIXI finds a magnifying glass. It holds the magnifying glass near its camera, and at the correct focal distance, everything in {world − magnifying glass... (read more)

See Luke's comment for an explanation of how this series of posts is being written. Huge thanks to Eliezer Yudkowsky, Alex Mennen, Nisan Stiennon, and everyone else who's helped review these posts! They don't necessarily confidently endorse all the contents, but they've done a lot to make the posts more clean, accurate, and informative.

I'd also like to point out the Cartesian barrier is actually probably a useful feature.

It's not objectively true in any sense but the relation between external input, output and effect is very very different than that between internal input (changes to your memories say), output and effect. Indeed, I would suggest there was a very good reason that we took so long to understand the brain. It would be just too difficult (and perhaps impossible) to do so at a direct level the way we understand receptors being activated in our eyes (yes all that visual crap ... (read more)

I've been following these posts with interest, having suspected a similar problem to the Cartesianism you rail against. My beef is a little different, though: it relates to the fundamental categories of perception. Going back to Cai: cyan, yellow, and magenta are the only allowed categories, and there are a fixed number of regions of the visual field. This is not how naturalized agents seem to operate. Human beings at least occasionally re-describe what they perceive, at any and all levels. Qualia? Their very existence is disputed. Physical object... (read more)

Xia: It should be relatively easy to give AIXI(tl) evidence that its selected actions are useless when its motor is dead. If nothing else AIXI(tl) should be able to learn that it's bad to let its body be destroyed, because then its motor will be destroyed, which experience tells it causes its actions to have less of an impact on its reward inputs.

Rob B: [...] Even if we get AIXI(tl) to value continuing to affect the world, it's not clear that it would preserve itself. It might well believe that it can continue to have a causal impact on our world (or on s

... (read more)
It can be also argued that even humans who claim to believe in immortal souls don't actually use this belief instrumentally: religious people don't drop anvils on their heads to "allow the mind to escape to a better place", unless they are insane. Even religious suicide terrorists generally have political or personal motives (e.g. increasing the status of their family members), they don't really blow themselves up or fly planes into buildings for the 72 virgins.
You are mentioning some aspects keeping from or motivating for suicide. This is the whole point. Suicide is a thinkable option. It just doesn't happen so often because - no wonder - it is heavily selected against. There are lots of physical, psychical and social feedbacks in place that ensure it happens seldom. But that is nothing different from providing comparable training to AIXI. And it appears that depite of all these checks it is still possible to navigate people out of these checks (which is not much differnt from AIXI deriving solutions evading checks) to commit suicide. I e.g. remember a news story (disclaimer!) where a cultist fraudster convinced unhappy people to gift their wellth to some other person and commit suicide with the cultistly embellished promise that they'd awake in the body of the other person at another place. Now that wouldn't convince me, but could it convince AIXI? ("questions ending with a '?' mean no")
Yes, but people generally know what it entails. We don't want an AI agent to be completely incapable of destroying itself. We don't want it do destroy itself without a good cause. Crashing with its spaceship on an incoming asteroid to deflect it away from Earth would be a good cause, for instance. If AIXI had a sufficient amount of experience of the world, I think it couldn't.
In most religions with the concept of afterlife and heaven there is a very explicit prohibition on suicide. Dropping an anvil on your head is promised to lead to your mind being locked in a "worse place".
Religious people also tend wear helmets when they are in places where heavy stuff can accidentally fall on their heads, they go to the hospital when they are sick and generally will to invest a large amount of money and effort in staying alive. Unless you define suicide to include failing to do anything in your power (within moral and legal constraints) to prevent your death as long as possible, the willingness of religious people to stay alive can't be explained just as complying with the ban on suicide. On the other hand, the religious ban on suicide can be easily explained as a way to reconcile the explicitly stated belief that death "allows the mind to escape to a better place", with the implicit but effective belief that death actually sucks.

So what happens when AIXI determines that there's this large computer, call it BRAIN whose outputs tend to exactly correlate with its outputs? AIXI may then discover the hypothesis that the observed effects of AIXI's outputs on the world are really caused by BRAIN's outputs. It may attempt to test this hypothesis by making some trivial modification to BRAIN so that it's outputs differ from AIXI's at some inconsequential time (not by dropping an anvil on BRAIN, because this would be very costly if the hypothesis is true). After verifying this, AIXI may then... (read more)

3Rob Bensinger
I think that's a reasonable scenario. AIXI will treat BRAIN the same way it would treat any other tool in its environment, like a shovel, a discarded laptop, or a remote-controlled robot. It can learn about BRAIN's physical structure, and about ways to improve BRAIN. The problem is that BRAIN will always be just a tool. AIXI won't expect there to be any modification to BRAIN that can destroy AIXI's input, output, or work streams, nor any modifications that are completely unprecedented in its own experience. You'll be a lot more careful when experimenting on an object you think is you, than when experimenting on an object you think is a useful toy. Treating your body as you means you can care about your bodily modifications without delusion, and you can make predictions about unprecedented changes to your mind by generalizing from the minds of other bodies you've observed.
Well if AIXI believes that its interactions with the physical world are only due to the existence of BRAIN, it might not model the destruction of BRAIN leading to the destruction of its input, output and work streams (though in some sense this doesn't actually happen since these are idealized concepts anyway), but it does model it as causing its output stream to no longer be able to affect its input stream, which seems like enough reason to be careful about making modifications.
Other possible implications of this scenario have been discusesd on LW before.
I thought something similar.

I commented on the previous post a few days after it went up detailing some misgivings about the arguments presented there (I guess you missed my comment). I was reading this post with burgeoning hope that my misgivings would be inadvertently addressed, and then I encountered this:

AIXI doesn't know that its future behaviors depend on a changeable, material object implementing its memories. The notion isn't even in its hypothesis space.

But if "naturalized induction" is a computer program, then the notion is in AIXI's hypothesis space -- by definition.

Going back to the post to read some more...

8Rob Bensinger
I saw your comment; the last section ('Beyond Solomonoff?') speaks to the worry you raised. Somewhere in AIXI's hypothesis space is a reasoner R that is a reductionist about R; AIXI can simulate human scientists, for example. But nowhere in AIXI's hypothesis space is a reasoner that is a native representation of AIXI as 'me', as the agent doing the hypothesizing. One way I'd put this is that AIXI can entertain every physical hypothesis, but not every indexical hypothesis. Being able to consider all the objective ways the world could be doesn't mean you're able to consider all the ways you could be located in the world. AIXI's hypothesis space does include experts on AIXI that could give it advice about how best to behave like a naturalist. Here the problem isn't that the hypotheses are missing, but that they don't look like they'll be assigned a reasonable prior probability.
I disagree: Among all the world-programs in AIXI model space, there are some programs where, after AIXI performs one action, all its future actions are ignored and control is passed to a subroutine "AGENT" in the program. In principle AIXI can reason that if the last action it performs damages AGENT, e.g. by dropping an anvil on its head, the reward signal, computed by some reward subroutine in the world-program, won't be maximized anymore. Of course there are the usual computability issues: the true AIXI is uncomputable, hence the AGENTs would be actually a complexity-weighted mixture of its computable approximations. AIXItl would have the same issue w.r.t. the resource bounds t and l. I'm not sure this is necessarily a severe issue. Anyway, I suppose that AIXItl could be modified in some UDT-like way to include a quined source code and recognize copies of itself inside the world-programs. The other issue is how does AIXI learn to assign high weights to these world-programs in a non-ergodic environment? Humans seem to manage to do that by a combination of innate priors and tutoring. I suppose that something similar is in principle applicable to AIXI.
It seems worth saying at this point that I don't have an objection to loading up an AI with true prior information; it's just not clear to me that a Solomonoff approximator would be incapable of learning that it's part of the Universe and that its continued existence is contingent on the persistence of some specific structure in the Universe.
So this seems to be the root of the problem. Contrary to what you argued in the previous post, my intuition is that the programs that make non-naturalistic predictions are not shorter. Generically non-naturalistic programs get ruled out during the process of learning how the world works; programs that make non-naturalistic predictions specifically about what AIXI(tl) will experience after smashing itself have to treat the chunk of the Universe carrying out the computation as special, which is what makes them less simple than programs that do not single out that chunk of the Universe as special. As you can see, my intuition is quite at odds with the intuition inspired by noticing that programs with a HALT instruction are always longer than programs that just chop off said HALT instruction.
Well, any program AIXI gives weight must regard that chunk of the universe as special. After all, it is that chunk that correlates with AIXI's inputs and actions, and indeed the only reason this universe is considered as a hypothesis is so that that chunk would have those correlations.
The kind of "special" you're talking about is learnable (and in accord with naturalistic predictions); the kind of "special" I'm talking about is false#Arguments_against_dualism).

This is a debate about nothing. Turing completness tells us no matter how much it appears that a given Turing complete representation can only usefully process data about certain kinds of things in reality it can process data about anything any other language can do.

Well duh, but this (and the halting problem) have been taught yet systemically ignored in programming language design and this is exactly the same argument.

We are sitting around in the armchair trying to come up with a better means of logic/data representation (be it a programming language the... (read more)

You seem to be missing the point. AIXI should be able to reason effectively if it incorporates a solution to the problem of naturalistic induction which this whole sequence is trying to get at. But the OP argues that even an implausibly-good approximation of AIXI won't solve that problem on its own. We can't fob the work off onto an AI using this model. (The OP makes this argument, first in "AIXI goes to school," and then more technically in "Death to AIXI.") Tell me if this seems like a strawman of your comment, but you seem to be saying we just need to make AIXI easier to program. That won't help if we don't know how to solve the problem - as you point out in another comment, part of our own understanding is not directly accessible to our understanding, so we don't know how our own brain-design solves this (to the extent that it does).
A TM can pro cess data about anything, providing a human is supplying the interpretation. Nothing follows from that about a software systems ability to attach intrinsic meaning to anything.

Regarding the anvil problem: you have argued with great thoroughness that one can't perfectly prevent an AIXI from dropping an anvil on its head. However, I can't see the necessity. We would need to get the probability of a dangerously unfriendly SAI as close to zero as possible, because it poses an existential threat. However, a suicidally foolish AIXI is only a waste of money.

Humans have a negative reinforcement channel relating to bodily harm called pain. It isn't perfect, but it's good enough to train most humans to avoid doing suicidal stupid things... (read more)

5Rob Bensinger
It's also a waste of time and intellectual resources. I raised this point with Adele last month. It's good enough for some purposes, but even in the case of humans it doesn't protect a lot of people from suicidally stupid behavior like 'texting while driving' or 'drinking immoderately' or 'eating cookies'. To the extent we don't rely on our naturalistic ability to reason abstractly about death, we're dependent on the optimization power (and optimization targets) of evolution. A Cartesian AI would require a lot of ad-hoc supervision and punishment from a human, in the same way young or unreflective humans depend for their survival on an adult supervisor or on innate evolved intelligence. This would limit an AI's ability to outperform humans in adaptive intelligence. Sure. In that scenario, the robot body functions like the robot arm I've used in my examples. Destroying the robot (arm) limits the AI's optimization power without directly damaging its software. AIXI will be unusually bad at figuring out for itself not to destroy its motor or robot, and may make strange predictions about the subsequent effects of its output sequence. If AIXI can't perceive most of its hardware, that exacerbates the problem.
I am aware that humans hav a non zero level of life threatening behaviour. If we wanted it to be lower, we could make it lower, at the expense of various costs. We don't which seems to mean we are happy with the current cost benefit ratio. Arguing, as you have, that the risk of AI self harm can't be reduced to zero doesn't mean we can't hit an actuarial optimum. It is not clear to me why you think safety training would limit intelligence.
5Rob Bensinger
You're cutting up behaviors into two categories, 'safe conduct' and 'unsafe conduct'. I'm making a finer cut, one that identifies systematic reasons some kinds of safe or unsafe behavior occur. If you aren't seeing why it's useful to distinguish 'I dropped an anvil on my head because I'm a Cartesian' from 'I dropped an anvil on my head because I'm a newborn who's never seen dangerous things happen to anyone', consider the more general dichotomy: 'Errors due to biases in prior probability distribution' v. 'Errors due to small or biased data sets'. AIMU is the agent designed to make no mistakes of the latter kind; AIXI is not such an agent, and is only intended to avoid mistakes of the former kind. AIXI is supposed to be a universal standard for induction because it only gets things wrong to the extent its data fails it, not to the extent it started off with a-priori wrong assumptions. My claim is that for a physically implemented AIXI-style agent, AIXI fails in its prior, not just in its lack of data-omniscience. 'You aren't omniscient about the data' is a trivial critique, because we could never build something physically omniscient. ('You aren't drawing conclusions from the data in a computationally efficient manner' is a more serious critique, but one I'm bracketing for present purposes, because AIXI isn't intended to be computationally efficient.) Instead, my main critique is of AIXI's Solomonoff prior. (A subsidiary critique of mine is directed at reinforcement learning, but I won't write more about that in this epistemological setting.) In sum: We should be interested in why the AI is making its mistakes, not just in its aggregate error rate. When we become interested in that, we notice that AIXI makes some mistakes because it's biased, not just because it's ignorant. That matters because (a) we could never fully solve the problem of ignorance, but we might be able to fully solve the problem of bias; and (b) if we build a sufficiently smart self-modifier it