Yeah, the existence of classification into ‘future’ and ‘past’ and ‘future’ not causing ‘past’, and what is exactly ‘future’, those are—ideally—a matter of the model of physics employed. Currently known physics already doesn’t quite work like this—it’s not just the future that can’t cause the present, but anything outside the past lightcone.
All those decision theory discussions leave me with a strong impression that ‘decision theory’ is something which is applied almost solely to the folk physics. As an example of a formalized decision making process, we have AIXI, which doesn’t really do what philosophers say either CDT or EDT does.
Actually, I think AIXI is basically CDT-like, and I suspect that it would two-box on Newcomb’s problem.
At a highly abstract level, the main difference between AIXI and a CDT agent is that AIXI has a generalized way of modeling physics (but it has a built-in assumption of forward causality), whereas the CDT agent needs you to tell it what the physics is in order to make a decision.
The optimality of the AIXI algorithm is predicated on viewing itself as a “black box” as far as its interactions with the environment are concerned, which is more or less what the CDT agent does when it makes a decision.
Actually, I think AIXI is basically CDT-like, and I suspect that it would two-box on Newcomb’s problem.
AIXI is a machine learning (hyper-)algorithm, hence we can’t expect it to perform better than a random coin toss on a one-shot problem.
If you repeatedly pose Newcomb’s problem to an AIXI agent, it will quickly learn to one-box. Trivially, AIXI doesn’t model the problem acausal structure in any way. For AIXI, this is just a matter of setting a bit and getting a reward, and AIXI will easily figuring out that setting its decision bit to “one-box” yields an higher expected reward that setting it to “two-box”. In fact, you don’t even need an AIXI agent to do that: any reinforcement learning toy agent will be able to do that.
The problem you’re discussing is not Newcomb’s problem; it’s a different problem that you’ve decided to apply the same name to.
It is a crucial part of the setup of Newcomb’s problem that the agent is presented with significant evidence about the nature of the problem. This applies to AIXI as well; at the beginning of the problem AIXI needs to be presented with observations that give it very strong evidence about Omega and about the nature of the problem setup. From Wikipedia: ”By the time the game begins, and the player is called upon to choose which boxes to take, the prediction has already been made, and the contents of box B have already been determined. That is, box B contains either $0 or $1,000,000 before the game begins, and once the game begins even the Predictor is powerless to change the contents of the boxes. Before the game begins, the player is aware of all the rules of the game, including the two possible contents of box B, the fact that its contents are based on the Predictor’s prediction, and knowledge of the Predictor’s infallibility. The only information withheld from the player is what prediction the Predictor made, and thus what the contents of box B are.”
It seems totally unreasonable to withhold information from AIXI that would be given to any other agent facing the Newcomb’s problem scenario.
That would require the AIXI agent to have been pretrained to understand English (or some language as expressive as English) and have some experience at solving problems given a verbal explanation of the rules.
In this scenario, the AIXI internal program ensemble concentrates its probability mass on programs which associate each pair of one English specification and one action to a predicted reward. Given the English specification, AIXI computes the expected reward for each action and outputs the action that maximizes the expected reward.
Note that in principle this can implement any computable decision theory. Which one it would choose depend on the agent history and the intrinsic bias of its UTM. It can be CDT, EDT, UDT, or, more likely, some approximation of them that worked well for the agent so far.
That would require the AIXI agent to have been pretrained to understand English (or some language as expressive as English) and have some experience at solving problems given a verbal explanation of the rules.
I don’t think someone posing Newcomb’s problem would be particularly interested in excuses like “but what if the agent only speaks French!?”
Obviously as part of the setup of Newcomb’s problem AIXI has to be provided with an epistemic background that is comparable to that of its intended target audience.
This means it doesn’t just have to be familiar with English, it has to be familiar with the real world, because Newcomb’s problem takes place in the context of the real world (or something very much like it).
I think you’re confusing two different scenarios:
Someone training an AIXI agent to output problem solutions given problem specifications as inputs.
Someone actually physically putting an AIXI agent into the scenario stipulated by Newcomb’s problem.
The second one is Newcomb’s problem; the first is the “what is the optimal strategy for Newcomb’s problem?” problem.
It’s the second one I’m arguing about in this thread, and it’s the second one that people have in mind when they bring up Newcomb’s problem.
Then AIXI ensemble will be dominated by programs which associate “real world” percepts and actions to predicted rewards.
The point is that there is no way, short of actually running the (physically impossible) experiment, that we can tell whether the behavior of this AIXI agent will be consistent with CDT, EDT, or something else entirely.
Would it be a valid instructional technique to give someone (particularly someone congenitally incapable of learning any other way) the opportunity to try out a few iterations of the ‘game’ Omega is offering, with clearly denominated but strategically worthless play money in place of the actual rewards?
The main issue with that is that Newcomb’s problem is predicated on the assumption that you prefer getting a million dollars to getting a thousand dollars. For the play money iterations, that assumption would not hold.
The second issue with iterating Newcomb’s more generally is that it gives the agent an opportunity to precommit to one-boxing. The problem is more interesting and more difficult if you face it without having had that opportunity.
For the play money iterations, that assumption would not hold.
Why not? People can get pretty competitive even when there’s nothing really at stake, and current-iteration play money is a proxy for future-iteration real money.
I’m not sure it really makes an assumption of causality, let alone a forward one. (Apart from the most rudimentary notion that actions determine future input) . Facing an environment with two manipulators seemingly controlled by it, it wont have a hang up over assuming that it equally controls both. Indeed it has no reason to privilege one. Facing an environment with particular patterns under its control, it will assume it controls instances of said pattern. It doesn’t view itself as anything at all. It has inputs and outputs, it builds a model of whats inbetween from the experience, if there are two idenical instances of it, it learns a weird model.
Edit: and what it would do in Newcombs, itll one box some and two box some and learn to one box. Or at least, the variation that values information will.
First of all, for any decision problem it’s an implicit assumption that you are given sufficient information to have a very high degree of certainty about the circumstances of the problem. If presented with the appropriate evidence, AIXI should be convinced of this. Indeed, given its nature as an “optimal sequence-predictor”, it should take far less evidence to convince AIXI than it would take to convince a human.
You are correct that if it was presented Newcomb’s problem repeatedly then in the long run it should eventually try one-boxing, but if it’s highly convinced it could take a very long time before it’s worth it for AIXI to try it.
Now, as for an assumption of causality, the model that AIXI has of the agent/environment interaction is based on an assumption that both of them are chronological Turing machines—see the description here. I’m reasonably sure this constitutes an assumption of forward causality.
Similarly, what AIXI would do in Newcomb’s problem depends very specifically on its notion of what exactly it can control. Just as a CDT agent does, AIXI should understand that whether or not the opaque box contains a million dollars is already predetermined; in fact, given that AIXI is a universal sequence predictor it should be relatively trivial for it to work out whether the box is empty or full. Given that, AIXI should calculate that it is optimal for it to two-box, so it will two-box and get $1000. For AIXI, Newcomb’s problem should essentially boil down to Agent Simulates Predictor.
Ultimately, the AIXI agent makes the same mistake that CDT makes—it fails to understand that its actions are ultimately controlled not by the agent itself, but by the output of the abstract AIXI equation, which is a mathematical construct that is accessible not just to AIXI, but the rest of the world as well. The design of the AIXI algorithm is inherently flawed because it fails to recognize this; ultimately this is the exact same error that CDT makes.
Granted, this doesn’t answer the interesting question of “what does AIXI do if it predicts Newcomb’s problem in advance?”, because before Omega’s prediction AIXI has an opportunity to causally affect that prediction.
I’m reasonably sure this constitutes an assumption of forward causality.
What it doesn’t do, is make an assumption that there must be physical sequence of dominoes falling on each other from one singular instance of it, to the effect.
AIXI should understand that whether or not the opaque box contains a million dollars is already predetermined; in fact, given that AIXI is a universal sequence predictor it should be relatively trivial for it to work out whether the box is empty or full.
Not at all. It can’t self predict. We assume that the predictor actually runs AIXI equation.
Ultimately, it doesn’t know what’s in the boxes, and it doesn’t assume that what’s in the boxes is already well defined (there’s certainly codes where it is not), and it can learn it controls contents of the box in precisely the same manner as it has to learn that it controls it’s own robot arm or what ever is it that it controls. Ultimately it can do exactly same output->predictor->box contents as it does for output->motor controller->robot arm. Indeed if you don’t let it observe ‘its own’ robot arm, and only let it observe the box, that’s what it controls. It has no more understanding that this box labelled ‘AIXI’ is the output of what it controls, than it has about the predictor’s output.
It is utterly lacking this primate confusion over something ‘else’ being the predictor. The predictor is representable in only 1 way, and that’s an extra counter factual insertion of actions into the model.
If I was to follow your line of reasoning, then CDT also one-boxes on Newcomb’s problem, because CDT can also just believe that its action causes the prediction. That goes against the whole point of the Newcomb setup—the idea is that the agent is given sufficient evidence to conclude, with a high degree of confidence, that the contents of the boxes are already determined before it chooses whether to one-box or two-box.
AIXI doesn’t assume that the causality is made up of a “physical sequence of dominoes falling”, but that doesn’t really matter. We’ve stated as part of the problem setup that Newcomb’s problem does, in fact, work that way, and a setup where Omega changes the contents of the boxes in advance, rather than doing it after the fact via some kind of magic, is obviously far simpler, and hence far more probable given a Solomonoff prior.
As for the predictor, it doesn’t need to run the full AIXI equation in order to make a good prediction. It just needs to conclude that due to the evidence AIXI will assign high probability to the obviously simpler, non-magical explanation, and hence AIXI will conclude that the contents of the box are predetermined, and hence AIXI will two-box.
There is no need for Omega to actually compute the (uncomputable) AIXI equation. It could simply take the simple chain of reasoning that I’ve outlined above. Moreover, it would be trivially easy for AIXI to follow Omega’s chain of reasoning, and hence predict (correctly) that the box is, in fact, empty, and walk away with only $1000.
If I was to follow your line of reasoning, then CDT also one-boxes on Newcomb’s problem, because CDT can also just believe that its action causes the prediction. That goes against the whole point of the Newcomb setup—the idea is that the agent is given sufficient evidence to conclude, with a high degree of confidence, that the contents of the boxes are already determined before it chooses whether to one-box or two-box.
We’ve stated as part of the problem setup that Newcomb’s problem does, in fact, work that way, and a setup where Omega changes the contents of the boxes in advance, rather than doing it after the fact via some kind of magic, is obviously far simpler, and hence far more probable given a Solomonoff prior.
Again, folk physics. You make your action available to your world model at the time t where t is when you take that action. You propagate the difference your action makes (to avoid re-evaluating everything). So you need back in time magic.
Let’s look at the equation here: http://www.hutter1.net/ai/uaibook.htm . You have a world model that starts at some arbitrary point well in the past (e.g. big bang), which proceeds from that past into the present, and which takes the list of past actions and the current potential action as an input. Action which is available to the model of the world since it’s very beginning. When evaluating potential action ‘take 1 box’, the model has money in the first box, when evaluating potential action ‘take 2 boxes’, the model doesn’t have money in the first box, and it doesn’t do any fancy reasoning about the relation between those models and how those models can and can’t differ. It just doesn’t perform this time saving optimization of ‘let first box content be x, if i take 2 boxes, i get x+1000 > x’.
Why do you need “back in time magic”, exactly? That’s a strictly more complex world model than the non-back-in-time-magic version. If Solomonoff induction results in a belief in the existence of back-in-time magic when what’s happening is just perfectly normal physics, this would be a massive failure in Solomonoff induction itself. Fortunately, no such thing occurs; Solomonoff induction works just fine.
I’m arguing that, because the box already either contains the million or does not, AIXI will (given a reasonable but not particularly large amount of evidence) massively downweight models that do not correctly describe this aspect of reality. It’s not doing any kind of “fancy reasoning” or “time-saving optimization”, it’s simply doing Solomonoff induction, and dong it correctly.
Then it can, for experiment’ sake, take 2 boxes if theres something in the first box, and take 1 otherwise. The box contents are supposedly a result of computing AIXI and as such are not computable; or for a bounded approximation, not approximable. You’re breaking your own hypothetical and replacing the predictor (which would have to perform hypercomputation) with something that incidentally coincides. AIXI responds appropriately.
edit: to stpop talking to one another: AIXI does not know if there’s money in the first box. The TM where AIXI is 1boxing is an entireliy separate TM from one where AIXI is 2boxing. AIXI does not in any way represent any facts about the relation between those models, such as ‘both have same thing in the first box’.
edit2: and , it is absoloutely correct to take 2 boxes if you don’t know anything about the predictor. AIXI represents the predictor as the surviving TMs using the choice action value as omega’s action to put/not put money in the box. AIXI does not preferentially self identify with the AIXI formula inside the robot that picks boxes, over AIXI formula inside ‘omega’.
Besides that, if you can’t even make a reasoned guess as to what AIXI would do in a given situation, then AIXI itself is pretty useless even as a theoretical concept, isn’t it?
Omega doesn’t have to actually evaluate the AIXI formula exactly; it can simply reason logically to work out what AIXI will do without performing those calculations. Sure, AIXI itself can’t take those shortcuts, but Omega most definitely can. As such, there is no need for Omega to perform hypercomputation, because it’s pretty easy to establish AIXI’s actions to a very high degree of accuracy using the arguments I’ve put forth above. Omega doesn’t have to be a “perfect predictor” at all.
In this case, AIXI is quite easily able to predict the chain of reasoning Omega takes, and so it can easily work out what the contents of the box are. This straightforwardly results in AIXI two-boxing, and because it’s so straightforward it’s quite easy for Omega to predict this, and so Omega only fills one box.
The problem with AIXI is not that it preferentially self-identifies with the AIXI formula inside the robot that picks boxes vs the “AIXI formula inside Omega”. The problem with AIXI is that it doesn’t self-identify with the AIXI formula at all.
One could argue that the simple predictor is “punishing” AIXI for being AIXI, but this is really just the same thing as the CDT agent who thinks Omega is punishing them for being “rational”. The point of this example is that if the AIXI algorithm were to output “one-box” instead of “two-box” for Newcomb’s problem, then it would get a million dollars. Instead, it only gets $1000.
Well, to make an object-level observation, it’s not entirely clear to me what it means for AIXI to occupy the epistemic state required by the problem definition. The “hypotheses” of AIXI are general sequence predictor programs rather than anything particularly realist. So while present program state can only depend on AIXI’s past actions, and not future actions, nothing stops a hypothesis from including a “thunk” that is only evaluated when the program receives the input describing AIXI’s actual action. In fact, as long as no observations or rewards depend on the missing information, there’s no need to even represent the “actual” contents of the boxes. Whether that epistemic state falls within the problem’s precondition seems like a matter of definition.
If you restrict AIXI’s hypothesis state to explicit physics simulations (with the hypercomputing part of AIXI treated as a black box, and decision outputs monkeypatched into a simulated control wire), then your argument does follow, I think; the whole issue of Omega’s prediction is just seen as some “physics stuff” happening, where Omega “does some stuff” and then fills the boxes, and AIXI then knows what’s in the boxes and it’s a simple decision to take both boxes.
But, if the more complicated “lazily-evaluating” sort of hypotheses gain much measure, then AIXI’s decision starts actually depending on its simulation of Omega, and then the above argument doesn’t really work and trying to figure out what actually happens could require actual simulation of AIXI or at least examination of the specific hypothesis space AIXI is working in.
So I suppose there’s a caveat to my post above, which is that if AIXI is simulating you, then it’s not necessarily so easy to “approximately guess” what AIXI would do (since it might depend on your approximate guess...). In that way, having logically-omniscient AIXI play kind of breaks the Newcomb’s Paradox game, since it’s not so easy to make Omega the “perfect predictor” he needs to be, and you maybe need to think about how Omega actually works.
I think it’s implicit in the Newcomb’s problem scenario that it takes place within the constraints of the universe as we know it. Obviously we have to make an exception for AIXI itself, but I don’t see a reason to make any further exceptions after that point. Additionally, it is explicitly stated in the problem setup that the contents of the box are supposed to be predetermined, and that the agent is made aware of this aspect of the setup. As far as the epistemic states are concerned, this would imply that AIXI has been presented with a number of prior observations that provide very strong evidential support for this fact.
I agree that AIXI’s universe programs are general Turing machines rather than explicit physics simulations, but I don’t think that’s a particularly big problem. Unless we’re talking about a particularly immature AIXI agent, it should already be aware of the obvious physics-like nature of the real world; it seems to me that the majority of AIXI’s probability mass should be occupied by physics-like Turing machines rather than by thunking. Why would AIXI come up with world programs that involve Omega making money magically appear or disappear after being presented significant evidence to the contrary?
I can agree that in the general case it would be rather difficult indeed to predict AIXI, but in many specific instances I think it’s rather straightforward. In particular, I think Newcomb’s problem is one of those cases.
I guess that in general Omega could be extremely complex, but unless there is a reason Omega needs to be that complex, isn’t it much more sensible to interpret the problem in a way that is more likely to comport with our knowledge of reality? Insofar as there exist simpler explanations for Omega’s predictive power, those simpler explanations should be preferred.
I guess you could say that AIXI itself cannot exist in our reality and so we need to reinterpret the problem in that context, but that seems like a flawed approach to me. After all, the whole point of AIXI is to reason about its performance relative to other agents, so I don’t think it makes sense to posit a different problem setup for AIXI than we would for any other agent.
If AIXI has been presented with sufficient evidence that the Newcomb’s problem works as advertised, then it must be assigning most of its model probability mass to programs where the content of the box, however internally represented, is correlated to the next decision. Such programs exist in the model ensemble, hence the question is how much probability mass does AIXI assign to them. If it not enough to dominate its choice, then by definition AIXI has not been presented with enough evidence.
What do you mean by “programs where the content of the box, however internally represented, is correlated to the next decision”? Do you mean world programs that output $1,000,000 when the input is “one-box” and output $1000 when the input is “two-box”? That seems to contradict the setup of Newcomb’s to me; in order for Newcomb’s problem to work, the content of the box has to be correlated to the actual next decision, not to counterfactual next decisions that don’t actually occur.
As such, as far as I can see it’s important for AIXI’s probability mass to focus down to models where the box already contains a million dollars and/or models where the box is already empty, rather than models in which the contents of the box are determined by the input to the world program at the moment AIXI makes its decision.
AIXI world programs have no inputs, they just run and produce sequences of triples in the form: (action, percept, reward).
So, let’s say AIXI has been just subjected to Newcomb’s problem. Assuming that the decision variable is always binary (“OneBox” vs “TwoBox”), of all the programs which produce a sequence consistent with the observed history, we distinguish five classes of programs, depending on the next triple they produce: 1: (“OneBox”, “Opaque box contains $1,000,000“, 1,000,000) 2: (“TwoBox”, “Opaque box is empty”, 1,000) 3: (“OneBox”, “Opaque box is empty”, 0) 4: (“TwoBox”, “Opaque box contains $1,000,000”, 1,001,000) 5: Anything else (eg. (“OneBox”, “A pink elephant appears”, 42)).
Class 5 should have a vanishing probability, since we assume that the agent already knows physics. Therefore: E(“OneBox”) = (1,000,000 p(class1) + 0 p(class3)) / (p(class1) + p(class3)) E(“TwoBox”) = (1,000 p(class2) + 1,001,000 p(class4)) / (p(class2) + p(class4))
Classes 1 and 2 are consistent with the setup of Newcomb’s problem, while classes 3 and 4 aren’t. Hence I would say that if AIXI has been presented with enough evidence to believe that it is facing Newcomb’s problem, then by definition of “enough evidence”, p(class1) >> p(class3) and p(class2) >> p(class4), implying that AIXI will OneBox.
AIXI world programs have no inputs, they just run and produce sequences of triples in the form: (action, percept, reward).
No, that isn’t true. See, for example, page 7 of this article. The environments (q) accept inputs from the agent and output the agent’s percepts.
As such (as per my discussion with private_messaging), there are only three relevant classes of world programs: (1) Opaque box contains $1,000,000 (2) Opaque box is empty (3) Contents of the box are determined by my action input
For any and all such environment programs that are consistent with AIXI’s observations to date, AIXI will evaluate the reward for both the OneBox and TwoBox actions. As long as classes (1) and (2) win out over class (3), which they should due to being simpler, AIXI will determine that the E(TwoBox) > E(OneBox) and therefore AIXI will TwoBox. In fact, as long as AIXI is smart enough to predict Omega’s reasoning, world programs of type (2) should win out over type (1) as well, and so AIXI will already be pretty sure that the opaque box is empty when it two-boxes.
The environments (q) accept inputs from the agent and output the agent’s percepts.
Yes, but the programs that AIXI maintains internally in its model ensemble are defined as input-less programs that generate all the possible histories. AIXI filters them for the one observed history and then evaluates the expected (discounted) reward over the future histories, for each possible choice of its next action. Anyway, that’s a technical detail.
As long as classes (1) and (2) win out over class (3), which they should due to being simpler
How can they be simpler, given that you have explained to AIXI what Newcomb’s problem is and provided it with enough evidence so that it really believes that it is going to face it?
Maybe Newcomb’s problem is simply inconceivable to AIXI, in a way that no amount of evidence can ever lead it to expect that the content of the box, and thus the reward, is correlated to its action. That’s a possibility, but I find it not very plausible: AIXI world programs contain embeddings of all human minds, and all super-human computable AIs. If we assume that the agent is experienced, world programs embedding these very very smart AIs will get most of probability mass, since they are very good sequence predictors. So if a human can understand Newcomb’s problem, I think that a super-human AI would understand it as well.
Anyway, if we stipulate that it is indeed possible to provide AIXI with enough evidence that it is facing Newcomb’s problem, then it seems to me that it will OneBox.
Maybe Newcomb’s problem is simply inconceivable to AIXI, in a way that no amount of evidence can ever lead it to expect that the content of the box, and thus the reward, is correlated to its action.
AIXI does recognise this correlation; it two-boxes and with a reasonable amount of evidence it also believes (correctly) that Omega predicted it would two-box.
That’s a possibility, but I find it not very plausible: AIXI world programs contain embeddings of all human minds, and all super-human computable AIs. If we assume that the agent is experienced, world programs embedding these very very smart AIs will get most of probability mass, since they are very good sequence predictors. So if a human can understand Newcomb’s problem, I think that a super-human AI would understand it as well.
The problem is that AIXI cannot recognise the kinds of models in which AIXI’s own action and Omega’s prediction of its action have a common cause (i.e. the AIXI equation). A better agent would be capable of recognising that dependency.
If you always exclude certain kinds of models then it doesn’t matter how smart you are, some explanations are simply never going to occur to you.
If you always exclude certain kinds of models then it doesn’t matter how smart you are, some explanations are simply never going to occur to you.
Actually, these models exist in AIXI world program ensemble. In order to support your point, you have to argue that they are more complex than models which make an incorrect prediction, no matter how much evidence for Newcomb’s problem AIXI has been presented with.
Yes, but the programs that AIXI maintains internally in its model ensemble are defined as input-less programs that generate all the possible histories.
Please clarify this and/or give a reference. Every time I’ve seen the equation AIXI’s actions are inputs to the environment program.
How can they be simpler, given that you have explained to AIXI what Newcomb’s problem is and provided it with enough evidence so that it really believes that it is going to face it?
The point of Newcomb’s problem is that the contents of the box are already predetermined; it’s stipulated that as part of the problem setup you are given enough evidence of this. In general, any explanation that involves AIXI’s action directly affecting the contents of the box will be more complex because it bypasses the physics-like explanation that AIXI would have for everything else.
When I am facing Newcomb’s problem I don’t believe that the box magically changes contents as the result of my action—that would be stupid. I believe that the box already has the million dollars because I’m predictably a one-boxer, and then I one-box.
Similarly, if AIXI is facing Newcomb’s then it should, without a particularly large amount of evidence, also narrow its environment programs down to ones that already either contains the million, and ones that already do not.
EDIT: Wait, perhaps we agree re. the environment programs.
AIXI filters them for the one observed history and then evaluates the expected (discounted) reward over the future histories, for each possible choice of its next action.
Yes, for each possible choice. As such, if AIXI has an environment program “q” in which Omega already predicted one-boxing and put the million dollars in, AIXI will check the outcome of OneBox as well as the outcome of TwoBox with that same “q”.
Please clarify this and/or give a reference. Every time I’ve seen the equation AIXI’s actions are inputs to the environment program.
Eq 22 in the paper you linked, trace the definitions back to eq. 16, which describes Solomonoff induction. It uses input-less programs to obtain the joint probability distribution, then it divides it by the marginal distribution to obtain the conditional probability distribution it needs.
(Anyway, Hutter’s original papers are somewhat difficult to read due to their heavy notation, I find Shane Legg’s PhD thesis more readable.)
The point of Newcomb’s problem is that the contents of the box are already predetermined; it’s stipulated that as part of the problem setup you are given enough evidence of this. In general, any explanation that involves AIXI’s action directly affecting the contents of the box will be more complex because it bypasses the physics-like explanation that AIXI would have for everything else.
If you tell AIXI: “Look, the transparent box contains $1,000 and the opaque box may contain $0 or $1,000,000. Do you want to take the content only of the opaque box or both?”, then AIXI will two-box, just as you would. Clearly the scenario where there is no Omega and the content of the opaque box is independent on your action is simpler than Newcomb’s problem.
But if you convince AIXI that it’s actually facing Newcomb’s problem, then its surviving world-programs must model the action of Omega somewhere in their “physics modules”. The simplest way of doing that is probably to assume that there is some physical variable which determines AIXI next action (remember, the world programs predict actions as well as the inputs), and Omega can observe it and use it to set the content of the opaque box. Or maybe they can assume that Omega has a time machine or something. Different programs in the ensemble will model Omega in a different way, but the point is that in order to be epistemically correct, the probability mass of programs that model Omega must be greater than the probability mass of programs that don’t.
Eq 22 in the paper you linked, trace the definitions back to eq. 16, which describes Solomonoff induction.
It uses input-less programs to obtain the joint probability distribution, then it divides it by the marginal distribution to obtain the conditional probability distribution it needs.
Nope, the environment q is a chronological program; it takes AIXI’s action sequence and outputs an observation sequence, with the restriction that observations cannot be dependent upon future actions.
Basically, it is assumed that the universal Turing machine U is fed both the environment program q, and AIXI’s action sequence y, and outputs AIXI’s observation sequence x by running the program q with input y. Quoting from the paper I linked: ”Reversely, if q already is a binary string we define q(y):=U(q,y)”
In the paper I linked, see Eq. 21:
%20=%20\sum_{q:q(y_{1:k})=x_{1:k}}2%5E{-l(q)}) and the the term
) from Eq. 22.
In other words, any program q that matches AIXI’s observations to date when given AIXI’s actions to date will be part of the ensemble. In order to evaluate different future action sequences, AIXI then evaluates the different future actions it could take by feeding them to its program ensemble, and summing over different possible future rewards conditional on the environments that output those rewards.
If you tell AIXI: “Look, the transparent box contains $1,000 and the opaque box may contain $0 or $1,000,000. Do you want to take the content only of the opaque box or both?”, then AIXI will two-box, just as you would.
Clearly the scenario where there is no Omega and the content of the opaque box is independent on your action is simpler than Newcomb’s problem.
The CDT agent can correctly argue that Omega already left the million dollars out of the box when the CDT agent was presented the choice, but that doesn’t mean that it’s correct to be a CDT agent. My argument is that AIXI suffers from the same flaw, and so a different algorithm is needed.
But if you convince AIXI that it’s actually facing Newcomb’s problem, then its surviving world-programs must model the action of Omega somewhere in their “physics modules”.
Correct. My point is that AIXI’s surviving world-programs boil down to “Omega predicted I would two-box, and didn’t put the million dollars in”, but it’s the fault of the AIXI algorithm that this happens.
The simplest way of doing that is probably to assume that there is some physical variable which determines AIXI next action (remember, the world programs predict actions as well as the inputs), and Omega can observe it and use it to set the content of the opaque box. Or maybe they can assume that Omega has a time machine or something.
As per the AIXI equations, this is incorrect. AIXI cannot recognize the presence of a physical variable determining its next action because for any environment program AIXI’s evaluation stage is always going to try both the OneBox and TwoBox actions. Given the three classes of programs above, the only way AIXI can justify one-boxing is if the class (3) programs, in which its action somehow causes the contents of the box, win out.
“Reversely, if q already is a binary string we define q(y):=U(q,y)”
Ok, missed that. I don’t think it matters to the rest of the argument, though.
As per the AIXI equations, this is incorrect. AIXI cannot recognize the presence of a physical variable determining its next action because for any environment program AIXI’s evaluation stage is always going to try both the OneBox and TwoBox actions. Given the three classes of programs above, the only way AIXI can justify one-boxing is if the class (3) programs, in which its action somehow causes the contents of the box, win out.
An environment program can just assume a value for the physical variable and then abort by failing to halt if the next action doesn’t match it. Or it can assume that the physical simulation branches at time t0, when Omega prepares the box, simulate each branch it until t < t1, when the next AIXI action occurs, and then kill off the branch corresponding to the wrong action. Or, as it has already been proposed by somebody else, it could internally represent the physical world as a set of constraints and then run a constraint solver on it, without the need of performing a step-by-step chronological simulation.
So it seems that there are plenty of environment programs that can represent the action of Omega without assuming that it violates the known laws of physics. But even if it had to, what is the problem? AIXI doesn’t assume that the laws of physics forbid retro-causality.
An environment program can just assume a value for the physical variable and then abort by failing to halt if the next action doesn’t match it.
Why would AIXI come up with something like that? Any such program is clearly more complex than one that does the same thing but doesn’t fail to halt.
Or it can assume that the physical simulation branches at time t0, when Omega prepares the box, simulate each branch it until t < t1, when the next AIXI action occurs, and then kill off the branch corresponding to the wrong action.
Once again, possible but unnecessarily complex to explain AIXI’s observations.
Or, as it has already been proposed by somebody else, it could internally represent the physical world as a set of constraints and then run a constraint solver on it, without the need of performing a step-by-step chronological simulation.
Sure, but the point is that those constraints would still be physics-like in nature. Omega’s prediction accuracy is much better explained by constraints that are physics-like rather than an extra constraint that says “Omega is always right”. if you assume a constraint of the latter kind, you are still forced to explain all the other aspects of Omega—things like Omega walking, Omega speaking, and Omega thinking, or more generally Omega doing all those things that ze does. Also, if Omega isn’t always right, but is instead right only 99% of the time, then the constraint-based approach is penalized further.
So it seems that there are plenty of environment programs that can represent the action of Omega without assuming that it violates the known laws of physics. But even if it had to, what is the problem? AIXI doesn’t assume that the laws of physics forbid retro-causality.
It doesn’t assume that, no, but because it assumes that its observations cannot be affected by its future actions AIXI is still very much restricted in that regard.
My point is a simple one:
If AIXI is going to end-up one-boxing, the simplest model of Omega will be one that used its prediction method and already predicted that AIXI would one-box.
If AIXI is going to end up two-boxing, the simplest model of Omega will be one that used its prediction method and already predicted that AIXI would two-box.
However, if Omega predicted one-boxing and AIXI realized that this was the case, AIXI would still evaluate that the two-boxing action results in AIXI getting more money than the one-boxing action, which means that AIXI would two-box.
As long as Omega is capable of reaching this relatively simple logical conclusion, Omega thereby knows that a prediction of one-boxing would turn out to be wrong, and hence Omega should predict two-boxing; this will, of course, turn out to be correct.
The kinds of models you’re suggesting, with branching etc. are significantly more complex and don’t really serve to explain anything.
It doesn’t assume that, no, but because it assumes that its observations cannot be affected by its future actions AIXI is still very much restricted in that regards.
But this doesn’t matter for Newcomb’s problem, since AIXI observes the content of the opaque box only after it has made its decision.
However, if Omega predicted one-boxing and AIXI realized that this was the case, AIXI would still evaluate that the two-boxing action results in AIXI getting more money than the one-boxing action, which means that AIXI would two-box.
Which means that the epistemic model was flawed with high probability. You are insisting that the flawed model is simpler that the correct one. This may be true for certain states of evidence where AIXI is not convinced that Omega works as advertised, but you haven’t shown that this is true for all possible states of evidence.
The kinds of models you’re suggesting, with branching etc. are significantly more complex and don’t really serve to explain anything.
They may be more complex only up to a small constant overhead (how many bits does it take to include a condition “if OmegaPrediction != NextAction then loop forever”?), therefore, a constant amount of evidence should be sufficient to select them.
Which means that the epistemic model was flawed with high probability.
You are insisting that the flawed model is simpler that the correct one. This may be true for certain states of evidence where AIXI is not convinced that Omega works as advertised, but you haven’t shown that this is true for all possible states of evidence.
Yes, AIXI’s epistemic model will be flawed. This is necessarily true because AIXI is not capable of coming up with the true model of Newcomb’s problem, which is one in which its action and Omega’s prediction of its action share a common cause. Since AIXI isn’t capable of having a self-model, the only way it could possibly replicate the behaviour of the true model is by inserting retrocausality and/or magic into its environment.
They may be more complex only up to a small constant overhead (how many bits does it take to include a condition “if OmegaPrediction != NextAction then loop forever”?), therefore, a constant amount of evidence should be sufficient to select them.
I’m not even sure AIXI is capable of considering programs of this kind, but even if it is, what kind of evidence can AIXI have received that would justify the condition “if OmegaPrediction != NextAction then loop forever”? What evidence would justify such a model over a strictly simpler version without that condition?
Essentially, you’re arguing that rather than coming up with a correct model of its environment (e.g. one in which Omega makes a prediction on the basis of the AIXI equation), AIXI will somehow make up for its inability to self-model by coming up with an inaccurate and obviously false retrocausal/magical model of its environment instead.
However, I don’t see why this would be the case. It’s quite clear that either Omega has already predicted one-boxing, or Omega has already predicted two-boxing. At the very least, the evidence should narrow things down to models of either kind, although I think that AIXI should easily have sufficient evidence to work out which of them is actually true (that being the two-boxing one).
I agree that AIXI’s universe programs are general Turing machines rather than explicit physics simulations, but I don’t think that’s a particularly big problem. Unless we’re talking about a particularly immature AIXI agent, it should already be aware of the obvious physics-like nature of the real world; it seems to me that the majority of AIXI’s probability mass should be occupied by physics-like Turing machines rather than by thunking. Why would AIXI come up with world programs that involve Omega making money magically appear or disappear after being presented significant evidence to the contrary?
The problem is not “programs that make money magically (dis)appear for the box after the fact” but rather programs that don’t explicitly represent the presence or nonpresence of money at all until it is known. For example, a constraint solver that seeks a proof of AIXI’s observations when they are called for (using a logic that expresses normal physics). This gives all the right answers, and is fairly simple but does allow the content of the box to be controlled by the decision.
Such models would generally not offer good explanations for why Omega is so good at predicting all those other agents who aren’t AIXI, and would be penalized for this. On the other hand, any model that explains Omega’s general predictive power would be made more complex by adding a special case just for AIXI.
I don’t understand what you mean by “a constraint solver that seeks a proof of AIXI’s observations when they are called for.” Can you explain it further?
A proof system that starts with some axioms describing the physical world (excluding the AIXI machine itself), and when run with input a_1 .. a_m being AIXI’s actions so far, plugs them in as axioms about AIXI’s control wires, and attempts to prove a statement of the form ‘AIXI’s input wire observes a 1 at time t’ or ‘AIXI’s input wire observes a 0 at time t’. And returns the first answer it finds.
It takes, as input, a description of the agent it’s predicting; typically source code, but in the case of AIXI, it gets the AIXI equation and a sequence of prior observations for AIXI.
As for what it does, it spends some period of time (maybe a very long one) on whatever kind of deductive and/or inductive reasoning it chooses to do in order to establish with a reasonable level of confidence what the agent it’s trying to predict will do.
Yes, AIXI being uncomputable means that Omega can’t simply run the equation for itself, but there is no need for a perfect prediction here. On the whole, it just needs to be able to come up with a well-reasoned argument for why AIXI will take a particular action, or perhaps run an approximation of AIXI for a while. Moreover, anyone in this thread arguing for either one-boxing or two-boxing has already implicitly agreed with this assumption.
Yes, AIXI being uncomputable means that Omega can’t simply run the equation for itself, but there is no need for a perfect prediction here. On the whole, it just needs to be able to come up with a well-reasoned argument for why AIXI will take a particular action, or perhaps run an approximation of AIXI for a while.
This opens up the possibility that AIXI figures out that Omega is going to mispredict it, which would make TwoBoxing the best decision.
Moreover, anyone in this thread arguing for either one-boxing or two-boxing has already implicitly agreed with this assumption.
I think it is generally assumed that, even if Omega is not a perfect predictor, the agent can’t outsmart it and predict its errors. But if Omega is computable and the agent is uncomputable, this doesn’t necessarily hold true.
Moreover, anyone in this thread arguing for either one-boxing or two-boxing has already implicitly agreed with this assumption.
I’m not so sure this is true now. People in this thread arguing that AIXI does something at least have the advantage that AIXI’s decision is not going to depend on how they do the arguing. The fact that AIXI can simulate Omega with perfect fidelity (assuming Omega is not also a hypercomputer) and will make its decision based on the simulation seems like it might impact Omega’s ability to make a good prediction.
Omega doesn’t have to be a “perfect predictor” at all.
So you don’t predict anything, just put nothing in the first box, and advertise this fact clearly enough for the agent making the choice.
Newcomb’s original problem did not include the clause ‘by the way, there’s nothing in the first box’. You’re adding that clause by making additional assertions regarding what AIXI knows about “Omega”.
In this case, AIXI is quite easily able to predict the chain of reasoning Omega takes
There’s a truly crazy amount of misunderstandings with regards to what Solomonoff Induction can learn about the world, on LW.
Let’s say you run AIXI, letting it oversee some gigabytes of webcam data, at your location. You think AIXI can match the exact location of raindrops on your roof, hours in advance? You think AIXI is going to know all about me—the DNA I have, how may I construct a predictor, etc?
So you don’t predict anything, just put nothing in the first box, and advertise this fact clearly enough for the agent making the choice.
A version of the problem in which Omega is predictable is hardly the same thing as a version of the problem in which the first box is always empty. Other algorithms get the million dollars; it’s just that AIXI does not. Moreover, AIXI is not being punished simply for being AIXI; AIXI not getting the million dollars is a direct consequence of the output of the AIXI algorithm.
Newcomb’s original problem did not include the clause ‘by the way, there’s nothing in the first box’. You’re adding that clause by making additional assertions regarding what AIXI knows about “Omega”.
Of course it didn’t include that clause; it would be a rather stupid problem if it did include that clause. On the other hand, what is in the statement of Newcomb’s problem is “By the time the game begins, and the player is called upon to choose which boxes to take, the prediction has already been made, and the contents of box B have already been determined.” Moreover, it is quite clearly stated that the agent playing the game is made fully aware of this fact.
If we stipulate, for the sake of argument, that AIXI cannot work out the contents of the opaque box, AIXI still fails and two-boxes. By the problem statement AIXI should already be convinced that the contents of the boxes are predetermined. Consequently, the vast majority of weight in AIXI’s distribution over world models should be held by models in which AIXI’s subsequent action has no effect on the contents of the box, and so AIXI will rather straightforwardly calculate two-boxing to have higher utility. Moreover, it’s easy for Omega to deduce this, and so the first box will be empty, and so AIXI gets $1000.
Setting the stipulation aside, I still think it should be pretty easy for AIXI to deduce that the box is empty. Given Omega’s astounding predictive success it is far more likely that Omega has a non-trivial capacity for intelligent reasoning and uses this reasoning capacity with a goal of making accurate predictions. As such, I would be surprised if an Omega-level predictor was not able to come across the simple argument I gave above. Of course, as I said above, it doesn’t really matter if AIXI can’t deduce the contents of the box; AIXI two-boxes and loses either way.
There’s a truly crazy amount of misunderstandings with regards to what Solomonoff Induction can learn about the world, on LW.
Let’s say you run AIXI, letting it oversee some gigabytes of webcam data, at your location. You think AIXI can match the exact location of raindrops on your roof, hours in advance? You think AIXI is going to know all about me—the DNA I have, how may I construct a predictor, etc?
AIXI not getting the million dollars is a direct consequence of the output of the AIXI algorithm.
Really? I thought your predictor didn’t evaluate the algorithm, so how is that a ‘direct consequence’?
By the problem statement AIXI should already be convinced that the contents of the boxes are predetermined.
Yeah, and in the Turing machine provided with the tape where the action is “choose 1 box” (the tape is provided at the very beginning), the content of the box is predetermined to have 1 million, while in the entirely different Turing machine provided with the tape where the action is “choose 2 boxes”, the box is predetermined to have nothing. What is so hard to get about it? Those are two entirely different Turing machines, in different iterations of the argmax loop. Are you just selectively ignoring the part of the statement where the predictor, you know, is actually being correct?
edit: as I said, it’s a word problem, only suitable for sloshy and faulty word reasoning using folk physics. You end up ignoring some part of the problem statement.
Really? I thought your predictor didn’t evaluate the algorithm, so how is that a ‘direct consequence’?
The predictor doesn’t have to fully evaluate the algorithm to be able to reason about the algorithm.
Yeah, and in the Turing machine provided with the tape where the action is “choose 1 box” (the tape is provided at the very beginning), the content of the box is predetermined to have 1 million, while in the entirely different Turing machine provided with the tape where the action is “choose 2 boxes”, the box is predetermined to have nothing. What is so hard to get about it? Those are two entirely different Turing machines, in different iterations of the argmax loop. Are you just selectively ignoring the part of the statement where the predictor, you know, is actually being correct?
Nowhere in the problem statement does it say that Omega is necessarily always correct. If it’s physically or logically impossible, Newcomb’s problem is basically just asking “would you prefer a million dollars or a thousand dollars.” The whole point of Newcomb’s problem is that Omega is just very, very good at predicting you.
Anyways, I think you’re misunderstanding the AIXI equation. If there are two Turing machines that are consistent with all observations to date, then both of those Turing machines would be evaluated in the one-boxing argmax iteration, and both would be evaluated in the two-boxing argmax iteration as well. There is no possible reason that either world machine would be excluded from either iteration.
As such, if in one of those Turing machines the box is predetermined to have 1 million, then it’s pretty obvious that when given the input “two-box” that Turing machine will output $1,001,000. More generally there would of course be infinitely many such Turing machines, but nonetheless the expected value over those machines will be very nearly that exact amount.
What exactly is the reason you’re suggesting for AIXI excluding the million-dollar Turing machines when it considers the two-boxing action? Where in the AIXI equation does this occur?
There is no possible reason that either world machine would be excluded from either iteration.
This is getting somewhere.
AIXI does S.I. multiple times using multiple machines differing in what they have on the extra actions tape (where the list of actions AIXI will ever take is written). All the machines used to evaluate the consequence of 1-boxing have different extra actions tape from all the machines used to evaluate the consequences of 2 boxing.
“where U is a universal (monotone Turing) machine executing q given a1..am.”
The U used for one boxing is different U from U used for two boxing, running the same q (which can use the action from the extra tape however it wants; to alter things that happen at the big bang, if it sees fit).
What exactly is the reason you’re suggesting for AIXI excluding the million-dollar Turing machines when it considers the two-boxing action? Where in the AIXI equation does this occur?
With regards to the content of the boxes, there are 3 relevant types of program. One is ‘there’s nothing in the box’, other is ‘there’s a million in the box’, but the third, and this is where it gets interesting, is ‘a bit from the extra input tape determines if there’s money in the box’. Third type can in principle be privileged over repeated observation of correct prediction as it does not have to duplicate the data provided on the third tape for the predictions to be correct all the time.
The third type evaluates to money in the box when the action (provided on the actions tape, which is available to the machine from the beginning) is to take 1 box, and evaluates to no money in the box when the action is to take 2 boxes.
If AIXI learns or is pre-set to know that there’s prediction of the decision happening, I take it as meaning that the third type of machine acquires sufficient weight. edit: and conversely, if the AIXI is not influenced by the program that reads from the actions tape to determine the movements of the ‘predictor’, I take it as AIXI being entirely ignorant of the predicting happening.
edit: clearer language regarding the extra actions tape
edit2: and to clarify further, there’s machines where a bit of information in q specifies that “predictor” has/hasn’t put money in the box, and there’s machines where a bit in the another tape, a1...am , determines this. Because it’s not doing any sort of back in time logic (the a1..am is here from the big bang), the latter are not that apriori improbable and can be learned just fine.
The U used for one boxing is different U from U used for two boxing, running the same q (which can use the action from the extra tape however it wants; to alter things that happen at the big bang, if it sees fit).
The U is always the same U; it’s a universal Turing machine. It takes as its input a world program q and a sequence of AIXI’s actions.
With regards to the content of the boxes, there are 3 relevant types of program. One is ‘there’s nothing in the box’, other is ‘there’s a million in the box’, but the third, and this is where it gets interesting, is ‘a bit from the extra input tape determines if there’s money in the box’. Third type can in principle be privileged over repeated observation of correct prediction as it does not have to duplicate the data provided on the third tape for the predictions to be correct all the time.
OK, yeah. I agree about the three types of program, but as far as I can see the third type of program basically corresponds to Omega being a faker and/or magical.
If AIXI learns or is pre-set to know that there’s prediction of the decision happening, I take it as meaning that the third type of machine acquires sufficient weight. edit: and conversely, if the AIXI is not influenced by the program that reads from the actions tape to determine the movements of the ‘predictor’, I take it as AIXI being entirely ignorant of the predicting happening.
I don’t see how this interpretation of the problem makes sense. What you’re saying is not a prediction at all, it’s simply direct causation from the action to the contents of the box. By contrast, it’s inherent in the term “prediction” that the prediction happens before the thing it’s trying to predict, and therefore that the box already either contains a million dollars or does not contain a million dollars.
Let me pose a more explicit form of Newcomb’s problem by way of clarification. I don’t think it changes anything relevant as compared to the standard version of the problem; the main point is to explicitly communicate the problem setup and describes a way for AIXI to reach the epistemic state that is posited as part of Newcomb’s problem.
Omega apppears to AIXI, presents it with the two boxes, and gives its usual long explanation of the problem setup, as well as presenting some it usual evidence that there is no kind of “trick” involved here.
However, before AIXI is allowed to make its decision, it is offered the opportunity to watch Omega run the game for 1000 other agents. AIXI gets to see Omega putting the money into the box in advance, and then it gets to watch the boxes the entire time. It also gets to see the player come in, it sees Omega introduce the game (the same way every time), and then watches the player make their decision, and watches them open the box and observes the contents. 1000 out of 1000 times (or maybe 999 out of 1000 if you prefer), it turns out that Omega correctly predicted the agent’s action.
Now, finally, it’s AIXI’s turn to make its own decision. All the time it was watching Omega run the other games, it has been watching the boxes that Omega originally set up for AIXI—they’ve remained completely untouched.
As far as I can see, this is a perfectly reasonable way to realize the problem setup for Newcomb’s problem.
OK, yeah. I agree about the three types of program, but as far as I can see the third type of program basically corresponds to Omega being a faker and/or magical.
Well, the way I see it, within the deterministic hypothetical that I 1-box, at the big bang the universe is in the initial state such that I 1-box, and within the deterministic hypothetical that I 2-box, at the big bang the universe is in the initial state such that I 2-box. A valid predictor looks at the initial state and determines what I will do, before I actually do it.
Exactly the same with AIXI, which sets up hypotheticals with different initial states (which is does by adding an universal constant of what it’s going to hypothetically do (the extra tape), which is a very, very clever hack it has to employ to avoid needing to model itself correctly), and can have (or not have) a predictor which uses the initial state—distinct—to determine what AIXI will do before it does that. It correctly captures the fact that initial states which result in different actions are different, even though the way it does so is rather messy and looks ugly.
edit: i.e. to me it seems that there’s nothing fake about the predictor looking at the world’s initial state and concluding that the agent will opt to one-box. It looks bad when for the sake of formal simplicity you’re just writing in the initial state ‘I will one box’ and then have the model of your body read that and one-box, but it seems to me it’s wrong up to a constant and not more wrong than TM using some utterly crazy tag system to run a world simulator.
OK, I think I’ve just answered your question in my response to your other comment, but I’ll give a brief version here.
If there is a bit corresponding to AIXI’s future action, then by AIXI’s assumptions that bit must not be observable to AIXI until after it takes its actions. As such, models of this sort must involve some reason why the bit is observable to Omega, but not observable to AIXI; models where the information determining Omega’s prediction is also observable to AIXI will be significantly simpler.
The U is always the same U; it’s a universal Turing machine. It takes as its input a world program q and a sequence of AIXI’s actions.
In the sense of the language of “where U is a universal (monotone Turing) machine executing q given a1..am.”. (I unilaterally resorted to using same language as Hutter to minimize confusion—seems like a reasonable thing for two argues to adopt...).
I don’t see how this interpretation of the problem makes sense. What you’re saying is not a prediction at all, it’s simply direct causation from the action to the contents of the box.
Well, it certainly made sense to me when I gone to the store today, that if the world is deterministic, then at the big bang, it was already predetermined entirely that I would have gone to the store today. And that in the alternative that I don’t go to the store, as a simple matter of me not changing any physical laws (or, even more ridiculously, me changing the way mathematics works), it must be the case that the at the big bang, the deterministic universe was set up so that I don’t go to the store today.
A simple matter of consistency of the laws of physics within the hypothetical universe requires that two hypothetical deterministic universes with different outcomes can’t have different initial state. It’s called a prediction because it occurs earlier in the simulation history than the actual action does.
Within the hypothetical universe where I go to the store, some entity looking at that initial state of that universe, could conclude—before today—that I go to the store. I don’t see how the hell that is ‘direct causation’ from me going to the store. Me going to the store is caused by the initial state, the prediction is caused by the initial state. In AIXI’s hypothetical where it takes 1 box, it taking 1 box is caused by the initial state of the Turing machine. It literally sets up the initial state of the TM so that it ends up picking 1 box (by the way of putting it’s picking 1 box on an extra tape, or appending it at the end of the program). Not by a later intervention, which would make no sense and be folk physics that’s wrong. The prediction, likewise, can be caused by the initial state.
Re: your set up.
If the agents are complicated and fairly opaque (if they aren’t its not necessarily reasonable to assume what predictor does for them would be what predictor does for a complicated and opaque agent), and if agents are one or two boxing basically at an uniform random (AIXI won’t learn much if they all 2-box), there’s a string with thousand ones and zeroes, and it’s repetition, which gives up to a 2^-1000 penalty to the representations where those are independently encoded using 2000 bits rather than 1000 bits.
Surely AIXI will compress those two same bitstrings into one bitstring somehow.
Now, you can of course decrease the number of cases or prediction accuracy so that AIXI is not provided sufficient information to learn the predictor’s behaviour.
In the sense of the language of “where U is a universal (monotone Turing) machine executing q given a1..am.”. (I unilaterally resorted to using same language as Hutter to minimize confusion—seems like a reasonable thing for two argues to adopt...).
I think you’re misunderstanding the language here; the inputs of a Turing machine are not part of the Turing machine. The program “q” and the actions “a1...am” are both inputs to U. That said, I guess it doesn’t matter if you have two different Us, because they will always compute the same output given the same inputs anyway.
In AIXI’s hypothetical where it takes 1 box, it taking 1 box is caused by the initial state of the Turing machine. It literally sets up the initial state of the TM so that it ends up picking 1 box (by the way of putting it’s picking 1 box on an extra tape, or appending it at the end of the program). Not by a later intervention, which would make no sense and be folk physics that’s wrong. The prediction, likewise, can be caused by the initial state.
According to AIXI’s assumption of “chronological Turing machines”, this isn’t quite right. If the bit was simply encoded into the “initial state” of the universe, then AIXI could potentially observe the state of that bit before it actually takes its action. Any models where that bit influences AIXI’s observations prior to acting would directly violate Hutter’s assumption; world programs of this kind are explicitly forbidden from occupying any of AIXI’s probability mass.
Now, I’ll grant that this is subtly different from an assumption of “forward causality” because the chronological assumption specifically applies to AIXI’s subjective past, rather than the actual past. However, I would argue that models in which AIXI’s action bit causes the contents of the box without affecting its past observations would necessarily be more complex. In order for such a model to satisfy AIXI’s assumptions, the action bit needs to do one of two things: 1) Magic (i.e. makes stuff appear or disappear within the box). 2) The action bit would need to be entangled with the state of the universe in just the right way; it would just so happen that Omega can observe that action bit but AIXI cannot observe it until after it takes its action.
It seems to me that Solomonoff induction will penalise both kinds of “action-caused” models quite heavily, because they offer a poor description of the problem. If the action bit was truly part of the state of the universe, it seems rather unlikely that Omega would be able to observe it while AIXI would not.
Re: your set up.
If the agents are complicated and fairly opaque (if they aren’t its not necessarily reasonable to assume what predictor does for them would be what predictor does for a complicated and opaque agent), and if agents are one or two boxing basically at an uniform random (AIXI won’t learn much if they all 2-box), [...]
I wouldn’t assume the agents are one-boxing or two-boxing at uniform random, that would be pretty stupid since Omega would be unable to predict them. Typical versions of Newcomb’s problem stipulate that when Omega thinks you’ll pick randomly it won’t put the million dollars in. Rather, it would be better to say that the agents are picked from some pool of agents, and it turns out that AIXI gets to witness reasonable proportions of both two-boxers and one-boxers.
[...] there’s a string with thousand ones and zeroes, and it’s repetition, which gives up to a 2^-1000 penalty to the representations where those are independently encoded using 2000 bits rather than 1000 bits.
Surely AIXI will compress those two same bitstrings into one bitstring somehow.
I completely agree! Given the enormous 1000-bit penalty, AIXI should determine that the problem is quite well described by a “common cause” explanation—that is, the actions of the individual actions and Omega’s prediction are both determined in advance by the same factors.
In fact, I would go even further than that; AIXI should be able to duplicate Omega’s feat and quickly come up with a universe model that predicts the agents as well as or better than Omega. When AIXI observed Omega playing the game it had access to the same information about the agents that Omega did, and so whatever the source of Omega’s predictive accuracy, AIXI should be able to replicate it.
More generally, I would argue a “common cause” explanation is implicit in Newcomb’s problem, and I think that AIXI should be able to deduce reasonable models of this without making such direct observations.
In any case, once AIXI comes upon this kind of explanation (which I think is really implicit in the setup of Newcomb’s problem), AIXI is doomed. Models in which AIXI’s future action bit manages to be observable to Omega without being observable to AIXI will be significantly more complicated than models in which Omega’s prediction is determined by information that AIXI has already observed.
The most obvious such model is the one I suggested before—Omega simply reasons in a relatively abstract way about the AIXI equation itself. All of this information is information that is accessible to AIXI in advance, and hence it cannot be dependent upon the future action bit.
As such, AIXI should focus in on world models where the box already contains a million dollars, or already does not. Since AIXI will determine that it’s optimal to two-box in both kinds of world, AIXI will two-box, and since this is a pretty simple line of reasoning Omega will predict that AIXI will two-box, and hence AIXI gets $1000.
It seems to me that Solomonoff induction will penalise both kinds of “action-caused” models quite heavily, because they offer a poor description of the problem.
What’s quite heavily? It seems to me that you can do that in under a hundred bits, and few hundred bits of information are not that hard to acquire. If I throw a die, and it does a couple dozen bounces, it’s influenced by the thermal noise and quantum fluctuations, there’s about 2.58 bits of information that is new even to the true magical AIXI. There’s kilobits if not megabits that can be learned from e.g. my genome (even if AIXI sat for years watching a webcam and browsing the internet beforehand, it still won’t be able to predict the quantum random—it’s either non deterministic, or multiverse and you don’t know where in the multiverse you are). AIXI presumably got a webcam or other high throughput input device, too, so it may be able to privilege some hundreds bits penalized hypothesis (rule out all simpler ones) in a fraction of a second.
It strikes me as you are thinking up a fairly specific environment where the observations do not provide enough actual information. If I observe a software predictor predict a bunch of simple programs with source that I know and can fully evaluate myself, that wouldn’t come close to convincing me it is going to predict my human decision, either. And I’m fundamentally more similar to those programs than AIXI is to agents it can predict.
What’s about my example environment, where AIXI lives in a house, plays Newcomb’s problem many times, and sometimes wants to get a lot of money, and sometimes doesn’t want to get a lot of money for various reasons, e.g. out of e.g. a fear of increasing the risk of burglary if it has too much money, or the fear of going to jail on money laundering charges, or what ever. Every time, the presence of money in the first box is correlated with the actual decision AIXI makes. This has to go on for long enough, of course, until the inherently unlikely hypothesis of being predicted by something, gets privileged.
Why would AIXI privilege a magical hypothesis like that one when there are other hypotheses are strictly simpler and explain the world better? If Omega is capable of predicting AIXI reasonably accurately without the kind of magic you’re proposing, why would AIXI possibly come up with a magical explanation that involves Omega having some kind of privileged, back-in-time access to AIXI’s action which has absolutely no impact on AIXI’s prior observations!?
As for your example environment, Iterated Newcomb’s problem != Newcomb’s problem, and the problem isn’t even Newcomb’s problem to begin with if AIXI doesn’t always want the million dollars. As far as I can tell, though, you’re just trying to come up with a setup in which Omega really needs to be retrocausal or magical, rather than just really good at predicting.
In other words, AIXI’s action is predetermined by the AIXI equation. A model where Omega predicts AIXI’s action on the basis of the AIXI equation is strictly simpler than a model involving a single bit of information that is entangled with Omega and yet somehow doesn’t leak out into the universe and remains invisible to AIXI until AIXI opens the box (or boxes).
Unless the universe and/or Omega really is magical, AIXI’s observations should obviously favour the real explanation over the magical one.
and yet somehow doesn’t leak out into the universe and remains invisible to AIXI until AIXI opens the box (or boxes).
This is kind of stipulated in the problem, the box being opaque, no? What does this leak look like, other than box being in some way not opaque?
We could talk of Superman Decision Theory here and how it necessarily two boxes because he has x-ray vision and sees into the box :)
Unless the universe and/or Omega really is magical, AIXI’s observations should obviously favour the real explanation over the magical one.
You keep asserting this, but I don’t see why that’s true. Let’s suppose that I design something. I have a bit of quantum shot noise in my synapses, the precise way how I implement something probably takes a lot of information to describe. Kilobits, megabits even. Meanwhile, the body of AIXI’s robot is magical—it’s reading from the action tape to set voltages on some wires or some such. So there’s some code to replicate...
And with regards to having a real explanation available, I think it falls under the purview of box not being opaque enough. It’s akin to having a videotape of omega putting or not putting money into the box.
I think I see a better way to clarify my original remark. There is a pretty easy way to modify AIXI to do CDT. Exclude machines that read Ai before printing Oi . (And it’s very possible that some approximations or other variations by Hutter did that. I’d certainly do that if I were making an approximation. I’d need to re-read him to make totally sure he didn’t have that somewhere) I think we can both agree that if you don’t do this, you can one-box without having backwards in time causation in your model, unlike CDT (and if you do, you can’t, like CDT).
In any case, my main point is that the one boxing and two boxing depends to the way of doing physics, and given that we’re arguing about different environments yielding different ways of doing physics, I think we agree on that point.
edit: also I think I can write a variation of AIXI that matches my decisionmaking more closely. I’d just require the TM to print actions on a tape, matching the hypothetical actions. Up to a constant difference in program lengths, so it’s not worse than a choice of a TM. (I’d prefer that not to screw up probabilities though, even if its up to a constant, I need to think how edit: actually quite easy to renormalize that away… I’m far too busy right now with other stuff though). Also using some symbolic package to approximately evaluate it, evading some of can’t model oneself traps.
There could be an universe that used some ahead of time indirect evaluation to tell me in advance what action I am going to take, and with me not taking another action out of spite. I don’t know for sure our universe isn’t this—I just have a somewhat low prior for that.
This is kind of stipulated in the problem, the box being opaque, no? What does this leak look like, other than box being in some way not opaque?
No, “the box is opaque” is very different to “you have absolutely no way of working out whether the box contains the million dollars”. For example, if I’m playing Newcomb’s problem then no matter how opaque the box is I’m already pretty sure it contains a million dollars, and when I proceed to one-box every time I’m almost always right. Are you saying I’m forbidden from being able to play Newcomb’s problem?
We could talk of Superman Decision Theory here and how it necessarily two boxes because he has x-ray vision and sees into the box :)
If “Superman Decision Theory” sees into the box and necessarily two-boxes, then Superman Decision Theory is doing it wrong, because SDT is always going to get $1000 and not a million dollars.
You keep asserting this, but I don’t see why that’s true. Let’s suppose that I design something. I have a bit of quantum shot noise in my synapses, the precise way how I implement something probably takes a lot of information to describe. Kilobits, megabits even. Meanwhile, the body of AIXI’s robot is magical—it’s reading from the action tape to set voltages on some wires or some such.
AIXI doesn’t need to work out the precise way Omega is implemented, it just needs to find programs that appear to function the same way. If the quantum noise has no impact on Omega’s actual predictions then it’s not going to matter as far as AIXI’s hypotheses about how Omega predicts AIXI are concerned.
As far as I can see, “the box is not opaque enough” translates to “AIXI knows too much about the real world!” Well, how is it a good thing if your decision theory performs worse when it has a more realistic model of the world?
It seems to me that the more the AIXI agent works out about the world, the more physics-like its programs should become, and consequently the less it will be able to come up with the kinds of explanations in which its actions cause things in the past.
Yes, world programs which involve reverse causality that just happens to be unobservable to AIXI until after it takes its own action are permissible Turing machines for AIXI, but the more information AIXI gathers, the higher the complexity penalty on those kinds of programs will be.
Why? Because, by necessity, AIXI’s action bit would have to be treated as a special case. If Omega predicts all those other agents in the same way, a bit of code that says “ah, but if Omega is facing AIXI, then Omega should access the a_5 bit from AIXI’s action tape instead of doing what Omega normally does” is simply an unnecessary complexity penalty that doesn’t help to explain anything about AIXI’s past observations of Omega.
So, the more mature AIXI gets, the more CDT-like it becomes.
No, “the box is opaque” is very different to “you have absolutely no way of working out whether the box contains the million dollars”.
I take the box being opaque to mean that the contents of the box do not affect my sensory input, and by extension that I don’t get to e.g. watch a video of omega putting money in the box, or do some forensic equivalent.
For example, if I’m playing Newcomb’s problem then no matter how opaque the box is I’m already pretty sure it contains a million dollars
Really? What if Omega is a program, which you know predicts outputs of other simple programs written in C++, Java, and Python, and it been fed your raw DNA as a description, ’cause you’re human?
What if you just know the exact logic Omega is using?
(Besides, decision theories tend to agree that you should pretend online that you one-box)
If “Superman Decision Theory” sees into the box and necessarily two-boxes, then Superman Decision Theory is doing it wrong, because SDT is always going to get $1000 and not a million dollars.
No, you just adapt the Newcomb’s “opaque box” in an obtuse way. Superman’s facing an entirely different decision problem from the Newcomb’s that you face.
Why? Because, by necessity, AIXI’s action bit would have to be treated as a special case. If Omega predicts all those other agents in the same way, a bit of code that says “ah, but if Omega is facing AIXI, then Omega should access the a_5 bit from AIXI’s action tape instead of doing what Omega normally does” is simply an unnecessary complexity penalty that doesn’t help to explain anything about AIXI’s past observations of Omega.
I think you’re just describing a case where AIXI fails to learn anything from other agents because they’re too different from the AIXI. What’s about my scenario where AIXI plays Newcomb’s multiple times, sometimes wanting more money and sometimes not? The program reading a_5 also appears to work right.
It seems to me that the more the AIXI agent works out about the world, the more physics-like its programs should become, and consequently the less it will be able to come up with the kinds of explanations in which its actions cause things in the past.
Well, given that predictors for AIXI are non existent, that should be the case.
edit: actually, what’s your reasons for one-boxing?
edit2: also I think this way of seeing the world—where your actions are entirely unlinked to the past—is a western phenomenon, some free will philosophy stuff. A quarter of my cultural background is quite fatalist in the outlook, so I see my decisions as the consequences of the laws of physics acting on the initial world state, and given same ‘random noise’, different decision by me implies both different future and different past.
I take the box being opaque to mean that the contents of the box do not affect my sensory input,
Yep, that’s what the box being opaque means—the contents of the box have no causal effect on your perceptions.
and by extension that I don’t get to e.g. watch a video of omega putting money in the box, or do some forensic equivalent.
Nope. Watching the video would contradict this principle as well, because you would still effectively be seeing the contents of the box.
What IS allowed by Newcomb’s problem, however, is coming to the conclusion that the contents of the box and your perceptions of Omega have a common cause in terms of how Omega functions or acts. You are then free to use that reasoning to work out what the contents of the box could be.
Your interpretation of Newcomb’s problem basically makes it incoherent. For example, let’s say I’m a CDT agent and I believe Omega predicted me correctly. Then, at the moment I make my decision to two-box, but before I actually see the contents of the opaque box, I already know that the opaque box is empty. Does this mean that the box is not “opaque”, by your reasoning?
Really? What if Omega is a program, which you know predicts outputs of other simple programs written in C++, Java, and Python, and it been fed your raw DNA as a description, ’cause you’re human?
If I don’t think Omega is able to predict me, then it’s not Newcomb’s problem, is it? Even if we assume that the Omega program is capable of predicting humans, DNA is not that likely to be sufficient evidence for it to be able to make good predictions.
What if you just know the exact logic Omega is using?
Well, then it obviously depends on what that exact logic is.
I think you’re just describing a case where AIXI fails to learn anything from other agents because they’re too different from the AIXI. What’s about my scenario where AIXI plays Newcomb’s multiple times, sometimes wanting more money and sometimes not? The program reading a_5 also appears to work right.
First of all, as I said previously, if AIXI doesn’t want the money then the scenario is not Newcomb’s. Also, I don’t think the a_5 reading program will end up being the simplest explanation even in that scenario. The program would need to use something like a_5, a_67, a_166, a_190 and a_222 in each instance of Newcomb’s problem respectively. Rather than a world program with a generic “get inputs from AIXI” subroutine, you need a world program with a “recognize Newcomblike problems and use the appropriate bits” subroutine; there is still a complexity penalty.
Unless you’re trying to make a setup in which Omega necessarily works by magic, then given sufficient evidence of reality at large magic is always going to be penalised. Given that reality at large works in a non-magical way, explanations that bootstrap your model of reality at large are always going to be simpler than explanations that have to add extraneous elements of “magic” to the model.
Besides, if Omega is just plain magical, then Newcomb’s problem boils down to “is a million bigger than a thousand?”
Well, given that predictors for AIXI are non existent, that should be the case.
Of course there can be predictors for AIXI. I can, for example, predict with a high degree of confidence that if AIXI knows what chess is and it wants to beat me at chess, it’s going to beat me. Also, if AIXI wants to maximise paperclips, I can easily predict that there are going to be a lot of paperclips.
edit: actually, what’s your reasons for one-boxing?
By being the kind of person who one-boxes, I end up with a million dollars instead of a thousand.
edit2: also I think this way of seeing the world—where your actions are entirely unlinked to the past—is a western phenomenon, some free will philosophy stuff. A quarter of my cultural background is quite fatalist in the outlook, so I see my decisions as the consequences of the laws of physics acting on the initial world state, and given same ‘random noise’, different decision by me implies both different future and different past.
Um, the “libertarian free will” perspective is mostly what I’m arguing against here. The whole problem with CDT is that it takes that perspective, and, in concluding that its action is not in any way caused by its past, it ends up with only $1000. My point is that AIXI ultimately suffers from the same problem; it assumes that it has this magical kind of free will when it actually does not, and also ends up with $1000.
Yep, that’s what the box being opaque means—the contents of the box have no causal effect on your perceptions.
Yeah, and then you kept stipulating that the model where Omega has read the action tape and then put or not put money into the box, but it didn’t leak onto sensory input, is very unlikely, and I noted that it’s stipulated in the problem statement that the box contents do not leak onto sensory input.
My point is that AIXI ultimately suffers from the same problem; it assumes that it has this magical kind of free will when it actually does not, and also ends up with $1000.
Let’s say AIXI lives inside the robot named Alice. According to every model employed by AIXI, the robot named Alice has pre-committed, since the beginning of time, to act out a specific sequence of actions. How the hell that assumes magical free will I don’t know. edit: and note that you can exclude machines which had read the action before printing matching sensory data, to actually ensure magical free will. I’m not even sure, maybe some variations by Hutter do just that.
edit:
Of course there can be predictors for AIXI. I can, for example, predict with a high degree of confidence that if AIXI knows what chess is and it wants to beat me at chess, it’s going to beat me. Also, if AIXI wants to maximise paperclips, I can easily predict that there are going to be a lot of paperclips.
That’s just abstruse. We both know what I mean.
By being the kind of person who one-boxes, I end up with a million dollars instead of a thousand.
Well, you’re just pre-committed to 1-box, then. The omegas that don’t know you’re pre-committed to 1-box (e.g. don’t trust you, can’t read your pre-committments, etc) would put nothing there, though, which you might be motivated to think about if its e.g. 10 millions vs 1 million, or 2 millions vs 1 million. (I wonder if one boxing is dependent on inflation...)
edit: let’s say I am playing the omega, and you know I know this weird trick for predicting you on the cheap.… you can work out what’s in the first box, can’t you? If you want money and don’t care of proving omega wrong out of spite, I can simply put nothing in the first box, and count on you to figure that out. You might have committed to the situation with 1 million vs 1 thousand, but I doubt you committed to 1000000 vs 999999 . You say you one box, fine, you get nothing—a rare time Omega is wrong.
edit2: a way to actually do Newcomb’s in real life, by the way. Take poor but not completely stupid people, make it 1000 000 vs 999 999 , and you can be almost always right. You can also draw some really rich people who you believe don’t really care and would 1-box for fun, and put a million in the first box for those, and be almost always right about both types of the case.
Yeah, and then you kept stipulating that the model where Omega has read the action tape and then put or not put money into the box, but it didn’t leak onto sensory input, is very unlikely, and I noted that it’s stipulated in the problem statement that the box contents do not leak onto sensory input.
The two situations are quite different. Any complexity penalty for the non-leaking box has already been paid via AIXI’s observations of the box and the whole Newcomb’s setup; the opaqueness of the box just boils down to normal reality.
On the other hand, your “action bit” model in which Omega reads AIXI’s action tape is associated with a significant complexity penalty because of the privileged nature of the situation—why specifically Omega, and not anyone else? Why does Omega specifically access that one bit, and not one of the other bits?
The more physics-like and real AIXI’s Turing machines get, the more of a penalty will be associated with Turing machines that need to incorporate a special case for a specific event.
Let’s say AIXI lives inside the robot named Alice. According to every model employed by AIXI, the robot named Alice has pre-committed, since the beginning of time, to act out a specific sequence of actions. How the hell that assumes magical free will I don’t know.
edit: and note that you can exclude machines which had read the action before printing matching sensory data, to actually ensure magical free will. I’m not even sure, maybe some variations by Hutter do just that.
AIXI as defined by Hutter (not just some “variation”) has a foundational assumption that an action at time t cannot influence AIXI’s perceptions at times 1..t-1. This is entirely incompatible with a model of Alice where she has pre-commited since the beginning of time, because such an Alice would be able to discover her own pre-commitment before she took the action in question. AIXI, on the other hand, explicitly forbids world models where that can happen.
That’s just abstruse. We both know what I mean.
No, I don’t. My point is that although you can’t predict AIXI in the general case, there are still many cases where AIXI can be predicted with relative ease. My argument is still that Newcomb’s problem is one of those cases (and that AIXI two-boxes).
As for all of your scenarios with different Omegas or different amounts of money, obviously a major factor is how accurate I think Omega’s predictions are. If ze has only been wrong one time in a million, and this includes people who have been one-boxing as well, why should I spend much time thinking about the possibility that I could be the one time he gets it wrong?
Similarly, if you’re playing Omega and you don’t have a past history of correctly predicting one-boxing vs one-boxers, then yes, I two-box. However, that scenario isn’t Newcomb’s problem. For it to be Newcomb’s problem, Omega has to have a history of correctly predicting one-boxers as well as two-boxers.
I suspect the unspecified implementation of Omega hides assumptions if not contradictions. Let me propose a more concrete version:
The problem is presented by Conservative Finite Omega (CFO), who works by pulling the agent’s source code, simulating it for a long but finite time, and putting $1,000,000 in the opaque box iff the simulation is determined to definitely one-box. The agent never walks away with the full $1,001,000, though the agent does sometimes walk away with $0.
So, assuming AIXI is confident in accurate models of how CFO works, CFO will simulate AIXI, which requires it to simulate AIXI’s (accurate) simulation of CFO—endless recursion. AIXI ‘wins’ the timeout war (correctly predicting CFO’s timeout), concludes that CFO has left the opaque box empty, and two-boxes.
You could look at that outcome as AIXI being penalized for being too smart. You could also say that an even smarter agent would prepend ‘if (facing CFO-like situation) then return one-box’ to its source code. Fundamentally, the specification of AIXI cannot conceive of its source code being an output; it’s baked into the assumptions that the explicit output bits are the only outputs.
Sure, I don’t necessarily blame the AIXI equation when it’s facing a relatively stupid Omega in that kind of situation.
However, consider “More Intelligent Finite Omega”, who pulls the agent’s source code and uses an approximate theorem-proving approach until it determines, with high confidence, what AIXI is going to do. Assuming that AIXI has received sufficient evidence to be reasonably confident in its model of MIFO, MIFO can reason like this:
AIXI will be able to accurately simulate me, therefore it will either have determined that the box is already empty, or already full.
Given either of those two models, AIXI will calculate that the best action is to two-box.
Consequently, AIXI will two-box. and then MIFO will leave the opaque box empty, and its prediction will have been correct. Moreover, MIFO had no other choice; if it were put the money in the opaque box, AIXI would still have two-boxed, and MIFO’s prediction would have been incorrect.
If you’re allowed to make the assumption that AIXI is confident in its model of CFO and CFO knows this, then I can make the same assumption about MIFO.
I think you’re right. At first I was worried (here and previously in the thread) that the proof that AIXI would two-box was circular, but I think it works out if you fill in the language about terminating turing machines and stuff. I was going to write up my formalization, but once I went through it in my head your proof suddenly looked too obviously correct to be worth expanding.
Oh, and if the special circumstances offered to AIXI are a problem w.r.t. allowing tricky explanations, I think that should be resolvable.
For example, getting to watch other agents play before making your own decision is the “VIP treatment”. Omega decides whether or not to give this treatment by transparently by generating a pseudorandom integer between 1 and 2000, and it gives the VIP treatment if and only if that integer comes up as a 1. The player gets to directly observe the random number generation mechanism while Omega is using it.
Out of the 1000 agents who also got to play Omega’s game while AIXI was watching, a 1 came up on the 965th game. When that happened, that agent got to watch 1000 games, although in those 1000 a 1 didn’t come up at all. Since AIXI was still waiting around, it got to watch those extra 1000 games before watching the last 35 games of its VIP allocation. Of the 2000 games played, Omega made 2000 correct predictions.
If 2000/2000 is too unlikely and inherently results in weird explanations, we can adjust the number down appropriately e.g. 1900/2000.
Actually, it’s a bit of challenge making an environment where AIXI learns about the predictor.
I think I have one. AIXI lives in a house, it has 100$, it gets reward any cycle there’s at least 1$ in the house, and it plays Newcomb’s repeatedly. Money are delivered to the house. So it doesn’t necessarily always want a million dollar. So first it grabs $1000 from the transparent box (it doesn’t know about the predictor) and immediately spends them on ordering a better door, because it has hypotheses concerning possible burglary, where the house would be set on fire, and no money will be left at all. Then, it doesn’t have the door yet, and it doesn’t want extra money because it can attract theft, so it one boxes, but gets a million.
It keeps one and two boxing as it’s waiting and receives various security upgrades to it’s house, sets up secure money pick ups for the banks, offshore accounts, and so on and so forth. And the predictor turns out to be always correct. So it is eventually dominated by TMs that use the one vs two boxing bit of data from the a1...am tape to specify what the hand of the predictor is doing with a million dollars when simulating the past. So at some point if it wants a million dollars, it one boxes, and if it doesn’t, it two boxes.
Yeah, the existence of classification into ‘future’ and ‘past’ and ‘future’ not causing ‘past’, and what is exactly ‘future’, those are—ideally—a matter of the model of physics employed. Currently known physics already doesn’t quite work like this—it’s not just the future that can’t cause the present, but anything outside the past lightcone.
All those decision theory discussions leave me with a strong impression that ‘decision theory’ is something which is applied almost solely to the folk physics. As an example of a formalized decision making process, we have AIXI, which doesn’t really do what philosophers say either CDT or EDT does.
Actually, I think AIXI is basically CDT-like, and I suspect that it would two-box on Newcomb’s problem.
At a highly abstract level, the main difference between AIXI and a CDT agent is that AIXI has a generalized way of modeling physics (but it has a built-in assumption of forward causality), whereas the CDT agent needs you to tell it what the physics is in order to make a decision.
The optimality of the AIXI algorithm is predicated on viewing itself as a “black box” as far as its interactions with the environment are concerned, which is more or less what the CDT agent does when it makes a decision.
AIXI is a machine learning (hyper-)algorithm, hence we can’t expect it to perform better than a random coin toss on a one-shot problem.
If you repeatedly pose Newcomb’s problem to an AIXI agent, it will quickly learn to one-box.
Trivially, AIXI doesn’t model the problem acausal structure in any way. For AIXI, this is just a matter of setting a bit and getting a reward, and AIXI will easily figuring out that setting its decision bit to “one-box” yields an higher expected reward that setting it to “two-box”.
In fact, you don’t even need an AIXI agent to do that: any reinforcement learning toy agent will be able to do that.
The problem you’re discussing is not Newcomb’s problem; it’s a different problem that you’ve decided to apply the same name to.
It is a crucial part of the setup of Newcomb’s problem that the agent is presented with significant evidence about the nature of the problem. This applies to AIXI as well; at the beginning of the problem AIXI needs to be presented with observations that give it very strong evidence about Omega and about the nature of the problem setup. From Wikipedia:
”By the time the game begins, and the player is called upon to choose which boxes to take, the prediction has already been made, and the contents of box B have already been determined. That is, box B contains either $0 or $1,000,000 before the game begins, and once the game begins even the Predictor is powerless to change the contents of the boxes. Before the game begins, the player is aware of all the rules of the game, including the two possible contents of box B, the fact that its contents are based on the Predictor’s prediction, and knowledge of the Predictor’s infallibility. The only information withheld from the player is what prediction the Predictor made, and thus what the contents of box B are.”
It seems totally unreasonable to withhold information from AIXI that would be given to any other agent facing the Newcomb’s problem scenario.
That would require the AIXI agent to have been pretrained to understand English (or some language as expressive as English) and have some experience at solving problems given a verbal explanation of the rules.
In this scenario, the AIXI internal program ensemble concentrates its probability mass on programs which associate each pair of one English specification and one action to a predicted reward. Given the English specification, AIXI computes the expected reward for each action and outputs the action that maximizes the expected reward.
Note that in principle this can implement any computable decision theory. Which one it would choose depend on the agent history and the intrinsic bias of its UTM.
It can be CDT, EDT, UDT, or, more likely, some approximation of them that worked well for the agent so far.
I don’t think someone posing Newcomb’s problem would be particularly interested in excuses like “but what if the agent only speaks French!?” Obviously as part of the setup of Newcomb’s problem AIXI has to be provided with an epistemic background that is comparable to that of its intended target audience. This means it doesn’t just have to be familiar with English, it has to be familiar with the real world, because Newcomb’s problem takes place in the context of the real world (or something very much like it).
I think you’re confusing two different scenarios:
Someone training an AIXI agent to output problem solutions given problem specifications as inputs.
Someone actually physically putting an AIXI agent into the scenario stipulated by Newcomb’s problem.
The second one is Newcomb’s problem; the first is the “what is the optimal strategy for Newcomb’s problem?” problem.
It’s the second one I’m arguing about in this thread, and it’s the second one that people have in mind when they bring up Newcomb’s problem.
Then AIXI ensemble will be dominated by programs which associate “real world” percepts and actions to predicted rewards.
The point is that there is no way, short of actually running the (physically impossible) experiment, that we can tell whether the behavior of this AIXI agent will be consistent with CDT, EDT, or something else entirely.
Would it be a valid instructional technique to give someone (particularly someone congenitally incapable of learning any other way) the opportunity to try out a few iterations of the ‘game’ Omega is offering, with clearly denominated but strategically worthless play money in place of the actual rewards?
The main issue with that is that Newcomb’s problem is predicated on the assumption that you prefer getting a million dollars to getting a thousand dollars. For the play money iterations, that assumption would not hold.
The second issue with iterating Newcomb’s more generally is that it gives the agent an opportunity to precommit to one-boxing. The problem is more interesting and more difficult if you face it without having had that opportunity.
Why not? People can get pretty competitive even when there’s nothing really at stake, and current-iteration play money is a proxy for future-iteration real money.
I’m not sure it really makes an assumption of causality, let alone a forward one. (Apart from the most rudimentary notion that actions determine future input) . Facing an environment with two manipulators seemingly controlled by it, it wont have a hang up over assuming that it equally controls both. Indeed it has no reason to privilege one. Facing an environment with particular patterns under its control, it will assume it controls instances of said pattern. It doesn’t view itself as anything at all. It has inputs and outputs, it builds a model of whats inbetween from the experience, if there are two idenical instances of it, it learns a weird model.
Edit: and what it would do in Newcombs, itll one box some and two box some and learn to one box. Or at least, the variation that values information will.
First of all, for any decision problem it’s an implicit assumption that you are given sufficient information to have a very high degree of certainty about the circumstances of the problem. If presented with the appropriate evidence, AIXI should be convinced of this. Indeed, given its nature as an “optimal sequence-predictor”, it should take far less evidence to convince AIXI than it would take to convince a human. You are correct that if it was presented Newcomb’s problem repeatedly then in the long run it should eventually try one-boxing, but if it’s highly convinced it could take a very long time before it’s worth it for AIXI to try it.
Now, as for an assumption of causality, the model that AIXI has of the agent/environment interaction is based on an assumption that both of them are chronological Turing machines—see the description here. I’m reasonably sure this constitutes an assumption of forward causality.
Similarly, what AIXI would do in Newcomb’s problem depends very specifically on its notion of what exactly it can control. Just as a CDT agent does, AIXI should understand that whether or not the opaque box contains a million dollars is already predetermined; in fact, given that AIXI is a universal sequence predictor it should be relatively trivial for it to work out whether the box is empty or full. Given that, AIXI should calculate that it is optimal for it to two-box, so it will two-box and get $1000. For AIXI, Newcomb’s problem should essentially boil down to Agent Simulates Predictor.
Ultimately, the AIXI agent makes the same mistake that CDT makes—it fails to understand that its actions are ultimately controlled not by the agent itself, but by the output of the abstract AIXI equation, which is a mathematical construct that is accessible not just to AIXI, but the rest of the world as well. The design of the AIXI algorithm is inherently flawed because it fails to recognize this; ultimately this is the exact same error that CDT makes.
Granted, this doesn’t answer the interesting question of “what does AIXI do if it predicts Newcomb’s problem in advance?”, because before Omega’s prediction AIXI has an opportunity to causally affect that prediction.
What it doesn’t do, is make an assumption that there must be physical sequence of dominoes falling on each other from one singular instance of it, to the effect.
Not at all. It can’t self predict. We assume that the predictor actually runs AIXI equation.
Ultimately, it doesn’t know what’s in the boxes, and it doesn’t assume that what’s in the boxes is already well defined (there’s certainly codes where it is not), and it can learn it controls contents of the box in precisely the same manner as it has to learn that it controls it’s own robot arm or what ever is it that it controls. Ultimately it can do exactly same output->predictor->box contents as it does for output->motor controller->robot arm. Indeed if you don’t let it observe ‘its own’ robot arm, and only let it observe the box, that’s what it controls. It has no more understanding that this box labelled ‘AIXI’ is the output of what it controls, than it has about the predictor’s output.
It is utterly lacking this primate confusion over something ‘else’ being the predictor. The predictor is representable in only 1 way, and that’s an extra counter factual insertion of actions into the model.
You need to notice and justify changing the subject.
If I was to follow your line of reasoning, then CDT also one-boxes on Newcomb’s problem, because CDT can also just believe that its action causes the prediction. That goes against the whole point of the Newcomb setup—the idea is that the agent is given sufficient evidence to conclude, with a high degree of confidence, that the contents of the boxes are already determined before it chooses whether to one-box or two-box.
AIXI doesn’t assume that the causality is made up of a “physical sequence of dominoes falling”, but that doesn’t really matter. We’ve stated as part of the problem setup that Newcomb’s problem does, in fact, work that way, and a setup where Omega changes the contents of the boxes in advance, rather than doing it after the fact via some kind of magic, is obviously far simpler, and hence far more probable given a Solomonoff prior.
As for the predictor, it doesn’t need to run the full AIXI equation in order to make a good prediction. It just needs to conclude that due to the evidence AIXI will assign high probability to the obviously simpler, non-magical explanation, and hence AIXI will conclude that the contents of the box are predetermined, and hence AIXI will two-box.
There is no need for Omega to actually compute the (uncomputable) AIXI equation. It could simply take the simple chain of reasoning that I’ve outlined above. Moreover, it would be trivially easy for AIXI to follow Omega’s chain of reasoning, and hence predict (correctly) that the box is, in fact, empty, and walk away with only $1000.
Again, folk physics. You make your action available to your world model at the time t where t is when you take that action. You propagate the difference your action makes (to avoid re-evaluating everything). So you need back in time magic.
Let’s look at the equation here: http://www.hutter1.net/ai/uaibook.htm . You have a world model that starts at some arbitrary point well in the past (e.g. big bang), which proceeds from that past into the present, and which takes the list of past actions and the current potential action as an input. Action which is available to the model of the world since it’s very beginning. When evaluating potential action ‘take 1 box’, the model has money in the first box, when evaluating potential action ‘take 2 boxes’, the model doesn’t have money in the first box, and it doesn’t do any fancy reasoning about the relation between those models and how those models can and can’t differ. It just doesn’t perform this time saving optimization of ‘let first box content be x, if i take 2 boxes, i get x+1000 > x’.
Why do you need “back in time magic”, exactly? That’s a strictly more complex world model than the non-back-in-time-magic version. If Solomonoff induction results in a belief in the existence of back-in-time magic when what’s happening is just perfectly normal physics, this would be a massive failure in Solomonoff induction itself. Fortunately, no such thing occurs; Solomonoff induction works just fine.
I’m arguing that, because the box already either contains the million or does not, AIXI will (given a reasonable but not particularly large amount of evidence) massively downweight models that do not correctly describe this aspect of reality. It’s not doing any kind of “fancy reasoning” or “time-saving optimization”, it’s simply doing Solomonoff induction, and dong it correctly.
Then it can, for experiment’ sake, take 2 boxes if theres something in the first box, and take 1 otherwise. The box contents are supposedly a result of computing AIXI and as such are not computable; or for a bounded approximation, not approximable. You’re breaking your own hypothetical and replacing the predictor (which would have to perform hypercomputation) with something that incidentally coincides. AIXI responds appropriately.
edit: to stpop talking to one another: AIXI does not know if there’s money in the first box. The TM where AIXI is 1boxing is an entireliy separate TM from one where AIXI is 2boxing. AIXI does not in any way represent any facts about the relation between those models, such as ‘both have same thing in the first box’.
edit2: and , it is absoloutely correct to take 2 boxes if you don’t know anything about the predictor. AIXI represents the predictor as the surviving TMs using the choice action value as omega’s action to put/not put money in the box. AIXI does not preferentially self identify with the AIXI formula inside the robot that picks boxes, over AIXI formula inside ‘omega’.
If you have to perform hypercomputation to even approximately guess what AIXI would do, then this conversation would seem like a waste of time/
Precisely.
Besides that, if you can’t even make a reasoned guess as to what AIXI would do in a given situation, then AIXI itself is pretty useless even as a theoretical concept, isn’t it?
Omega doesn’t have to actually evaluate the AIXI formula exactly; it can simply reason logically to work out what AIXI will do without performing those calculations. Sure, AIXI itself can’t take those shortcuts, but Omega most definitely can. As such, there is no need for Omega to perform hypercomputation, because it’s pretty easy to establish AIXI’s actions to a very high degree of accuracy using the arguments I’ve put forth above. Omega doesn’t have to be a “perfect predictor” at all.
In this case, AIXI is quite easily able to predict the chain of reasoning Omega takes, and so it can easily work out what the contents of the box are. This straightforwardly results in AIXI two-boxing, and because it’s so straightforward it’s quite easy for Omega to predict this, and so Omega only fills one box.
The problem with AIXI is not that it preferentially self-identifies with the AIXI formula inside the robot that picks boxes vs the “AIXI formula inside Omega”. The problem with AIXI is that it doesn’t self-identify with the AIXI formula at all.
One could argue that the simple predictor is “punishing” AIXI for being AIXI, but this is really just the same thing as the CDT agent who thinks Omega is punishing them for being “rational”. The point of this example is that if the AIXI algorithm were to output “one-box” instead of “two-box” for Newcomb’s problem, then it would get a million dollars. Instead, it only gets $1000.
Well, to make an object-level observation, it’s not entirely clear to me what it means for AIXI to occupy the epistemic state required by the problem definition. The “hypotheses” of AIXI are general sequence predictor programs rather than anything particularly realist. So while present program state can only depend on AIXI’s past actions, and not future actions, nothing stops a hypothesis from including a “thunk” that is only evaluated when the program receives the input describing AIXI’s actual action. In fact, as long as no observations or rewards depend on the missing information, there’s no need to even represent the “actual” contents of the boxes. Whether that epistemic state falls within the problem’s precondition seems like a matter of definition.
If you restrict AIXI’s hypothesis state to explicit physics simulations (with the hypercomputing part of AIXI treated as a black box, and decision outputs monkeypatched into a simulated control wire), then your argument does follow, I think; the whole issue of Omega’s prediction is just seen as some “physics stuff” happening, where Omega “does some stuff” and then fills the boxes, and AIXI then knows what’s in the boxes and it’s a simple decision to take both boxes.
But, if the more complicated “lazily-evaluating” sort of hypotheses gain much measure, then AIXI’s decision starts actually depending on its simulation of Omega, and then the above argument doesn’t really work and trying to figure out what actually happens could require actual simulation of AIXI or at least examination of the specific hypothesis space AIXI is working in.
So I suppose there’s a caveat to my post above, which is that if AIXI is simulating you, then it’s not necessarily so easy to “approximately guess” what AIXI would do (since it might depend on your approximate guess...). In that way, having logically-omniscient AIXI play kind of breaks the Newcomb’s Paradox game, since it’s not so easy to make Omega the “perfect predictor” he needs to be, and you maybe need to think about how Omega actually works.
I think it’s implicit in the Newcomb’s problem scenario that it takes place within the constraints of the universe as we know it. Obviously we have to make an exception for AIXI itself, but I don’t see a reason to make any further exceptions after that point. Additionally, it is explicitly stated in the problem setup that the contents of the box are supposed to be predetermined, and that the agent is made aware of this aspect of the setup. As far as the epistemic states are concerned, this would imply that AIXI has been presented with a number of prior observations that provide very strong evidential support for this fact.
I agree that AIXI’s universe programs are general Turing machines rather than explicit physics simulations, but I don’t think that’s a particularly big problem. Unless we’re talking about a particularly immature AIXI agent, it should already be aware of the obvious physics-like nature of the real world; it seems to me that the majority of AIXI’s probability mass should be occupied by physics-like Turing machines rather than by thunking. Why would AIXI come up with world programs that involve Omega making money magically appear or disappear after being presented significant evidence to the contrary?
I can agree that in the general case it would be rather difficult indeed to predict AIXI, but in many specific instances I think it’s rather straightforward. In particular, I think Newcomb’s problem is one of those cases.
I guess that in general Omega could be extremely complex, but unless there is a reason Omega needs to be that complex, isn’t it much more sensible to interpret the problem in a way that is more likely to comport with our knowledge of reality? Insofar as there exist simpler explanations for Omega’s predictive power, those simpler explanations should be preferred.
I guess you could say that AIXI itself cannot exist in our reality and so we need to reinterpret the problem in that context, but that seems like a flawed approach to me. After all, the whole point of AIXI is to reason about its performance relative to other agents, so I don’t think it makes sense to posit a different problem setup for AIXI than we would for any other agent.
If AIXI has been presented with sufficient evidence that the Newcomb’s problem works as advertised, then it must be assigning most of its model probability mass to programs where the content of the box, however internally represented, is correlated to the next decision.
Such programs exist in the model ensemble, hence the question is how much probability mass does AIXI assign to them. If it not enough to dominate its choice, then by definition AIXI has not been presented with enough evidence.
What do you mean by “programs where the content of the box, however internally represented, is correlated to the next decision”? Do you mean world programs that output $1,000,000 when the input is “one-box” and output $1000 when the input is “two-box”? That seems to contradict the setup of Newcomb’s to me; in order for Newcomb’s problem to work, the content of the box has to be correlated to the actual next decision, not to counterfactual next decisions that don’t actually occur.
As such, as far as I can see it’s important for AIXI’s probability mass to focus down to models where the box already contains a million dollars and/or models where the box is already empty, rather than models in which the contents of the box are determined by the input to the world program at the moment AIXI makes its decision.
AIXI world programs have no inputs, they just run and produce sequences of triples in the form: (action, percept, reward).
So, let’s say AIXI has been just subjected to Newcomb’s problem. Assuming that the decision variable is always binary (“OneBox” vs “TwoBox”), of all the programs which produce a sequence consistent with the observed history, we distinguish five classes of programs, depending on the next triple they produce:
1: (“OneBox”, “Opaque box contains $1,000,000“, 1,000,000)
2: (“TwoBox”, “Opaque box is empty”, 1,000)
3: (“OneBox”, “Opaque box is empty”, 0)
4: (“TwoBox”, “Opaque box contains $1,000,000”, 1,001,000)
5: Anything else (eg. (“OneBox”, “A pink elephant appears”, 42)).
Class 5 should have a vanishing probability, since we assume that the agent already knows physics.
Therefore:
E(“OneBox”) = (1,000,000 p(class1) + 0 p(class3)) / (p(class1) + p(class3))
E(“TwoBox”) = (1,000 p(class2) + 1,001,000 p(class4)) / (p(class2) + p(class4))
Classes 1 and 2 are consistent with the setup of Newcomb’s problem, while classes 3 and 4 aren’t.
Hence I would say that if AIXI has been presented with enough evidence to believe that it is facing Newcomb’s problem, then by definition of “enough evidence”, p(class1) >> p(class3) and p(class2) >> p(class4), implying that AIXI will OneBox.
EDIT: math.
No, that isn’t true. See, for example, page 7 of this article. The environments (q) accept inputs from the agent and output the agent’s percepts.
As such (as per my discussion with private_messaging), there are only three relevant classes of world programs:
(1) Opaque box contains $1,000,000
(2) Opaque box is empty
(3) Contents of the box are determined by my action input
For any and all such environment programs that are consistent with AIXI’s observations to date, AIXI will evaluate the reward for both the OneBox and TwoBox actions. As long as classes (1) and (2) win out over class (3), which they should due to being simpler, AIXI will determine that the E(TwoBox) > E(OneBox) and therefore AIXI will TwoBox. In fact, as long as AIXI is smart enough to predict Omega’s reasoning, world programs of type (2) should win out over type (1) as well, and so AIXI will already be pretty sure that the opaque box is empty when it two-boxes.
Yes, but the programs that AIXI maintains internally in its model ensemble are defined as input-less programs that generate all the possible histories.
AIXI filters them for the one observed history and then evaluates the expected (discounted) reward over the future histories, for each possible choice of its next action.
Anyway, that’s a technical detail.
How can they be simpler, given that you have explained to AIXI what Newcomb’s problem is and provided it with enough evidence so that it really believes that it is going to face it?
Maybe Newcomb’s problem is simply inconceivable to AIXI, in a way that no amount of evidence can ever lead it to expect that the content of the box, and thus the reward, is correlated to its action.
That’s a possibility, but I find it not very plausible: AIXI world programs contain embeddings of all human minds, and all super-human computable AIs. If we assume that the agent is experienced, world programs embedding these very very smart AIs will get most of probability mass, since they are very good sequence predictors. So if a human can understand Newcomb’s problem, I think that a super-human AI would understand it as well.
Anyway, if we stipulate that it is indeed possible to provide AIXI with enough evidence that it is facing Newcomb’s problem, then it seems to me that it will OneBox.
AIXI does recognise this correlation; it two-boxes and with a reasonable amount of evidence it also believes (correctly) that Omega predicted it would two-box.
The problem is that AIXI cannot recognise the kinds of models in which AIXI’s own action and Omega’s prediction of its action have a common cause (i.e. the AIXI equation). A better agent would be capable of recognising that dependency.
If you always exclude certain kinds of models then it doesn’t matter how smart you are, some explanations are simply never going to occur to you.
Actually, these models exist in AIXI world program ensemble. In order to support your point, you have to argue that they are more complex than models which make an incorrect prediction, no matter how much evidence for Newcomb’s problem AIXI has been presented with.
Please clarify this and/or give a reference. Every time I’ve seen the equation AIXI’s actions are inputs to the environment program.
The point of Newcomb’s problem is that the contents of the box are already predetermined; it’s stipulated that as part of the problem setup you are given enough evidence of this. In general, any explanation that involves AIXI’s action directly affecting the contents of the box will be more complex because it bypasses the physics-like explanation that AIXI would have for everything else.
When I am facing Newcomb’s problem I don’t believe that the box magically changes contents as the result of my action—that would be stupid. I believe that the box already has the million dollars because I’m predictably a one-boxer, and then I one-box.
Similarly, if AIXI is facing Newcomb’s then it should, without a particularly large amount of evidence, also narrow its environment programs down to ones that already either contains the million, and ones that already do not.
EDIT: Wait, perhaps we agree re. the environment programs.
Yes, for each possible choice. As such, if AIXI has an environment program “q” in which Omega already predicted one-boxing and put the million dollars in, AIXI will check the outcome of OneBox as well as the outcome of TwoBox with that same “q”.
Eq 22 in the paper you linked, trace the definitions back to eq. 16, which describes Solomonoff induction.
It uses input-less programs to obtain the joint probability distribution, then it divides it by the marginal distribution to obtain the conditional probability distribution it needs.
(Anyway, Hutter’s original papers are somewhat difficult to read due to their heavy notation, I find Shane Legg’s PhD thesis more readable.)
If you tell AIXI: “Look, the transparent box contains $1,000 and the opaque box may contain $0 or $1,000,000. Do you want to take the content only of the opaque box or both?”, then AIXI will two-box, just as you would.
Clearly the scenario where there is no Omega and the content of the opaque box is independent on your action is simpler than Newcomb’s problem.
But if you convince AIXI that it’s actually facing Newcomb’s problem, then its surviving world-programs must model the action of Omega somewhere in their “physics modules”.
The simplest way of doing that is probably to assume that there is some physical variable which determines AIXI next action (remember, the world programs predict actions as well as the inputs), and Omega can observe it and use it to set the content of the opaque box. Or maybe they can assume that Omega has a time machine or something.
Different programs in the ensemble will model Omega in a different way, but the point is that in order to be epistemically correct, the probability mass of programs that model Omega must be greater than the probability mass of programs that don’t.
Nope, the environment q is a chronological program; it takes AIXI’s action sequence and outputs an observation sequence, with the restriction that observations cannot be dependent upon future actions. Basically, it is assumed that the universal Turing machine U is fed both the environment program q, and AIXI’s action sequence y, and outputs AIXI’s observation sequence x by running the program q with input y. Quoting from the paper I linked:
”Reversely, if q already is a binary string we define q(y):=U(q,y)”
In the paper I linked, see Eq. 21:
and the the term
In other words, any program q that matches AIXI’s observations to date when given AIXI’s actions to date will be part of the ensemble. In order to evaluate different future action sequences, AIXI then evaluates the different future actions it could take by feeding them to its program ensemble, and summing over different possible future rewards conditional on the environments that output those rewards.
The CDT agent can correctly argue that Omega already left the million dollars out of the box when the CDT agent was presented the choice, but that doesn’t mean that it’s correct to be a CDT agent. My argument is that AIXI suffers from the same flaw, and so a different algorithm is needed.
Correct. My point is that AIXI’s surviving world-programs boil down to “Omega predicted I would two-box, and didn’t put the million dollars in”, but it’s the fault of the AIXI algorithm that this happens.
As per the AIXI equations, this is incorrect. AIXI cannot recognize the presence of a physical variable determining its next action because for any environment program AIXI’s evaluation stage is always going to try both the OneBox and TwoBox actions. Given the three classes of programs above, the only way AIXI can justify one-boxing is if the class (3) programs, in which its action somehow causes the contents of the box, win out.
Ok, missed that. I don’t think it matters to the rest of the argument, though.
An environment program can just assume a value for the physical variable and then abort by failing to halt if the next action doesn’t match it.
Or it can assume that the physical simulation branches at time t0, when Omega prepares the box, simulate each branch it until t < t1, when the next AIXI action occurs, and then kill off the branch corresponding to the wrong action.
Or, as it has already been proposed by somebody else, it could internally represent the physical world as a set of constraints and then run a constraint solver on it, without the need of performing a step-by-step chronological simulation.
So it seems that there are plenty of environment programs that can represent the action of Omega without assuming that it violates the known laws of physics. But even if it had to, what is the problem? AIXI doesn’t assume that the laws of physics forbid retro-causality.
Why would AIXI come up with something like that? Any such program is clearly more complex than one that does the same thing but doesn’t fail to halt.
Once again, possible but unnecessarily complex to explain AIXI’s observations.
Sure, but the point is that those constraints would still be physics-like in nature. Omega’s prediction accuracy is much better explained by constraints that are physics-like rather than an extra constraint that says “Omega is always right”. if you assume a constraint of the latter kind, you are still forced to explain all the other aspects of Omega—things like Omega walking, Omega speaking, and Omega thinking, or more generally Omega doing all those things that ze does. Also, if Omega isn’t always right, but is instead right only 99% of the time, then the constraint-based approach is penalized further.
It doesn’t assume that, no, but because it assumes that its observations cannot be affected by its future actions AIXI is still very much restricted in that regard.
My point is a simple one: If AIXI is going to end-up one-boxing, the simplest model of Omega will be one that used its prediction method and already predicted that AIXI would one-box. If AIXI is going to end up two-boxing, the simplest model of Omega will be one that used its prediction method and already predicted that AIXI would two-box.
However, if Omega predicted one-boxing and AIXI realized that this was the case, AIXI would still evaluate that the two-boxing action results in AIXI getting more money than the one-boxing action, which means that AIXI would two-box. As long as Omega is capable of reaching this relatively simple logical conclusion, Omega thereby knows that a prediction of one-boxing would turn out to be wrong, and hence Omega should predict two-boxing; this will, of course, turn out to be correct.
The kinds of models you’re suggesting, with branching etc. are significantly more complex and don’t really serve to explain anything.
But this doesn’t matter for Newcomb’s problem, since AIXI observes the content of the opaque box only after it has made its decision.
Which means that the epistemic model was flawed with high probability.
You are insisting that the flawed model is simpler that the correct one. This may be true for certain states of evidence where AIXI is not convinced that Omega works as advertised, but you haven’t shown that this is true for all possible states of evidence.
They may be more complex only up to a small constant overhead (how many bits does it take to include a condition “if OmegaPrediction != NextAction then loop forever”?), therefore, a constant amount of evidence should be sufficient to select them.
Yes, AIXI’s epistemic model will be flawed. This is necessarily true because AIXI is not capable of coming up with the true model of Newcomb’s problem, which is one in which its action and Omega’s prediction of its action share a common cause. Since AIXI isn’t capable of having a self-model, the only way it could possibly replicate the behaviour of the true model is by inserting retrocausality and/or magic into its environment.
I’m not even sure AIXI is capable of considering programs of this kind, but even if it is, what kind of evidence can AIXI have received that would justify the condition “if OmegaPrediction != NextAction then loop forever”? What evidence would justify such a model over a strictly simpler version without that condition?
Essentially, you’re arguing that rather than coming up with a correct model of its environment (e.g. one in which Omega makes a prediction on the basis of the AIXI equation), AIXI will somehow make up for its inability to self-model by coming up with an inaccurate and obviously false retrocausal/magical model of its environment instead.
However, I don’t see why this would be the case. It’s quite clear that either Omega has already predicted one-boxing, or Omega has already predicted two-boxing. At the very least, the evidence should narrow things down to models of either kind, although I think that AIXI should easily have sufficient evidence to work out which of them is actually true (that being the two-boxing one).
The problem is not “programs that make money magically (dis)appear for the box after the fact” but rather programs that don’t explicitly represent the presence or nonpresence of money at all until it is known. For example, a constraint solver that seeks a proof of AIXI’s observations when they are called for (using a logic that expresses normal physics). This gives all the right answers, and is fairly simple but does allow the content of the box to be controlled by the decision.
Such models would generally not offer good explanations for why Omega is so good at predicting all those other agents who aren’t AIXI, and would be penalized for this. On the other hand, any model that explains Omega’s general predictive power would be made more complex by adding a special case just for AIXI.
I don’t understand what you mean by “a constraint solver that seeks a proof of AIXI’s observations when they are called for.” Can you explain it further?
A proof system that starts with some axioms describing the physical world (excluding the AIXI machine itself), and when run with input
a_1 .. a_mbeing AIXI’s actions so far, plugs them in as axioms about AIXI’s control wires, and attempts to prove a statement of the form ‘AIXI’s input wire observes a 1 at timet’ or ‘AIXI’s input wire observes a 0 at timet’. And returns the first answer it finds.Alternatively, what about a version of Newcomb’s problem where the predictor’s source code is shown to AIXI before it makes its decision?
What would the source code of an Omega able to predict an AIXI look like?
It won’t have source code per se, but one can posit the existence of a halting oracle without generating an inconsistency.
It takes, as input, a description of the agent it’s predicting; typically source code, but in the case of AIXI, it gets the AIXI equation and a sequence of prior observations for AIXI.
As for what it does, it spends some period of time (maybe a very long one) on whatever kind of deductive and/or inductive reasoning it chooses to do in order to establish with a reasonable level of confidence what the agent it’s trying to predict will do.
Yes, AIXI being uncomputable means that Omega can’t simply run the equation for itself, but there is no need for a perfect prediction here. On the whole, it just needs to be able to come up with a well-reasoned argument for why AIXI will take a particular action, or perhaps run an approximation of AIXI for a while. Moreover, anyone in this thread arguing for either one-boxing or two-boxing has already implicitly agreed with this assumption.
This opens up the possibility that AIXI figures out that Omega is going to mispredict it, which would make TwoBoxing the best decision.
I think it is generally assumed that, even if Omega is not a perfect predictor, the agent can’t outsmart it and predict its errors. But if Omega is computable and the agent is uncomputable, this doesn’t necessarily hold true.
I’m not so sure this is true now. People in this thread arguing that AIXI does something at least have the advantage that AIXI’s decision is not going to depend on how they do the arguing. The fact that AIXI can simulate Omega with perfect fidelity (assuming Omega is not also a hypercomputer) and will make its decision based on the simulation seems like it might impact Omega’s ability to make a good prediction.
So you don’t predict anything, just put nothing in the first box, and advertise this fact clearly enough for the agent making the choice.
Newcomb’s original problem did not include the clause ‘by the way, there’s nothing in the first box’. You’re adding that clause by making additional assertions regarding what AIXI knows about “Omega”.
There’s a truly crazy amount of misunderstandings with regards to what Solomonoff Induction can learn about the world, on LW.
Let’s say you run AIXI, letting it oversee some gigabytes of webcam data, at your location. You think AIXI can match the exact location of raindrops on your roof, hours in advance? You think AIXI is going to know all about me—the DNA I have, how may I construct a predictor, etc?
A version of the problem in which Omega is predictable is hardly the same thing as a version of the problem in which the first box is always empty. Other algorithms get the million dollars; it’s just that AIXI does not. Moreover, AIXI is not being punished simply for being AIXI; AIXI not getting the million dollars is a direct consequence of the output of the AIXI algorithm.
Of course it didn’t include that clause; it would be a rather stupid problem if it did include that clause. On the other hand, what is in the statement of Newcomb’s problem is “By the time the game begins, and the player is called upon to choose which boxes to take, the prediction has already been made, and the contents of box B have already been determined.” Moreover, it is quite clearly stated that the agent playing the game is made fully aware of this fact.
If we stipulate, for the sake of argument, that AIXI cannot work out the contents of the opaque box, AIXI still fails and two-boxes. By the problem statement AIXI should already be convinced that the contents of the boxes are predetermined. Consequently, the vast majority of weight in AIXI’s distribution over world models should be held by models in which AIXI’s subsequent action has no effect on the contents of the box, and so AIXI will rather straightforwardly calculate two-boxing to have higher utility. Moreover, it’s easy for Omega to deduce this, and so the first box will be empty, and so AIXI gets $1000.
Setting the stipulation aside, I still think it should be pretty easy for AIXI to deduce that the box is empty. Given Omega’s astounding predictive success it is far more likely that Omega has a non-trivial capacity for intelligent reasoning and uses this reasoning capacity with a goal of making accurate predictions. As such, I would be surprised if an Omega-level predictor was not able to come across the simple argument I gave above. Of course, as I said above, it doesn’t really matter if AIXI can’t deduce the contents of the box; AIXI two-boxes and loses either way.
No, I don’t think that.
Really? I thought your predictor didn’t evaluate the algorithm, so how is that a ‘direct consequence’?
Yeah, and in the Turing machine provided with the tape where the action is “choose 1 box” (the tape is provided at the very beginning), the content of the box is predetermined to have 1 million, while in the entirely different Turing machine provided with the tape where the action is “choose 2 boxes”, the box is predetermined to have nothing. What is so hard to get about it? Those are two entirely different Turing machines, in different iterations of the argmax loop. Are you just selectively ignoring the part of the statement where the predictor, you know, is actually being correct?
edit: as I said, it’s a word problem, only suitable for sloshy and faulty word reasoning using folk physics. You end up ignoring some part of the problem statement.
The predictor doesn’t have to fully evaluate the algorithm to be able to reason about the algorithm.
Nowhere in the problem statement does it say that Omega is necessarily always correct. If it’s physically or logically impossible, Newcomb’s problem is basically just asking “would you prefer a million dollars or a thousand dollars.” The whole point of Newcomb’s problem is that Omega is just very, very good at predicting you.
Anyways, I think you’re misunderstanding the AIXI equation. If there are two Turing machines that are consistent with all observations to date, then both of those Turing machines would be evaluated in the one-boxing argmax iteration, and both would be evaluated in the two-boxing argmax iteration as well. There is no possible reason that either world machine would be excluded from either iteration.
As such, if in one of those Turing machines the box is predetermined to have 1 million, then it’s pretty obvious that when given the input “two-box” that Turing machine will output $1,001,000. More generally there would of course be infinitely many such Turing machines, but nonetheless the expected value over those machines will be very nearly that exact amount.
What exactly is the reason you’re suggesting for AIXI excluding the million-dollar Turing machines when it considers the two-boxing action? Where in the AIXI equation does this occur?
This is getting somewhere.
AIXI does S.I. multiple times using multiple machines differing in what they have on the extra actions tape (where the list of actions AIXI will ever take is written). All the machines used to evaluate the consequence of 1-boxing have different extra actions tape from all the machines used to evaluate the consequences of 2 boxing.
From
http://www.hutter1.net/ai/uaibook.htm
“where U is a universal (monotone Turing) machine executing q given a1..am.”
The U used for one boxing is different U from U used for two boxing, running the same q (which can use the action from the extra tape however it wants; to alter things that happen at the big bang, if it sees fit).
With regards to the content of the boxes, there are 3 relevant types of program. One is ‘there’s nothing in the box’, other is ‘there’s a million in the box’, but the third, and this is where it gets interesting, is ‘a bit from the extra input tape determines if there’s money in the box’. Third type can in principle be privileged over repeated observation of correct prediction as it does not have to duplicate the data provided on the third tape for the predictions to be correct all the time.
The third type evaluates to money in the box when the action (provided on the actions tape, which is available to the machine from the beginning) is to take 1 box, and evaluates to no money in the box when the action is to take 2 boxes.
If AIXI learns or is pre-set to know that there’s prediction of the decision happening, I take it as meaning that the third type of machine acquires sufficient weight. edit: and conversely, if the AIXI is not influenced by the program that reads from the actions tape to determine the movements of the ‘predictor’, I take it as AIXI being entirely ignorant of the predicting happening.
edit: clearer language regarding the extra actions tape
edit2: and to clarify further, there’s machines where a bit of information in q specifies that “predictor” has/hasn’t put money in the box, and there’s machines where a bit in the another tape, a1...am , determines this. Because it’s not doing any sort of back in time logic (the a1..am is here from the big bang), the latter are not that apriori improbable and can be learned just fine.
The U is always the same U; it’s a universal Turing machine. It takes as its input a world program q and a sequence of AIXI’s actions.
OK, yeah. I agree about the three types of program, but as far as I can see the third type of program basically corresponds to Omega being a faker and/or magical.
I don’t see how this interpretation of the problem makes sense. What you’re saying is not a prediction at all, it’s simply direct causation from the action to the contents of the box. By contrast, it’s inherent in the term “prediction” that the prediction happens before the thing it’s trying to predict, and therefore that the box already either contains a million dollars or does not contain a million dollars.
Let me pose a more explicit form of Newcomb’s problem by way of clarification. I don’t think it changes anything relevant as compared to the standard version of the problem; the main point is to explicitly communicate the problem setup and describes a way for AIXI to reach the epistemic state that is posited as part of Newcomb’s problem.
Omega apppears to AIXI, presents it with the two boxes, and gives its usual long explanation of the problem setup, as well as presenting some it usual evidence that there is no kind of “trick” involved here. However, before AIXI is allowed to make its decision, it is offered the opportunity to watch Omega run the game for 1000 other agents. AIXI gets to see Omega putting the money into the box in advance, and then it gets to watch the boxes the entire time. It also gets to see the player come in, it sees Omega introduce the game (the same way every time), and then watches the player make their decision, and watches them open the box and observes the contents. 1000 out of 1000 times (or maybe 999 out of 1000 if you prefer), it turns out that Omega correctly predicted the agent’s action.
Now, finally, it’s AIXI’s turn to make its own decision. All the time it was watching Omega run the other games, it has been watching the boxes that Omega originally set up for AIXI—they’ve remained completely untouched.
As far as I can see, this is a perfectly reasonable way to realize the problem setup for Newcomb’s problem.
Ohh, and I forgot to address this:
Well, the way I see it, within the deterministic hypothetical that I 1-box, at the big bang the universe is in the initial state such that I 1-box, and within the deterministic hypothetical that I 2-box, at the big bang the universe is in the initial state such that I 2-box. A valid predictor looks at the initial state and determines what I will do, before I actually do it.
Exactly the same with AIXI, which sets up hypotheticals with different initial states (which is does by adding an universal constant of what it’s going to hypothetically do (the extra tape), which is a very, very clever hack it has to employ to avoid needing to model itself correctly), and can have (or not have) a predictor which uses the initial state—distinct—to determine what AIXI will do before it does that. It correctly captures the fact that initial states which result in different actions are different, even though the way it does so is rather messy and looks ugly.
edit: i.e. to me it seems that there’s nothing fake about the predictor looking at the world’s initial state and concluding that the agent will opt to one-box. It looks bad when for the sake of formal simplicity you’re just writing in the initial state ‘I will one box’ and then have the model of your body read that and one-box, but it seems to me it’s wrong up to a constant and not more wrong than TM using some utterly crazy tag system to run a world simulator.
OK, I think I’ve just answered your question in my response to your other comment, but I’ll give a brief version here.
If there is a bit corresponding to AIXI’s future action, then by AIXI’s assumptions that bit must not be observable to AIXI until after it takes its actions. As such, models of this sort must involve some reason why the bit is observable to Omega, but not observable to AIXI; models where the information determining Omega’s prediction is also observable to AIXI will be significantly simpler.
In the sense of the language of “where U is a universal (monotone Turing) machine executing q given a1..am.”. (I unilaterally resorted to using same language as Hutter to minimize confusion—seems like a reasonable thing for two argues to adopt...).
Well, it certainly made sense to me when I gone to the store today, that if the world is deterministic, then at the big bang, it was already predetermined entirely that I would have gone to the store today. And that in the alternative that I don’t go to the store, as a simple matter of me not changing any physical laws (or, even more ridiculously, me changing the way mathematics works), it must be the case that the at the big bang, the deterministic universe was set up so that I don’t go to the store today.
A simple matter of consistency of the laws of physics within the hypothetical universe requires that two hypothetical deterministic universes with different outcomes can’t have different initial state. It’s called a prediction because it occurs earlier in the simulation history than the actual action does.
Within the hypothetical universe where I go to the store, some entity looking at that initial state of that universe, could conclude—before today—that I go to the store. I don’t see how the hell that is ‘direct causation’ from me going to the store. Me going to the store is caused by the initial state, the prediction is caused by the initial state. In AIXI’s hypothetical where it takes 1 box, it taking 1 box is caused by the initial state of the Turing machine. It literally sets up the initial state of the TM so that it ends up picking 1 box (by the way of putting it’s picking 1 box on an extra tape, or appending it at the end of the program). Not by a later intervention, which would make no sense and be folk physics that’s wrong. The prediction, likewise, can be caused by the initial state.
Re: your set up.
If the agents are complicated and fairly opaque (if they aren’t its not necessarily reasonable to assume what predictor does for them would be what predictor does for a complicated and opaque agent), and if agents are one or two boxing basically at an uniform random (AIXI won’t learn much if they all 2-box), there’s a string with thousand ones and zeroes, and it’s repetition, which gives up to a 2^-1000 penalty to the representations where those are independently encoded using 2000 bits rather than 1000 bits.
Surely AIXI will compress those two same bitstrings into one bitstring somehow.
Now, you can of course decrease the number of cases or prediction accuracy so that AIXI is not provided sufficient information to learn the predictor’s behaviour.
I think you’re misunderstanding the language here; the inputs of a Turing machine are not part of the Turing machine. The program “q” and the actions “a1...am” are both inputs to U. That said, I guess it doesn’t matter if you have two different Us, because they will always compute the same output given the same inputs anyway.
According to AIXI’s assumption of “chronological Turing machines”, this isn’t quite right. If the bit was simply encoded into the “initial state” of the universe, then AIXI could potentially observe the state of that bit before it actually takes its action. Any models where that bit influences AIXI’s observations prior to acting would directly violate Hutter’s assumption; world programs of this kind are explicitly forbidden from occupying any of AIXI’s probability mass.
Now, I’ll grant that this is subtly different from an assumption of “forward causality” because the chronological assumption specifically applies to AIXI’s subjective past, rather than the actual past. However, I would argue that models in which AIXI’s action bit causes the contents of the box without affecting its past observations would necessarily be more complex. In order for such a model to satisfy AIXI’s assumptions, the action bit needs to do one of two things:
1) Magic (i.e. makes stuff appear or disappear within the box).
2) The action bit would need to be entangled with the state of the universe in just the right way; it would just so happen that Omega can observe that action bit but AIXI cannot observe it until after it takes its action.
It seems to me that Solomonoff induction will penalise both kinds of “action-caused” models quite heavily, because they offer a poor description of the problem. If the action bit was truly part of the state of the universe, it seems rather unlikely that Omega would be able to observe it while AIXI would not.
I wouldn’t assume the agents are one-boxing or two-boxing at uniform random, that would be pretty stupid since Omega would be unable to predict them. Typical versions of Newcomb’s problem stipulate that when Omega thinks you’ll pick randomly it won’t put the million dollars in. Rather, it would be better to say that the agents are picked from some pool of agents, and it turns out that AIXI gets to witness reasonable proportions of both two-boxers and one-boxers.
I completely agree! Given the enormous 1000-bit penalty, AIXI should determine that the problem is quite well described by a “common cause” explanation—that is, the actions of the individual actions and Omega’s prediction are both determined in advance by the same factors.
In fact, I would go even further than that; AIXI should be able to duplicate Omega’s feat and quickly come up with a universe model that predicts the agents as well as or better than Omega. When AIXI observed Omega playing the game it had access to the same information about the agents that Omega did, and so whatever the source of Omega’s predictive accuracy, AIXI should be able to replicate it.
More generally, I would argue a “common cause” explanation is implicit in Newcomb’s problem, and I think that AIXI should be able to deduce reasonable models of this without making such direct observations.
In any case, once AIXI comes upon this kind of explanation (which I think is really implicit in the setup of Newcomb’s problem), AIXI is doomed. Models in which AIXI’s future action bit manages to be observable to Omega without being observable to AIXI will be significantly more complicated than models in which Omega’s prediction is determined by information that AIXI has already observed.
The most obvious such model is the one I suggested before—Omega simply reasons in a relatively abstract way about the AIXI equation itself. All of this information is information that is accessible to AIXI in advance, and hence it cannot be dependent upon the future action bit.
As such, AIXI should focus in on world models where the box already contains a million dollars, or already does not. Since AIXI will determine that it’s optimal to two-box in both kinds of world, AIXI will two-box, and since this is a pretty simple line of reasoning Omega will predict that AIXI will two-box, and hence AIXI gets $1000.
What’s quite heavily? It seems to me that you can do that in under a hundred bits, and few hundred bits of information are not that hard to acquire. If I throw a die, and it does a couple dozen bounces, it’s influenced by the thermal noise and quantum fluctuations, there’s about 2.58 bits of information that is new even to the true magical AIXI. There’s kilobits if not megabits that can be learned from e.g. my genome (even if AIXI sat for years watching a webcam and browsing the internet beforehand, it still won’t be able to predict the quantum random—it’s either non deterministic, or multiverse and you don’t know where in the multiverse you are). AIXI presumably got a webcam or other high throughput input device, too, so it may be able to privilege some hundreds bits penalized hypothesis (rule out all simpler ones) in a fraction of a second.
It strikes me as you are thinking up a fairly specific environment where the observations do not provide enough actual information. If I observe a software predictor predict a bunch of simple programs with source that I know and can fully evaluate myself, that wouldn’t come close to convincing me it is going to predict my human decision, either. And I’m fundamentally more similar to those programs than AIXI is to agents it can predict.
What’s about my example environment, where AIXI lives in a house, plays Newcomb’s problem many times, and sometimes wants to get a lot of money, and sometimes doesn’t want to get a lot of money for various reasons, e.g. out of e.g. a fear of increasing the risk of burglary if it has too much money, or the fear of going to jail on money laundering charges, or what ever. Every time, the presence of money in the first box is correlated with the actual decision AIXI makes. This has to go on for long enough, of course, until the inherently unlikely hypothesis of being predicted by something, gets privileged.
Why would AIXI privilege a magical hypothesis like that one when there are other hypotheses are strictly simpler and explain the world better? If Omega is capable of predicting AIXI reasonably accurately without the kind of magic you’re proposing, why would AIXI possibly come up with a magical explanation that involves Omega having some kind of privileged, back-in-time access to AIXI’s action which has absolutely no impact on AIXI’s prior observations!?
As for your example environment, Iterated Newcomb’s problem != Newcomb’s problem, and the problem isn’t even Newcomb’s problem to begin with if AIXI doesn’t always want the million dollars. As far as I can tell, though, you’re just trying to come up with a setup in which Omega really needs to be retrocausal or magical, rather than just really good at predicting.
In other words, AIXI’s action is predetermined by the AIXI equation. A model where Omega predicts AIXI’s action on the basis of the AIXI equation is strictly simpler than a model involving a single bit of information that is entangled with Omega and yet somehow doesn’t leak out into the universe and remains invisible to AIXI until AIXI opens the box (or boxes).
Unless the universe and/or Omega really is magical, AIXI’s observations should obviously favour the real explanation over the magical one.
This is kind of stipulated in the problem, the box being opaque, no? What does this leak look like, other than box being in some way not opaque?
We could talk of Superman Decision Theory here and how it necessarily two boxes because he has x-ray vision and sees into the box :)
You keep asserting this, but I don’t see why that’s true. Let’s suppose that I design something. I have a bit of quantum shot noise in my synapses, the precise way how I implement something probably takes a lot of information to describe. Kilobits, megabits even. Meanwhile, the body of AIXI’s robot is magical—it’s reading from the action tape to set voltages on some wires or some such. So there’s some code to replicate...
And with regards to having a real explanation available, I think it falls under the purview of box not being opaque enough. It’s akin to having a videotape of omega putting or not putting money into the box.
I think I see a better way to clarify my original remark. There is a pretty easy way to modify AIXI to do CDT. Exclude machines that read Ai before printing Oi . (And it’s very possible that some approximations or other variations by Hutter did that. I’d certainly do that if I were making an approximation. I’d need to re-read him to make totally sure he didn’t have that somewhere) I think we can both agree that if you don’t do this, you can one-box without having backwards in time causation in your model, unlike CDT (and if you do, you can’t, like CDT).
In any case, my main point is that the one boxing and two boxing depends to the way of doing physics, and given that we’re arguing about different environments yielding different ways of doing physics, I think we agree on that point.
edit: also I think I can write a variation of AIXI that matches my decisionmaking more closely. I’d just require the TM to print actions on a tape, matching the hypothetical actions. Up to a constant difference in program lengths, so it’s not worse than a choice of a TM. (I’d prefer that not to screw up probabilities though, even if its up to a constant, I need to think how edit: actually quite easy to renormalize that away… I’m far too busy right now with other stuff though). Also using some symbolic package to approximately evaluate it, evading some of can’t model oneself traps.
There could be an universe that used some ahead of time indirect evaluation to tell me in advance what action I am going to take, and with me not taking another action out of spite. I don’t know for sure our universe isn’t this—I just have a somewhat low prior for that.
No, “the box is opaque” is very different to “you have absolutely no way of working out whether the box contains the million dollars”. For example, if I’m playing Newcomb’s problem then no matter how opaque the box is I’m already pretty sure it contains a million dollars, and when I proceed to one-box every time I’m almost always right. Are you saying I’m forbidden from being able to play Newcomb’s problem?
If “Superman Decision Theory” sees into the box and necessarily two-boxes, then Superman Decision Theory is doing it wrong, because SDT is always going to get $1000 and not a million dollars.
AIXI doesn’t need to work out the precise way Omega is implemented, it just needs to find programs that appear to function the same way. If the quantum noise has no impact on Omega’s actual predictions then it’s not going to matter as far as AIXI’s hypotheses about how Omega predicts AIXI are concerned.
As far as I can see, “the box is not opaque enough” translates to “AIXI knows too much about the real world!” Well, how is it a good thing if your decision theory performs worse when it has a more realistic model of the world?
It seems to me that the more the AIXI agent works out about the world, the more physics-like its programs should become, and consequently the less it will be able to come up with the kinds of explanations in which its actions cause things in the past. Yes, world programs which involve reverse causality that just happens to be unobservable to AIXI until after it takes its own action are permissible Turing machines for AIXI, but the more information AIXI gathers, the higher the complexity penalty on those kinds of programs will be.
Why? Because, by necessity, AIXI’s action bit would have to be treated as a special case. If Omega predicts all those other agents in the same way, a bit of code that says “ah, but if Omega is facing AIXI, then Omega should access the a_5 bit from AIXI’s action tape instead of doing what Omega normally does” is simply an unnecessary complexity penalty that doesn’t help to explain anything about AIXI’s past observations of Omega.
So, the more mature AIXI gets, the more CDT-like it becomes.
I take the box being opaque to mean that the contents of the box do not affect my sensory input, and by extension that I don’t get to e.g. watch a video of omega putting money in the box, or do some forensic equivalent.
Really? What if Omega is a program, which you know predicts outputs of other simple programs written in C++, Java, and Python, and it been fed your raw DNA as a description, ’cause you’re human?
What if you just know the exact logic Omega is using?
(Besides, decision theories tend to agree that you should pretend online that you one-box)
No, you just adapt the Newcomb’s “opaque box” in an obtuse way. Superman’s facing an entirely different decision problem from the Newcomb’s that you face.
I think you’re just describing a case where AIXI fails to learn anything from other agents because they’re too different from the AIXI. What’s about my scenario where AIXI plays Newcomb’s multiple times, sometimes wanting more money and sometimes not? The program reading a_5 also appears to work right.
Well, given that predictors for AIXI are non existent, that should be the case.
edit: actually, what’s your reasons for one-boxing?
edit2: also I think this way of seeing the world—where your actions are entirely unlinked to the past—is a western phenomenon, some free will philosophy stuff. A quarter of my cultural background is quite fatalist in the outlook, so I see my decisions as the consequences of the laws of physics acting on the initial world state, and given same ‘random noise’, different decision by me implies both different future and different past.
Yep, that’s what the box being opaque means—the contents of the box have no causal effect on your perceptions.
Nope. Watching the video would contradict this principle as well, because you would still effectively be seeing the contents of the box.
What IS allowed by Newcomb’s problem, however, is coming to the conclusion that the contents of the box and your perceptions of Omega have a common cause in terms of how Omega functions or acts. You are then free to use that reasoning to work out what the contents of the box could be.
Your interpretation of Newcomb’s problem basically makes it incoherent. For example, let’s say I’m a CDT agent and I believe Omega predicted me correctly. Then, at the moment I make my decision to two-box, but before I actually see the contents of the opaque box, I already know that the opaque box is empty. Does this mean that the box is not “opaque”, by your reasoning?
If I don’t think Omega is able to predict me, then it’s not Newcomb’s problem, is it? Even if we assume that the Omega program is capable of predicting humans, DNA is not that likely to be sufficient evidence for it to be able to make good predictions.
Well, then it obviously depends on what that exact logic is.
First of all, as I said previously, if AIXI doesn’t want the money then the scenario is not Newcomb’s. Also, I don’t think the a_5 reading program will end up being the simplest explanation even in that scenario. The program would need to use something like a_5, a_67, a_166, a_190 and a_222 in each instance of Newcomb’s problem respectively. Rather than a world program with a generic “get inputs from AIXI” subroutine, you need a world program with a “recognize Newcomblike problems and use the appropriate bits” subroutine; there is still a complexity penalty.
Unless you’re trying to make a setup in which Omega necessarily works by magic, then given sufficient evidence of reality at large magic is always going to be penalised. Given that reality at large works in a non-magical way, explanations that bootstrap your model of reality at large are always going to be simpler than explanations that have to add extraneous elements of “magic” to the model.
Besides, if Omega is just plain magical, then Newcomb’s problem boils down to “is a million bigger than a thousand?”
Of course there can be predictors for AIXI. I can, for example, predict with a high degree of confidence that if AIXI knows what chess is and it wants to beat me at chess, it’s going to beat me. Also, if AIXI wants to maximise paperclips, I can easily predict that there are going to be a lot of paperclips.
By being the kind of person who one-boxes, I end up with a million dollars instead of a thousand.
Um, the “libertarian free will” perspective is mostly what I’m arguing against here. The whole problem with CDT is that it takes that perspective, and, in concluding that its action is not in any way caused by its past, it ends up with only $1000. My point is that AIXI ultimately suffers from the same problem; it assumes that it has this magical kind of free will when it actually does not, and also ends up with $1000.
Yeah, and then you kept stipulating that the model where Omega has read the action tape and then put or not put money into the box, but it didn’t leak onto sensory input, is very unlikely, and I noted that it’s stipulated in the problem statement that the box contents do not leak onto sensory input.
Let’s say AIXI lives inside the robot named Alice. According to every model employed by AIXI, the robot named Alice has pre-committed, since the beginning of time, to act out a specific sequence of actions. How the hell that assumes magical free will I don’t know. edit: and note that you can exclude machines which had read the action before printing matching sensory data, to actually ensure magical free will. I’m not even sure, maybe some variations by Hutter do just that.
edit:
That’s just abstruse. We both know what I mean.
Well, you’re just pre-committed to 1-box, then. The omegas that don’t know you’re pre-committed to 1-box (e.g. don’t trust you, can’t read your pre-committments, etc) would put nothing there, though, which you might be motivated to think about if its e.g. 10 millions vs 1 million, or 2 millions vs 1 million. (I wonder if one boxing is dependent on inflation...)
edit: let’s say I am playing the omega, and you know I know this weird trick for predicting you on the cheap.… you can work out what’s in the first box, can’t you? If you want money and don’t care of proving omega wrong out of spite, I can simply put nothing in the first box, and count on you to figure that out. You might have committed to the situation with 1 million vs 1 thousand, but I doubt you committed to 1000000 vs 999999 . You say you one box, fine, you get nothing—a rare time Omega is wrong.
edit2: a way to actually do Newcomb’s in real life, by the way. Take poor but not completely stupid people, make it 1000 000 vs 999 999 , and you can be almost always right. You can also draw some really rich people who you believe don’t really care and would 1-box for fun, and put a million in the first box for those, and be almost always right about both types of the case.
The two situations are quite different. Any complexity penalty for the non-leaking box has already been paid via AIXI’s observations of the box and the whole Newcomb’s setup; the opaqueness of the box just boils down to normal reality. On the other hand, your “action bit” model in which Omega reads AIXI’s action tape is associated with a significant complexity penalty because of the privileged nature of the situation—why specifically Omega, and not anyone else? Why does Omega specifically access that one bit, and not one of the other bits? The more physics-like and real AIXI’s Turing machines get, the more of a penalty will be associated with Turing machines that need to incorporate a special case for a specific event.
AIXI as defined by Hutter (not just some “variation”) has a foundational assumption that an action at time t cannot influence AIXI’s perceptions at times 1..t-1. This is entirely incompatible with a model of Alice where she has pre-commited since the beginning of time, because such an Alice would be able to discover her own pre-commitment before she took the action in question. AIXI, on the other hand, explicitly forbids world models where that can happen.
No, I don’t. My point is that although you can’t predict AIXI in the general case, there are still many cases where AIXI can be predicted with relative ease. My argument is still that Newcomb’s problem is one of those cases (and that AIXI two-boxes).
As for all of your scenarios with different Omegas or different amounts of money, obviously a major factor is how accurate I think Omega’s predictions are. If ze has only been wrong one time in a million, and this includes people who have been one-boxing as well, why should I spend much time thinking about the possibility that I could be the one time he gets it wrong?
Similarly, if you’re playing Omega and you don’t have a past history of correctly predicting one-boxing vs one-boxers, then yes, I two-box. However, that scenario isn’t Newcomb’s problem. For it to be Newcomb’s problem, Omega has to have a history of correctly predicting one-boxers as well as two-boxers.
I suspect the unspecified implementation of Omega hides assumptions if not contradictions. Let me propose a more concrete version: The problem is presented by Conservative Finite Omega (CFO), who works by pulling the agent’s source code, simulating it for a long but finite time, and putting $1,000,000 in the opaque box iff the simulation is determined to definitely one-box. The agent never walks away with the full $1,001,000, though the agent does sometimes walk away with $0.
So, assuming AIXI is confident in accurate models of how CFO works, CFO will simulate AIXI, which requires it to simulate AIXI’s (accurate) simulation of CFO—endless recursion. AIXI ‘wins’ the timeout war (correctly predicting CFO’s timeout), concludes that CFO has left the opaque box empty, and two-boxes.
You could look at that outcome as AIXI being penalized for being too smart. You could also say that an even smarter agent would prepend ‘if (facing CFO-like situation) then return one-box’ to its source code. Fundamentally, the specification of AIXI cannot conceive of its source code being an output; it’s baked into the assumptions that the explicit output bits are the only outputs.
Sure, I don’t necessarily blame the AIXI equation when it’s facing a relatively stupid Omega in that kind of situation.
However, consider “More Intelligent Finite Omega”, who pulls the agent’s source code and uses an approximate theorem-proving approach until it determines, with high confidence, what AIXI is going to do. Assuming that AIXI has received sufficient evidence to be reasonably confident in its model of MIFO, MIFO can reason like this:
AIXI will be able to accurately simulate me, therefore it will either have determined that the box is already empty, or already full.
Given either of those two models, AIXI will calculate that the best action is to two-box.
Consequently, AIXI will two-box.
and then MIFO will leave the opaque box empty, and its prediction will have been correct. Moreover, MIFO had no other choice; if it were put the money in the opaque box, AIXI would still have two-boxed, and MIFO’s prediction would have been incorrect.
If you’re allowed to make the assumption that AIXI is confident in its model of CFO and CFO knows this, then I can make the same assumption about MIFO.
I think you’re right. At first I was worried (here and previously in the thread) that the proof that AIXI would two-box was circular, but I think it works out if you fill in the language about terminating turing machines and stuff. I was going to write up my formalization, but once I went through it in my head your proof suddenly looked too obviously correct to be worth expanding.
Oh, and if the special circumstances offered to AIXI are a problem w.r.t. allowing tricky explanations, I think that should be resolvable.
For example, getting to watch other agents play before making your own decision is the “VIP treatment”. Omega decides whether or not to give this treatment by transparently by generating a pseudorandom integer between 1 and 2000, and it gives the VIP treatment if and only if that integer comes up as a 1. The player gets to directly observe the random number generation mechanism while Omega is using it.
Out of the 1000 agents who also got to play Omega’s game while AIXI was watching, a 1 came up on the 965th game. When that happened, that agent got to watch 1000 games, although in those 1000 a 1 didn’t come up at all. Since AIXI was still waiting around, it got to watch those extra 1000 games before watching the last 35 games of its VIP allocation. Of the 2000 games played, Omega made 2000 correct predictions.
If 2000/2000 is too unlikely and inherently results in weird explanations, we can adjust the number down appropriately e.g. 1900/2000.
Actually, it’s a bit of challenge making an environment where AIXI learns about the predictor.
I think I have one. AIXI lives in a house, it has 100$, it gets reward any cycle there’s at least 1$ in the house, and it plays Newcomb’s repeatedly. Money are delivered to the house. So it doesn’t necessarily always want a million dollar. So first it grabs $1000 from the transparent box (it doesn’t know about the predictor) and immediately spends them on ordering a better door, because it has hypotheses concerning possible burglary, where the house would be set on fire, and no money will be left at all. Then, it doesn’t have the door yet, and it doesn’t want extra money because it can attract theft, so it one boxes, but gets a million.
It keeps one and two boxing as it’s waiting and receives various security upgrades to it’s house, sets up secure money pick ups for the banks, offshore accounts, and so on and so forth. And the predictor turns out to be always correct. So it is eventually dominated by TMs that use the one vs two boxing bit of data from the a1...am tape to specify what the hand of the predictor is doing with a million dollars when simulating the past. So at some point if it wants a million dollars, it one boxes, and if it doesn’t, it two boxes.