Suppose you needed to assign non-zero probability to any way things could conceivably turn out to be, given humanity’s rather young and confused state—enumerate all the hypotheses a superintelligent AI should ever be able to arrive at, based on any sort of strange world it might find by observation of Time-Turners or stranger things. How would you enumerate the hypothesis space of all the coherently-thinkable worlds we could remotely maybe possibly be living in, including worlds with Stable Time Loops and even stranger features?
Hmmm. Causal universes are a bit like integers; there’s an infinite number of them, but they pale as compared to thenumber of numbers as a whole.
Mostly-causal universes with some time-travel elements are more like rational numbers; there’s more than we’re ever going to use, and it looks at first like it covers all possibilities except for a few strange outliers, like pi or the square root of two.
But there’s vastly, vastly more irrational numbers than rational numbers; to the point where, if you had to pick a truly random number, it would almost certainly be irrational. Yet, aside from a few special cases (such as pi), irrational numbers are hardly even considered, never mind used; we try to approximate the universe in terms of rational numbers only. (Though a rational number can be arbitrarily close to any given number).
Irrational numbers are also uncountable, and I imagine that I’ll end up in similar trouble trying to enumerate all the universes that could exist, given “Stable Time Loops and even stranger features”.
Given that, there’s only one reasonable way to handle the situation; I need to assign some probability to “stranger things” without being able to describe, or to know, what those stranger things are.
The possibilities that I can consider include:
Physics as we know it is entirely and absolutely correct (v. low probability)
Physics as we know it is an extremely good approximation to reality (reasonable probability)
The real laws of the universe are understandable by human minds (surprisingly high probability)
Stranger Things (added to the three above potions, adds up to 100%)
Alternatively:
The universe is entirely causal (fairly low probability)
The universe is almost entirely causal, with one or more rare and esoteric acausal features (substantially higher probability, maybe four or five times as high as the above option)
The local causality observed is merely a statistical fluke in a mostly acausal universe (extremely low probability)
Stranger Things (whatever probability remains)
The reason why the second is higher than the first, is simply that there are so many more possible universes in which the second would be true (but not the first) in which the observations observed to date would nonetheless be true. The problem with these categorisations is that, in every case, the highest probability seems to be reserved for Stranger Things...
I almost went with that answer, and didn’t ask. But then I thought about trade with future agents who have different resources and values than we do—resources and values which will be heavily influenced by what we do today. The structure seems to be at least as similar as self-consistent solutions in plasma physics.
Lowenheim-Skolem is going to give you trouble, unless “coherently-thinkable” is meant of as a subtantive restriction. You might be able to enumerate finitely-axiomatisable models, up to isomorphism, up to aleph-w, if you limit yourself to k-categorical theories, for k < aleph-w, though. Then you could use Will’s strategy and enumerate axioms.
Edit: I realised I’m being pointlessly obscure.
The Upwards Lowenheim-Skolem means that, for every set of axioms in your list, you’ll have multiple (non-isomorphic) models.
You might avoid this if “coherantly thinkable” was taken to mean “of small cardinality”.
If you didn’t enjoy this restriction, you could, for any given set of axioms, enumerate the k-categorical models of that set of axioms—or at least enumerate the models of whose cardinality can be expressed as 2^2^...2^w, for some finite number of 2′s. This is because k-categoriciticy means you’ll only have one model of each cardinality, up to isomorphism.
So then you just enumerate all the possible countable combinations of axioms, and you have an enumeration of all countably axiomatisable, k-categorical, models.
I don’t think it’s unfair to put some restrictions on the universes you want to describe. Sure, reality could be arbitrarily weird—but if the universe cannot even be approximated within a number of bits much larger than the number of neurons (or even atoms, quarks, whatever), “rationality” has lost anyway.
(The obvious counterexample is that previous generations would have considered different classes of universes unthinkable in this fashion.)
Sure, reality could be arbitrarily weird—but if the universe cannot even be approximated within a number of bits much larger than the number of neurons (or even atoms, quarks, whatever), “rationality” has lost anyway.
Why? If the universe has features that our current computers can’t approximate, maybe we could use those features to build better computers.
Enumerate mathematical objects by representing them in a description language and enumerating all strings. Look for structures that are in some sense indistinguishable from “you”. (taboo “you”, and solve a few philosphical problems along the way). There’s your set of possible universes. Distribute probability in some way.
Bayesian inference falls out by aggregating sets of possible worlds, and talking about total probability.
In the same stroke with whch you solve the “you”-identification problem, solve the value-identification problem so that you can distribute utility over possible worlds, too. Excercising the logical power to actually observe the worlds that involve you on a close enough level will involve some funky shit where you end up determining/observing your entire future utility-maximizing policy/plan. This will involve crazy recursion and turning this whole thing inside-out, and novel work in math on programs deducing their own output. (see TDT, UDT, and whatever solves their problems).
Approximating this thing will be next to impossible, but we have an existence proof by example (humans), so get to it. (we don’t have prrof that lawful recursion is possible, though, if I understand correctly)
Our current half-assed version of the inference thing (Solominoff Induction) uses Turing Machines (ick) as the description language, and P’= 2^(-L), where L is the length of the strings describing the universes (that’s an improper prior, but renorm handles that quick).
We have proofs that P’ = 1 does not work (no free lunch (or is that not the right one here...)), and we can pack all of our degrees of freedom into the design of the description language if we choose the length prior. (Or is that almost all? Proof, anyone?)
This leaves just the design of the description langauge. Computable programming languages seem OK, but all have unjustified inductive bias. Basically we have to figure out which one is a close approximation for our prior. Turing machines don’t seem particularly priveledged in this respect.
EDIT: Bolded the Tl;dr.
EDIT: Downvotes? WTF? Can we please have a norm that people can speculate freely in meditation threads without being downvoted? At least point out flaws… If it’s not about logical flaws, I don’t know what it is, and the downvote carries very nearly no information.
“Non-zero probability” doesn’t seem like quite the right word. If a parameter describing the way things could conceivably turn out to be can take, say, arbitrary real values, then we really want “non-zero probability density.” (It’s mathematically impossible to assign non-zero probability to each of uncountably many disjoint hypotheses because they can’t add to 1.)
The first answer that occurred to me was “enumerate all Turing machines” but I’m worried because it seems pretty straightforward to coherently think up a universe that can’t be described by a Turing machine (either because Turing machines aren’t capable of doing computations with infinite-precision real numbers or because they can’t solve the halting problem). More generally I’m worried that “coherently-thinkable” implies “not necessarily describable using math,” and that would make me sad.
can’t be described by a Turing machine… because Turing machines aren’t capable of doing computations with infinite-precision real numbers
I think you can get around that by defining “describe” to mean “for some tolerance t greater than zero, simulate with accuracy within t”. Since computable numbers are dense in the reals, for any t > 0 there will always be a Turing machine that can do the job.
The halting problem is insuperable, though. Universes with initial conditions or dynamics that depend on, e.g., Chaitin’s constant are coherently thinkable but not computable.
I don’t think your first point solves the problem. If the universe is exponentially sensitive to initial conditions, then even arbitrarily small inaccuracies in initial conditions make any simulation exponentially worse with time.
The function exp(x—K) grows exponentially in x, but is nevertheless really, really small for any x << K. Unbounded resources for computing means that the analogue of K may be made as large as necessary to satisfy any fixed tolerance t.
Yes, for a fixed amount of time. I should have made that explicit in my definition of “describe”: for some tolerance t greater than zero, simulate results at time T with accuracy within t. Then for any t > 0 and any T there will always be a Turing machine that can do the job.
this is my first time approaching a meditation, and I’ve actually only now decided to de-lurk and interact with the website.
One way to enumerate them would be, as CCC has just pointed out, with integers where irrationality denotes acausal worlds and rationality denotes causal worlds.
This however doesn’t leaves space for Stranger Things; I suppose we could use the alphabet for that.
1
If, however, and like I think, you mean enumerate as “order in which the simulation for universes can be run” then all universes would have a natural number assigned to them, and they could be arranged in order of complexity; this would mean our own universe would be fairly early in the numbering, if causal universes are indeed simpler than acausal ones, if I’ve understood things correctly.
This would mean we’d have a big gradient of “universes which I can run with a program” followed by a gradient of “universes which I can find by sifting through all possible states with an algorithm” and weirder stuff elsewhere (it’s weird; thus it’s magic, and I don’t know how it works; thus it can be simpler or more complex because I don’t know how it works).
In the end, the difference between causal and acausal universes is that one asks you only the starting state, while the other discriminates between all states and binds them together.
AAAAANNNNNNNND I’ve lost sight of the original question. Dammit.
I mean the class of causal worlds be dense in the class of worlds, where worlds consists of causal and acausal worlds. The same way we understand a lot of things in functional analysis: prove the result for the countable case, prove that taking compactifications/completions preserves the property, and then you have it for all separable spaces.
One way to enumerate them would be, as CCC has just pointed out, with integers where irrationality denotes acausal worlds and rationality denotes causal worlds.
This however doesn’t leaves space for Stranger Things;
Well, I admit that I had originally considered that Stranger Things would most likely be either causal or acausal; I can’t really imagine anything that’s neither, given that the words are direct opposites.
In the case of a Stranger Thing that’s strange enough that I can’t imagine it, we could always fall back on the non-real complex numbers, which are neither rational nor irrational (and are numerous enough to make real numbers look trivial by comparison).
Well, I appear to be somewhat confused. Here is the logic that I’m using so far:
If:
1: A hypothesis space can contain mathematical constants,
2: Those mathematical constants can be irrational numbers,
3: The hypothesis space allows those mathematical constants to set to any irrational number,
4: And the set of irrational numbers cannot be ennumerated.
Then:
5: A list of hypothesis spaces is impossible to enumerate.
So If I assume 5 is incorrect (and that it is possible to enumerate the list) I seem to either have put together something logically invalid or one of my premises is wrong. I would suspect it is premise 3 because it seems to be a bit less justifiable then the others.
On the other hand, it’s possible premise 3 is correct, my logic is valid, and this is a rhetorical question where the answer is intended to be “That’s impossible to enumerate.”
I think the reason that I am confused is likely because I’m having a hard time figuring out where to proceed from here.
If you ever plan on talking about your hypothesis, you need to be able to describe it in a language with a finite alphabet (such as English or a programming language). There are only countably many things you can say in a language with a finite alphabet, so there are only countably many hypotheses you can even talk about (unambiguously).
This means that if there are constants floating around which can have arbitrary real values, then you can’t talk about all but countably many of those values. (What you can do instead is, for example, specify them to arbitrary but finite precision.)
If you ever plan on talking about your hypothesis, you need to be able to describe it in a language with a finite alphabet (such as English or a programming language). There are only countably many things you can say in a language with a finite alphabet, so there are only countably many hypotheses you can even talk about (unambiguously).
Only if you live in a universe where you’re limited to writing finitely many symbols in finite space and time.
If I lived in such a universe, then it seems like I could potentially entertain uncountably many disjoint hypotheses about something, all of which I could potentially write down and potentially distinguish from one another. But I wouldn’t be able to assign more than countably many of them nonzero probability (because otherwise they couldn’t add to 1) as long as I stuck to real numbers. So it seems like I would have to revisit that particular hypothesis in Cox’s theorem…
It looks like you’re right, but let’s not give up there. How could we parametrize the hypothesis space, given that the parameters may be real numbers (or maybe even higher precision than that).
Well I suppose starting with the assumption that my superintelligent AI is merely turing complete, I think that we can only say our AI has “hypothesis about the world” if it has a computable model of the world. Even if the world weren’t computable, any non-computable model would be useless to our AI, and the best it could do is a computable approximation. Stable time loops seem computable through enumeration as you show in the post.
Now, if you claim that my assumption that the AI is computable is flawed, well then I give up. I truly have no idea how to program an AI more powerful than turing complete.
Even if the world weren’t computable, any non-computable model would be useless to our AI, and the best it could do is a computable approximation.
Again, what distinguishes a “turing oracle” from a finite oracle with a bound well above the realizable size of a computer in the universe? They are indistinguishable hypotheses. Giving a turing complete AI a turing oracle doesn’t make it capable of understanding anything more than turing complete models. The turing-transcendant part must be an integral part of the AI for it to have non-turing-complete hypotheses about the universe, and I have no idea what a turing-transcendant language looks like and even less of an idea of how to program in it.
I don’t see how this changes the possible sense-data our AI could expect. Again, what’s the difference between infinitely many computations being performed in finite time and only the computations numbered up to a point too large for the AI to query being calculated?
If you can give me an example of a universe for which the closest turing machine model will not give indistinguishable sense-data to the AI, then perhaps this conversation can progress.
Well, for starters, an AI living in a universe where infinitely many computations can be performed in finite time can verify the responses a Turing oracle gives it. So it can determine that it lives in a universe with Turing oracles (in fact it can itself be a Turing oracle), which is not what an AI living in this universe would determine (as far as I know).
As mentioned below, we you’d need to make infinitely many queries to the Turing oracle. But even if you could, that wouldn’t make a difference.
Again, even if there was a module to do infinitely many computations, the code I wrote still couldn’t tell the difference between that being the case, and this module being a really good computable approximation of one. Again, it all comes back to the fact that I am programming my AI on a turing complete computer. Unless I somehow (personally) develop the skills to program trans-turing-complete computers, then whatever I program is only able to comprehend something that is turing complete. I am sitting down to write the AI right now, and so regardless of what I discover in the future, I can’t program my turing complete AI to understand anything beyond that. I’d have to program a trans-turing complete computer now, if I ever hoped for it to understand anything beyond turing completeness in the future.
Ah, I see. I think we were answering different questions. (I had this feeling earlier but couldn’t pin down why.) I read the original question as being something like “what kind of hypotheses should a hypothetical AI hypothetically entertain” whereas I think you read the original question as being more like “what kind of hypotheses can you currently program an AI to entertain.” Does this sound right?
I was reading a lesswrong post and I found this paragraph which lines up with what I was trying to say
Some boxes you really can’t think outside. If our universe really is Turing computable, we will never be able to concretely envision anything that isn’t Turing-computable—no matter how many levels of halting oracle hierarchy our mathematicians can talk about, we won’t be able to predict what a halting oracle would actually say, in such fashion as to experimentally discriminate it from merely computable reasoning.
I don’t think that’s different, unless it can also make infinitely many queries of the Turing oracle in finite time. Or make one query of a program of infinite length. In any case, I think it needs to perform infinite communication with the oracle.
I’ll grant that it seems likely that a universe with infinite computation capability will also have infinite communication capability using the same primitives, but I don’t think it’s a logical requirement.
The hypothesis that should interest an AI are not necessarily limited to those it can compute but to those it could test. A hypothesis is useless if it does not tell us something about how the world looks when it’s true as opposed to when it’s false. So if there is a way for the AI to interact with the world such that it expects different probabilities of outcomes depending on whether the (possibly uncomputable) hypothesis holds or not then it is something worth having a symbol for, even if the exact dynamics of this universe cannot be computed.
Let’s consider the case of our AI encountering a Turing Oracle. Two possible hypotheses of the AI could be A = This is in fact a Turing Oracle and for every program P it will output either the time until halting or 0 if no halting, and B = This is not a Turing Oracle but some computable machine Q. The AI could feed the supposed oracle a number of programs and if it was told any of them would halt it could try to run them for the specified number of steps to see if they did indeed halt. After each program had halted it would have to increase it’s probability that this was in fact a Turing Oracle using Bayes’ Theorem and estimates of the probabilities of guessing this right, or computationally deriving these numbers. If it did this for long enough and this was in fact a Turing Oracle it would gain higher and higher certainty of this fact.
What is it that the AI is doing? We can view the whole above process as a program which given one of a limited set of experimental outcomes outputs the probability that this experimental outcome would be the real one if H held. In the case of the Turing Oracle above the set of outcomes is the set of pairs (P,n) where P is a program and n a positive integer, and the program will output 1 if P halts after n steps and 0 otherwise. I think this captures in full generality all possibilities a computable agent would be able to recognise.
What if the AI later on gains some extra computational capacity which makes it non-computable? Say for example that it finds a Turing Oracle in like in the above example and integrates it into its main processor. But this is essentially everything that could happen: for the AI to become uncomputable, it would have to integrate an uncomputable physical process into its own processing. But for the AI to know it was actually uncomputable and not only incorporating the results of some computational process it didn’t recognise it would have to preform above test. So when it now preforms some uncomputable test on a new process we can see this simply as the composite of the tests of the original and the new process viewing all the message passing between the uncomputable processes as a part of the experimental setup rather than internal computation.
Hrmm… Well, if the AI is computable, it can only ever arrive at computable hypotheses, so we can enumerate them with any complete program specification language. I feel like I want to say that anything that isn’t computable, doesn’t matter. What I mean is, if the AI encounters something that is truly outside of its computable hypothesis space, then there’s nothing it can do about it. For concreteness:
TL;DR for paragraph below: our FAI encounters an Orb, which seems to randomly display red or green, and which our FAI really really wants to model accurately.
Say that our successful superintelligent FAI, in its preliminary probing of the local cosmos, has encountered another alien species with its own FAI. HumanFAI and AlienFAI come to an agreement to share the universe equitably. But HumanFAI finds out about a smallish spherical region of space, impervious to all probes. Every second, the Orb (apparently) randomly changes to emit either green light or red light. Many important things causally depend on which color the Orb displays; for some reason the entire alien culture is very causally dependent on the Orb, and the aliens still causally interact with the HumanFAI’s domain. That is, the utility of the universe under HumanFAI’s utility function is causally affected by the Orb. Thus, the AI cares about the Orb, as in it wants to model it accurately.
However, try as it might, our poor computable AI cannot do even an epsilon better than random in predicting the Orb. This is because the Orb is not computable. My point is that this is not distinguishable from a problem that the AI just can’t solve given its current resources. If the prediction problem is really hard, but nevertheless the AI can gain useful information about how the aliens will behave… then either the AI is modelling the alien species, plus a Truly Random variable (in the classical statistics sense) for the Orb, or the AI can do better than random at predicting the Orb.
Therefore, if our AI ever encountered something Truly Random or otherwise Really Weird (that is, something that is coherent in whatever way it has to be in order to be real, but not computable), then the AI would not and could not do better than it would by just reacting as though it was a problem too hard to solve, and modelling it as a random variable. For things that seemed random or weird but that were actually computable, the AI would naturally (if we’ve done our job) become smart enough or think long enough to solve the problem, or at least work around it. For things that were really truly unpredictable by a computable hypothesis, the same thing would happen. It’s just a special case where the AI never gets around to solving it.
Declaring uncomputable things irrelevant hopefully isn’t too crippling in practice; the universe looks computable, Time Turners can maybe be brute-forced, etc. Now, that doesn’t really answer the question. What do we do about uncomputable universes? Again, nothing… except if there is a chance of hypercomputation. But even if an AI is trying to somehow harness a hypercomputation to do better than chance at dealing with an uncomputable facet of reality, it still has to do figure out how to do the harnessing using its current computable hypotheses and the rest of its computable self.
In other words, hypercomputation isn’t a special case. It’s still a part of reality that correlates in some way with another part of reality (right? I’m, like, totally out of my depth here, but as long as we’re speculating...). The AI can notice and use this, while still only working from computable hypotheses. It should do this naturally, even operating under computable hypotheses, if it sees some way of expanding its (hyper)computational abilities.
TL;DR: whether or not the universe is computable, the AI can’t do better than computable hypotheses. The differences between reality and the best hypotheses that the AI can muster will be unavoidable, since the AI is computable. It can harness hypercomputation, but it still does so working from its computable hypotheses. Unless we program an uncomputable AI. Are you trying to ask how the AI should write an uncomputable extension of itself if it encounters hypercomputation?
Since I’m asking about a superintelligent AI’s model of the world, and the world of an AI is digital input and output, I first enumerate all possible programs, then enumerate all input strings of finite length, then count diagonally over both.
Then I convert the bits into ASCII, compile them in LOLCODE (since I’m already doing this for the lulz), and throw out the ones that give me compiler errors or duplicates.
Then I sum over the countable number of computable things using inverse squares and divide by pi squared over six (minus whatever I’ve thrown out).
I hope you didn’t want this information to be compiled in a way that is at all helpful to anyone, ever.
But if you did, I guess I might attempt to organize the information as the set of graphs on N vertices for all natural numbers N, or attempt to classify the set of categories of modules of objects with models in Grothendieck’s second universe, so that I could do all possible linear algebra. And then I would say that if I can’t use linear algebra the object I’m studying doesn’t have local consistency and so it doesn’t make sense to think about it as a continuous universe and I no longer no what thought means so I have more important issues to deal with.
This seems an odd question to ask in the comments like this. I know how I’d go about figuring out the answer, but it involves doing lots and lots of really hard math. Coming up with an answer i thee 5 minutes that anyone is going to realistically spend on this seems almost disrespectful, and certainly not very productive.
I think the problem of enumerating these possibilities is impossible. You should notice that even the conventional possibility, quantum field theory somehow modified to have gravity and cosmology, is incomplete. It describes a mathematical construct, but it doesn’t describe how our experiences fit into that construct. It’s possible that just by looking at this mathematical object in a different way, you can find a different universe. That’s why this point-of-view information is actually important. Looking just at the possibilities where the universe is computable, enumerating Turing machines looks sufficient, but it is not. Turing machines don’t describe where we should look for ourselves in them, which is the most important part of the business. If we allow this, we should also allow the universe to be described by finite binary strings, which at times code for a Turing machine where we can be found in a certain point of view, but at other times code for various more powerful modes of computation. We can even say there is only one possibility, the totality of mathematical objects being the universe, which we can find ourselves in in very many different ways (this is the Tegmark level 4 multiverse theory).
So we can’t truly enumerate all the possibilities, even assuming a casual universe, since a casual diagram isn’t really capable of fully describing a possibility. It might be reasonable at certain times to enumerate these things anyways, and deal with this degeneracy in a ad hoc way. In that case, there would be nothing wrong with also making an ad hoc assumption along the lines of saying that the universe must be Turing computable (in which case you can simply list Turing machines).
Meditation:
Suppose you needed to assign non-zero probability to any way things could conceivably turn out to be, given humanity’s rather young and confused state—enumerate all the hypotheses a superintelligent AI should ever be able to arrive at, based on any sort of strange world it might find by observation of Time-Turners or stranger things. How would you enumerate the hypothesis space of all the coherently-thinkable worlds we could remotely maybe possibly be living in, including worlds with Stable Time Loops and even stranger features?
Hmmm. Causal universes are a bit like integers; there’s an infinite number of them, but they pale as compared to thenumber of numbers as a whole.
Mostly-causal universes with some time-travel elements are more like rational numbers; there’s more than we’re ever going to use, and it looks at first like it covers all possibilities except for a few strange outliers, like pi or the square root of two.
But there’s vastly, vastly more irrational numbers than rational numbers; to the point where, if you had to pick a truly random number, it would almost certainly be irrational. Yet, aside from a few special cases (such as pi), irrational numbers are hardly even considered, never mind used; we try to approximate the universe in terms of rational numbers only. (Though a rational number can be arbitrarily close to any given number).
Irrational numbers are also uncountable, and I imagine that I’ll end up in similar trouble trying to enumerate all the universes that could exist, given “Stable Time Loops and even stranger features”.
Given that, there’s only one reasonable way to handle the situation; I need to assign some probability to “stranger things” without being able to describe, or to know, what those stranger things are.
The possibilities that I can consider include:
Physics as we know it is entirely and absolutely correct (v. low probability)
Physics as we know it is an extremely good approximation to reality (reasonable probability)
The real laws of the universe are understandable by human minds (surprisingly high probability)
Stranger Things (added to the three above potions, adds up to 100%)
Alternatively:
The universe is entirely causal (fairly low probability)
The universe is almost entirely causal, with one or more rare and esoteric acausal features (substantially higher probability, maybe four or five times as high as the above option)
The local causality observed is merely a statistical fluke in a mostly acausal universe (extremely low probability)
Stranger Things (whatever probability remains)
The reason why the second is higher than the first, is simply that there are so many more possible universes in which the second would be true (but not the first) in which the observations observed to date would nonetheless be true. The problem with these categorisations is that, in every case, the highest probability seems to be reserved for Stranger Things...
Rationals and integers are both coutable! This is one of my favorite not-often-taught-in-elementary-schools but easily-explainable-to-elementary-school-students math facts. And they, the rationals, make a pretty tree: http://mathlesstraveled.com/2008/01/07/recounting-the-rationals-part-ii-fractions-grow-on-trees/
That’s one of my favorite mathematical constructions! Also see Ford circles.
If this universe contains agents who engage in acausal trade, does that make it partially acausal?
Nope. It’s just a terrible name.
I almost went with that answer, and didn’t ask. But then I thought about trade with future agents who have different resources and values than we do—resources and values which will be heavily influenced by what we do today. The structure seems to be at least as similar as self-consistent solutions in plasma physics.
Agents can make choices that enforce global logical constraints, using computational devices that run on local causality.
Thanks, I feel like I grok this answer: There may be higher order acausal structures in the universe, but they run on a causal substrate.
By ‘acausal trade’, do you mean:
Trading based on a present expectation of the future (such as trading in pork futures)
or
Trading based on data from the actual future
The first is causal (but does not preclude the possibility of the universe containing other acausal effects), the second is acausal.
Nope.
A start is to choose some language for writing down axiom lists for formal systems, and a measure on strings in that language.
Lowenheim-Skolem is going to give you trouble, unless “coherently-thinkable” is meant of as a subtantive restriction. You might be able to enumerate finitely-axiomatisable models, up to isomorphism, up to aleph-w, if you limit yourself to k-categorical theories, for k < aleph-w, though. Then you could use Will’s strategy and enumerate axioms.
Edit: I realised I’m being pointlessly obscure.
The Upwards Lowenheim-Skolem means that, for every set of axioms in your list, you’ll have multiple (non-isomorphic) models.
You might avoid this if “coherantly thinkable” was taken to mean “of small cardinality”.
If you didn’t enjoy this restriction, you could, for any given set of axioms, enumerate the k-categorical models of that set of axioms—or at least enumerate the models of whose cardinality can be expressed as 2^2^...2^w, for some finite number of 2′s. This is because k-categoriciticy means you’ll only have one model of each cardinality, up to isomorphism.
So then you just enumerate all the possible countable combinations of axioms, and you have an enumeration of all countably axiomatisable, k-categorical, models.
I don’t think it’s unfair to put some restrictions on the universes you want to describe. Sure, reality could be arbitrarily weird—but if the universe cannot even be approximated within a number of bits much larger than the number of neurons (or even atoms, quarks, whatever), “rationality” has lost anyway.
(The obvious counterexample is that previous generations would have considered different classes of universes unthinkable in this fashion.)
Why? If the universe has features that our current computers can’t approximate, maybe we could use those features to build better computers.
Enumerate mathematical objects by representing them in a description language and enumerating all strings. Look for structures that are in some sense indistinguishable from “you”. (taboo “you”, and solve a few philosphical problems along the way). There’s your set of possible universes. Distribute probability in some way.
Bayesian inference falls out by aggregating sets of possible worlds, and talking about total probability.
In the same stroke with whch you solve the “you”-identification problem, solve the value-identification problem so that you can distribute utility over possible worlds, too. Excercising the logical power to actually observe the worlds that involve you on a close enough level will involve some funky shit where you end up determining/observing your entire future utility-maximizing policy/plan. This will involve crazy recursion and turning this whole thing inside-out, and novel work in math on programs deducing their own output. (see TDT, UDT, and whatever solves their problems).
Approximating this thing will be next to impossible, but we have an existence proof by example (humans), so get to it. (we don’t have prrof that lawful recursion is possible, though, if I understand correctly)
Our current half-assed version of the inference thing (Solominoff Induction) uses Turing Machines (ick) as the description language, and P’= 2^(-L), where L is the length of the strings describing the universes (that’s an improper prior, but renorm handles that quick).
We have proofs that P’ = 1 does not work (no free lunch (or is that not the right one here...)), and we can pack all of our degrees of freedom into the design of the description language if we choose the length prior. (Or is that almost all? Proof, anyone?)
This leaves just the design of the description langauge. Computable programming languages seem OK, but all have unjustified inductive bias. Basically we have to figure out which one is a close approximation for our prior. Turing machines don’t seem particularly priveledged in this respect.
EDIT: Bolded the Tl;dr.
EDIT: Downvotes? WTF? Can we please have a norm that people can speculate freely in meditation threads without being downvoted? At least point out flaws… If it’s not about logical flaws, I don’t know what it is, and the downvote carries very nearly no information.
“Non-zero probability” doesn’t seem like quite the right word. If a parameter describing the way things could conceivably turn out to be can take, say, arbitrary real values, then we really want “non-zero probability density.” (It’s mathematically impossible to assign non-zero probability to each of uncountably many disjoint hypotheses because they can’t add to 1.)
The first answer that occurred to me was “enumerate all Turing machines” but I’m worried because it seems pretty straightforward to coherently think up a universe that can’t be described by a Turing machine (either because Turing machines aren’t capable of doing computations with infinite-precision real numbers or because they can’t solve the halting problem). More generally I’m worried that “coherently-thinkable” implies “not necessarily describable using math,” and that would make me sad.
I think you can get around that by defining “describe” to mean “for some tolerance t greater than zero, simulate with accuracy within t”. Since computable numbers are dense in the reals, for any t > 0 there will always be a Turing machine that can do the job.
The halting problem is insuperable, though. Universes with initial conditions or dynamics that depend on, e.g., Chaitin’s constant are coherently thinkable but not computable.
What about a universe with really mean laws of physics, like gravity that acts in reverse on particles whose masses aren’t computable numbers?
How is that different than “within accuracy t, these particles have those computable masses, but gravity acts backwards on them”?
The intention of my example was that you couldn’t tell for a given particle which direction gravity went.
Wouldn’t you just need one additional bit of information for each particle as an initial condition to make this computable again?
I don’t think your first point solves the problem. If the universe is exponentially sensitive to initial conditions, then even arbitrarily small inaccuracies in initial conditions make any simulation exponentially worse with time.
The function exp(x—K) grows exponentially in x, but is nevertheless really, really small for any x << K. Unbounded resources for computing means that the analogue of K may be made as large as necessary to satisfy any fixed tolerance t.
For a fixed amount of time. What if you wanted to simulate a universe that runs forever?
Yes, for a fixed amount of time. I should have made that explicit in my definition of “describe”: for some tolerance t greater than zero, simulate results at time T with accuracy within t. Then for any t > 0 and any T there will always be a Turing machine that can do the job.
this is my first time approaching a meditation, and I’ve actually only now decided to de-lurk and interact with the website.
One way to enumerate them would be, as CCC has just pointed out, with integers where irrationality denotes acausal worlds and rationality denotes causal worlds.
This however doesn’t leaves space for Stranger Things; I suppose we could use the alphabet for that. 1 If, however, and like I think, you mean enumerate as “order in which the simulation for universes can be run” then all universes would have a natural number assigned to them, and they could be arranged in order of complexity; this would mean our own universe would be fairly early in the numbering, if causal universes are indeed simpler than acausal ones, if I’ve understood things correctly.
This would mean we’d have a big gradient of “universes which I can run with a program” followed by a gradient of “universes which I can find by sifting through all possible states with an algorithm” and weirder stuff elsewhere (it’s weird; thus it’s magic, and I don’t know how it works; thus it can be simpler or more complex because I don’t know how it works).
In the end, the difference between causal and acausal universes is that one asks you only the starting state, while the other discriminates between all states and binds them together.
AAAAANNNNNNNND I’ve lost sight of the original question. Dammit.
It would be nice if there was some topology where the causal worlds were dense in the acausal ones.
Why would that be nice?
Unfortunately, this strikes me as unlikely.
Yes, and I forgot to put it in.
Wait, causal worlds are dense IN acausal ones?
Is that a typo, and you meant “causal worlds were denser than acausal ones” or did I just lose a whole swath of conversation?
I mean the class of causal worlds be dense in the class of worlds, where worlds consists of causal and acausal worlds. The same way we understand a lot of things in functional analysis: prove the result for the countable case, prove that taking compactifications/completions preserves the property, and then you have it for all separable spaces.
Well, I admit that I had originally considered that Stranger Things would most likely be either causal or acausal; I can’t really imagine anything that’s neither, given that the words are direct opposites.
In the case of a Stranger Thing that’s strange enough that I can’t imagine it, we could always fall back on the non-real complex numbers, which are neither rational nor irrational (and are numerous enough to make real numbers look trivial by comparison).
Well, I appear to be somewhat confused. Here is the logic that I’m using so far:
If:
1: A hypothesis space can contain mathematical constants,
2: Those mathematical constants can be irrational numbers,
3: The hypothesis space allows those mathematical constants to set to any irrational number,
4: And the set of irrational numbers cannot be ennumerated.
Then:
5: A list of hypothesis spaces is impossible to enumerate.
So If I assume 5 is incorrect (and that it is possible to enumerate the list) I seem to either have put together something logically invalid or one of my premises is wrong. I would suspect it is premise 3 because it seems to be a bit less justifiable then the others.
On the other hand, it’s possible premise 3 is correct, my logic is valid, and this is a rhetorical question where the answer is intended to be “That’s impossible to enumerate.”
I think the reason that I am confused is likely because I’m having a hard time figuring out where to proceed from here.
If you ever plan on talking about your hypothesis, you need to be able to describe it in a language with a finite alphabet (such as English or a programming language). There are only countably many things you can say in a language with a finite alphabet, so there are only countably many hypotheses you can even talk about (unambiguously).
This means that if there are constants floating around which can have arbitrary real values, then you can’t talk about all but countably many of those values. (What you can do instead is, for example, specify them to arbitrary but finite precision.)
Only if you live in a universe where you’re limited to writing finitely many symbols in finite space and time.
Point.
If I lived in such a universe, then it seems like I could potentially entertain uncountably many disjoint hypotheses about something, all of which I could potentially write down and potentially distinguish from one another. But I wouldn’t be able to assign more than countably many of them nonzero probability (because otherwise they couldn’t add to 1) as long as I stuck to real numbers. So it seems like I would have to revisit that particular hypothesis in Cox’s theorem…
It looks like you’re right, but let’s not give up there. How could we parametrize the hypothesis space, given that the parameters may be real numbers (or maybe even higher precision than that).
Well I suppose starting with the assumption that my superintelligent AI is merely turing complete, I think that we can only say our AI has “hypothesis about the world” if it has a computable model of the world. Even if the world weren’t computable, any non-computable model would be useless to our AI, and the best it could do is a computable approximation. Stable time loops seem computable through enumeration as you show in the post.
Now, if you claim that my assumption that the AI is computable is flawed, well then I give up. I truly have no idea how to program an AI more powerful than turing complete.
Suppose the AI lives in a universe with Turing oracles. Give it one.
Again, what distinguishes a “turing oracle” from a finite oracle with a bound well above the realizable size of a computer in the universe? They are indistinguishable hypotheses. Giving a turing complete AI a turing oracle doesn’t make it capable of understanding anything more than turing complete models. The turing-transcendant part must be an integral part of the AI for it to have non-turing-complete hypotheses about the universe, and I have no idea what a turing-transcendant language looks like and even less of an idea of how to program in it.
Suppose the AI lives in a universe where infinitely many computations can be performed in finite time...
(I’m being mildly facetious here, but in the interest of casting the “coherently-thinkable” net widely.)
I don’t see how this changes the possible sense-data our AI could expect. Again, what’s the difference between infinitely many computations being performed in finite time and only the computations numbered up to a point too large for the AI to query being calculated?
If you can give me an example of a universe for which the closest turing machine model will not give indistinguishable sense-data to the AI, then perhaps this conversation can progress.
Well, for starters, an AI living in a universe where infinitely many computations can be performed in finite time can verify the responses a Turing oracle gives it. So it can determine that it lives in a universe with Turing oracles (in fact it can itself be a Turing oracle), which is not what an AI living in this universe would determine (as far as I know).
As mentioned below, we you’d need to make infinitely many queries to the Turing oracle. But even if you could, that wouldn’t make a difference.
Again, even if there was a module to do infinitely many computations, the code I wrote still couldn’t tell the difference between that being the case, and this module being a really good computable approximation of one. Again, it all comes back to the fact that I am programming my AI on a turing complete computer. Unless I somehow (personally) develop the skills to program trans-turing-complete computers, then whatever I program is only able to comprehend something that is turing complete. I am sitting down to write the AI right now, and so regardless of what I discover in the future, I can’t program my turing complete AI to understand anything beyond that. I’d have to program a trans-turing complete computer now, if I ever hoped for it to understand anything beyond turing completeness in the future.
Ah, I see. I think we were answering different questions. (I had this feeling earlier but couldn’t pin down why.) I read the original question as being something like “what kind of hypotheses should a hypothetical AI hypothetically entertain” whereas I think you read the original question as being more like “what kind of hypotheses can you currently program an AI to entertain.” Does this sound right?
Yes, I agree. I can imagine some reasoning being concieving of things that are trans-turing complete, but I don’t see how I could make an AI do so.
I was reading a lesswrong post and I found this paragraph which lines up with what I was trying to say
I don’t think that’s different, unless it can also make infinitely many queries of the Turing oracle in finite time. Or make one query of a program of infinite length. In any case, I think it needs to perform infinite communication with the oracle.
I’ll grant that it seems likely that a universe with infinite computation capability will also have infinite communication capability using the same primitives, but I don’t think it’s a logical requirement.
Yes, let’s replace “computations” with “actions,” I guess.
The hypothesis that should interest an AI are not necessarily limited to those it can compute but to those it could test. A hypothesis is useless if it does not tell us something about how the world looks when it’s true as opposed to when it’s false. So if there is a way for the AI to interact with the world such that it expects different probabilities of outcomes depending on whether the (possibly uncomputable) hypothesis holds or not then it is something worth having a symbol for, even if the exact dynamics of this universe cannot be computed.
Let’s consider the case of our AI encountering a Turing Oracle. Two possible hypotheses of the AI could be A = This is in fact a Turing Oracle and for every program P it will output either the time until halting or 0 if no halting, and B = This is not a Turing Oracle but some computable machine Q. The AI could feed the supposed oracle a number of programs and if it was told any of them would halt it could try to run them for the specified number of steps to see if they did indeed halt. After each program had halted it would have to increase it’s probability that this was in fact a Turing Oracle using Bayes’ Theorem and estimates of the probabilities of guessing this right, or computationally deriving these numbers. If it did this for long enough and this was in fact a Turing Oracle it would gain higher and higher certainty of this fact.
What is it that the AI is doing? We can view the whole above process as a program which given one of a limited set of experimental outcomes outputs the probability that this experimental outcome would be the real one if H held. In the case of the Turing Oracle above the set of outcomes is the set of pairs (P,n) where P is a program and n a positive integer, and the program will output 1 if P halts after n steps and 0 otherwise. I think this captures in full generality all possibilities a computable agent would be able to recognise.
What if the AI later on gains some extra computational capacity which makes it non-computable? Say for example that it finds a Turing Oracle in like in the above example and integrates it into its main processor. But this is essentially everything that could happen: for the AI to become uncomputable, it would have to integrate an uncomputable physical process into its own processing. But for the AI to know it was actually uncomputable and not only incorporating the results of some computational process it didn’t recognise it would have to preform above test. So when it now preforms some uncomputable test on a new process we can see this simply as the composite of the tests of the original and the new process viewing all the message passing between the uncomputable processes as a part of the experimental setup rather than internal computation.
Hrmm… Well, if the AI is computable, it can only ever arrive at computable hypotheses, so we can enumerate them with any complete program specification language. I feel like I want to say that anything that isn’t computable, doesn’t matter. What I mean is, if the AI encounters something that is truly outside of its computable hypothesis space, then there’s nothing it can do about it. For concreteness:
TL;DR for paragraph below: our FAI encounters an Orb, which seems to randomly display red or green, and which our FAI really really wants to model accurately.
However, try as it might, our poor computable AI cannot do even an epsilon better than random in predicting the Orb. This is because the Orb is not computable. My point is that this is not distinguishable from a problem that the AI just can’t solve given its current resources. If the prediction problem is really hard, but nevertheless the AI can gain useful information about how the aliens will behave… then either the AI is modelling the alien species, plus a Truly Random variable (in the classical statistics sense) for the Orb, or the AI can do better than random at predicting the Orb.
Therefore, if our AI ever encountered something Truly Random or otherwise Really Weird (that is, something that is coherent in whatever way it has to be in order to be real, but not computable), then the AI would not and could not do better than it would by just reacting as though it was a problem too hard to solve, and modelling it as a random variable. For things that seemed random or weird but that were actually computable, the AI would naturally (if we’ve done our job) become smart enough or think long enough to solve the problem, or at least work around it. For things that were really truly unpredictable by a computable hypothesis, the same thing would happen. It’s just a special case where the AI never gets around to solving it.
Declaring uncomputable things irrelevant hopefully isn’t too crippling in practice; the universe looks computable, Time Turners can maybe be brute-forced, etc. Now, that doesn’t really answer the question. What do we do about uncomputable universes? Again, nothing… except if there is a chance of hypercomputation. But even if an AI is trying to somehow harness a hypercomputation to do better than chance at dealing with an uncomputable facet of reality, it still has to do figure out how to do the harnessing using its current computable hypotheses and the rest of its computable self.
In other words, hypercomputation isn’t a special case. It’s still a part of reality that correlates in some way with another part of reality (right? I’m, like, totally out of my depth here, but as long as we’re speculating...). The AI can notice and use this, while still only working from computable hypotheses. It should do this naturally, even operating under computable hypotheses, if it sees some way of expanding its (hyper)computational abilities.
TL;DR: whether or not the universe is computable, the AI can’t do better than computable hypotheses. The differences between reality and the best hypotheses that the AI can muster will be unavoidable, since the AI is computable. It can harness hypercomputation, but it still does so working from its computable hypotheses. Unless we program an uncomputable AI. Are you trying to ask how the AI should write an uncomputable extension of itself if it encounters hypercomputation?
Since I’m asking about a superintelligent AI’s model of the world, and the world of an AI is digital input and output, I first enumerate all possible programs, then enumerate all input strings of finite length, then count diagonally over both.
Then I convert the bits into ASCII, compile them in LOLCODE (since I’m already doing this for the lulz), and throw out the ones that give me compiler errors or duplicates.
Then I sum over the countable number of computable things using inverse squares and divide by pi squared over six (minus whatever I’ve thrown out).
I hope you didn’t want this information to be compiled in a way that is at all helpful to anyone, ever.
But if you did, I guess I might attempt to organize the information as the set of graphs on N vertices for all natural numbers N, or attempt to classify the set of categories of modules of objects with models in Grothendieck’s second universe, so that I could do all possible linear algebra. And then I would say that if I can’t use linear algebra the object I’m studying doesn’t have local consistency and so it doesn’t make sense to think about it as a continuous universe and I no longer no what thought means so I have more important issues to deal with.
This seems an odd question to ask in the comments like this. I know how I’d go about figuring out the answer, but it involves doing lots and lots of really hard math. Coming up with an answer i thee 5 minutes that anyone is going to realistically spend on this seems almost disrespectful, and certainly not very productive.
Or I just misunderstood what you were asking.
Counter-meditation:
come up with an example of a “strange world” which could not “conceivably turn out to” include this one
construct a world that does include both
repeat
I think the problem of enumerating these possibilities is impossible. You should notice that even the conventional possibility, quantum field theory somehow modified to have gravity and cosmology, is incomplete. It describes a mathematical construct, but it doesn’t describe how our experiences fit into that construct. It’s possible that just by looking at this mathematical object in a different way, you can find a different universe. That’s why this point-of-view information is actually important. Looking just at the possibilities where the universe is computable, enumerating Turing machines looks sufficient, but it is not. Turing machines don’t describe where we should look for ourselves in them, which is the most important part of the business. If we allow this, we should also allow the universe to be described by finite binary strings, which at times code for a Turing machine where we can be found in a certain point of view, but at other times code for various more powerful modes of computation. We can even say there is only one possibility, the totality of mathematical objects being the universe, which we can find ourselves in in very many different ways (this is the Tegmark level 4 multiverse theory).
So we can’t truly enumerate all the possibilities, even assuming a casual universe, since a casual diagram isn’t really capable of fully describing a possibility. It might be reasonable at certain times to enumerate these things anyways, and deal with this degeneracy in a ad hoc way. In that case, there would be nothing wrong with also making an ad hoc assumption along the lines of saying that the universe must be Turing computable (in which case you can simply list Turing machines).