2) Treat differently mathematical knowledge that we learn by genuinely mathematical reasoning and by physical observation. In this case we know (D xor E) not by mathematical reasoning, but by physically observing a box whose state we believe to be correlated with D xor E. This may justify constructing a causal DAG with a node descending from D and E, so a counterfactual setting of D won’t affect
the setting of E.

Perhaps I’m misunderstanding you here, but D and E are Platonic computations. What does it mean to construct a causal DAG among Platonic computations? [EDIT: Ok, I may understand that a little better now; see my edit to my reply to (1).] Such a graph links together general mathematical facts, so the same issues arise as in (1), it seems to me: Do the links correspond to logical inference, or something else? What makes the graph acyclic? Is mathematical causality even coherent? And if you did have a module that can detect (presumably timeless) causal links among Platonic computations, then why not use that module directly to solve your decision problems?

Plus I’m not convinced that there’s a meaningful distinction between math knowledge that you gain by genuine math reasoning, and math knowledge that you gain by physical observation.

Let’s say, for instance, that I feed a particular conjecture to an automatic theorem prover, which tells me it’s true. Have I then learned that math fact by genuine mathematical reasoning (performed by the physical computer’s Platonic abstraction)? Or have I learned it by physical observation (of the physical computer’s output), and hence be barred from using that math fact for purposes of TDT’s logical-dependency-detection? Presumably the former, right? (Or else TDT will make even worse errors.)

But then suppose the predictor has simulated the universe sufficiently to establish that U (the universe’s algorithm, including physics and initial conditions) leads to there being $1M in the box in this situation. That’s a mathematical fact about U, obtained by (the simulator’s) mathematical reasoning. Let’s suppose that when the predictor briefs me, the briefing includes mention of this mathematical fact. So even if I keep my eyes closed and never physically see the $1M, I can rely instead on the corresponding mathematically derived fact.

(Or more straightforwardly, we can view the universe itself as a computer that’s performing mathematical reasoning about how U unfolds, in which case any physical observation is intrinsically obtained by mathematical reasoning.)

Logical uncertainty has always been more difficult to deal with than physical uncertainty; the problem with logical uncertainty is that if you analyze it enough, it goes away. I’ve never seen any really good treatment of logical uncertainty.

But if we depart from TDT for a moment, then it does seem clear that we need to have causelike nodes corresponding to logical uncertainty in a DAG which describes our probability distribution. There is no other way you can completely observe the state of a calculator sent to Mars and a calculator sent to Venus, and yet remain uncertain of their outcomes yet believe the outcomes are correlated. And if you talk about error-prone calculators, two of which say 17 and one of which says 18, and you deduce that the “Platonic answer” was probably in fact 17, you can see that logical uncertainty behaves in an even more causelike way than this.

So, going back to TDT, my hope is that there’s a neat set of rules for factoring our logical uncertainty in our causal beliefs, and that these same rules also resolve the sort of situation that you describe.

If you consider the notion of the correlated error-prone calculators, two returning 17 and one returning 18, then the most intuitive way to handle this would be to see a “Platonic answer” as its own causal node, and the calculators as error-prone descendants. I’m pretty sure this is how my brain is drawing the graph, but I’m not sure it’s the correct answer; it seems to me that a more principled answer would involve uncertainty about which mathematical fact affects each calculator—physically uncertain gates which determine which calculation affects each calculator.

For the (D xor E) problem, we know the behavior we want the TDT calculation to exhibit; we want (D xor E) to be a descendant node of D and E. If we view the physical observation of $1m as telling us the raw mathematical fact (D xor E), and then perform mathematical inference on D, we’ll find that we can affect E, which is not what we want. Conversely if we view D as having a physical effect, and E as having a physical effect, and the node D xor E as a physical descendant of D and E, we’ll get the behavior we want. So the question is whether there’s any principled way of setting this up which will yield the second behavior rather than the first, and also, presumably, yield epistemically correct behavior when reasoning about calculators and so on.

That’s if we go down avenue (2). If we go down avenue (1), then we give primacy to our intuition that if-counterfactually you make a different decision, this logically controls the mathematical fact (D xor E) with E held constant, but does not logically control E with (D xor E) held constant. While this does sound intuitive in a sense, it isn’t quite nailed down—after all, D is ultimately just as constant as E and (D xor E), and to change any of them makes the model equally inconsistent.

These sorts of issues are something I’m still thinking through, as I think I’ve mentioned, so let me think out loud for a bit.

In order to observe anything that you think has already been controlled by your decision—any physical thing in which a copy of D has already played a role—then (leaving aside the question of Omega’s strategy that simulated alternate versions of you to select a self-consistent problem, and whether this introduces conditional-strategy-dependence rather than just decision-dependence into the problem) there have to be other physical facts which combine with D to yield our observation.

Some of these physical facts may themselves be affected by mathematical facts, like an implemented computation of E; but the point is that in order to have observed anything controlled by D, we already had to draw a physical, causal diagram in which other nodes descended from D.

So suppose we introduce the rule that in every case like this, we will have some physical node that is affected by D, and if we can observe info that depends on D in any way, we’ll view the other mathematical facts as combining with D’s physical node. This is a rule that tells us not to draw the diagram with a physical node being determined by the mathematical fact D xor E, but rather to have a physical node determined by D, and then a physical descendent D xor E. (Which in this particular problem should descend from a physical node E that descends from the mathematical fact E, because the mathematical fact E is correlated with our uncertainty about other things, and a factored causal graph should have no remaining correlated sources of background uncertainty; but if E didn’t correlate to anything else in particular, we could just have D descending to (D xor E) via the (xor with E) rule.)

When I evaluate this proposed solution for ad-hoc-ness, it does admittedly look a bit ad-hoc, but it solves at least one other problem than the one I started with, and which I didn’t think of until now. Suppose Omega tells me that I make the same decision in the Prisoner’s Dilemma as Agent X. This does not necessarily imply that I should cooperate with Agent X. X and I could have made the same decision for different (uncorrelated) reasons, and Omega could have simply found out by simulating the two of us that X and I gave the same response. In this case, presumably defecting; but if I cooperated, X wouldn’t do anything differently. X is just a piece of paper with “Defect” written on it.

If I draw a causal diagram of how I came to learn this correlation from Omega, and I follow the rule of always drawing a causal boundary around the mathematical fact D as soon as it physically affects something, then, given the way Omega simulated both of us to observe the correlation, I see that D and X separately physically affected the correlation-checker node.

On the other hand, if I can analyze the two pieces of code D and X and see that they return the same output, without yet knowing the output, then this knowledge was obtained in a way that doesn’t involve D producing an output, so I don’t have to draw a hard causal boundary around that output.

If this works, the underlying principle that makes it work is something along the lines of “for D to control X, the correlation between our uncertainty about D and X has to emerge in a way that doesn’t involve anyone already computing D”. Otherwise D has no free will (said firmly tongue-in-cheek). I am not sure that this principle has any more elegant expression than the rule, “whenever, in your physical model of the universe, D finishes computing, draw a physical/causal boundary around that finished computation and have other things physically/causally descend from it”.

If this principle is violated then D ends up “correlated” to all sorts of other things we observe, like the price of fish and whether it’s raining outside, via the magic of xor.

If we go down avenue (1), then we give primacy to our intuition that if-counterfactually you make a different decision, this logically controls the mathematical fact (D xor E) with E held constant, but does not logically control E with (D xor E) held constant. While this does sound intuitive in a sense, it isn’t quite nailed down—after all, D is ultimately just as constant as E and (D xor E), and to change any of them makes the model equally inconsistent.

I agree this sounds intuitive. As I mentioned earlier, though, nailing this down is tantamount to circling back and solving the full-blown problem of (decision-supporting) counterfactual reasoning: the problem of how to distinguish which facts to “hold fixed”, and which to “let vary” for consistency with a counterfactual antecedent.

In any event, is the idea to try to build a separate graph for math facts, and use that to analyze “logical dependency” among the Platonic nodes in the original graph, in order to carry out TDT’s modified “surgical alteration” of the original graph? Or would you try to build one big graph that encompasses physical and logical facts alike, and then use Pearl’s decision procedure without further modification?

If we view the physical observation of $1m as telling us the raw mathematical fact (D xor E), and then perform mathematical inference on D, we’ll find that we can affect E, which is not what we want.

Wait, isn’t it decision-computation C—rather than simulation D—whose “effect” (in the sense of logical consequence) on E we’re concerned about here? It’s the logical dependents of C that get surgically altered in the graph when C gets surgically altered, right? (I know C and D are logically equivalent, but you’re talking about inserting a physical node after D, not C, so I’m a bit confused.)

I’m having trouble following the gist of avenue (2) at the moment. Even with the node structure you suggest, we can still infer E from C and from the physical node that matches (D xor E)—unless the new rule prohibits relying on that physical node, which I guess is the idea. But what exactly is the prohibition? Are we forbidden to infer any mathematical fact from any physical indicator of that fact? Or is there something in particular about node (D xor E) that makes it forbidden? (It would be circular to cite the node’s dependence on C in the very sense of “dependence” that the new rule is helping us to compute.)

Or would you try to build one big graph that encompasses physical and logical facts alike, and then use Pearl’s decision procedure without further modification?

I definitely want one big graph if I can get it.

Wait, isn’t it decision-computation C—rather than simulation D—whose “effect” (in the sense of logical consequence) on E we’re concerned about here?

Sorry, yes, C.

Even with the node structure you suggest, we can still infer E from C and from the physical node that matches (D xor E)—unless the new rule prohibits relying on that physical node, which I guess is the idea. But what exactly is the prohibition? Are we forbidden to infer any mathematical fact from any physical indicator of that fact?

No, but whenever we see a physical fact F that depends on a decision C/D we’re still in the process of making plus Something Else (E), then we express our uncertainty in the form of a causal graph with directed arrows from C to D, D to F, and E to F. Thus when we compute a counterfactual on C, we find that F changes, but E does not.

No, but whenever we see a physical fact F that depends on a decision C/D we’re still in the process of making plus Something Else (E),

Wait, F depends on decision computation C in what sense of “depends on”? It can’t quite be the originally defined sense (quoted from your email near the top of the OP), since that defines dependency between Platonic computations, not between a Platonic computation and a physical fact. Do you mean that D depends on C in the original sense, and F in turn depends on D (and on E) in a different sense?

then we express our uncertainty in the form of a causal graph with directed arrows from C to D, D to F, and E to F.

Ok, but these arrows can’t be used to define the relevant sense of dependency above, since the relevant sense of dependency is what tells us we need to draw the arrows that way, if I understand correctly.

Sorry to keep being pedantic about the meaning of “depends”; I know you’re in thinking-out-loud mode here. But the theory gives wildly different answers depending (heh) on how that gets pinned down.

In my view, the chief form of “dependence” that needs to be discriminated is inferential dependence and causal dependence. If earthquakes cause burglar alarms to go off, then we can infer an earthquake from a burglar alarm or infer a burglar alarm from an earthquake. Logical reasoning doesn’t have the kind of directionality that causation does—or at least, classical logical reasoning does not—there’s no preferred form between ~A->B, ~B->A, and A \/ B.

The link between the Platonic decision C and the physical decision D might be different from the link between the physical decision D and the physical observation F, but I don’t know of anything in the current theory that calls for treating them differently. They’re just directional causal links. On the other hand, if C mathematically implies a decision C-2 somewhere else, that’s a logical implication that ought to symmetrically run backward to ~C-2 → ~C, except of course that we’re presumably controlling/evaluating C rather than C-2.

Thinking out loud here, the view is that your mathematical uncertainty ought to be in one place, and your physical uncertainty should be built on top of your mathematical uncertainty. The mathematical uncertainty is a logical graph with symmetric inferences, the physical uncertainty is a directed acyclic graph. To form controlling counterfactuals, you update the mathematical uncertainty, including any logical inferences that take place in mathland, and watch it propagate downward into the physical uncertainty. When you’ve already observed facts that physically depend on mathematical decisions you control but you haven’t yet made and hence whose values you don’t know, then those observations stay in the causal, directed, acyclic world; when the counterfactual gets evaluated, they get updated in the Pearl, directional way, not the logical, symmetrical inferential way.

Okay, then we have a logical link from C-platonic to D-platonic, and causal links descending from C-platonic to C-physical, E-platonic to E-physical, and D-platonic to D-physical to F-physical = D-physical xor E-physical. The idea being that when we counterfactualize on C-platonic, we update D-platonic and its descendents, but not E-platonic or its descendents.

I suppose that as written, this requires a rule, “for purposes of computing counterfactuals, keep in the causal graph rather than the logical knowledge base, any mathematical knowledge gained by observing a fact descended from your decision-output or any logical implications of your decision-output”. I could hope that this is a special case of something more elegant, but it would only be hope.

Ok. I think it would be very helpful to sketch, all in one place, what TDT2 (i.e., the envisioned avenue-2 version of TDT) looks like, taking care to pin down any needed sense of “dependency”. And similarly for TDT1, the avenue-1 version. (These suggestions may be premature, I realize.)

When you use terms like “draw a hard causal boundary” I’m forced to imagine you’re actually drawing these things on the back of a cocktail napkin somewhere using some sorts of standard symbols. Are there such standards, and do you have such diagrams scanned in online somewhere?

ETA: A note for future readers: Eliezer below is referring to Judea Pearl (simply “Pearl” doesn’t convey much via google-searching, though I suppose “pearl causality” does at the moment)

Hmm… Pearl uses a lot of diagrams but they all seem pretty ad-hoc. Just the sorts of arrows and dots and things that you’d use to represent any graph (in the mathematics sense). Should I infer from this description that the answer is, “No, there isn’t a standard”?

I was picturing something like a legend that would tell someone, “Use a dashed line for a causal boundary, and a red dotted line to represent a logical inference, and a pink squirrel to represent postmodernism”

Um… I’m not sure there’s much I can say to that beyond “Read Probabilistic Reasoning in Intelligent Systems, or Causality”.

Pearl’s system is not ad-hoc. It is very not ad-hoc. It has a metric fuckload of math backing up the simple rules. But Pearl’s system does not include logical uncertainty. I’m trying to put logical uncertainty into it, while obeying the rules. This is a work in progress.

Pearl’s system is not ad-hoc. It is very not ad-hoc. It has a metric fuckload of math backing up the simple rules.

Thomblake’s observation may be that while Pearl’s system is extremely rigorous the diagrams used do not give an authoritative standard style for diagram drawing.

I’m rereading past discussions to find insights. This jumped out at me:

Suppose Omega tells me that I make the same decision in the Prisoner’s Dilemma as Agent X. This does not necessarily imply that I should cooperate with Agent X.

I was referring to the example Eliezer gives with your opponent being a DefectBot, in which case cooperating makes Omega’s claim false, which may just mean that you’d make your branch of the thought experiment counterfactual, instead of convincing DefectBot to cooperate:

X is just a piece of paper with “Defect” written on it.

which may just mean that you’d make your branch of the thought experiment counterfactual

So? That doesn’t hurt my utility in reality. I would cooperate because that wins if agent X is correlated with me, and doesn’t lose otherwise.

Winning is about how alternatives you choose between compare. By cooperating against a same-action DefectBot, you are choosing nonexistence over a (D,D), which is not obviously a neutral choice.

I don’t think this is how it works. Particular counterfactual instances of you can’t influence whether they are counterfactual or exist in some stronger sense. They can only choose whether there are more real instances with identical experiences (and their choices can sometimes acausally influence what happens with real instances, which doesn’t seem to be the case here since the real you will choose defect either way as predicted by Omega). Hypothetical instances don’t lose anything by being in the branch that chooses the opposite of what the real you chooses unless they value being identical to the real you, which IMO would be silly.

Particular counterfactual instances of you can’t influence whether they are counterfactual or exist in some stronger sense.

What can influence things like that? Whatever property of a situation can mark it as counterfactual (more precisely, given by a contradictory specification, or not following from a preceding construction, assumed-real past state for example), that property could as well be a decision made by an agent present in that situation. There is nothing special about agents or their decisions.

Why do you think something can influence it? Whether you choose to cooperate or defect, you can always ask both “what would happen if I cooperated?” and “what would happen if I defected?”. In as far as being counterfactual makes sense the alternative to being the answer to “what would happen if I cooperated?” is being the answer to “what would happen if I defected?”, even if you know that the real you defects.

Compare Omega telling you that your answer will be the the same as the Nth digit of Pi. That doesn’t you allow to choose the Nth digit of Pi.

Winning is about how alternatives you choose between compare. By cooperating against a same-action DefectBot, you are choosing nonexistence over a (D,D), which is not obviously a neutral choice.

This becomes a (relatively) straightforward matter of working out where the (potentially counterfactual—depending what you choose) calculation is being performed to determine exactly what this ‘nonexistence’ means. Since this particular thought experiment doesn’t seem to specify any other broader context I assert that cooperate is clearly the correct option. Any agent which doesn’t cooperate is broken.

Basically, if you ever find yourself in this situation then you don’t matter. It’s your job to play chicken with the universe and not exist so the actual you can win.

I don’t see this argument making sense. Omega’s claim reduces to neglibible chances that a choice of Defection will be advantageous for me, because Omega’s claim makes it of neglible probability that either (D,C) or (C, D) will be realized. So I can only choose between the worlds of (C, C) and (D, D). Which means that the Cooperation world is advantageous, and that I should Cooperate.

In contrast, if Omega had claimed that we’d make the opposite decisions, then I’d only have to choose between the worlds of (D, C) or (C, D) -- with the worlds of (C, C) and (D, D) now having negligible probability. In which case, I should, of course, Defect.

The reasons for the correlation between me and Agent X are irrelevant when the fact of their correlation is known.

Agent X is a piece of paper with “Defect” written on it.

Sorry, was this intended as part of the problem statement, like “Omega tells you that agent X is a DefectBot that will play the same as you”? If yes, then ok. But if we don’t know what agent X is, then I don’t understand why a DefectBot is apriori more probable than a CooperateBot. If they are equally probable, then it cancels out (edit: no it doesn’t, it actually makes cooperating a better choice, thx ArisKatsaris). And there’s also the case where X is a copy of you, where cooperating does help. So it seems to be a better choice overall.

This is a rule that tells us not to draw the diagram with a physical node being determined by the mathematical fact D xor E, but rather to have a physical node determined by D, and then a physical descendent D xor E...

When I evaluate this proposed solution for ad-hoc-ness, it does admittedly look a bit ad-hoc, but it solves at least one other problem than the one I started with, and which I didn’t think of until now. Suppose Omega tells me that I make the same decision in the Prisoner’s Dilemma as Agent X. This does not necessarily imply that I should cooperate with Agent X. X and I could have made the same decision for different (uncorrelated) reasons, and Omega could have simply found out by simulating the two of us that X and I gave the same response. In this case, presumably defecting; but if I cooperated, X wouldn’t do anything differently. X is just a piece of paper with “Defect” written on it.

If X isn’t like us, we can’t “control” X by making a decision similar to what we would want X to output*. We shouldn’t go from being an agent that defects in the prisoner’s dilemma with Agent X when told we “make the same decision in the Prisoner’s Dilemma as Agent X” to being one that does not defect, just as we do not unilaterally switch from natural to precision bidding when in contract bridge a partner opens with two clubs (which signals a good hand under precision bidding, and not under natural bidding).

However, there do exist agents who should cooperate every time they hear they “make the same decision in the Prisoner’s Dilemma as Agent X”, those who have committed to cooperate in such cases. In some such cases, they are up against pieces of paper on which “cooperate” is written (too bad they didn’t have a more discriminating algorithm/clear Omega), in others, they are up against copies of themselves or other agents whose output depends on what Omega tells them. In any case, many agents should cooperate when they hear that.

Yes? No?

Why shouldn’t one be such an agent? Do we know ahead of time that we are likely to be up against pieces of paper with “cooperate” on them, and Omega would tell unhelpfully tell us we “make the same decision in the Prisoner’s Dilemma as Agent X” in all such cases, though if we had a different strategy we could have gotten useful information and defected in that case?

*Other cases include us defecting to get X to cooperate, and others where X’s play depends on ours, but this is the natural case to use when considering if the Agent X’s action depends on ours, a not strategically incompetent Agent X that has a strategy at least as good as always defecting or cooperating and does not try to condition his cooperating on our defecting or the like.

Perhaps I’m misunderstanding you here, but D and E are Platonic computations. What does it mean to construct a causal DAG among Platonic computations? [EDIT: Ok, I may understand that a little better now; see my edit to my reply to (1).] Such a graph links together general mathematical facts, so the same issues arise as in (1), it seems to me: Do the links correspond to logical inference, or something else? What makes the graph acyclic? Is mathematical causality even coherent? And if you did have a module that can detect (presumably timeless) causal links among Platonic computations, then why not use that module directly to solve your decision problems?

Plus I’m not convinced that there’s a meaningful distinction between math knowledge that you gain by genuine math reasoning, and math knowledge that you gain by physical observation.

Let’s say, for instance, that I feed a particular conjecture to an automatic theorem prover, which tells me it’s true. Have I then learned that math fact by genuine mathematical reasoning (performed by the physical computer’s Platonic abstraction)? Or have I learned it by physical observation (of the physical computer’s output), and hence be barred from using that math fact for purposes of TDT’s logical-dependency-detection? Presumably the former, right? (Or else TDT will make even worse errors.)

But then suppose the predictor has simulated the universe sufficiently to establish that U (the universe’s algorithm, including physics and initial conditions) leads to there being $1M in the box in this situation. That’s a mathematical fact about U, obtained by (the simulator’s) mathematical reasoning. Let’s suppose that when the predictor briefs me, the briefing includes mention of this mathematical fact. So even if I keep my eyes closed and never physically see the $1M, I can rely instead on the corresponding mathematically derived fact.

(Or more straightforwardly, we can view the universe itself as a computer that’s performing mathematical reasoning about how U unfolds, in which case any physical observation is intrinsically obtained by mathematical reasoning.)

Logical uncertainty has always been more difficult to deal with than physical uncertainty; the problem with logical uncertainty is that if you

analyze it enough, itgoes away. I’ve never seen any really good treatment of logical uncertainty.But if we depart from TDT for a moment, then it does seem clear that we need to have

causelike nodescorresponding to logical uncertainty in a DAG which describes our probability distribution. There is no other way you can completely observe the state of a calculator sent to Mars and a calculator sent to Venus, and yet remain uncertain of their outcomes yet believe the outcomes are correlated. And if you talk about error-prone calculators, two of which say 17 and one of which says 18, and you deduce that the “Platonic answer” was probably in fact 17, you can see that logical uncertainty behaves in an even more causelike way than this.So, going back to TDT, my hope is that there’s a neat set of rules for factoring our logical uncertainty in our causal beliefs, and that these same rules also resolve the sort of situation that you describe.

If you consider the notion of the correlated error-prone calculators, two returning 17 and one returning 18, then the most intuitive way to handle this would be to see a “Platonic answer” as its own causal node, and the calculators as error-prone descendants. I’m pretty sure this is how my brain is drawing the graph, but I’m not sure it’s the correct answer; it seems to me that a more principled answer would involve uncertainty about

whichmathematical fact affects each calculator—physically uncertain gates which determine which calculation affects each calculator.For the (D xor E) problem, we know the behavior we

wantthe TDT calculation to exhibit; we want (D xor E) to be a descendant node of D and E. If we view the physical observation of $1m as telling us the raw mathematical fact (D xor E), and then perform mathematical inference on D, we’ll find that we can affect E, which is not what we want. Conversely if we view D as having a physical effect, and E as having a physical effect, and the node D xor E as a physical descendant of D and E, we’ll get the behavior we want. So the question is whether there’s any principled way of setting this up which will yield the second behavior rather than the first, and also, presumably, yield epistemically correct behavior when reasoning about calculators and so on.That’s if we go down avenue (2). If we go down avenue (1), then we give primacy to our intuition that if-counterfactually you make a different decision, this logically controls the mathematical fact (D xor E) with E held constant, but does not logically control E with (D xor E) held constant. While this does sound intuitive in a sense, it isn’t quite nailed down—after all, D is ultimately just as constant as E and (D xor E), and to change any of them makes the model equally inconsistent.

These sorts of issues are something I’m still thinking through, as I think I’ve mentioned, so let me think out loud for a bit.

In order to observe anything that you think has already been controlled by your decision—any physical thing in which a copy of D has already played a role—then (leaving aside the question of Omega’s strategy that simulated alternate versions of you to select a self-consistent problem, and whether this introduces conditional-strategy-dependence rather than just decision-dependence into the problem) there have to be

other physical factswhich combine with D to yield our observation.Some of these physical facts may themselves be affected by mathematical facts, like an implemented computation of E; but the point is that in order to have observed anything controlled by D, we already had to draw a

physical, causaldiagram in which other nodesdescendedfrom D.So suppose we introduce the rule that in every case like this, we will have some physical node that is affected by D, and if we can observe info that depends on D in any way, we’ll view the other mathematical facts as combining with D’s physical node. This is a rule that tells us not to draw the diagram with a physical node being determined by the mathematical fact D xor E, but rather to have a physical node determined by D, and then a physical descendent D xor E. (Which in this particular problem should descend from a physical node E that descends from the mathematical fact E, because the mathematical fact E is correlated with our uncertainty about other things, and a factored causal graph should have no remaining correlated sources of background uncertainty; but if E didn’t correlate to anything else in particular, we could just have D descending to (D xor E) via the (xor with E) rule.)

When I evaluate this proposed solution for ad-hoc-ness, it does admittedly look a bit ad-hoc, but it solves at least

oneother problem than the one I started with, and which I didn’t think of until now. Suppose Omega tells me that I make the same decision in the Prisoner’s Dilemma as Agent X.This does not necessarily imply that I should cooperate with Agent X.X and I could have made the same decision for different (uncorrelated) reasons, and Omega could have simply found out by simulating the two of us that X and I gave the same response. In this case, presumably defecting; but if I cooperated, X wouldn’t do anything differently. X is just a piece of paper with “Defect” written on it.If I draw a causal diagram of how I came to learn this correlation from Omega, and I follow the rule of always drawing a causal boundary around the mathematical fact D as soon as it physically affects something, then, given the way Omega simulated both of us to observe the correlation, I see that D and X separately physically affected the correlation-checker node.

On the other hand, if I can analyze the two pieces of code D and X and see that they return the same output, without yet knowing the output, then this knowledge was obtained in a way that doesn’t involve D producing an output, so I don’t have to draw a hard causal boundary around that output.

If this works, the underlying principle that makes it work is something along the lines of “for D to control X, the correlation between our uncertainty about D and X has to emerge in a way that doesn’t involve anyone already computing D”. Otherwise D has no free will (said firmly tongue-in-cheek). I am not sure that this principle has any more elegant expression than the rule, “whenever, in your physical model of the universe, D

finishescomputing, draw a physical/causal boundary around that finished computation and have other things physically/causally descend from it”.If this principle is violated then D ends up “correlated” to all sorts of other things we observe, like the price of fish and whether it’s raining outside, via the magic of xor.

I agree this sounds intuitive. As I mentioned earlier, though, nailing this down is tantamount to circling back and solving the full-blown problem of (decision-supporting) counterfactual reasoning: the problem of how to distinguish which facts to “hold fixed”, and which to “let vary” for consistency with a counterfactual antecedent.

In any event, is the idea to try to build a separate graph for math facts, and use that to analyze “logical dependency” among the Platonic nodes in the original graph, in order to carry out TDT’s modified “surgical alteration” of the original graph? Or would you try to build one big graph that encompasses physical and logical facts alike, and then use Pearl’s decision procedure without further modification?

Wait, isn’t it decision-computation C—rather than simulation D—whose “effect” (in the sense of logical consequence) on E we’re concerned about here? It’s the logical dependents of C that get surgically altered in the graph when C gets surgically altered, right? (I know C and D are logically equivalent, but you’re talking about inserting a physical node after D, not C, so I’m a bit confused.)

I’m having trouble following the gist of avenue (2) at the moment. Even with the node structure you suggest, we can still infer E from C and from the physical node that matches (D xor E)—unless the new rule prohibits relying on that physical node, which I guess is the idea. But what exactly is the prohibition? Are we forbidden to infer any mathematical fact from any physical indicator of that fact? Or is there something in particular about node (D xor E) that makes it forbidden? (It would be circular to cite the node’s dependence on C in the very sense of “dependence” that the new rule is helping us to compute.)

I definitely want one big graph if I can get it.

Sorry, yes, C.

No, but whenever we see a

physicalfact F that depends on a decision C/D we’re still in the process of making plus Something Else (E), then we express our uncertainty in the form of acausalgraph with directed arrows from C to D, D to F, and E to F. Thus when we compute acounterfactualon C, we find that F changes, but E does not.Wait, F depends on decision computation C in what sense of “depends on”? It can’t quite be the originally defined sense (quoted from your email near the top of the OP), since that defines dependency between Platonic computations, not between a Platonic computation and a physical fact. Do you mean that D depends on C in the original sense, and F in turn depends on D (and on E) in a different sense?

Ok, but these arrows can’t be used to define the relevant sense of dependency above, since the relevant sense of dependency is what tells us we need to draw the arrows that way, if I understand correctly.

Sorry to keep being pedantic about the meaning of “depends”; I know you’re in thinking-out-loud mode here. But the theory gives wildly different answers depending (heh) on how that gets pinned down.

In my view, the chief form of “dependence” that needs to be discriminated is inferential dependence and causal dependence. If earthquakes

causeburglar alarms to go off, then we caninferan earthquake from a burglar alarm orinfera burglar alarm from an earthquake. Logical reasoning doesn’t have the kind of directionality that causation does—or at least, classical logical reasoning does not—there’s no preferred form between ~A->B, ~B->A, and A \/ B.The link between the Platonic decision C and the physical decision D might be different from the link between the physical decision D and the physical observation F, but I don’t know of anything in the current theory that calls for treating them differently. They’re just directional causal links. On the other hand, if C mathematically implies a decision C-2 somewhere else, that’s a logical implication that ought to symmetrically run backward to ~C-2 → ~C, except of course that we’re presumably controlling/evaluating C rather than C-2.

Thinking out loud here, the view is that your mathematical uncertainty ought to be in one place, and your physical uncertainty should be built on top of your mathematical uncertainty. The mathematical uncertainty is a logical graph with symmetric inferences, the physical uncertainty is a directed acyclic graph. To form controlling counterfactuals, you update the mathematical uncertainty, including any logical inferences that take place in mathland, and watch it propagate downward into the physical uncertainty. When you’ve already observed facts that physically depend on mathematical decisions you control but you haven’t yet made and hence whose values you don’t know, then those observations stay in the causal, directed, acyclic world; when the counterfactual gets evaluated, they get updated in the Pearl, directional way, not the logical, symmetrical inferential way.

No, D was the Platonic simulator. That’s why the nature of the C->D dependency is crucial here.

Okay, then we have a logical link from C-platonic to D-platonic, and causal links descending from C-platonic to C-physical, E-platonic to E-physical, and D-platonic to D-physical to F-physical = D-physical xor E-physical. The idea being that when we counterfactualize on C-platonic, we update D-platonic and its descendents, but not E-platonic or its descendents.

I suppose that as written, this requires a rule, “for purposes of computing counterfactuals, keep in the causal graph rather than the logical knowledge base, any mathematical knowledge gained by observing a fact descended from your decision-output or any logical implications of your decision-output”. I could hope that this is a special case of something more elegant, but it would only be hope.

Ok. I think it would be very helpful to sketch, all in one place, what TDT2 (i.e., the envisioned avenue-2 version of TDT) looks like, taking care to pin down any needed sense of “dependency”. And similarly for TDT1, the avenue-1 version. (These suggestions may be premature, I realize.)

When you use terms like “draw a hard causal boundary” I’m forced to imagine you’re actually drawing these things on the back of a cocktail napkin somewhere using some sorts of standard symbols. Are there such standards, and do you have such diagrams scanned in online somewhere?

ETA: A note for future readers: Eliezer below is referring to Judea Pearl (simply “Pearl” doesn’t convey much via google-searching, though I suppose “pearl causality” does at the moment)

Read Pearl. I

thinkhis online intros should give you a good idea of what the cocktail napkin looks like.Hmm… Pearl uses a lot of diagrams but they all seem pretty ad-hoc. Just the sorts of arrows and dots and things that you’d use to represent any graph (in the mathematics sense). Should I infer from this description that the answer is, “No, there isn’t a standard”?

I was picturing something like a legend that would tell someone, “Use a dashed line for a causal boundary, and a red dotted line to represent a logical inference, and a pink squirrel to represent postmodernism”

Um… I’m not sure there’s much I can say to that beyond “Read Probabilistic Reasoning in Intelligent Systems, or Causality”.

Pearl’s system is not ad-hoc. It is very not ad-hoc. It has a metric fuckload of math backing up the simple rules. But Pearl’s system does not include logical uncertainty. I’m trying to put logical uncertainty into it, while obeying the rules. This is a work in progress.

Thomblake’s observation may be that while Pearl’s system is extremely rigorous the diagrams used do not give an authoritative standard style for diagram drawing.

That’s correct—I was looking for a standard style for diagram drawing.

I’d just like to register a general approval of specifying that one’s imaginary units are

metric.FWIW

I’m rereading past discussions to find insights. This jumped out at me:

Do you still believe this?

Playing chicken with Omega may result in you becoming counterfactual.

Why is cooperation more likely to qualify as “playing chicken” than defection here?

I was referring to the example Eliezer gives with your opponent being a DefectBot, in which case cooperating makes Omega’s claim false, which may just mean that you’d make your branch of the thought experiment counterfactual, instead of convincing DefectBot to cooperate:

So? That doesn’t hurt my utility in reality. I would cooperate because that wins if agent X is correlated with me, and doesn’t lose otherwise.

Winning is about how alternatives you choose between compare. By cooperating against a same-action DefectBot, you are choosing nonexistence over a (D,D), which is not

obviouslya neutral choice.I don’t think this is how it works. Particular counterfactual instances of you can’t influence whether they are counterfactual or exist in some stronger sense. They can only choose whether there are more real instances with identical experiences (and their choices can sometimes acausally influence what happens with real instances, which doesn’t seem to be the case here since the real you will choose defect either way as predicted by Omega). Hypothetical instances don’t lose anything by being in the branch that chooses the opposite of what the real you chooses unless they value being identical to the real you, which IMO would be silly.

What can influence things like that? Whatever property of a situation can mark it as counterfactual (more precisely, given by a contradictory specification, or not following from a preceding construction, assumed-real past state for example), that property could as well be a decision made by an agent present in that situation. There is nothing special about agents or their decisions.

Why do you think something can influence it? Whether you choose to cooperate or defect, you can always ask both “what would happen if I cooperated?” and “what would happen if I defected?”. In as far as being counterfactual makes sense the alternative to being the answer to “what would happen if I cooperated?” is being the answer to “what would happen if I defected?”, even if you know that the real you defects.

Compare Omega telling you that your answer will be the the same as the Nth digit of Pi. That doesn’t you allow to choose the Nth digit of Pi.

This becomes a (relatively) straightforward matter of working out where the (potentially counterfactual—depending what you choose) calculation is being performed to determine exactly what this ‘nonexistence’ means. Since this particular thought experiment doesn’t seem to specify any other broader context I assert that cooperate

isclearly the correct option. Any agent which doesn’t cooperate is broken.Basically, if you ever find yourself in this situation then you don’t matter. It’s your job to play chicken with the universe and not exist so the actual you can win.

Agent X is a piece of paper with “Defect” written on it. I defect against it. Omega’s claim is true and does not imply that I should cooperate.

I don’t see this argument making sense. Omega’s claim reduces to neglibible chances that a choice of Defection will be advantageous for me, because Omega’s claim makes it of neglible probability that either (D,C) or (C, D) will be realized. So I can only choose between the worlds of (C, C) and (D, D). Which means that the Cooperation world is advantageous, and that I

shouldCooperate.In contrast, if Omega had claimed that we’d make the opposite decisions, then I’d only have to choose between the worlds of (D, C) or (C, D) -- with the worlds of (C, C) and (D, D) now having negligible probability. In which case, I should, of course, Defect.

The reasons for the correlation between me and Agent X are irrelevant when the

factof their correlation is known.Sorry, was this intended as part of the problem statement, like “Omega tells you that agent X is a DefectBot that will play the same as you”? If yes, then ok. But if we don’t know what agent X is, then I don’t understand why a DefectBot is apriori more probable than a CooperateBot. If they are equally probable, then it cancels out (

edit:no it doesn’t, it actually makes cooperating a better choice, thx ArisKatsaris). And there’s also the case where X is a copy of you, where cooperating does help. So it seems to be a better choice overall.There is also a case where X is an anticopy (performs opposite action), which argues for defecting in the same manner.

Edit: This reply is wrong.No it doesn’t. If X is an anticopy, the situation can’t be real and your action doesn’t matter.

Why can’t it be real?

Because Omega has told you that X’s action is the same as yours.

OK.

If X isn’t like us, we can’t “control” X by making a decision similar to what we would want X to output*. We shouldn’t go from being an agent that defects in the prisoner’s dilemma with Agent X when told we “make the same decision in the Prisoner’s Dilemma as Agent X” to being one that does not defect, just as we do not unilaterally switch from natural to precision bidding when in contract bridge a partner opens with two clubs (which signals a good hand under precision bidding, and not under natural bidding).

However, there do exist agents who should cooperate every time they hear they “make the same decision in the Prisoner’s Dilemma as Agent X”, those who have committed to cooperate in such cases. In some such cases, they are up against pieces of paper on which “cooperate” is written (too bad they didn’t have a more discriminating algorithm/clear Omega), in others, they are up against copies of themselves or other agents whose output depends on what Omega tells them. In any case, many agents should cooperate when they hear that.

Yes? No?

Why shouldn’t one be such an agent? Do we know ahead of time that we are likely to be up against pieces of paper with “cooperate” on them, and Omega would tell unhelpfully tell us we “make the same decision in the Prisoner’s Dilemma as Agent X” in all such cases, though if we had a different strategy we could have gotten useful information and

defectedin that case?*Other cases include us defecting to get X to cooperate, and others where X’s play depends on ours, but this is the natural case to use when considering if the Agent X’s action depends on ours, a not strategically incompetent Agent X that has a strategy at least as good as always defecting or cooperating and does not try to condition his cooperating on our defecting or the like.