According to Ingredients of Timeless Decision Theory, when you set up a factored causal graph for TDT, "You treat your choice as determining the result of the logical computation, and hence all instantiations of that computation, and all instantiations of other computations dependent on that logical computation", where "the logical computation" refers to the TDT-prescribed argmax computation (call it C) that takes all your observations of the world (from which you can construct the factored causal graph) as input, and outputs an action in the present situation.

I asked Eliezer to clarify what it means for another logical computation D to be either the same as C, or "dependent on" C, for purposes of the TDT algorithm. Eliezer answered:

For D to depend on C means that if C has various logical outputs, we can infer new logical facts about D's logical output in at least some cases, relative to our current state of non-omniscient logical knowledge. A nice form of this is when supposing that C has a given exact logical output (not yet known to be impossible) enables us to infer D's exact logical output, and this is true for every possible logical output of C. Non-nice forms would be harder to handle in the decision theory but we might perhaps fall back on probability distributions over D.

I replied as follows (which Eliezer suggested I post here).

If that's what TDT means by the logical dependency between Platonic computations, then TDT may have a serious flaw.

Consider the following version of the transparent-boxes scenario. The predictor has an infallible simulator D that predicts whether I one-box here [EDIT: if I see $1M]. The predictor also has a module E that computes whether the ith digit of pi is zero, for some ridiculously large value of i that the predictor randomly selects. I'll be told the value of i, but the best I can do is assign an a priori probability of .1 that the specified digit is zero.

...reasoning under logical uncertainty using limited computing power... is another huge unsolved open problem of AI. Human mathematicians had this whole elaborate way of believing that the Taniyama Conjecture implied Fermat's Last Theorem at a time when they didn't know whether the Taniyama Conjecture was true or false; and we seem to treat this sort of implication in a rather different way than '2=1 implies FLT', even though the material implication is equally valid.

*Good and Real*) are: the predictor conducts a simulation that tentatively presumes there will be $1M in the large box, and then puts $1M in the box (for real) iff that simulation showed one-boxing. Thus, if the large box turns out to be

*empty*, there is no requirement for that to be predictive of the agent's choice under those circumstances. The present variant is the same, except that (D xor E) determines the $1M, instead of just D. (Sorry, I should have said this to begin with, instead of assuming it as background knowledge.)

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.

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