This seems related in spirit to the fact that time is only partially ordered in physics as well. You could even use special relativity to make a model for concurrency ambiguity in parallel computing: each processor is a parallel worldline, detecting and sending signals at points in spacetime that are spacelike-separated from when the other processors are doing these things. The database follows some unknown worldline, continuously broadcasts its contents, and updates its contents when it receives instructions to do so. The set of possible ways that the processors and database end up interacting should match the parallel computation model. This makes me think that intuitions about time that were developed to be consistent with special relativity should be fine to also use for computation.
Wikipedia claims that every sequence is Turing reducible to a random one, giving a positive answer to the non-resource-bounded version of any question of this form. There might be a resource-bounded version of this result as well, but I'm not sure.
Fisherian runaway doesn't make any sense to me.
Suppose that each individual in a species of a given sex has some real-valued variable , which is observable by the other sex. Suppose that, absent considerations about sexual selection by potential mates for the next generation, the evolutionarily optimal value for is 0. How could we end up with a positive feedback loop involving sexual selection for positive values of , creating a new evolutionary equilibrium with an optimal value when taking into account sexual selection? First the other sex ends up with some smaller degree of selection for positive values of (say selecting most strongly for ). If sexual selection by the next generation of potential mates were the only thing that mattered, then the optimal value of to select for is , since that's what everyone else is selecting for. That's stability, not positive feedback. But sexual selection by the next generation of potential mates isn't the only thing that matters; by stipulation, different values of have effects on evolutionary fitness other than through sexual selection, with values closer to being better. So, when choosing a mate, one must balance the considerations of sexual selection by the next generation (for which is optimal) and other considerations (for which is optimal), leading to selection for mates with being evolutionarily optimal. That's negative feedback. How do you get positive feedback?
I know this was tagged as humor, but taking it seriously anyway,
I'm skeptical that breeding octopuses for intelligence would yield much in the way of valuable insights for AI safety, since octopuses and humans have so much in common that AGI wouldn't. That said, it's hard to rule out that uplifting another species could reveal some valuable unknown unknowns about general intelligence, so I unironically think this is a good reason to try it.
Another, more likely to pay off, benefit to doing this would be as a testbed for genetically engineering humans for higher intelligence (which also might have benefits for AI safety under long-timelines assumptions). I also think it would just be really cool from a scientific perspective.
One example of a class of algorithms that can solve its own halting problem is the class of primitive recursive functions. There's a primitive recursive function that takes as input a description of a primitive recursive function and input and outputs if halts, and otherwise: this program is given by , because all primitive recursive functions halt on all inputs. In this case, it is that does not exist.
I think should exist, at least for classical bits (which as others have pointed out, is all that is needed), for any reasonably versatile model of computation. This is not so for , since primitive recursion is actually an incredibly powerful model of computation; any program that you should be able to get an output from before the heat death of the universe can be written with primitive recursion, and in some sense, primitive recursion is ridiculous overkill for that purpose.
If a group decides something unanimously, and has the power to do it, they can do it. That would take them outside the formal channels of the EU (or in another context of NATO) but I do not see any barrier to an agreement to stop importing Russian gas followed by everyone who agreed to it no longer importing Russian gas. Hungary would keep importing, but that does not seem like that big a problem.
If politicians can blame Hungary for their inaction, then this partially protects them from being blamed by voters for not doing anything. But it doesn't protect them at all from being blamed for high fuel prices if they stop importing it from Russia. So they have incentives not to find a solution to this problem.
If you have a 10-adic integer, and you want to reduce it to a 5-adic integer, then to know its last n digits in base 5, you just need to know what it is modulo . If you know what it is modulo , then you can reduce it module , so you only need to look at the last n digits in base 10 to find its last n digits in base 5. So a base-10 integer ending in ...93 becomes a base-5 integer ending in ...33, because 93 mod 25 is 18, which, expressed in base 5, is 33.
The Chinese remainder theorem tells us that we can go backwards: given a 5-adic integer and a 2-adic integer, there's exactly one 10-adic integer that reduces to each of them. Let's say we want the 10-adic integer that's 1 in base 5 and -1 in base 2. The last digit is the digit that's 1 mod 5 and 1 mod 2 (i.e. 1). The last 2 digits are the number from 0 to 99 that's 1 mod 25 and 3 mod 4 (i.e. 51). The last 3 digits are the number from 0 to 999 that's 1 mod 125 and 7 mod 8 (i.e. 751). And so on.
(My intuition tells me that if you choose a subset of the primes dividing the base, you can somehow obtain from that a value of in a way that maps "none" to 1, and "all" to -1, and the remaining combinations to the irrational integers. No more specific ideas.)
That is correct! In base (for distinct primes ), an integer is determined by an integer in each of the bases , essentially by the Chinese remainder theorem. In other words, . In prime base, 1 and -1 are the only two square roots of 1. In arbitrary base, a number squares to 1 iff the number it reduces to in each prime factor of the base also squares to 1, and there's 2 options for each of these factors.
It sounds to me like, in the claim "deep learning is uninterpretable", the key word in "deep learning" that makes this claim true is "learning", and you're substituting the similar-sounding but less true claim "deep neural networks are uninterpretable" as something to argue against. You're right that deep neural networks can be interpretable if you hand-pick the semantic meanings of each neuron in advance and carefully design the weights of the network such that these intended semantic meanings are correct, but that's not what deep learning is. The other things you're comparing it to that are often called more interpretable than deep learning are in fact more interpretable than deep learning, not (as you rightly point out) because the underlying structures they work with is inherently more interpretable, but because they aren't machine learning of any kind.