Typo: should be "Gell-Mann"
I figured out the encoding, but I expressed the algorithm for computing the decoding in different language from you. My algorithm produces equivalent outputs but is substantially uglier. I wanted to leave a note here in case anyone else had the same solution.
Alt phrasing of the solution:
Each expression (i.e. a well-formed string of brackets) has a "degree", which is defined as the the number of well-formed chunks that the encoding can be broken up into. Some examples: [], [[]], and [-[][]] have degree one, [][], -[][[]], and [][[][]] have degree two, etc.
Here's a special case: the empty string maps to 0, i.e. decode("") = 0
When an encoding has degree one, you take off the outer brackets and do 2^decode(the enclosed expression), defined recursively. So decode([]) = 2^decode("") = 2^0 = 1, decode([[]]) = 2^decode([]) = 2, etc.
Negation works as normal. So decode([-[]]) = 2^decode(-[]) = 2^(-decode([])) = 2^(-1) = 1/2
So now all we have to deal with is expressions with degree >1.
When an expression has degree >1, you compute its decoding as the product of decoding of the first subexpression and inc(decode(everything after the first subexpression)). I will define the "inc" function shortly.
So decode([][[]]) = decode([]) * inc(decode([[]]) = 1 * inc(2)
decode([[[]]][][[]]) = decode([[[]]]) * inc(decode([][[]])) = 4 * inc(decode([]) * inc(decode([[]]))) = 4 * inc(1 * inc(2))
What is inc()? inc() is a function that computes a prime factorization of a number and the increments (from one prime to the next) all the prime bases. So inc(10) = inc(2 * 5) = 3 * 7 = 21, and inc(36) = inc(2^2 * 3^2) = 3^2 * 5^2 = 225. But inc() doesn't just take in integers, it can take in any number representable as a product of primes raised to powers. So inc(2^(1/2) * 3^(-1)) = 3^(1/2) * 5^(-1) = sqrt(3)/5. I asked the language models whether there's a standard name for the set of numbers definable in this way, and they didn't have ideas.
On point 6, "Humanity can survive an unaligned superintelligence": In this section, I initially took you to be making a somewhat narrow point about humanity's safety if we develop aligned superintelligence and humanity + the aligned superintelligence has enough resources to out-innovate and out-prepare a misaligned superintelligence. But I can't tell if you think this conditional will be true, i.e. whether you think the existential risk to humanity from AI is low due to this argument. I infer from this tweet of yours that AI "kill[ing] us all" is not among your biggest fears about AI, which suggests to me that you expect the conditional to be true—am I interpreting you correctly?
To make a clarifying point (which will perhaps benefit other readers): you're using the term "scheming" in a different sense from how Joe's report or Ryan's writing uses the term, right?
I assume your usage is in keeping with your paper here, which is definitely different from those other two writers' usages. In particular, you use the term "scheming" to refer to a much broader set of failure modes. In fact, I think you're using the term synonymously with Joe's "alignment-faking"—is that right?
People interested in working on these sorts of problems should consider applying to Open Phil's request for proposals: https://www.openphilanthropy.org/request-for-proposals-technical-ai-safety-research/
I think the link in footnote two goes to the wrong place?
1I haven't read the paper, but based only on the phrase you quote, I assume it's referring to hacks like the one shown here: https://arxiv.org/pdf/2210.10760#19=&page=19.0
Do you think that cyber professionals would take multiple hours to do the tasks with 20-40 min first-solve times? I'm intuitively skeptical.
One (edit: minor) component of my skepticism is that someone told me that the participants in these competitions are less capable than actual cyber professionals, because the actual professionals have better things to do than enter competitions. I have no idea how big that selection effect is, but it at least provides some countervailing force against the selection effect you're describing.
For more discussion of the hard cases of exploration hacking, readers should see the comments of this post.