I mean, it makes sense. If sky-net rules over everything, then Prince Harry will not be prince anymore.

Yes

3. This allows us to compute the eigenvalues of the Fourier transform exactly, and hence its spectral norm.

Furthermore, we show that the mixing time of the random walk for our family is tight.

2. Remarkably, this Fourier transform is governed entirely by the standard representation of the group. Using tools from group and representation theory, the entire analysis collapses to that single representation.

Main proof technique.

We couple two machines from this family as a random walk on the group S_n x S_n and show that, with high probability, they are indistinguishable. How?

1. Indistinguishability is controlled by the spectral norm of the Fourier transform of the walk’s single-step distribution.

We construct a large randomized family of shuffling machines. Each machine shuffles its input using only transpositions. By flipping a random coin to decide whether to apply or ignore each transposition, we obtain a randomized family with the desired properties.

We show that SQ hardness can be established when both the alphabet size and input length are polynomial in the number of states.

We prove the first SQ hardness result for learning semiautomata under the uniform distribution over input words and initial states, without relying on parity gadgets or adversarial inputs. The hardness is structural, it arises purely from the transition structure, not from hard languages.

Done.

2) Can a neural network discover instructions for performing multiplication itself?

The answer to the first question is yes, with high probability and up to some arbitrary, predetermined precision (see the quoted post).

Link to the paper: arxiv.org/abs/2502.16763
Link to the repository: github.com/opallab/bina...

Learning to execute arithmetic exactly, with high probability, can be quite expensive. In the plot, 'ensemble complexity' refers to the number of independently trained models required to achieve exact learning with high probability. ell is the number of bits per number in the input.

Learning to execute arithmetic exactly, with high probability, can be quite expensive. In the plot, 'ensemble complexity' refers to the number of independently trained models required to achieve exact learning with high probability. ell is the number of bits per number in the input.

I never understood the point of trams. They're slow and expensive. I've been to two cities that built them while I was there, Edinburgh and Athens, and in both cases, the projects were born out of corruption. Especially in Edinburgh, it was a disaster. en.wikipedia.org/wiki/Edinbur...

Thanks, the connection to formal languages is quite interesting. I have a section in the repo regarding formal languages but it's small mainly because it's not a topic that I am familiar with. I will add them!