The million dollar question is if it's possible to construct a theory of computation where the input language itself is automatically differentiable, and where the execution semantics are also so. Perhaps a continuous spatial automata that has been proven to be Turing complete.

I see no reason why you couldn't make a differentiable virtual machine! Getting it to learn may be quite tricky tho.

"We extend the capabilities of neural networks by coupling them to external memory resources, which they can interact with by attentional processes. The combined system is analogous to a Turing Machine or Von Neumann architecture but is differentiable end-to-end, allowing it to be efficiently trained with gradient descent. Preliminary results demonstrate that Neural Turing Machines can infer simple algorithms such as copying, sorting, and associative recall from input and output examples." https://arxiv.org/pdf/1410.5401.pdf

Seems like it should work. A fully connected network would encode all permutations. Interesting.

Yeah, the trick is choosing the right one...