Yes, but that’s for a functional notion of equivalence—i.e. it’s about whether the two TMs have the same input-output behavior. The notion of equivalence I’m looking at is not just about same input-output, but also structurally-similar computations. Intuitively, I’m asking whether they’re computing the same function in the same way.
(In fact, circumventing the undecidability issue is a key part of why I’m formulating the problem like this in the first place. So you’re definitely asking the right question here.)
Isn’t separability of arbitrary Turing machines equivalent to the Halting problem and therefore undecidable?
Yes, but that’s for a functional notion of equivalence—i.e. it’s about whether the two TMs have the same input-output behavior. The notion of equivalence I’m looking at is not just about same input-output, but also structurally-similar computations. Intuitively, I’m asking whether they’re computing the same function in the same way.
(In fact, circumventing the undecidability issue is a key part of why I’m formulating the problem like this in the first place. So you’re definitely asking the right question here.)