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That's pretty cool, but isn't any computable function can be computed via FHE, so I'm not sure the differentiable part is necessary.


Any program which you apply FHE to needs to be expressed as a circuit, which implies that the time taken to run a computation needs to be fixed in advance. It's therefore impossible to express a branch instruction (or "if" statement, if you prefer).

The circuits are built out of "+" and "×" gates, which are enough to express any polynomial. In turn, these are enough to approximate any continuous function (Weierstrass's approximation theorem). In turn, every computable function on the real numbers is a continuous function - so FHE is very powerful.


> In turn, every computable function on the real numbers is a continuous function

That doesn't seem right. Consider the function f(x: ℝ) = 1 if x ≥ 0, 0 otherwise. That's computable but not continuous.


That's uncomputable because equality of real numbers is undecidable. Think infinite strings of digits.




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