In order to satisfy this definition, 𝔼¹ needs to know every particular fact 𝔼 knows. It would be nice to have a definition that got at the heart of the matter while relaxing this requirement.
I don’t think your condition gets around this requirement. Suppose that Y is a bit that 𝔼 knows and 𝔼¹ does not, Z[0] and Z[1] are two hard-to-estimate quantities (that 𝔼¹ and 𝔼² know but 𝔼 does not), and that X=Z[Y].
In order to satisfy this definition, 𝔼¹ needs to know every particular fact 𝔼 knows. It would be nice to have a definition that got at the heart of the matter while relaxing this requirement.
I don’t think your condition gets around this requirement. Suppose that Y is a bit that 𝔼 knows and 𝔼¹ does not, Z[0] and Z[1] are two hard-to-estimate quantities (that 𝔼¹ and 𝔼² know but 𝔼 does not), and that X=Z[Y].