Computers can efficiently calculate anything that can possibly be calculated
This is pretty much the P=NP problem though, which hasn't been proven (and many believe it'll eventually be proven in the negative), with 'efficient' being defined as polynomial time.
If something can only ever be calculated in exponential (or worse) time, then a computer which can calculate it in exponential time I'd say is still efficient, so it would be true whether or not P=NP. When I say that a computer is efficient what I really mean is that it's not exponentially worse than any other possible way of arriving at the result. I should have worded it better, sorry.
This is pretty much the P=NP problem though, which hasn't been proven (and many believe it'll eventually be proven in the negative), with 'efficient' being defined as polynomial time.
If something can only ever be calculated in exponential (or worse) time, then a computer which can calculate it in exponential time I'd say is still efficient, so it would be true whether or not P=NP. When I say that a computer is efficient what I really mean is that it's not exponentially worse than any other possible way of arriving at the result. I should have worded it better, sorry.