Integrating using with the usual “strict” binary representation actually requires a modulus of continuity. So this result is very surprising from that perspective. I had never seen such a practical example of the non-computable equivalence of these two representations of computable reals.

I always thought that interval arithmetic was just a way to achieve better certainty, at a reasonable price. Now I understand that it is actually more powerful than floating-point arithmetic in a very concrete sense.

I wonder if there is a more practical version of this. More precisely, is there a nice example of an integration problem for which there is a P-time algorithm to approximate it within an interval of length but computing the -th bit is NP-hard? (I suspect it’s not too difficult code a lot of things as integrals of trigonometric polynomials, but I don’t recall ever seeing that.)

I think that the idea is that a computable real number is (among other equivalent characterizations) a computable function which takes an and returns a rational so that .

Using a representation with digits is equivalent to this, but just using , for example, isn’t. This is because if you’re given a method for computing the binary expansion of a real number in , you could, for example, be able to confidently assert after looking at one bit either that or that , which is not something that you’re guaranteed to be able to do with a computable real number in .

Or have I completely misunderstood your objection?