Section: New Results

Computations

Participant : Mathieu Hoyrup.

Inversion of computable functions

We strengthen the preceding result by making $P$ and $Q$ computable. This result is a particular case of a more general problem. In many situations an operator $F\to Y$ can be computed but can hardly be reversed: given $F\left(x\right)$, $x$ cannot always we recovered (computed) even when $F$ is one-to-one. We introduce a strong notion of discontinuity for the inverse of $F$ and prove that it entails the existence of a non-computable $x\in X$ such that $F\left(x\right)$ is computable. Our result on the ergodic decomposition can be derived by applying our general result to the operator $F(P,Q)=P+Q$ which is computable but difficult to reverse. At the same time we prove a significant improvement of a classical result of Pour-El and Richards [67] about the computability of linear operators. The paper [26] is currently submitted.

Computability and measure theory.

We study the constructive content of the Radon-Nikodym theorem, show that it is not computable in general and precisely locate its non-computability in the Weihrauch lattice. The paper [15] appeared in the first issue of the new journal Computability.