Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 358 / 3 / c

For inexact information in the SCI hierarchy, replace every exact evaluation $\lambda \in \Lambda$ by a family of admissible approximations ${\lambda }_m$ satisfyingAn algorithm must converge for every admissible choice of approximations, not merely for one favored encoding.

For continuous nonsingular maps $F:X\to X$, take the evaluations to be arbitrary point queries. At precision $m$, a query at $x\in X$ returns any $y$ satisfyingThus the information set contains all triples $(x,m,y)$ satisfying this inequality. This is a perfect measurement device for a dynamical system: it can sample any state, at any requested accuracy, with no fixed noise floor. The finite-information rule still requires each terminating computation to make only finitely many such measurements.