Robustness — undecidability needs infinite precision

Parent: relaxing-undecidability

The undecidability of dynamical-system questions (reachability, convergence) is non-robust: it survives only with exact real arithmetic. Add any positive noise and it collapses to a decidable problem.

Perturbed machines lose their power (Asarin–Bouajjani, 2001)

A Turing machine embedded in a continuous dynamical system relies on infinitely fine distinctions in the state. Under any perturbation of radius ε>0, those distinctions blur, the machine can no longer store an unbounded tape faithfully, and reachability for the ε-perturbed system becomes decidable.

Geometrically: the undecidable instances sit on a measure-zero knife-edge; robust instances (answer unchanged under small perturbation) are decidable. Undecidability is a property of the idealization, not of any physically-realizable system.

Control tie: real controllers are designed to be robust (gain/phase margins, ISS) — partly for disturbance rejection, but it also means their behavior is certifiable/decidable. Robustness is what makes the δ and certificate relaxations bite: a robustly-convergent system has a Lyapunov certificate with margin to spare.