Measuring robustness of dynamical systems. Relating time and space to length and precision

Abstract

Verification of discrete time or continuous time dynamical systems over the reals is known to be undecidable. It is however known that undecidability does not hold for various classes of systems: if robustness is defined as the fact that reachability relation is stable under infinitesimal perturbation, then their reachability relation is decidable. In other words, undecidability implies sensitivity under infinitesimal perturbation, a property usually not expected in systems considered in practice, and hence can be seen (somehow informally) as an artefact of the theory, that always assumes exactness. In a similar vein, it is known that, while undecidability holds for logical formulas over the reals, it does not hold when considering delta-undecidability: one must determine whether a property is true, or $δ$-far from being true. We first extend the previous statements to a theory for general (discrete time, continuous-time, and even hybrid) dynamical systems, and we relate the two approaches. We also relate robustness to some geometric properties of reachability relation. But mainly, when a system is robust, it then makes sense to quantify at which level of perturbation. We prove that assuming robustness to polynomial perturbations on precision leads to reachability verifiable in complexity class PSPACE, and even to a characterization of this complexity class. We prove that assuming robustness to polynomial perturbations on time or length of trajectories leads to similar statements, but with PTIME. It has been recently unexpectedly shown that the length of a solution of a polynomial ordinary differential equation corresponds to a time of computation: PTIME corresponds to solutions of polynomial differential equations of polynomial length. Our results argue that the answer is given by precision: space corresponds to the involved precision.