Reasoning about dynamical systems evolving over the reals is well-known to lead to undecidability. In particular, it is known that there cannot be reachability decision procedures for first-order theories over the reals extended with even very basic functions, or for logical theories that reason about real-valued functions, or decision procedures for state reachability. This mostly comes from the fact that reachability for dynamical systems over the reals is fundamentally undecidable, as Turing machines can be embedded into (even very simple) dynamical systems.

However, various results in the literature have shown that decision procedures exist when restricting to robust systems, with a suitably-chosen notion of robustness. In particular, it has been established in the field of verification that if the state reachability is not sensitive to infinitesimal perturbations, then decision procedures for state reachability exist. In the context of logical theories over the reals, it has been established that decision procedures exist if we focus on properties not sensitive to arbitrarily small perturbations. For example by considering properties that are either true or δ-far from being true, for some δ > 0.

In this article, we first propose a unified theory explaining in a uniform framework these statements, that were established in different contexts.

More fundamentally, while all these statements are only about computability issues, we also consider complexity theory aspects. We prove that robustness to some precision is inherently related to the complexity of the decision procedure. When a system is robust, it makes sense to quantify at which level of perturbation it is. We prove that assuming robustness to a polynomial perturbation on precision leads to a characterisation of PSPACE. We prove that assuming robustness to polynomial perturbation on time or length leads to similar statements for PTIME.

In other words, precision on computations is inherently related to space complexity, while length or time of trajectories, is intrinsically related to time complexity. These statements can also be interpreted in relation to several recent results about the computational power of analogue models of computation.