🤖 AI Summary
This study investigates the intrinsic connection between finite-time reachability in dynamical systems and computational complexity. By introducing a family of decision problems termed “telic problems,” which formalize reachability under coarse-grained state-space representations, the work establishes—via polynomial-time reductions and topological entropy analysis—the first direct link between algorithmic time lower bounds (e.g., exponential time) and positive topological entropy in dynamical systems. The paper proposes a novel paradigm for classifying dynamical systems based on algorithmic complexity: if the telic problem associated with a system is solvable only by exponential-time algorithms, then the system necessarily exhibits positive topological entropy. This result provides a computationally grounded characterization of complexity in dynamical systems, bridging theoretical computer science and dynamical systems theory.
📝 Abstract
We begin development of a method for studying dynamical systems using concepts from computational complexity theory. We associate families of decision problems, called telic problems, to dynamical systems of a certain class. These decision problems formalize finite-time reachability questions for the dynamics with respect to natural coarse-grainings of state space. Our main result shows that complexity-theoretic lower bounds have dynamical consequences: if a system admits a telic problem for which every decider runs in time $2^{\Omega(n)}$, then it must have positive topological entropy. This result and others lead to methods for classifying dynamical systems through proving bounds on the runtime of algorithms solving their associated telic problems, or by constructing polynomial-time reductions between telic problems coming from distinct dynamical systems.