Finite abstractions are discrete models of dynamical systems, such that the set of abstraction trajectories contains all system trajectories. There is a consensus that abstractions suffer from a scalability bottleneck: to obtain a sufficient ``accuracy" (how closely the abstraction models the system), the required abstraction size often becomes very large, rendering computation infeasible. This is accentuated for complex, high-dimensional systems, due to the curse of dimensionality. Yet, after decades of research, there are no formal results on the size-accuracy tradeoff of abstractions. Here, we derive a statistical, quantitative theory of the size-accuracy tradeoff of abstractions of autonomous deterministic systems and uncover fundamental limits on their scalability, through rate-distortion theory---the information theory of lossy compression. Abstractions are viewed as encoder-decoder pairs, encoding trajectories of dynamical systems. Rate measures abstraction size, while distortion describes accuracy, defined as the spatial average deviation between abstract trajectories and system ones. We obtain a fundamental lower bound on the minimum achievable abstraction distortion, given the system dynamics and the abstraction size; and vice-versa a lower bound on the minimum required size, for given distortion. The bound depends on the complexity of the dynamics, through trajectory entropy and a constant depending on the trajectory manifold's geometry. We demonstrate its tightness on some dynamical systems. Finally, we showcase how this new theory enables constructing minimal abstractions, optimizing the size-accuracy tradeoff, through an example on a chaotic system.

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