An Information Theory of Finite Abstractions and their Fundamental Scalability Limits

Finite abstractions are discrete approximations of dynamical systems, such that the set of abstraction trajectories contains all system trajectories. There is a consensus that abstractions suffer from the curse of dimensionality: for the same ``accuracy" (how closely the abstraction represents the system), the abstraction size scales poorly with system dimensions. And yet, after decades of research on abstractions, there are no formal results on their accuracy-size tradeoff. In this work, we derive a statistical, quantitative theory of abstractions' accuracy-size tradeoff 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 size, for given distortion. The bound depends on the complexity of the dynamics, through trajectory entropy. 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.