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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, in a formal sense, 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 concerning 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 branch of information theory studying lossy compression. Abstractions are viewed as encoder-decoder pairs, encoding trajectories of dynamical systems in a higher-dimensional ambient space. Rate represents abstraction size, while distortion describes abstraction accuracy, defined as the spatial average deviation between abstract trajectories and system ones. We obtain a fundamental lower bound on the minimum abstraction distortion, given the system dynamics and a threshold on abstraction size. The bound depends on the complexity of the dynamics, through generalized entropy. We demonstrate the bound's tightness on certain dynamical systems. Finally, we showcase how the developed theory can be employed to construct optimal abstractions, in terms of the size-accuracy tradeoff, through an example on a chaotic system.