Inferring the Dynamic, Quantifying Physical Complexity

Through its formalization of inductive inference, computational learning theory provides a foundation for the inverse problem of chaotic data analysis: inferring the deterministic equations of motion underlying observed random behavior in physical systems. Integrating the geometric and statistical techniques of dynamical systems with learning theory provides a framework for consistently, although not absolutely, distinguishing between deterministic chaos and extrinsic fluctuations at a given level of computational resources. Two approaches to the inverse problem, estimating symbolic equations of motion and reconstructing minimal automata from chaotic data series, are reviewed from this point of view. With an inferred model dynamic the dynamical entropies and dimensions can be estimated. More interestingly, its structural properties give a measure of the intrinsic computational complexity of the underlying process.