Exact complexity: The spectral decomposition of intrinsic computation

• We provide exact, closed-form expressions for a hidden stationary process' intrinsic computation.
• These include information measures such as the excess entropy, transient information, and synchronization information and the entropy-rate finite-length approximations.
• The method uses an epsilon-machine's mixed-state presentation.
• The spectral decomposition of the mixed-state presentation relies on the recent development of meromorphic functional calculus for nondiagonalizable operators.

We give exact formulae for a wide family of complexity measures that capture the organization of hidden nonlinear processes. The spectral decomposition of operator-valued functions leads to closed-form expressions involving the full eigenvalue spectrum of the mixed-state presentation of a process's ϵ-machine causal-state dynamic. Measures include correlation functions, power spectra, past-future mutual information, transient and synchronization informations, and many others. As a result, a direct and complete analysis of intrinsic computation is now available for the temporal organization of finitary hidden Markov models and nonlinear dynamical systems with generating partitions and for the spatial organization in one-dimensional systems, including spin systems, cellular automata, and complex materials via chaotic crystallography.