  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
epsilon-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.
