Statistical complexity dimension of the Mother of All Processes

Determine the statistical complexity dimension d_mu of the Mother of All Processes M defined by M = U_pi(⊗_{M ∈ L} M), where L is the library of epsilon-machine presentations of stationary processes and pi is the chosen mixture distribution over components. Quantify d_mu for this multistationary process’ mixed-state measure.

Background

The paper defines the Mother of All Processes (MOAP) as a hidden multistationary process obtained by applying the mixed-state operator U_pi to the tensor product over a library of epsilon-machines that enumerate stationary processes. In the simplified paper, the authors consider the mixture of all one-state and two-state epsilon-machines with a uniform mixture distribution.

The statistical complexity dimension d_mu, introduced in prior work, quantifies the fractal or dimensional properties of the distribution over causal states (mixed-state measure). Establishing d_mu for the MOAP would characterize the geometric and structural complexity inherent in the transient mixed-state set, which appears to fill a complicated fractal measure within the simplex of state distributions.