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Computational mechanics can prove that the causal states obtained through the <math>\epsilon</math>-machine have three important characteristics: "maximum predictability", "minimum statistical complexity", and "minimum randomness", and it is verified that it is optimal in a certain sense. In addition, the author introduces a hierarchical machine reconstruction algorithm that can calculate causal states and <math>\epsilon</math>-machines from observational data. Although this algorithm may not be applicable to all scenarios, the author takes chaotic dynamics, hidden Markov models, and cellular automata as examples and gives numerical calculation results and corresponding machine reconstruction paths <ref name="The_calculi_of_emergence">{{cite journal|author1=Crutchfield, J.P|title=The calculi of emergence: computation, dynamics and induction|journal=Physica D: Nonlinear Phenomena|year=1994|volume=75|issue=1-3|page=11-54|url=https://www.sciencedirect.com/science/article/abs/pii/0167278994902739}}</ref>.

Computational mechanics can prove that the causal states obtained through the <math>\epsilon</math>-machine have three important characteristics: '''maximum predictability ''', '''minimum statistical complexity ''', and '''minimum randomness ''', and it is verified that it is optimal in a certain sense. In addition, the author introduces a hierarchical machine reconstruction algorithm that can calculate causal states and <math>\epsilon</math>-machines from observational data. Although this algorithm may not be applicable to all scenarios, the author takes chaotic dynamics, hidden Markov models, and cellular automata as examples and gives numerical calculation results and corresponding machine reconstruction paths <ref name="The_calculi_of_emergence">{{cite journal|author1=Crutchfield, J.P|title=The calculi of emergence: computation, dynamics and induction|journal=Physica D: Nonlinear Phenomena|year=1994|volume=75|issue=1-3|page=11-54|url=https://www.sciencedirect.com/science/article/abs/pii/0167278994902739}}</ref>.