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Mathematical treatment of the canonical finite state machine for the Ising model: $ε$-machine

The complete framework for the $ε$-machine construction of the one dimensional Ising model is presented correcting previous mistakes on the subject. The approach follows the known treatment of the Ising model as a Markov random field, where usually the local characteristic are obtained from the stochastic matrix, the problem at hand needs the inverse relation, or how to obtain the stochastic matrix from the local characteristics, which are given via the transfer matrix treatment. The obtained expressions allow to perform complexity-entropy analysis of particular instance of the Ising model. Three examples are discussed: the 1/2-spin nearest neighbor and next nearest neighbor Ising model, and the persistent biased random walk.