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Many roads to synchrony: Natural time scales and their algorithms

  • Author(s): James, RG
  • Mahoney, JR
  • Ellison, CJ
  • Crutchfield, JP
  • et al.
Abstract

We consider two important time scales - the Markov and cryptic orders - that monitor how an observer synchronizes to a finitary stochastic process. We show how to compute these orders exactly and that they are most efficiently calculated from the ε-machine, a process's minimal unifilar model. Surprisingly, though the Markov order is a basic concept from stochastic process theory, it is not a probabilistic property of a process. Rather, it is a topological property and, moreover, it is not computable from any finite-state model other than the ε-machine. Via an exhaustive survey, we close by demonstrating that infinite Markov and infinite cryptic orders are a dominant feature in the space of finite-memory processes. We draw out the roles played in statistical mechanical spin systems by these two complementary length scales. © 2014 American Physical Society.

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