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 -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 -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.
8 More- Received 23 December 2010
- Revised 20 December 2013
DOI:https://doi.org/10.1103/PhysRevE.89.042135
©2014 American Physical Society