Many roads to synchrony: natural time scales and their algorithms.
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.
Complexity Sciences Center and Department of Physics, University of California at Davis, One Shields Avenue, Davis, California 95616, USA and Department of Computer Science, University of Colorado at Boulder, Boulder, Colorado 80309, USA.
Ryan G
RG
James
School of Natural Sciences, University of California, Merced, California 95344, USA.
John R
JR
Mahoney
Complexity Sciences Center and Department of Physics, University of California at Davis, One Shields Avenue, Davis, California 95616, USA and Center for Complexity and Collective Computation, Wisconsin Institute for Discovery, University of Wisconsin-Madison, Madison, Wisconsin 53715, USA.
Christopher J
CJ
Ellison
Complexity Sciences Center and Department of Physics, University of California at Davis, One Shields Avenue, Davis, California 95616, USA and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA.
James P
JP
Crutchfield