Many Roads to Synchrony: Natural Time Scales and Their Algorithms

TL;DR

This paper introduces methods to compute the Markov and cryptic orders of stochastic processes from epsilon-machines, revealing their significance in understanding process synchronization and their prevalence in finite-memory systems.

Contribution

It provides exact algorithms for calculating these orders from epsilon-machines and demonstrates their importance and ubiquity in finite-memory processes.

Findings

Markov and cryptic orders are key to understanding process synchronization.

These orders are most efficiently computed from epsilon-machines.

Infinite orders are common in finite-memory processes.

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 epsilon-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 epsilon-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.