On the Synchronization Rate for e-machines

TL;DR

This paper investigates the synchronization and prediction rate constants of epsilon-machines, providing polynomial-time approximation methods for these constants based on the number of machine states.

Contribution

It introduces polynomial-time algorithms to approximate the synchronization and prediction rate constants of epsilon-machines.

Findings

Approximation of synchronization rate constant in polynomial time.

Approximation of prediction rate constant in polynomial time.

Provides bounds and methods for measuring synchronization efficiency.

Abstract

It is known, that an $\epsilon$-machine is either exactly or asymptotically synchronizing. In the exact case, the observer can infer the current machine state after observing $L$ generated symbols with probability $1-a^L$ where $0 \leq a<1$ is a so-called synchronization rate constant. In the asymptotic case, the probability of the correct prediction the current machine state after observing $L$ generated symbols tends to $1$ exponentially fast as $1-b^L$ for $0<b<1$ and the infimum of such $b$ is a so-called prediction rate constant. Hence the synchronization and prediction rate constants serve as natural measures of synchronization for $\epsilon$-machines. In the present work we show how to approximate these constants in polynomial time in terms of the number of machine states.