Information Accessibility and Cryptic Processes

TL;DR

This paper introduces a systematic expansion of crypticity, a measure of how inaccessible a process's internal states are, revealing a hierarchy of processes and enabling precise analysis of information in complex systems.

Contribution

It develops an exact hierarchy of k-cryptic processes and links crypticity to excess entropy, advancing understanding of information accessibility in stationary processes.

Findings

Finite-state processes can have infinite crypticity.

The crypticity expansion is exact for finite and infinite cases.

An efficient, exact expansion of excess entropy for finite-order cryptic processes.

Abstract

We give a systematic expansion of the crypticity--a recently introduced measure of the inaccessibility of a stationary process's internal state information. This leads to a hierarchy of k-cryptic processes and allows us to identify finite-state processes that have infinite crypticity--the internal state information is present across arbitrarily long, observed sequences. The crypticity expansion is exact in both the finite- and infinite-order cases. It turns out that k-crypticity is complementary to the Markovian finite-order property that describes state information in processes. One application of these results is an efficient expansion of the excess entropy--the mutual information between a process's infinite past and infinite future--that is finite and exact for finite-order cryptic processes.