Information Accessibility and Cryptic Processes

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.