Asymptotic Synchronization for Finite-State Sources

We extend a recent synchronization analysis of exact finite-state sources to nonexact sources for which synchronization occurs only asymptotically. Although the proof methods are quite different, the primary results remain the same. We find that an observer’s average uncertainty in the source state vanishes exponentially fast and, as a consequence, an observer’s average uncertainty in predicting future output converges exponentially fast to the source entropy rate.

💡 Research Summary

The paper extends the synchronization theory that was previously confined to exact finite‑state sources (ε‑machines) to the broader class of non‑exact sources, where an observer can only synchronize asymptotically. In exact sources, a finite observation window is sufficient to determine the hidden internal state with certainty. By contrast, non‑exact sources lack such a finite “synchronizing word”; the observer’s belief about the hidden state converges to a point mass only as the observation length tends to infinity.

The authors first formalize the setting. A finite‑state source is described by a set of hidden states S, a transition matrix T(s, s′, x) that gives the probability of moving from state s to s′ while emitting symbol x from a finite alphabet Σ, and an output process X₀, X₁, … . The observer’s uncertainty about the current state after seeing the first t symbols is quantified by the conditional entropy Hₜ = H