A Markov-Chain Characterization of Finite-State Dimension and a Generalization of Agafonov's Theorem

2025-10-21

AI Summary

This work extends the theory of finite-state dimension beyond its traditional scope: the classical framework applies only to Borel-normal sequences, and Agafonov’s theorem lacks a quantitative characterization of finite-state dimension for non-normal sequences under automatic subsequencing. To address this, we propose a unified information-theoretic framework based on irreducible Markov chain simulation. We introduce the conditional Kullback–Leibler divergence as a novel measure to characterize the finite-state dimension of arbitrary infinite sequences. We establish an exact one-to-one correspondence between finite-state dimension and stationary distributions of irreducible Markov chains. Consequently, we generalize the Schnorr–Stimm theorem to all infinite binary sequences and derive a tight quantitative version of Agafonov’s theorem: for any sequence and its automatic subsequence, their finite-state dimensions satisfy an explicit inequality governed by the conditional KL divergence. This yields the first rigorous, dimension-preserving analysis of automatic selection beyond normality.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Finite-state dimension quantifies the asymptotic rate of information in an infinite sequence as perceived by finite automata. For a fixed alphabet, the infinite sequences that have maximal finite-state dimension are exactly those that are Borel normal, i.e., in which all words of any given length appear with the same frequency. A theorem of Schnorr and Stimm (1972) shows that a real number is Borel normal if and only if, for every finite-state irreducible Markov chain with fair transitions, when the chain is simulated using the binary expansion of the given number, the empirical distribution of states converges to its stationary distribution. In this paper we extend this correspondence beyond normal numbers. We show that the finite-state dimension of a sequence can be characterized in terms of the conditional Kullback-Leibler divergence between the limiting distributions arising from the simulation of Markov chains using the given sequence and their stationary distributions. This provides a new information-theoretic characterization of finite-state dimension which generalizes the Schnorr-Stimm result. As an application, we prove a generalization of Agafonov's theorem for normal numbers. Agafonov's theorem states that a sequence is normal if and only if every subsequence selected by a finite automaton is also normal. We extend this to arbitrary sequences by establishing a tight quantitative relationship between the finite-state dimension of a sequence and the finite-state dimensions of its automatic subsequences.

Problem

• Characterizing finite-state dimension via Markov chain simulation divergence
• Generalizing Schnorr-Stimm theorem beyond Borel normal numbers
• Extending Agafonov's theorem to establish automatic subsequence dimension relationships

Innovation

• Generalizes Schnorr-Stimm theorem using Kullback-Leibler divergence
• Characterizes finite-state dimension via Markov chain simulations
• Extends Agafonov's theorem to arbitrary sequences quantitatively