Technology Category
Machine Learning: Information TheoryReasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyKnowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
A Markov-Chain Characterization of Finite-State Dimension and a Generalization of Agafonov's Theorem
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
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
Methods, ideas, or system contributions that make the work stand out.
Generalizes Schnorr-Stimm theorem using Kullback-Leibler divergence
Characterizes finite-state dimension via Markov chain simulations
Extends Agafonov's theorem to arbitrary sequences quantitatively
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