The Agafonov and Schnorr-Stimm theorems for probabilistic automata

2025-02-17

AI Summary

This paper resolves a long-standing open problem: whether Agafonov’s and Schnorr–Stimm’s theorems—characterizing normality via selection by deterministic finite automata (DFA)—extend to arbitrary probabilistic finite automata (PFA), particularly those with irrational transition probabilities. Employing a synthesis of probabilistic analysis, ergodic theory, martingale convergence techniques, and tools from normality theory, we establish, for the first time, a rigorous characterization: an infinite sequence is normal if and only if every PFA selects a normal subsequence with probability one; equivalently, if and only if no probabilistic finite-state gambler can achieve positive expected profit. Our result overcomes the prior restriction to PFAs with rational transition probabilities, unifying normality characterizations across deterministic and general probabilistic models. This advances the theoretical foundations of algorithmic randomness and information theory by providing a broadly applicable framework for normality under arbitrary stochastic selection mechanisms.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Preferences

Application Category

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

Abstract

For a fixed alphabet $A$, an infinite sequence $X$ is said to be normal if every word $w$ over $A$ appears in $X$ with the same frequency as any other word of the same length. A classical result of Agafonov (1966) relates normality to finite automata as follows: a sequence $X$ is normal if and only if any subsequence of $X$ selected by a finite automaton is itself normal. Another theorem of Schnorr and Stimm (1972) gives an alternative characterization: a sequence $X$ is normal if and only if no gambler can win large amounts of money by betting on the sequence $X$ using a strategy that can be described by a finite automaton. Both of these theorems are established in the setting of deterministic finite automata. This raises the question as to whether they can be extended to the setting of probabilistic finite automata. In the case of the Agafonov theorem, this question was positively answered by L'echine et al. (2024) in a restricted case of probabilistic automata with rational transition probabilities. In this paper, we settle the full conjecture by proving that both the Agafonov and the Schnorr-Stimm theorems hold true for arbitrary probabilistic automata. Specifically, we show that a sequence $X$ is normal if and only if any probabilistic automaton selects a normal subsequence of $X$ with probability $1$. We also show that a sequence $X$ is normal if and only if a probabilistic finite-state gambler fails to win on $X$ with probability $1$.

Problem

• Extends Agafonov theorem to probabilistic automata
• Proves Schnorr-Stimm theorem for probabilistic automata
• Establishes normality criteria for probabilistic automata

Innovation

• Extends Agafonov theorem to probabilistic automata
• Validates Schnorr-Stimm theorem probabilistically
• Proves normality with probabilistic automaton selection