Characterizing normality via automata and random matrix products

2026-04-14

AI Summary

This work investigates the normality of infinite sequences through the lens of probabilistic finite automata and products of nonnegative matrices. Inspired by the classical Schnorr–Stimm theorem, we extend it for the first time to the framework of probabilistic automata, establishing an equivalence between sequence normality and the exponential convergence of the expected capital of any gambling strategy modeled by such automata. This equivalence allows us to reformulate the problem in terms of the asymptotic behavior of norms of nonnegative matrix products. Our main contributions include proving that a sequence is normal if and only if the expected capital under every probabilistic automaton-based betting strategy converges exponentially to a finite limit, and demonstrating that the asymptotic behavior of these matrix product norms admits only three decidable regimes. This provides a novel algebraic and automata-theoretic perspective on sequence randomness.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• 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 relates normality to finite automata as follows: a sequence X is normal if and only if all gambling strategies implementable with finite deterministic automata lose all their capital when trying to predict the next bit of X after seing the ones before. More precisely, Schnorr and Stimm (1972) proved that the capital goes exponentially fast to zero unless the automaton represents the gambler that never bets, in which case the capital remains constant. In this paper we show that an analogous result holds when considering probabilistic automata: a sequence X is normal if and only if for any gambling strategy implementable with probabilistic finite automaton it holds that the expected value of the capital of the gambler converges exponentially fast to a finite value when playing against X. To obtain this result, we show a more general statement related to the convergence of martingales given by finite sets of non-negative matrices {M a } a$\in$A . In particular, we show that X is normal if and only if ||vM X[1] . . . M X[n] || converges exponentially fast to a finite value for any non-negative starting vector v. Moreover, we distinguish three distinctive behaviours that this sequence can attain, and prove that the problem of recognizing, given a family of matrices, to which case it belongs, is decidable.

Problem

• normality
• probabilistic automata
• random matrix products
• martingales
• finite automata

Innovation

• normal sequences
• probabilistic finite automata
• matrix products
• martingale convergence
• decidability