The Agafonov and Schnorr-Stimm theorems for probabilistic automata

TL;DR

The paper addresses extending the classical normality characterizations (Agafonov and Schnorr–Stimm) from deterministic finite automata to probabilistic finite automata. It adopts a reduction strategy to deterministic cases by coupling with an independent random source and leverages Bernoulli-measure normality concepts to transfer results from deterministic selectors and gamblers to PFAs. The main contributions prove that a sequence $X$ is normal iff any probabilistic automaton selects a normal subsequence with probability $1$, and iff no probabilistic finite-state gambler wins on $X$ with probability $1$. It also establishes a probabilistic Schnorr–Stimm dichotomy and shows the equivalence of $oldsymbol{ u}$-normality and $oldsymbol{ u}$-block normality, expanding the theory of normal sequences under random automata. The results close the open conjecture by Léchine et al. for arbitrary PFAs and pave the way for future work on stronger automata like pushdowns.

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 Léchine 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 $1$.

Theorems & Definitions (35)

• theorem 1: Agafonov Agafonov1968
• theorem 2: Schnorr-Stimm SchnorrS1972
• theorem 3: Léchine et al. LechineSS2024
• Definition 1
• Definition 2
• theorem 4: Agafonov theorem for Bernoulli measures SeillerS2020
• theorem 5: Schnorr-Stimm theorem for Bernoulli measures
• proof
• theorem 6: Schnorr-Stimm dichotomy theorem SchnorrS1972
• theorem 7: Schnorr-Stimm dichotomy theorem for Bernoulli measures