Characterizing normality via automata and random matrix products

TL;DR

This paper extends classical automata-based characterizations of normal sequences to probabilistic automata, linking sequence normality to exponential convergence of expected gambler capital and matrix product behaviors.

Contribution

It introduces a probabilistic automata framework for characterizing normality and proves the decidability of classifying matrix product convergence behaviors.

Findings

Normal sequences correspond to exponential convergence of gambler capital in probabilistic automata.

The convergence of matrix products determines sequence normality.

The classification of matrix product behaviors is decidable.

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.