Finite-state independence

TL;DR

This paper introduces a finite-state automata-based notion of independence for infinite words, explores its measure-theoretic properties, and investigates its implications for normality and word joins.

Contribution

It defines a new finite-state independence concept, proves measure 1 for independent pairs, and analyzes normality preservation under independence.

Findings

Set of independent pairs has Lebesgue measure 1

Join of two independent normal words is normal

Independence of two normal words is not guaranteed by their join

Abstract

In this work we introduce a notion of independence based on finite-state automata: two infinite words are independent if no one helps to compress the other using one-to-one finite-state transducers with auxiliary input. We prove that, as expected, the set of independent pairs of infinite words has Lebesgue measure 1. We show that the join of two independent normal words is normal. However, the independence of two normal words is not guaranteed if we just require that their join is normal. To prove this we construct a normal word $x_1x_2x_3\ldots$ where $x_{2n}=x_n$ for every $n$.