  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 $n$.
