Normality and two-way automata

We prove that two-way transducers (both deterministic and non-deterministic) cannot compress normal numbers. To achieve this, we first show that it is possible to generalize compressibility from one-way transducers to two-way transducers. These results extend a known result: normal infinite words are exactly those that cannot be compressed by lossless finite-state transducers, and, more generally, by bounded-to-one non-deterministic finite-state transducers. We also argue that such a generalization cannot be extended to two-way transducers with unbounded memory, even in the simple form of a single counter.