Normality and automata

• We prove a non-real-time 1-counter transducers cannot compress any normal number.
• We prove a real-time k-counter transducers cannot compress any normal number.
• We prove there exist pushdown transducers that can compress a normal number.
• We prove normality is preserved by suffix selection by finite automata.

We prove that finite-state transducers with injective behavior, deterministic or not, real-time or not, with no extra memory or a single counter, cannot compress any normal word. We exhaust all combinations of determinism, real-time, and additional memory in the form of counters or stacks, identifying which models can compress normal words. The case of deterministic push-down transducers is the only one still open. We also present results on the preservation of normality by selection with finite automata. Complementing Agafonov's theorem for prefix selection, we show that suffix selection preserves normality. However, there are simple two-sided selection rules that do not.