Context-Free Sequences

A sequence w over a finite alphabet A is generated by a uniform automaton if there exists an automaton labelled on {0, . . . , k − 1} for some k > 1 and recognizing for each output a in A the set of positions of a in w expressed in base k. Automatic sequences are generated by finite automata. By considering pushdown automata instead of finite ones, we generate exactly the context-free sequences. We distinguish the sub-families of unambiguous, deterministic, realtime deterministic context-free sequences associated with the corresponding families of pushdown automata. We study the closure under shift, product, morphisms, inverse substitutions and various extractions of these four families of context-free sequences. Additionally, we show that only using multiplicatively dependent bases yields the same set of context-free sequences.