Mix-automatic sequences

Abstract

Mix-automatic sequences form a proper extension of the class of automatic sequences, and arise from a generalization of finite state automata where the input alphabet is state-dependent. In this paper we compare the class of mix-automatic sequences with the class of morphic sequences. For every polynomial π we construct a mix-automatic sequence whose subword complexity exceeds π. This stands in contrast to automatic and morphic sequences which are known to have at most quadratic subword complexity. We then adapt the notion of k-kernels to obtain a characterization of mix-automatic sequences, and employ this notion to construct morphic sequences that are not mix-automatic. © 2013 Springer-Verlag Berlin Heidelberg.