We prove that a sequence satisfying a certain symmetry property is
2-regular in the sense of Allouche and Shallit, i.e., the Z-module
generated by its 2-kernel is finitely generated. We apply this theorem to
develop a general approach for studying the â„“-abelian complexity of
2-automatic sequences. In particular, we prove that the period-doubling word
and the Thue--Morse word have 2-abelian complexity sequences that are
2-regular. Along the way, we also prove that the 2-block codings of these
two words have 1-abelian complexity sequences that are 2-regular.Comment: 44 pages, 2 figures; publication versio