A Generalization of Automatic Sequences

Abstract We generalize the uniform tag sequences of Cobham, which arise as images of fixed points of k -uniform homomorphisms, to the case where the homomorphism ϕ is not necessarily uniform, but rather satisfies the analogue of an algebraic equation. We show that these sequences coincide with 1. (a) the class of sequences accepted by a finite automaton with “generalized digits” as input, and 2. (b) generalizations of the “locally catenative formula” of Rozenberg and Lindenmayer. Examples include the infinite Fibonacci word, which is generated as the fixed point of the homomorphism ϕ( a ) = ab , ϕ( b ) = a , and sequences of Rauzy and De Bruijn.