The Frobenius problem for homomorphic embeddings of languages into the integers

Let S be a map from a language L to the integers satisfying S(vw)=S(v)+S(w) for all words v,w from the language. The classical Frobenius problem asks whether the complement of S(L) in the natural numbers will be infinite or finite, and in the latter case the value of the largest element in this complement. This is also known as the ‘coin’-problem, and L is the full language consisting of all words over a finite alphabet. We solve the Frobenius problem for the golden mean language, any Sturmian language and the Thue-Morse language. We also consider two-dimensional embeddings.