Cardinality problems of compositions of morphisms and inverse morphisms

Abstract

We show that we cannot effectively determine whether, for morphismsα _i ,β _i ,card (uα ⁻¹₀ α ₁) =card(uβ ⁻¹₀ β ₁) for all wordsu over the domain alphabets of the two given compositions. In contrast it is decidable for morphismsα _i ,β _i and a regular setR whethercard(uα ₀ α ⁻¹₁ ) =card(uβ ₀ β ⁻¹₁ ) for all wordsu inR. In order to prove the latter result we give a characterization of the multiplicity functions of simple finite automata by using cardinalities of compositions of the above form. Finally, we show that the above decidability result also holds when we consider rational functions rather than morphisms.