Modulo quantifiers over functional vocabularies extending addition

We show that first order logic (FO) and first order logic extended with modulo counting quantifiers (FOMOD) over purely functional vocabularies which extend addition, satisfy the Crane beach property (CBP) if the logic satisfies a normal form (called positional normal form). This not only shows why logics over the addition vocabulary have the CBP but also gives new CBP results, for example for the vocabulary which extends addition with the exponentiation function. The above results can also be viewed from the perspective of circuit complexity. Showing the existence of regular languages not definable in FOMOD[<,+,×] is equivalent to the separation of the circuit complexity classes ACC and NC. Our theorem shows that a weaker logic , namely, FOMOD[<,+, 2] cannot define all regular languages.