Non-definability of Languages by Generalized First-order Formulas over (N,+)

We consider first-order logic with monoidal quantifiers over words. We show that all languages with a neutral letter, definable using the addition predicate are also definable with the order predicate as the only numerical predicate. Let S be a subset of monoids. Let L be the logic closed under quantification over the monoids in S. Then we prove that L[<;,+] and L[<;] define the same neutral letter languages. Our result can be interpreted as the Crane Beach conjecture to hold for the logic L[<;,+]. As a consequence we get the result of Roy and Straubing that FO+MOD[<;,+] collapses to FO+MOD[<;]. For cyclic groups, we answer an open question of Roy and Straubing, proving that MOD[<;,+] collapses to MOD[<;]. Our result also shows that multiplication as a numerical predicate is necessary for Barrington´s theorem to hold and also to simulate majority quantifiers. All these results can be viewed as separation results for highly uniform circuit classes. For example we separate FO[<;,+]-uniform CC0 from FO[<;,+]-uniform ACC0.