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

Abstract

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.