Definability of Languages by Generalized First-Order Formulas over $(\mathbb{N},+)$

Abstract

We consider an extension of first-order logic by modular quantifiers of a fixed modulus $q$. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the number of 1's is divisible by a prime $p$ that does not divide $q$. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity class $ACC$ from $NC^1$. Thus our theorem can be viewed as proving a highly uniform version of the conjecture.