Languages Defined with Modular Counting Quantifiers

Abstract

We prove that a regular language defined by a boolean combination of generalized Σ 1 -sentences built using modular counting quantifiers can be defined by a boolean combination of Σ 1 -sentences in which only regular numerical predicates appear. The same statement, with “Σ 1 ” replaced by “first-order,” is equivalent to the conjecture that the nonuniform circuit complexity class ACC is strictly contained in NC 1 . The argument introduces some new techniques, based on a combination of semigroup theory and Ramsey theory, which may shed some light on the general case.