First Order Extensions of Residue Classes and Uniform Circuit Complexity

The first order logic Ring(0,+, ∗, <) for finite residue class rings with order is presented, and extensions of this logic with generalized quantifiers are given. It is shown that this logic and its extensions capture DLOGTIME-uniform circuit complexity classes ranging from AC to TC. Separability results are obtained for the hierarchy of these logics when order is not present, and for Ring(0,+, ∗, <) from the unordered version. These separations are obtained using tools from class field theory, adapting notions as the spectra of polynomials over finite fields to sets of sentences in this logic of finite rings, and studying asymptotic measures of these sets such as their relative densities. This framework of finite rings with order provides new algebraic tools and a novel perspective for descriptive complexity.