Methods of Class Field Theory to Separate Logics over Finite Residue Classes and Circuit Complexity

Separations among the first order logic ${\cal R}ing(0,+,*)$ of finite residue class rings, its extensions with generalized quantifiers, and in the presence of a built-in order are shown, using algebraic methods from class field theory. These methods include classification of spectra of sentences over finite residue classes as systems of congruences, and the study of their $h$-densities over the set of all prime numbers, for various functions $h$ on the natural numbers. Over ordered structures the logic of finite residue class rings and extensions are known to capture DLOGTIME-uniform circuit complexity classes ranging from $AC^0$ to $TC^0$. Separating these circuit complexity classes is directly related to classifying the $h$-density of spectra of sentences in the corresponding logics of finite residue classes. We further give general conditions under which a logic over the finite residue class rings has a sentence whose spectrum has no $h$-density. One application of this result is that in ${\cal R}ing(0,+,*,<) + M$, the logic of finite residue class rings with built-in order and extended with the majority quantifier $M$, there are sentences whose spectrum have no exponential density.