# Logical Characterization of Algebraic Circuit Classes over Integral Domains

## TL;DR

This paper extends algebraic circuit classes over the reals to arbitrary infinite integral domains, providing logical characterizations and hierarchy results that unify and generalize classical complexity classes over various domains.

## Contribution

It introduces a generalized framework for algebraic circuits over integral domains and establishes logical characterizations and hierarchy theorems for these classes.

## Key findings

- Sets decided by constant-depth, polynomial-size circuits match definable first-order logic.
- Characterizations of $ ext{AC}_R$ and $ ext{NC}_R$ hierarchies are provided.
- Framework applies to Boolean $ ext{AC}$ and $ ext{NC}$ hierarchies as well.

## Abstract

We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the $\mathrm{AC}_{\mathbb{R}}$ and $\mathrm{NC}_{\mathbb{R}}$-classes for this setting. We give a theorem in the style of Immerman's theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer and we show characterizations for the $\mathrm{AC}_{R}$ and $\mathrm{NC}_{R}$ hierarchy. Those generalizations apply to the Boolean $\mathrm{AC}$ and $\mathrm{NC}$ hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains.