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.

翻译：我们针对任意无限整环，将Cucker与Meer提出的实数域代数电路构造进行适配推广，并在此设定下概括了$\mathrm{AC}_{\mathbb{R}}$与$\mathrm{NC}_{\mathbb{R}}$类。我们给出一个符合Immerman定理风格的结论：对于这些适配形式体系，由常数深度且多项式规模的电路所判定的集合，与通过适当适配一阶逻辑可定义的集合等价。此外，我们讨论了Durand、Haak与Vollmer提出的保护谓词逻辑的推广形式，并展示了$\mathrm{AC}_{R}$与$\mathrm{NC}_{R}$层次结构的刻画。这些推广同样适用于布尔$\mathrm{AC}$和$\mathrm{NC}$层次结构。最后，我们引入一种形式体系，用于比较上述部分复杂度类在底层整环不同情况下的性质。