We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that

and

appear as classes of this hierarchy. In this way, we unconditionally place

properly in a strict hierarchy of arithmetic classes within

. Furthermore, we show that some of our classes admit efficient approximation in the sense of FPRAS. We compare our classes with a hierarchy within

defined in a model-theoretic way by Saluja et al and argue that our approach is better suited to study arithmetic circuit classes such as

which can be descriptively characterized as a class in our framework.

中文翻译：

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#P功能的描述性复杂性：新视角

我们为算术计算的描述性复杂性方法引入了一个新框架。我们基于对一阶公式中自由函数变量的分配进行计数的思想来定义类的层次结构。我们完全确定了夹杂物结构并表明

和

显示为该层次结构的类。这样，我们无条件放置

正确地在严格的算术类层次结构中

。此外，我们证明了某些类在FPRAS的意义上接受了有效逼近。我们将类与内部的层次结构进行比较

由Saluja等人以模型理论的方式定义，并认为我们的方法更适合研究算术电路类，例如

在我们的框架中可以描述为一个类。