Akama et al. [1] introduced a hierarchical classification of first-order formulas for a hierarchical prenex normal form theorem in semi-classical arithmetic. In this paper, we give a justification for the hierarchical classification in a general context of first-order theories. To this end, we first formalize the standard transformation procedure for prenex normalization. Then we show that the classes \(\textrm{E}_k\) and \(\textrm{U}_k\) introduced in [1] are exactly the classes induced by \(\Sigma _k\) and \(\Pi _k\) respectively via the transformation procedure in any first-order theory.

中文翻译：

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Prenex 归一化和公式的层次分类

阿卡玛等人。[1] 引入了半经典算术中分层 Prenex 范式定理的一阶公式的分层分类。在本文中，我们在一阶理论的一般背景下给出了层次分类的理由。为此，我们首先将 prenex 标准化的标准转换程序形式化。然后我们证明[1]中引入的类\(\textrm{E}_k\)和\(\textrm{U}_k\)正是由\(\Sigma _k\)和\(\Pi导出的类_k\)分别通过任何一阶理论中的变换过程。