In this article, we propose a new classification of $\Sigma^0_2$ formulas under the realizability interpretation of many-one reducibility (i.e., Levin reducibility). For example, ${\sf Fin}$, the decision of being eventually zero for sequences, is many-one/Levin complete among $\Sigma^0_2$ formulas of the form $\exists n\forall m\geq n.\varphi(m,x)$, where $\varphi$ is decidable. The decision of boundedness for sequences ${\sf BddSeq}$ and for width of posets ${\sf FinWidth}$ are many-one/Levin complete among $\Sigma^0_2$ formulas of the form $\exists n\forall m\geq n\forall k.\varphi(m,k,x)$, where $\varphi$ is decidable. However, unlike the classical many-one reducibility, none of the above is $\Sigma^0_2$-complete. The decision of non-density of linear order ${\sf NonDense}$ is truly $\Sigma^0_2$-complete.

翻译：本文在可实现性解释下的多一可归约性（即Levin可归约性）框架下，提出了一种新的$\Sigma^0_2$公式分类方法。例如，序列最终为零的判定问题${\sf Fin}$，在形如$\exists n\forall m\geq n.\varphi(m,x)$（其中$\varphi$是可判定的）的$\Sigma^0_2$公式中是多一/Levin完全的。序列有界性判定问题${\sf BddSeq}$与偏序集宽度有限性判定问题${\sf FinWidth}$，在形如$\exists n\forall m\geq n\forall k.\varphi(m,k,x)$（其中$\varphi$是可判定的）的$\Sigma^0_2$公式中是多一/Levin完全的。然而，与经典多一可归约性不同，上述问题均非$\Sigma^0_2$完全的。线性序的非稠密性判定问题${\sf NonDense}$是真正的$\Sigma^0_2$完全问题。