The Caucal hierarchy contains graphs that can be obtained from finite directed graphs by alternately applying the unfolding operation and inverse rational mappings. The goal of this work is to check whether the hierarchy is closed under interpretations in logics extending the monadic second-order logic by the unbounding quantifier, $\mathsf{U}$ (saying that a subformula holds for arbitrarily large finite sets). We prove that by applying interpretations described in the MSO+${\mathsf{U}}^{\mathsf{fin}}$ logic (hence also in its fragment WMSO+$\mathsf{U}$) to graphs of the Caucal hierarchy we can only obtain graphs on the same level of the hierarchy. Conversely, interpretations described in the more powerful MSO+$\mathsf{U}$ logic can give us graphs with an undecidable MSO theory, hence outside of the Caucal hierarchy.

Caucal 层次结构包含可以通过交替应用展开操作和逆有理映射从有限有向图获得的图。这项工作的目标是检查层次结构是否在通过无界量词扩展一元二阶逻辑的逻辑解释下关闭，$\mathsf{你}$（说一个子公式适用于任意大的有限集）。我们证明，通过应用 MSO+ 中描述的解释${\mathsf{你}}^{\mathsf{鳍}}$ 逻辑（因此也在其片段 WMSO+$\mathsf{你}$) 对于 Caucal 层次结构的图，我们只能获得层次结构相同级别的图。相反，更强大的 MSO+ 中描述的解释$\mathsf{你}$ 逻辑可以为我们提供具有不可判定 MSO 理论的图，因此在 Caucal 层次结构之外。