We study the model-checking problem for recursion schemes: does the tree generated by a given higher-order recursion scheme satisfy a given logical sentence. The problem is known to be decidable for sentences of the MSO logic. We prove decidability for an extension of MSO in which we additionally have an unbounding quantifier U, saying that a subformula is true for arbitrarily large finite sets. This quantifier can be used only for subformulae in which all free variables represent finite sets (while an unrestricted use of the quantifier leads to undecidability). We also show that the logic has the properties of reflection and effective selection for trees generated by recursion schemes.

中文翻译：

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递归方案、MSO 逻辑和 U 量词

我们研究递归方案的模型检查问题：由给定的高阶递归方案生成的树是否满足给定的逻辑句子。已知该问题对于 MSO 逻辑的句子是可判定的。我们证明了 MSO 扩展的可判定性，其中我们还有一个无界量词 U，表示子公式对于任意大的有限集都成立。此量词只能用于所有自由变量都表示有限集的子公式（而不受限制地使用量词会导致不可判定性）。我们还表明，该逻辑对递归方案生成的树具有反射和有效选择的特性。