We study a new extension of the weak MSO logic, talking about boundedness. Instead of a previously considered quantifier U, expressing the fact that there exist arbitrarily large finite sets satisfying a given property, we consider a generalized quantifier U, expressing the fact that there exist tuples of arbitrarily large finite sets satisfying a given property. First, we prove that the new logic WMSO+U_tup is strictly more expressive than WMSO+U. In particular, WMSO+U_tup is able to express the so-called simultaneous unboundedness property, for which we prove that it is not expressible in WMSO+U. Second, we prove that it is decidable whether the tree generated by a given higher-order recursion scheme satisfies a given sentence of WMSO+K_tup.

中文翻译：

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通过元组量化扩展 WMSO+U 逻辑

我们研究弱 MSO 逻辑的新扩展，讨论有界性。我们考虑使用广义量词 U，表达存在满足给定属性的任意大有限集的元组这一事实，而不是先前考虑的量词 U，表示存在满足给定属性的任意大有限集这一事实。首先，我们证明新逻辑 WMSO+U_tup 严格上比 WMSO+U 更具表现力。特别是，WMSO+U_tup能够表达所谓的同时无界性性质，为此我们证明它在WMSO+U中是不可表达的。其次，我们证明给定的高阶递归方案生成的树是否满足给定的WMSO+K_tup语句是可判定的。