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

Extending the WMSO+U Logic With Quantification Over Tuples