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.

@article{abs-1810-04763,
  title = {Recursion Schemes, the MSO Logic, and the U quantifier},
  author = {Paweł Parys},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 16, Issue 1},
  year = {2020},
  doi = {10.23638/lmcs-16(1:20)2020}
}