We study the expressive power of successor-invariant first-order logic, which is an extension of first-order logic where the usage of an additional successor relation on the structure is allowed, as long as the validity of formulas is independent of the choice of a particular successor on finite structures. We show that when the degree is bounded, successor-invariant first-order logic is no more expressive than first-order logic.

@article{paperbot1284,
  title = {Successor-Invariant First-Order Logic on Classes of Bounded Degree},
  author = {Julien Grange},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 17, Issue 3},
  year = {2021},
  doi = {10.46298/lmcs-17(3:20)2021}
}