Successor-Invariant First-Order Logic on Classes of Bounded Degree

December 7, 2020

Julien Grange (INRIA Rennes)

Successor-invariant first-order logic is the extension of first-order logic where one has access to an additional successor relation on the elements of the structure, as long as the validity of formulas is independent on the choice of a particular successor.

It has been shown by Rossman that this formalism allows to express properties that are not FO-definable.
However, his separating example involves dense classes of structures, and the expressive power of successor-invariant first-order logic is an open question for sparse classes of structures.

We prove that when the degree is bounded, successor-invariant first-order logic is no more expressive than first-order logic.