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

TL;DR

This paper proves that successor-invariant first-order logic does not increase expressive power over first-order logic on structures with bounded degree, simplifying understanding of logical expressiveness in such classes.

Contribution

It establishes that successor-invariant first-order logic is equivalent to first-order logic on classes of structures with bounded degree, clarifying its expressive limitations.

Findings

Successor-invariant FO equals FO on bounded degree structures

Bounded degree structures limit the expressive power of successor-invariant logic

The result simplifies logical analysis on bounded degree classes

Abstract

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.