Order-Invariance in the Two-Variable Fragment of First-Order Logic

TL;DR

This paper investigates the expressive power of a two-variable, order-invariant fragment of first-order logic, showing it aligns with first-order logic on bounded degree structures, even with counting quantifiers.

Contribution

It demonstrates that order-invariant two-variable logic with counting quantifiers does not exceed the expressive power of first-order logic on bounded degree structures.

Findings

Order-invariant two-variable logic is equivalent to first-order logic on bounded degree structures.

Adding counting quantifiers does not increase expressive power in this setting.

The results hold for finite structures with bounded degree.

Abstract

We study the expressive power of the two-variable fragment of order-invariant first-order logic. This logic departs from first-order logic in two ways: first, formulas are only allowed to quantify over two variables. Second, formulas can use an additional binary relation, which is interpreted in the structures under scrutiny as a linear order, provided that the truth value of a sentence over a finite structure never depends on which linear order is chosen on its domain. We prove that on classes of structures of bounded degree, any property expressible in this logic is definable in first-order logic. We then show that the situation remains the same when we add counting quantifiers to this logic.