Finite-Degree Predicates and Two-Variable First-Order Logic

TL;DR

This paper demonstrates that for two-variable first-order logic on finite words, adding finite-degree predicates does not increase expressive power when a neutral letter is present, leading to a hierarchy separation.

Contribution

It proves that finite-degree predicates do not add expressive power in the presence of a neutral letter, and establishes the separation of the alternation hierarchy for this logic.

Findings

Finite-degree predicates do not extend definability with a neutral letter.

The alternation hierarchy of two-variable logic is separated on this signature.

Abstract

We consider two-variable first-order logic on finite words with a fixed number of quantifier alternations. We show that all languages with a neutral letter definable using the order and finite-degree predicates are also definable with the order predicate only. From this result we derive the separation of the alternation hierarchy of two-variable logic on this signature.