An effective characterization of the alternation hierarchy in two-variable logic

TL;DR

This paper characterizes the levels of the quantifier alternation hierarchy in two-variable first-order logic using identities, establishing their decidability and extending results to certain algebraic structures.

Contribution

It provides a novel algebraic characterization of the hierarchy levels and proves their decidability, advancing understanding of logical definability and algebraic properties.

Findings

Decidability of individual levels of the hierarchy

Algebraic identities characterize hierarchy levels

Decidability extends to two-sided semidirect products

Abstract

We characterize the languages in the individual levels of the quantifier alternation hierarchy of first-order logic with two variables by identities. This implies decidability of the individual levels. More generally we show that the two-sided semidirect product of a decidable variety with the variety J is decidable.