On logical hierarchies within FO^2-definable languages

TL;DR

This paper investigates the quantifier alternation hierarchy within FO^2-definable languages over linear orders, establishing its decidability and relating hierarchy levels to algebraic varieties via condensed rankers.

Contribution

It proves the decidability of the quantifier alternation hierarchy within FO^2[<] and links each level to algebraic varieties characterized by iterated deterministic and co-deterministic products.

Findings

Hierarchy levels are decidable within one unit.

Each level corresponds to specific algebraic varieties.

Condensed rankers refine previous ranker concepts.

Abstract

We consider the class of languages defined in the 2-variable fragment of the first-order logic of the linear order. Many interesting characterizations of this class are known, as well as the fact that restricting the number of quantifier alternations yields an infinite hierarchy whose levels are varieties of languages (and hence admit an algebraic characterization). Using this algebraic approach, we show that the quantifier alternation hierarchy inside FO^{2}[<] is decidable within one unit. For this purpose, we relate each level of the hierarchy with decidable varieties of languages, which can be defined in terms of iterated deterministic and co-deterministic products. A crucial notion in this process is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle languages of Schwentick, Th\'erien and Vollmer.