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

On logical hierarchies within FO^2-definable languages