On FO2 quantifier alternation over words

We show that each level of the quantifier alternation hierarchy within FO^2[<] – the 2-variable fragment of the first order logic of order on words – is a variety of languages. We then use the notion of condensed rankers, a refinement of the rankers defined by Weis and Immerman, to produce a decidable hierarchy of varieties which is interwoven with the quantifier alternation hierarchy – and conjecturally equal to it. It follows that the latter hierarchy is decidable within one unit: given a formula alpha in FO^2[<], one can effectively compute an integer m such that alpha is equivalent to a formula with at most m+1 alternating blocks of quantifiers, but not to a formula with only m-1 blocks. This is a much more precise result than what is known about the quantifier alternation hierarchy within FO[<], where no decidability result is known beyond the very first levels.

💡 Research Summary

The paper investigates the quantifier‑alternation hierarchy within the two‑variable fragment of first‑order logic over words, denoted FO²