Title: Going Higher in First-Order Quantifier Alternation Hierarchies on Words,
Abstract: We investigate quantifier alternation hierarchies in first-order logic on
finite words. Levels in these hierarchies are defined by counting the number of
quantifier alternations in formulas. We prove that one can decide membership of
a regular language in the levels $\mathcal{B}{\Sigma}_2$ (finite boolean
combinations of formulas having only one alternation) and ${\Sigma}_3$
(formulas having only two alternations and beginning with an existential
block). Our proofs work by considering a deeper problem, called separation,
which, once solved for lower levels, allows us to solve membership for higher
levels.