Separation for dot-depth two

Abstract

The dot-depth hierarchy of Brzozowski and Cohen is a classification of all first-order definable languages. It rose to prominence following the work of Thomas, who established an exact correspondence with the quantifier alternation hierarchy of first-order logic: each level contains languages that can be defined with a prescribed number of quantifier blocks. One of the most famous open problems in automata theory is to obtain membership algorithms for all levels in this hierarchy.