Separation for dot-depth two

The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free languages finite words. By a theorem McNaughton Papert, these are also first-order definable languages. rose to prominence following work Thomas, who proved an exact correspondence with quantifier alternation logic: each level in consists all that can be defined prescribed number blocks. One most famous open problems automata theory is settle whether membership problem decidable for level: it possible decide input regular language belongs this level? Despite significant research effort, by itself has only been solved low levels. A recent breakthrough was achieved replacing more general problem: separation. Given two languages, one there exists third investigated containing first disjoint from second. motivation that: (1) while difficult, separation rewarding (2) provides convenient framework (3) algorithms reductions lower We present algorithm two. While our prominent application, result general. consider family hierarchies includes dot-depth: concatenation hierarchies. They built via generic construction process. chooses initial class, basis, which lowest hierarchy. Further levels applying operations. Our main states any whose basis finite, one. In special case dot-depth, lifted using previously known results.