Hierarchies in transitive closure logic, stratified Datalog and infinitary logic

We establish a general hierarchy theorem for quantifier classes in the infinitary logic L∞ωωon finite structures. In particular, it is shown that no infinitary formula with bounded number of universal quantifiers can express the negation of a transitive closure.This implies the solution of several open problems in finite model theory: On finite structures, positive transitive closure logic is not closed under negation. More generally the hierarchy defined by interleaving negation and transitive closure operators is strict. This proves a conjecture of Immerman.We also separate the expressive power of several extensions of Datalog, giving new insight in the fine structure of stratified Datalog