Hierarchies in Transitive Closure Logic, Stratified Datalog and Infinitary Logic

We establish a general hierarchy theorem for quantiier classes in the innnitary logic L ! 1! on nite structures. In particular, it is shown that no innnitary formula with bounded number of universal quantiiers can express the negation of a transitive closure. This implies the solution of several open problems in nite model theory: On nite structures, positive transitive closure logic is not closed under negation. More generally the hierarchy de-ned 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 ne structure of stratiied Datalog.