Hierarchies in transitive closure logic, stratified Datalog and infinitary logic

Abstract

The authors establish a general hierarchy theorem for quantifier classes in the infinitary logic L/sub infinity omega //sup omega / on 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 N. Immerman (1987). The authors also separate the expressive power of several extensions of Datalog, giving new insight in the fine structure of stratified Datalog.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>