The Plogi and ACi−1 operators on the polynomial time hierarchy

In [CS-92] we studied the agreement of operators Plogi and ACi−1 acting on NP . In this article we extend this work to other classes of the polynomial time hierarchy. We show that on Σk , Π P k , ∆ P k and Θ P k -classes both operators have the same behaviour, but this coincidence does not seem to be true on other classes included in the PH hierarchy: we give a set A such that, relativized to A, Plogi(Plogj (NP )) is different from ACi−1(Plogj (NP )). As a result of these characterizations we show Plog(Θk ) = Θ P k , an equality that is useful to show lowness properties. In fact, we get easily the Θ-lowness results given by Long and Sheu in [LS-91]. Besides, we clarify the situation of the classes in L 2 for which their membership to L P,Θ 2 was not clear.