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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...

We show that some natural re nements of the Straubing and Brzozowski hierarchies correspond (via the so called leaf-languages) step by step to similar re nements of the polynomial-time hierarchy. This extends a result of H.-J. Burtschik and H. Vollmer on relationship between the Straubing and the polynomial hierarchies. In particular, this applies to the boolean hierarchy and the plus{hierarchy.

We show that if SAT does not have small circuits, then there must exist a small number of satisfiable formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if PNP[1] = PNP[2], then the polynomial-time hierarchy collapses to Sp2 ⊆ Σp2∩Πp2. Even showing that the hierarchy collapsed to Σp2 remained open p...

We characterize the collapse of Buss’ bounded arithmetic in terms of the prov­ able collapse of the polynomial time hierarchy. We include also some general model-theoretical investigations on fragments of bounded arithmetic.

We show that if the Boolean hierarchy collapses to level k, then the polynomial hierarchy collapses to BH3(k), where BH3(k) is the k th level of the Boolean hierarchy over Σ2 . This is an improvement over the known results [3], which show that the polynomial hierarchy would collapse to P [O(log . This result is significant in two ways. First, the theorem says that a deeper collapse of the Boole...