Immunity and simplicity for exact counting and other counting classes

Ko [26] and Bruschi [11] independently showed that, in some relativized world, PSPACE (in fact, +P) contains a set that is immune to the polynomial hierarchy (PH). In this paper, Ne study and settle the question of relativized separations with immunity for PH and the counting classes PP, C=P, and +P in all possible pairwise combinations. Our main result is that there is an oracle A relative to which C=P contains a set that is immune to BPP+P. In particular, this C=P-A set is immune to PHA and to +P-A. Strengthening results of Toran [48] and Green [18], we also show that, in suitable relativizations, NP contains a C=P-immune set, and +P contains a PPPH-immune set. This implies the existence of a C=P-B-simple set for some oracle B, which extends results of Balcazar et al. [2, 4]. Our proof technique requires a circuit lower bound for "exact counting" that is derived from Raeborov's [35] circuit lower bound for majority.