Immunity and Simplicity for Exact Counting and Other Counting Classes

Abstract

Ko [Ko90] and Bruschi [Bru92] 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, we 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 PH A and to ⊕P A . Strengthening results of Torán [Tor91] and Green [Gre91], we also show that, in suitable relativizations, NP contains a C =P-immune set, and ⊕P contains a PP PH -immune set. This implies the existence of a C =P B -simple set for some oracle B, which extends results of Balcázar et al. [Bal85,BR88], and provides the first example of a simple set in a class not known to be contained in PH. Our proof technique requires a circuit lower bound for "exact counting" that is derived from Razborov's [Raz87] circuit lower bound for majority.

Resource bounded immunity and simplicity

“…Of all complexity classes, we focus our study only on the classes lying within the polynomial hierarchy. Other classes, such as C = P and EXP, have been studied by, e.g., Rothe [43] and Schaefer and Fenner [45]. The goals of our investigation are to (i) analyze the behaviors of variants of immunity and simplicity notions, (ii) study the relationships between immunity and other complexity-theoretical notions, and (iii) explore new directions for better understandings of the polynomial hierarchy.…”

On computing the smallest four-coloring of planar graphs and non-self-reducible sets in P

“…And for C = C = P, the closure properties of C = P imply that raising NP-hardness to Θ p 2 -hardness is worthless (see [9]), but do not seem to suffice in any obvious way to yield the same claim for Δ p 2 -hardness. Again, there are relativized counterexamples for the inclusion Δ p 2 ⊆ C = Pand even relativizations that separate the entire polynomial hierarchy from C = P with immunity, see [18]. However, unlike for PP, these relativized separations do not give us any more insight regarding the value of raising…”