Immunity and Simplicity for Exact Counting and Other Counting Classes

Ko [RAIRO 24, 1990] and Bruschi [TCS 102, 1992] showed that in some relativized world, PSPACE (in fact, ParityP) 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 ParityP 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^{ParityP}. In particular, this C_{=}P^A set is immune to PH^{A} and ParityP^{A}. Strengthening results of Tor\'{a}n [J.ACM 38, 1991] and Green [IPL 37, 1991], we also show that, in suitable relativizations, NP contains a C_{=}P-immune set, and ParityP 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 Balc\'{a}zar et al. [SIAM J.Comp. 14, 1985; RAIRO 22, 1988] 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 [Mat. Zametki 41, 1987] lower bound for majority.Comment: 20 page

On bounded query machines

AbstractSimple proofs are given for each of the following results: (a) P = Pspace if and only if, for every set A, P(A) = Pquery(A) (Selman et al., 1983): (b) NP = Pspace if and only if, for every set A, NP(A) = NPquery(S) (Book, 1981); (c) PH = Pspace if and only if, for every set A, PH(A) = PQH(A) (Book and Wrathall, 1981); (c) PH = Pspace if and only if, for every set set S, PH(S) = PQH(S) = Pspace(S) (Balcázar et al., 1986; Long and Selman, 1986)

Relativizations of the P =? DNP Question for the BSS Model

We consider the uniform BSS model of computation where the machines can perform additions, multiplications, and tests of the form $xgeq 0$. The oracle machines can also check whether a tuple of real numbers belongs to a given oracle set ${cal O}$ or not. We construct oracles such that the classes P and DNP relative to these oracles are equal or not equal

Self-Specifying Machines

We study the computational power of machines that specify their own acceptance types, and show that they accept exactly the languages that \manyonesharp-reduce to NP sets. A natural variant accepts exactly the languages that \manyonesharp-reduce to P sets. We show that these two classes coincide if and only if \psone = \psnnoplusbigohone, where the latter class denotes the sets acceptable via at most one question to \sharpp followed by at most a constant number of questions to \np.Comment: 15 pages, to appear in IJFC