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

Boolean Operations, Joins, and the Extended Low Hierarchy

We prove that the join of two sets may actually fall into a lower level of the extended low hierarchy than either of the sets. In particular, there exist sets that are not in the second level of the extended low hierarchy, EL_2, yet their join is in EL_2. That is, in terms of extended lowness, the join operator can lower complexity. Since in a strong intuitive sense the join does not lower complexity, our result suggests that the extended low hierarchy is unnatural as a complexity measure. We also study the closure properties of EL_ and prove that EL_2 is not closed under certain Boolean operations. To this end, we establish the first known (and optimal) EL_2 lower bounds for certain notions generalizing Selman's P-selectivity, which may be regarded as an interesting result in its own right.Comment: 12 page

A Counterexample to the Generalized Linial-Nisan Conjecture

In earlier work, we gave an oracle separating the relational versions of BQP and the polynomial hierarchy, and showed that an oracle separating the decision versions would follow from what we called the Generalized Linial-Nisan (GLN) Conjecture: that "almost k-wise independent" distributions are indistinguishable from the uniform distribution by constant-depth circuits. The original Linial-Nisan Conjecture was recently proved by Braverman; we offered a $200 prize for the generalized version. In this paper, we save ourselves$200 by showing that the GLN Conjecture is false, at least for circuits of depth 3 and higher. As a byproduct, our counterexample also implies that Pi2P is not contained in P^NP relative to a random oracle with probability 1. It has been conjectured since the 1980s that PH is infinite relative to a random oracle, but the highest levels of PH previously proved separate were NP and coNP. Finally, our counterexample implies that the famous results of Linial, Mansour, and Nisan, on the structure of AC0 functions, cannot be improved in several interesting respects.Comment: 17 page

Quantum Computation Relative to Oracles

The study of the power and limitations of quantum computation remains a major challenge in complexity theory. Key questions revolve around the quantum complexity classes EQP, BQP, NQP, and their derivatives. This paper presents new relativized worlds in which (i) co-RP is not a subset of NQE, (ii) P=BQP and UP=EXP, (iii) P=EQP and RP=EXP, and (iv) EQP is not a subset of the union of Sigma{p}{2} and Pi{p}{2}. We also show a partial answer to the question of whether Almost-BQP=BQP

Complexity of optimizing over the integers

In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like "input", "size" and "complexity" in the context of general mathematical optimization, avoiding context dependent definitions which is one of the sources of difference in the treatment of complexity within continuous and discrete optimization. In the second part of the paper, we employ the language developed in the first part to study information theoretic and algorithmic complexity of {\em mixed-integer convex optimization}, which contains as a special case continuous convex optimization on the one hand and pure integer optimization on the other. We strive for the maximum possible generality in our exposition. We hope that this paper contains material that both continuous optimizers and discrete optimizers find new and interesting, even though almost all of the material presented is common knowledge in one or the other community. We see the main merit of this paper as bringing together all of this information under one unifying umbrella with the hope that this will act as yet another catalyst for more interaction across the continuous-discrete divide. In fact, our motivation behind Part I of the paper is to provide a common language for both communities

The Structure of logarithmic advice complexity classes

A nonuniform class called here Full-P/log, due to Ko, is studied. It corresponds to polynomial time with logarithmically long advice. Its importance lies in the structural properties it enjoys, more interesting than those of the alternative class P/log; specifically, its introduction was motivated by the need of a logarithmic advice class closed under polynomial-time deterministic reductions. Several characterizations of Full-P/log are shown, formulated in terms of various sorts of tally sets with very small information content. A study of its inner structure is presented, by considering the most usual reducibilities and looking for the relationships among the corresponding reduction and equivalence classes defined from these special tally sets.Postprint (published version

Resource Bounded Immunity and Simplicity

Revisiting the thirty years-old notions of resource-bounded immunity and simplicity, we investigate the structural characteristics of various immunity notions: strong immunity, almost immunity, and hyperimmunity as well as their corresponding simplicity notions. We also study limited immunity and simplicity, called k-immunity and feasible k-immunity, and their simplicity notions. Finally, we propose the k-immune hypothesis as a working hypothesis that guarantees the existence of simple sets in NP.Comment: This is a complete version of the conference paper that appeared in the Proceedings of the 3rd IFIP International Conference on Theoretical Computer Science, Kluwer Academic Publishers, pp.81-95, Toulouse, France, August 23-26, 200