Finding Irrefutable Certificates for S_2^p via Arthur and Merlin

We show that $S_2^psubseteq P^{prAM}$, where $S_2^p$ is the symmetric alternation class and $prAM$ refers to the promise version of the Arthur-Merlin class $AM$. This is derived as a consequence of our main result that presents an $FP^{prAM}$ algorithm for finding a small set of ``collectively irrefutable certificates\u27\u27 of a given $S_2$-type matrix. The main result also yields some new consequences of the hypothesis that $NP$ has polynomial size circuits. It is known that the above hypothesis implies a collapse of the polynomial time hierarchy ($PH$) to $S_2^psubseteq ZPP^{NP}$ (Cai 2007, K"obler and Watanabe 1998). Under the same hypothesis, we show that $PH$ collapses to $P^{prMA}$. We also describe an $FP^{prMA}$ algorithm for learning polynomial size circuits for $SAT$, assuming such circuits exist. For the same problem, the previously best known result was a $ZPP^{NP}$ algorithm (Bshouty et al. 1996)

Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework

Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds, that is, lower bounds on the size of approximations of an explicit function. In this paper, we provide a general framework for proving worst/average case lower bounds for circuits and upper bounds for #SAT that is built on ideas of Chen and Kabanets. A proof in such a framework goes as follows. One starts by fixing three parameters: a class of circuits, a circuit complexity measure, and a set of allowed substitutions. The main ingredient of a proof goes as follows: by going through a number of cases, one shows that for any circuit from the given class, one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount. This case analysis immediately implies an upper bound for #SAT. To~obtain worst/average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser/extractor for the class of sources defined by the set of substitutions under consideration. We show that many known proofs (of circuit size lower bounds and upper bounds for #SAT) fall into this framework. Using this framework, we prove the following new bounds: average case lower bounds of 3.24n and 2.59n for circuits over U_2 and B_2, respectively (though the lower bound for the basis B_2 is given for a quadratic disperser whose explicit construction is not currently known), and faster than 2^n #SAT-algorithms for circuits over U_2 and B_2 of size at most 3.24n and 2.99n, respectively. Here by B_2 we mean the set of all bivariate Boolean functions, and by U_2 the set of all bivariate Boolean functions except for parity and its complement

Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems

A frontier open problem in circuit complexity is to prove P^{NP} is not in SIZE[n^k] for all k; this is a necessary intermediate step towards NP is not in P_{/poly}. Previously, for several classes containing P^{NP}, including NP^{NP}, ZPP^{NP}, and S_2 P, such lower bounds have been proved via Karp-Lipton-style Theorems: to prove C is not in SIZE[n^k] for all k, we show that C subset P_{/poly} implies a "collapse" D = C for some larger class D, where we already know D is not in SIZE[n^k] for all k. It seems obvious that one could take a different approach to prove circuit lower bounds for P^{NP} that does not require proving any Karp-Lipton-style theorems along the way. We show this intuition is wrong: (weak) Karp-Lipton-style theorems for P^{NP} are equivalent to fixed-polynomial size circuit lower bounds for P^{NP}. That is, P^{NP} is not in SIZE[n^k] for all k if and only if (NP subset P_{/poly} implies PH subset i.o.- P^{NP}_{/n}). Next, we present new consequences of the assumption NP subset P_{/poly}, towards proving similar results for NP circuit lower bounds. We show that under the assumption, fixed-polynomial circuit lower bounds for NP, nondeterministic polynomial-time derandomizations, and various fixed-polynomial time simulations of NP are all equivalent. Applying this equivalence, we show that circuit lower bounds for NP imply better Karp-Lipton collapses. That is, if NP is not in SIZE[n^k] for all k, then for all C in {Parity-P, PP, PSPACE, EXP}, C subset P_{/poly} implies C subset i.o.-NP_{/n^epsilon} for all epsilon > 0. Note that unconditionally, the collapses are only to MA and not NP. We also explore consequences of circuit lower bounds for a sparse language in NP. Among other results, we show if a polynomially-sparse NP language does not have n^{1+epsilon}-size circuits, then MA subset i.o.-NP_{/O(log n)}, MA subset i.o.-P^{NP[O(log n)]}, and NEXP is not in SIZE[2^{o(m)}]. Finally, we observe connections between these results and the "hardness magnification" phenomena described in recent works

A Full Characterization of Quantum Advice

We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly is contained in PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice. Proving our main result requires combining a large number of previous tools -- including a result of Alon et al. on learning of real-valued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new ones. The main new tool is a so-called majority-certificates lemma, which is closely related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm in S, such that each fi is the unique function in S compatible with O(log|S|) input/output constraints.Comment: We fixed two significant issues: 1. The definition of YQP machines needed to be changed to preserve our results. The revised definition is more natural and has the same intuitive interpretation. 2. We needed properties of Local Hamiltonian reductions going beyond those proved in previous works (whose results we'd misstated). We now prove the needed properties. See p. 6 for more on both point

Oblivious symmetric alternation

1 Introduction The symmetric alternation class (Sp2) was introduced independently by Russelland Sundaram [16] and by Canetti [5]. The class S p 2 contains languages havingan interactive proof system of the following type. The proof system consists o