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)

Reducing the Number of Solutions of NP Functions

AbstractWe study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines, we nonetheless show that it often is possible to reduce the number of solutions of NP functions. For finite cardinality types, we give a sufficient condition for such solution reduction. We also give absolute and conditional necessary conditions for solution reduction, and in particular we show that in many cases solution reduction is impossible unless the polynomial hierarchy collapses

A study of one-turn quantum refereed games

This thesis studies one-turn quantum refereed games, which are abstract zero-sum games with two competing computationally unbounded quantum provers and a computationally bounded quantum referee. The provers send quantum states to the referee, who plugs the two states into his quantum circuit, measures the output of the circuit in the standard basis, and declares one of the two players as the winner depending on the outcome of the measurement. The complexity class QRG(1) comprises of those promise problems for which there exists a one-turn quantum refereed game such that one of the players wins with high probability for the yes-instances, and the other player wins with high probability for the no-instances, irrespective of the opponent’s strategy. QRG(1) is a generalization of QMA (or co-QMA), and can informally be viewed as QMA with a no-prover (or co-QMA with a yes-prover). We have given a full characterization of QRG(1), starting with appropriate definitions and known results, and building on to two new results about this class. Previously, the best known upper bound on QRG(1) was PSPACE. We have proved that if one of the provers is completely classical, sending a classical probability distribution instead of a quantum state, the new class, which we name CQRG(1), is contained in Ǝ · PP (non- deterministic polynomial-time operator applied to the class PP). We have also defined another restricted version of QRG(1) where both provers send quantum states, but the referee measures one of the quantum states first, and plugs the classical outcome into the measurement, along with the other prover’s quantum state, into a quantum circuit, before measuring the output of the quantum circuit in the standard basis. The new class, which we name MQRG(1), is contained in P · PP (the probabilistic polynomial time operator applied to PP). Ǝ · PP is contained in P · PP, which is, in turn, contained in PSPACE. Hence, our results give better containments than PSPACE for restricted versions of QRG(1)

Interactive proof system variants and approximation algorithms for optical networks

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1996.Includes bibliographical references (p. 111-121).by Ravi Sundaram.Ph.D