Circuit Lower Bounds for Average-Case MA

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9139)

Abstract

Santhanam (2007) proved that \(\mathbf {MA}/1\) does not have circuits of size \(n^k\). We translate his result to the average-case setting by proving that there is a constant a such that for any k, there is a language in \(\mathrm {Avg}_{ }\mathbf {MA}\) that cannot be solved by circuits of size \(n^k\) on more than the \(1 - \frac{1}{n^a}\) fraction of inputs.

In order to get rid of the non-uniform advice, we supply the inputs with the probability threshold that we use to determine the acceptance. This technique was used by Pervyshev (2007) for proving a time hierarchy for heuristic computations.

Notes

Acknowledgments

The author is grateful to Edward A. Hirsch for bringing the problem to his attention, to Dmitry Itsykson and anonymous referees for their comments that significantly improved the (initially unreadable) presentation.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Steklov Institute of Mathematics at St. PetersburgSt. PetersburgRussia

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