## Abstract

We study computational procedures that use both randomness and nondeterminism. The goal of this paper is to derandomize such procedures under the weakest possible assumptions. Our main technical contribution allows one to "boost" a given hardness assumption: We show that if there is a problem in EXP that cannot be computed by poly-size nondeterministic circuits then there is one which cannot be computed by poly-size circuits that make non-adaptive NP oracle queries. This in particular shows that the various assumptions used over the last few years by several authors to derandomize Arthur-Merlin games (i.e., show AM = NP) are in fact all equivalent. We also define two new primitives that we regard as the natural pseudorandom objects associated with approximate counting and sampling of NP-witnesses. We use the "boosting" theorem and hashing techniques to construct these primitives using an assumption that is no stronger than that used to derandomize AM. We observe that Cai's proof that rm {2} {NP} and the learning algorithm of Bshouty et al. can be seen as reductions to sampling that are not probabilistic. As a consequence they can be derandomized under an assumption which is weaker than the assumption that was previously known to suffice.

Original language | English |
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Pages (from-to) | 298-341 |

Number of pages | 44 |

Journal | Computational Complexity |

Volume | 15 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2006 |

### Bibliographical note

Funding Information:Some of R. Shaltiel’s work was done while at the Weizmann Institute and supported by the Koshland Scholarship. This research was also supported by BSF grant 2004329. C. Umans’ research was supported by NSF grant CCF-0346991, BSF grant 2004329, and an Alfred P. Sloan Research Fellowship. We thank Lance Fortnow, Oded Goldreich, Russell Impagliazzo, Rahul Santhanam, Amnon Ta-Shma and Salil Vadhan for helpful comments. We also thank the anonymous referees for numerous helpful comments and suggestions.

## Keywords

- Approximate counting
- Arthur Merlin games
- Derandomization
- Nondeterministic circuits
- Pseudorandomness

## ASJC Scopus subject areas

- Theoretical Computer Science
- Mathematics (all)
- Computational Theory and Mathematics
- Computational Mathematics