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computational complexity

, Volume 12, Issue 3–4, pp 85–130 | Cite as

Uniform hardness versus randomness tradeoffs for Arthur-Merlin games

  • Dan Gutfreund
  • Ronen Shaltiel
  • Amnon Ta-Shma
Original Article

Abstract

Impagliazzo and Wigderson proved a uniform hardness vs. randomness “gap theorem” for BPP. We show an analogous result for AM: Either Arthur-Merlin protocols are very strong and everything in \( \textrm{E = DTIME}(2^{O(n)}) \) can be proved to a subexponential time verifier, or else Arthur-Merlin protocols are weak and every language in AM has a polynomial time nondeterministic algorithm such that it is infeasible to come up with inputs on which the algorithm fails. We also show that if Arthur-Merlin protocols are not very strong (in the sense explained above) then \( \textrm{AM} \cap \textrm{coAM} = \textrm{NP} \cap \textrm{coNP} \)

Our technique combines the nonuniform hardness versus randomness tradeoff of Miltersen and Vinodchandran with “instance checking”. A key ingredient in our proof is identifying a novel “resilience” property of hardness vs. randomness tradeoffs.

Keywords.

Derandomization Arthur-Merlin games 

Mathematics Subject Classification (2000).

68Q15 

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Copyright information

© Birkhäuser-Verlag 2003

Authors and Affiliations

  1. 1.School of Computer Science and EngineeringThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Department of Applied Mathematics and Computer ScienceWeizmann Institute of ScienceRehovotIsrael
  3. 3.Computer Science DepartmentTel-Aviv UniversityTel-AvivIsrael

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