NDLI: Holographic Proofs and Derandomization

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	van Melkebeek, Dieter Santhanam, Rahul
Copyright Year	2005
Abstract	We derive a stronger consequence of $\mathsf{EXP}$ (deterministic exponential time) having polynomial-size circuits than was known previously, namely that for each language $L \in \mathsf{P}$ (polynomial time), and for each efficiently decidable error-correcting code E having nontrivial relative distance, there is a simulation of L in Merlin-Arthur polylogarithmic time that fools all deterministic polynomial-time adversaries for inputs that are codewords of E. Using the connection between circuit lower bounds and derandomization, we obtain uniform assumptions for derandomizing $\mathsf{BPP}$ (probabilistic polynomial time). Our results strengthen the space-randomness tradeoffs of Sipser [J. Comput. System Sci., 36 (1988), pp. 379--383], Nisan and Wigderson [J. Comput. System Sci.}, 49 (1994), pp. 149--167], and Lu [Comput. Complexity, 10 (2001), pp. 247--259]. We also consider a more quantitative notion of simulation, where the measure of success of the simulation is the fraction of inputs of a given length on which the simulation works. Among other results, we show that if there is no polynomial-time bound t such that $\mathsf{P}$ can be simulated well by Merlin-Arthur machines operating in time t, then for any $\epsilon > 0$ there is a simulation of $\mathsf{BPP}$ in $\mathsf{P}$ that works for all but $2^{n^{\epsilon}}$ inputs of length n. This is a uniform strengthening of a recent result of Goldreich and Wigderson [ Proceedings of the 6th International Workshop on Randomization and Approximation Techniques in Computer Science, 2002, pp. 209--223].Finally, we give an unconditional simulation of multitape Turing machines operating in probabilistic time t by Turing machines operating in deterministic time o(2t). We show similar results for randomized $\mathsf{NC}^{1}$ circuits.Our proofs are based on a combination of techniques in the theory of derandomization with results on holographic proofs.
Starting Page	59
Ending Page	90
Page Count	32
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539703438629
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	1
Volume Number	35
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-27
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems Modes of computation Arthur-Merlin games holographic proofs Complexity of computation derandomization Complexity classes
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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