Improving exhaustive search implies superpolynomial lower bounds
Proceedings of the forty-second ACM symposium on Theory of computing
Improved simulation of nondeterministic turing machines
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
Some results on average-case hardness within the polynomial hierarchy
FSTTCS'06 Proceedings of the 26th international conference on Foundations of Software Technology and Theoretical Computer Science
Improved simulation of nondeterministic Turing machines
Theoretical Computer Science
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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.