7 research outputs found
Derandomizing Arthur-Merlin Games using Hitting Sets
We prove that AM (and hence Graph Nonisomorphism) is in NPif for some epsilon > 0, some language in NE intersection coNE requires nondeterministiccircuits of size 2^(epsilon n). This improves recent results of Arvindand K¨obler and of Klivans and Van Melkebeek who proved the sameconclusion, but under stronger hardness assumptions, namely, eitherthe existence of a language in NE intersection coNE which cannot be approximatedby nondeterministic circuits of size less than 2^(epsilon n) or the existenceof a language in NE intersection coNE which requires oracle circuits of size 2^(epsilon n)with oracle gates for SAT (satisfiability).The previous results on derandomizing AM were based on pseudorandomgenerators. In contrast, our approach is based on a strengtheningof Andreev, Clementi and Rolim's hitting set approach to derandomization.As a spin-off, we show that this approach is strong enoughto give an easy (if the existence of explicit dispersers can be assumedknown) proof of the following implication: For some epsilon > 0, if there isa language in E which requires nondeterministic circuits of size 2^(epsilon n),then P=BPP. This differs from Impagliazzo and Wigderson's theorem"only" by replacing deterministic circuits with nondeterministicones
Certified Hardness vs. Randomness for Log-Space
Let be a language that can be decided in linear space and let
be any constant. Let be the exponential hardness
assumption that for every , membership in for inputs of
length~ cannot be decided by circuits of size smaller than .
We prove that for every function , computable
by a randomized logspace algorithm , there exists a deterministic logspace
algorithm (attempting to compute ), such that on every input of
length , the algorithm outputs one of the following:
1: The correct value .
2: The string: ``I am unable to compute because the hardness
assumption is false'', followed by a (provenly correct) circuit
of size smaller than for membership in for
inputs of length~, for some ; that is, a circuit that
refutes .
Our next result is a universal derandomizer for : We give a
deterministic algorithm that takes as an input a randomized logspace
algorithm and an input and simulates the computation of on ,
deteriministically. Under the widely believed assumption , the space
used by is at most (where is a constant depending
on~). Moreover, for every constant , if then the space used by is at most .
Finally, we prove that if optimal hitting sets for ordered branching programs
exist then there is a deterministic logspace algorithm that, given a black-box
access to an ordered branching program of size , estimates the
probability that accepts on a uniformly random input. This extends the
result of (Cheng and Hoza CCC 2020), who proved that an optimal hitting set
implies a white-box two-sided derandomization.Comment: Abstract shortened to fit arXiv requirement
Weak Random Sources, Hitting Sets, and BPP Simulations
We show how to simulate any BPP algorithm in polynomial time using a weak random source of r bits and min-entropy r fl for any fl ? 0. This follows from a more general result about sampling with weak random sources. Our result matches an information-theoretic lower bound and solves a question that has been open for some years. The previous best results were a polynomial time simulation of RP [Saks, Srinivasan and Zhou 1995] and a quasi-polynomial time simulation of BPP [Ta-Shma 1996]. Departing significantly from previous related works, we do not use extractors; instead, we use the OR-disperser of [Saks, Srinivasan, and Zhou 1995] in combination with a tricky use of hitting sets borrowed from [Andreev, Clementi, and Rolim 1996]. AMS Subject Classification: 68Q10, 11K45. Key Words and Phrases: Derandomization, Imperfect Sources of Randomness, Hitting Sets, Randomized Computations, Expander Graphs. Abbreviated Title: BPP Simulations using Weak Random Sources. 1 Introduction Randomi..