43 research outputs found

    Derandomizing Arthur-Merlin Games using Hitting Sets

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    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

    Instance-Wise Hardness Versus Randomness Tradeoffs for Arthur-Merlin Protocols

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    Tighter Connections between Derandomization and Circuit Lower Bounds

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    We tighten the connections between circuit lower bounds and derandomization for each of the following three types of derandomization: - general derandomization of promiseBPP (connected to Boolean circuits), - derandomization of Polynomial Identity Testing (PIT) over fixed finite fields (connected to arithmetic circuit lower bounds over the same field), and - derandomization of PIT over the integers (connected to arithmetic circuit lower bounds over the integers). We show how to make these connections uniform equivalences, although at the expense of using somewhat less common versions of complexity classes and for a less studied notion of inclusion. Our main results are as follows: 1. We give the first proof that a non-trivial (nondeterministic subexponential-time) algorithm for PIT over a fixed finite field yields arithmetic circuit lower bounds. 2. We get a similar result for the case of PIT over the integers, strengthening a result of Jansen and Santhanam [JS12] (by removing the need for advice). 3. We derive a Boolean circuit lower bound for NEXP intersect coNEXP from the assumption of sufficiently strong non-deterministic derandomization of promiseBPP (without advice), as well as from the assumed existence of an NP-computable non-empty property of Boolean functions useful for proving superpolynomial circuit lower bounds (in the sense of natural proofs of [RR97]); this strengthens the related results of [IKW02]. 4. Finally, we turn all of these implications into equivalences for appropriately defined promise classes and for a notion of robust inclusion/separation (inspired by [FS11]) that lies between the classical "almost everywhere" and "infinitely often" notions

    Hitting Sets Give Two-Sided Derandomization of Small Space

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    On Nonadaptive Security Reductions of Hitting Set Generators

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    One of the central open questions in the theory of average-case complexity is to establish the equivalence between the worst-case and average-case complexity of the Polynomial-time Hierarchy (PH). One general approach is to show that there exists a PH-computable hitting set generator whose security is based on some NP-hard problem. We present the limits of such an approach, by showing that there exists no exponential-time-computable hitting set generator whose security can be proved by using a nonadaptive randomized polynomial-time reduction from any problem outside AM ? coAM, which significantly improves the previous upper bound BPP^NP of Gutfreund and Vadhan (RANDOM/APPROX 2008 [Gutfreund and Vadhan, 2008]). In particular, any security proof of a hitting set generator based on some NP-hard problem must use either an adaptive or non-black-box reduction (unless the polynomial-time hierarchy collapses). To the best of our knowledge, this is the first result that shows limits of black-box reductions from an NP-hard problem to some form of a distributional problem in DistPH. Based on our results, we argue that the recent worst-case to average-case reduction of Hirahara (FOCS 2018 [Hirahara, 2018]) is inherently non-black-box, without relying on any unproven assumptions. On the other hand, combining the non-black-box reduction with our simulation technique of black-box reductions, we exhibit the existence of a "non-black-box selector" for GapMCSP, i.e., an efficient algorithm that solves GapMCSP given as advice two circuits one of which is guaranteed to compute GapMCSP

    Non-Disjoint Promise Problems from Meta-Computational View of Pseudorandom Generator Constructions

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    An Algorithmic Approach to Uniform Lower Bounds

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    Errorless Versus Error-Prone Average-Case Complexity

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    We consider the question of whether errorless and error-prone notions of average-case hardness are equivalent, and make several contributions. First, we study this question in the context of hardness for NP, and connect it to the long-standing open question of whether there are instance checkers for NP. We show that there is an efficient non-uniform non-adaptive reduction from errorless to error-prone heuristics for NP if and only if there is an efficient non-uniform average-case non-adaptive instance-checker for NP. We also suggest an approach to proving equivalence of the two notions of average-case hardness for PH. Second, we show unconditionally that error-prone average-case hardness is equivalent to errorless average-case hardness for P against NC¹ and for UP ∩ coUP against P. Third, we apply our results about errorless and error-prone average-case hardness to get new equivalences between hitting set generators and pseudo-random generators

    Certified Hardness vs. Randomness for Log-Space

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    Let L\mathcal{L} be a language that can be decided in linear space and let ϵ>0\epsilon >0 be any constant. Let A\mathcal{A} be the exponential hardness assumption that for every nn, membership in L\mathcal{L} for inputs of length~nn cannot be decided by circuits of size smaller than 2ϵn2^{\epsilon n}. We prove that for every function f:{0,1}{0,1}f :\{0,1\}^* \rightarrow \{0,1\}, computable by a randomized logspace algorithm RR, there exists a deterministic logspace algorithm DD (attempting to compute ff), such that on every input xx of length nn, the algorithm DD outputs one of the following: 1: The correct value f(x)f(x). 2: The string: ``I am unable to compute f(x)f(x) because the hardness assumption A\mathcal{A} is false'', followed by a (provenly correct) circuit of size smaller than 2ϵn2^{\epsilon n'} for membership in L\mathcal{L} for inputs of length~nn', for some n=Θ(logn)n' = \Theta (\log n); that is, a circuit that refutes A\mathcal{A}. Our next result is a universal derandomizer for BPLBPL: We give a deterministic algorithm UU that takes as an input a randomized logspace algorithm RR and an input xx and simulates the computation of RR on xx, deteriministically. Under the widely believed assumption BPL=LBPL=L, the space used by UU is at most CRlognC_R \cdot \log n (where CRC_R is a constant depending on~RR). Moreover, for every constant c1c \geq 1, if BPLSPACE[(log(n))c]BPL\subseteq SPACE[(\log(n))^{c}] then the space used by UU is at most CR(log(n))cC_R \cdot (\log(n))^{c}. 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 BB of size nn, estimates the probability that BB 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

    Pseudodistributions That Beat All Pseudorandom Generators (Extended Abstract)

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