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Certified Hardness vs. Randomness for Log-Space | IEEE Conference Publication | IEEE Xplore

Certified Hardness vs. Randomness for Log-Space


Abstract:

Let \mathcal{L} be a language that can be decided in linear space and let \epsilon \gt 0 be any constant. Let \mathcal{A} be the exponential hardness assumption tha...Show More

Abstract:

Let \mathcal{L} be a language that can be decided in linear space and let \epsilon \gt 0 be any constant. Let \mathcal{A} be the exponential hardness assumption that for every n, membership in \mathcal{L} for inputs of length n cannot be decided by circuits of size smaller than 2^{\epsilon n}. We prove that for every function f:\{0,1\}^{*} \rightarrow\{0,1\}, computable by a randomized logspace algorithm R, there exists a deterministic logspace algorithm D (attempting to compute f), such that on every input x of length n, the algorithm D outputs one of the following:1)The correct value f(x).2)The string: “I am unable to compute f(x) because the hardness assumption \mathcal{A} is false”, followed by a (provenly correct) circuit of size smaller than 2^{\epsilon n^{\prime}} for membership in \mathcal{L} for inputs of length n^{\prime}, for some n^{\prime}=\Theta(\log n); that is, a circuit that refutes \mathcal{A}. Moreover, D is explicitly constructed, given R.We note that previous works on the hardness-versus-randomness paradigm give derandomized algorithms that rely blindly on the hardness assumption. If the hardness assumption is false, the algorithms may output incorrect values, and thus a user cannot trust that an output given by the algorithm is correct. Instead, our algorithm D verifies the computation so that it never outputs an incorrect value. Thus, if D outputs a value for f(x), that value is certified to be correct. Moreover, if D does not output a value for f(x), it alerts that the hardness assumption was found to be false, and refutes the assumption.Our next result is a universal derandomizer for BPL (the class of problems solvable by bounded-error randomized logspace algorithms)1: We give a deterministic algorithm U that takes as an input a randomized logspace algorithm R and an input x and simulates the computation of R on x, deteriministically. Under the widely believed assumption \mathbf{BPL}=\mathbf{L}, the space us...
Date of Conference: 06-09 November 2023
Date Added to IEEE Xplore: 22 December 2023
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Conference Location: Santa Cruz, CA, USA

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