Using nondeterminism to amplify hardness

Healy, Alexander; Vadhan, Salil; Viola, Emanuele

doi:10.1145/1007352.1007389

Cited by 33 publications

(49 citation statements)

References 26 publications

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“…Such a generator would imply that every randomized algorithm can be derandomized with only a constantfactor increase in space (RL = L), and would also have a variety of other applications, such as in streaming algorithms [26], deterministic dimension reduction and SDP rounding [43,16], hashing [13], hardness amplification [23], almost k-wise independent permutations [27], and cryptographic pseudorandom generator constructions [21].…”

Section: Pseudorandom Generators For Space-bounded Computationmentioning

confidence: 99%

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Steinke¹,

Vadhan²,

Wan³

2017

Theory of Comput.

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Section: Pseudorandom Generators For Space-bounded Computationmentioning

confidence: 99%

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Steinke¹,

Vadhan²,

Wan³

2017

Theory of Comput.

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“…Nisan and Wigderson [302] observed that derandomization only requires a mildly explicit pseudorandom generator, and showed how to construct such generators based on the average-case hardness of E (Theorem 7.24). A variant of Open Problem 7.25 was posed in [202], who showed that it also would imply stronger results on hardness amplification; some partial negative results can be found in [214,320].…”

Section: Chapter Notes and Referencesmentioning

confidence: 99%

“…Contrast this bound with the XOR Lemma, where C is the (non-monotone) parity function and ensures that f is (1/2 − 1/2 Ω(k) )-hard. Healy, Vadhan, and Viola [202] showed how to derandomize O'Donnell's construction, so that the inputs x 1 , . . .…”

Section: Hardness Amplification In Npmentioning

confidence: 99%

Chapter 8 Notes and References

2011

Feynman-Kac-Type Theorems and Gibbs Measures on Path Space

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“…Specifically, we adopt the following notations and definitions from [HVV06]. The bias of a 0-1 random variable X is defined to be…”

Section: Bias and Noise Stability Following The Analysis In [O'd04hmentioning

confidence: 99%

Optimal Cryptographic Hardness of Learning Monotone Functions

Dachman-Soled

Lee

Malkin

et al. 2008

Automata, Languages and Programming

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Abstract. A wide range of positive and negative results have been established for learning different classes of Boolean functions from uniformly distributed random examples. However, polynomial-time algorithms have thus far been obtained almost exclusively for various classes of monotone functions, while the computational hardness results obtained to date have all been for various classes of general (nonmonotone) functions. Motivated by this disparity between known positive results (for monotone functions) and negative results (for nonmonotone functions), we establish strong computational limitations on the efficient learnability of various classes of monotone functions. We give several such hardness results which are provably almost optimal since they nearly match known positive results. Some of our results show cryptographic hardness of learning polynomial-size monotone circuits to accuracy only slightly greater than 1/2 + 1/ √ n; this accuracy bound is close to optimal by known positive results (Blum et al., FOCS '98). Other results show that under a plausible cryptographic hardness assumption, a class of constant-depth, sub-polynomialsize circuits computing monotone functions is hard to learn; this result is close to optimal in terms of the circuit size parameter by known positive results as well (Servedio, Information and Computation '04). Our main tool is a complexitytheoretic approach to hardness amplification via noise sensitivity of monotone functions that was pioneered by O'Donnell (JCSS '04).

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Using nondeterminism to amplify hardness

Cited by 33 publications

References 26 publications

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Chapter 8 Notes and References

Optimal Cryptographic Hardness of Learning Monotone Functions

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