Input-Oblivious Proof Systems and a Uniform Complexity Perspective on P/poly

Goldreich, Oded; Meir, Or

doi:10.1145/2799645

Cited by 3 publications

(1 citation statement)

References 24 publications

(32 reference statements)

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“…The problems of proving that classes such as P, NP, and P NP have fixed polynomial-size circuits are discussed in [Lip94,FSW09,GM15,Din15], and many absurd-looking consequences have been derived from propositions such as NP ⊂ SIZE(n k ) (of course, none of these have been proved to be actually contradictory).…”

Section: Np Does Not Have Fixed Polynomial-size Circuitsmentioning

confidence: 99%

Some Estimated Likelihoods for Computational Complexity

Williams¹

2019

Lecture Notes in Computer Science

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Section: Np Does Not Have Fixed Polynomial-size Circuitsmentioning

confidence: 99%

Some Estimated Likelihoods for Computational Complexity

Williams¹

2019

Lecture Notes in Computer Science

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On Exponential-Time Hypotheses, Derandomization, and Circuit Lower Bounds: Extended Abstract

Chen

Rothblum

Tell

et al. 2020

2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)

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On Exponential-time Hypotheses, Derandomization, and Circuit Lower Bounds

Chen

Rothblum

Tell

et al. 2023

J. ACM

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The Exponential-Time Hypothesis ( \mathtt {ETH} ) is a strengthening of the \mathcal {P} \ne \mathcal {NP} conjecture, stating that 3\text{-}\mathtt {SAT} on n variables cannot be solved in (uniform) time 2^{\epsilon \cdot n} , for some \epsilon \gt 0 . In recent years, analogous hypotheses that are “exponentially-strong” forms of other classical complexity conjectures (such as \mathcal {NP}\nsubseteq \mathcal {BPP} or co\mathcal {NP}\nsubseteq \mathcal {NP} ) have also been introduced, and have become widely influential. In this work, we focus on the interaction of exponential-time hypotheses with the fundamental and closely-related questions of derandomization and circuit lower bounds . We show that even relatively-mild variants of exponential-time hypotheses have far-reaching implications to derandomization, circuit lower bounds, and the connections between the two. Specifically, we prove that: (1) The Randomized Exponential-Time Hypothesis ( \mathsf {rETH} ) implies that \mathcal {BPP} can be simulated on “average-case” in deterministic (nearly-)polynomial-time (i.e., in time 2^{\tilde{O}(\log (n))}=n^{\mathrm{loglog}(n)^{O(1)}} ). The derandomization relies on a conditional construction of a pseudorandom generator with near-exponential stretch (i.e., with seed length \tilde{O}(\log (n)) ); this significantly improves the state-of-the-art in uniform “hardness-to-randomness” results, which previously only yielded pseudorandom generators with sub-exponential stretch from such hypotheses. (2) The Non-Deterministic Exponential-Time Hypothesis ( \mathsf {NETH} ) implies that derandomization of \mathcal {BPP} is completely equivalent to circuit lower bounds against \mathcal {E} , and in particular that pseudorandom generators are necessary for derandomization. In fact, we show that the foregoing equivalence follows from a very weak version of \mathsf {NETH} , and we also show that this very weak version is necessary to prove a slightly stronger conclusion that we deduce from it. Lastly, we show that disproving certain exponential-time hypotheses requires proving breakthrough circuit lower bounds. In particular, if \mathtt {CircuitSAT} for circuits over n bits of size \mathrm{poly}(n) can be solved by probabilistic algorithms in time 2^{n/\mathrm{polylog}(n)} , then \mathcal {BPE} does not have circuits of quasilinear size.

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Input-Oblivious Proof Systems and a Uniform Complexity Perspective on P/poly

Cited by 3 publications

References 24 publications

Some Estimated Likelihoods for Computational Complexity

Some Estimated Likelihoods for Computational Complexity

On Exponential-Time Hypotheses, Derandomization, and Circuit Lower Bounds: Extended Abstract

On Exponential-time Hypotheses, Derandomization, and Circuit Lower Bounds

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