Improved Bounds for Quantified Derandomization of Constant-Depth Circuits and Polynomials

Tell, Roei

doi:10.1007/s00037-019-00179-2

Cited by 7 publications

(4 citation statements)

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“…Comparison With Prior Work. Several papers [21,28,61,62] prove obtaining quantified derandomization "thresholds" for circuit classes such as AC 0 , TC 0 and AC 0 [⊕]. Most related to ours is [21,61], showing if there is a QD algorithm for…”

Section: Sharp Threshold For Qd Of Generalized Probabilistic Formulasmentioning

confidence: 91%

“…For a Boolean circuit class C and a function B : N → N, the QD problem for C with B exceptional inputs is the following: Given an n-input circuit C ∈ C that evaluates to 0 on at most B(n) inputs, deterministically output an n-bit string on which C evaluates to 1. 5 Note the standard derandomization problem has B(n) = 2 n /3, but studying the case of B(n) = 2 n α for α < 1 turns out to also be a very interesting and subtle problem [21,28,61,62], with implications for general derandomization.…”

Section: Sharp Threshold For Quantified Derandomizationmentioning

confidence: 99%

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Sharp threshold results for computational complexity

Chen

Jin

Williams

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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We establish several "sharp threshold" results for computational complexity. For certain tasks, we can prove a resource lower bound of n c for c ≥ 1 (or obtain an efficient circuit-analysis algorithm for n c size), there is strong intuition that a similar result can be proved for larger functions of n, yet we can also prove that replacing "n c " with "n c+ε " in our results, for any ε > 0, would imply a breakthrough n ω(1) lower bound. We first establish such a result for Hardness Magnification. We prove (among other results) that for some c, the Minimum Circuit Size Problem for (log n) c-size circuits on length-n truth tables (MCSP[(log n) c ]) does not have n 2−o(1)-size probabilistic formulas. We also prove that an n 2+ε lower bound for MCSP[(log n) c ] (for any ε > 0 and c ≥ 1) would imply major lower bound results, such as NP does not have n k-size formulas for all k, and #SAT does not have log-depth circuits. Similar results hold for time-bounded Kolmogorov complexity. Note that cubic size lower bounds are known for probabilistic De Morgan formulas (for other functions). Next we show a sharp threshold for Quantified Derandomization (QD) of probabilistic formulas: (a) For all α, ε > 0, there is a deterministic polynomial-time algorithm that finds satisfying assignments to every probabilistic formula of n 2−2α −ε size with at most 2 n α falsifying assignments. (b) If for some α, ε > 0, there is such an algorithm for probabilistic formulas of n 2−α +ε-size and 2 n α unsatisfying assignments, then a full derandomization of NC 1 follows: a deterministic poly-time algorithm additively approximating the acceptance probability of any polynomial-size formula. Consequently, NP does not have n k-size formulas, for all k. Finally we show a sharp threshold result for Explicit Obstructions, inspired by Mulmuley's notion of explicit obstructions from GCT. An explicit obstruction against S(n)-size formulas is a poly-time algorithm A such that A(1 n) outputs a list {(x i , f (x i))} i ∈[poly(n)] ⊆ {0, 1} n ×{0, 1}, and every S(n)-size formula F is inconsistent with the (partially defined) function f. We prove that for all ε > 0, there is an explicit obstruction against n 2−ε-size formulas, and prove that there is an explicit obstruction against n 2+ε-size formulas for some ε > 0 if and only if there is an explicit obstruction against all * L.C. and R.W. are supported by NSF CCF-1741615 and a Google Faculty Research Award. Portions of this work were completed while L.C. and R.W. were visiting the Simons Institute at UC Berkeley.

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Section: Sharp Threshold For Qd Of Generalized Probabilistic Formulasmentioning

confidence: 91%

Section: Sharp Threshold For Quantified Derandomizationmentioning

confidence: 99%

Sharp threshold results for computational complexity

Chen

Jin

Williams

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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“…Prior to this work, quantified derandomization algorithms have been constructed for AC 0 , for subclasses of AC 0 [⊕], for polynomials over F 2 that vanish rarely, and for a subclass of MA. On the other hand, reductions of standard derandomization to quantified derandomization are known for AC 0 , for AC 0 [⊕], for polynomials over large finite fields, and for the class AM (both the algorithms and the reductions appear in [GW14,Tel17]). In some cases, most notably for AC 0 , the parameters of the known quantified derandomization algorithms are very close to the parameters of quantified derandomization to which standard derandomization can be reduced (see [Tel17,Thms 1 & 2]).…”

Section: Quantified Derandomizationmentioning

confidence: 99%

“…To solve this problem, we use the following general technique that was introduced in our previous work [Tel17], which is called randomized tests. Loosely speaking, a lemma from our previous work implies the following: Assume that there exists a distribution T over tests {−1, 1} [n]\I → {−1, 1} such that for every fixed input z for which Φ↾ ρ I,z is n −100close to σ it holds that T(z) = −1, with high probability, and for every fixed input z for which Φ↾ ρ I,z is not n −10 -close to σ it holds that T(z) = 1, with high probability.…”

Section: Preserving the Closeness Of The Circuit To Its Approximationsmentioning

confidence: 99%

Quantified derandomization of linear threshold circuits

Tell

2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

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One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for T C 0 , the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for T C 0 . In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of T C 0 circuits of depth d > 2.Our first main result is a quantified derandomization algorithm for T C 0 circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a T C 0 circuit C over n input bits with depth d and n 1+exp(−d) wires, runs in almostpolynomial-time, and distinguishes between the case that C rejects at most 2 n 1−1/5d inputs and the case that C accepts at most 2 n 1−1/5d inputs. In fact, our algorithm works even when the circuit C is a linear threshold circuit, rather than just a T C 0 circuit (i.e., C is a circuit with linear threshold gates, which are stronger than majority gates).Our second main result is that even a modest improvement of our quantified derandomization algorithm would yield a non-trivial algorithm for standard derandomization of all of T C 0 , and would consequently imply that N E X P ⊆ T C 0 . Specifically, if there exists a quantified derandomization algorithm that gets as input a T C 0 circuit with depth d and n 1+O(1/d) wires (rather than n 1+exp(−d) wires), runs in time at most 2 n exp(−d) , and distinguishes between the case that C rejects at most 2 n 1−1/5d inputs and the case that C accepts at most 2 n 1−1/5d inputs, then there exists an algorithm with running time 2 n 1−Ω(1) for standard derandomization of T C 0 . 7 Quantified derandomization of depth-2 linear threshold circuits 35 8 Restrictions for sparse T C 0 circuits: A potential path towards N E X P ⊆ T C 0 38 Acknowledgements 38 Appendix A Quantified derandomization and lower bounds 43 Appendix B Proof of a technical claim from Section 6 44 i 1. The circuit C mapsn input bits ton =n · (1/ǫ) O(1/ρ) output bits.

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