Bootstrapping results for threshold circuits “just beyond” known lower bounds

Chen, Lijie; Tell, Roei

doi:10.1145/3313276.3316333

Cited by 26 publications

(27 citation statements)

References 30 publications

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“…Comparison With Prior Work. Several recent papers [19][20][21][53][54][55]] also establish hardness magnification results for problems such as MCSP and MKtP. Most recently, the reference [19] proves Theorem 1.3 for deterministic formula lower bounds of size n 3+ε ; we generalize their results for probabilistic formula lower bounds of size n 2+ε .…”

Section: Theorem 15 (Sub-quadratic Mktp and Mcsp Lower Bounds)mentioning

confidence: 82%

“…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%

“…However, in the case of general circuits, researchers have been stuck for decades on proving even a 6n-size lower bound for De Morgan circuits, 4 so an n 1+ε -size lower bound on MCSP seems far, relatively speaking. For another example, Allender and Koucky [9] (later refined by Chen and Tell [21]) show that if certain NC 1complete problems do not have n 1+ε size TC 0 -size circuits, then NC 1 does not have polynomial-size TC 0 circuits. However, there is still a gap between these results and the best known lower bounds for TC 0 [22,36].…”

Section: Theorem 15 (Sub-quadratic Mktp and Mcsp Lower Bounds)mentioning

confidence: 99%

“…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: Theorem 15 (Sub-quadratic Mktp and Mcsp Lower Bounds)mentioning

confidence: 82%

Section: Sharp Threshold For Qd Of Generalized Probabilistic Formulasmentioning

confidence: 91%

Section: Theorem 15 (Sub-quadratic Mktp and Mcsp Lower Bounds)mentioning

confidence: 99%

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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“…Hardness Magnification Against TC 0 Circuits. To get our results for TC 0 circuits (the last item of Theorem 1.1 and Theorem 1.6), we make use of the recent construction of error correcting codes computable by uniform and extremely sparse TC 0 circuits [CT19].…”

Section: Intuitionmentioning

confidence: 99%

Hardness Magnification for all Sparse NP Languages

Chen

Jin

Williams

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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In the Minimum Circuit Size Problem (MCSP[s(m)]), we ask if there is a circuit of size s(m) computing a given truth-table of length n = 2 m . Recently, a surprising phenomenon termed as hardness magnification by [Oliveira and Santhanam, FOCS 2018] was discovered for MCSP[s(m)] and the related problem MKtP of computing time-bounded Kolmogorov complexity. In [Oliveira and Santhanam, FOCS 2018], [Oliveira, Pich, and Santhanam, CCC 2019], and Williams, STOC 2019], it was shown that minor (n 1+ε -style) lower bounds for MCSP [2 o(m) ] or MKtP [2 o(m) ] would imply breakthrough circuit lower bounds such as NP ⊂ P /poly , NP ⊂ NC 1 , or EXP ⊂ P /poly .We consider the question: What is so special about MCSP and MKtP? Why do they admit this striking phenomenon? One simple property is that all variants of MCSP (and MKtP) considered in prior work are sparse languages. For example,-sparse. We show that there is a hardness magnification phenomenon for all equally-sparse NP languages. Formally, suppose there is an ε > 0 and a language L ∈ NP which is-sparse, and L / ∈ Circuit[n 1+ε ]. Then NP does not have n k -size circuits for all k. We prove analogous theorems for De Morgan formulas, B 2 -formulas, branching programs, AC 0 [6] and TC 0 circuits, and more: improving the state of the art in NP lower bounds against any of these models by an ε factor in the exponent would already imply NP lower bounds for all fixed polynomials. In fact, in our proofs it is not necessary to prove a (say) n 1+ε circuit size lower bound for L: one only has to prove a lower bound against n 1+ε -time n ε -space deterministic algorithms with n ε advice bits. Such lower bounds are well-known for non-sparse problems.Building on our techniques, we also show interesting new hardness magnifications for search-MCSP and search-MKtP (where one must output small circuits or short representations of strings), showing consequences such as ⊕P (or PP, PSPACE, and EXP) is not contained in P /poly (or NC 1 , AC 0 [6], or branching programs of polynomial size). For instance, if there is an ε > 0 such that search-MCSP[2 βm ] does not have De Morgan formulas of size n 3+ε for all constants β > 0, then ⊕P ⊂ NC 1 .

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On the Minimum Depth of Circuits with Linear Number of Wires Encoding Good Codes

Drucker,

2023

Lecture Notes in Computer Science

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Bootstrapping results for threshold circuits “just beyond” known lower bounds

Cited by 26 publications

References 30 publications

Sharp threshold results for computational complexity

Sharp threshold results for computational complexity

Hardness Magnification for all Sparse NP Languages

On the Minimum Depth of Circuits with Linear Number of Wires Encoding Good Codes

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