Feasibly constructive proofs of succinct weak circuit lower bounds

Müller, Moritz; Pich, Ján

doi:10.1016/j.apal.2019.102735

Cited by 19 publications

(35 citation statements)

References 55 publications

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“…This result can be seen as an analogue in circuit complexity of the hardness magnification phenomenon observed in proof complexity by Müller and Pich [17,Proposition 4.14].…”

Section: B Results and Techniquesmentioning

confidence: 55%

“…On the other hand, a form of hardness magnification has been recently observed by Müller and Pich [17,Proposition 4.14] in the context of proof complexity. In more detail, their argument establishes that barely non-trivial lower bounds for bounded-depth Frege systems for certain tautologies of interest imply super-polynomial lower bounds for Frege systems.…”

Section: Introduction a Overviewmentioning

confidence: 94%

“…Related work on magnification. In addition to the papers [15] and [17] mentioned above, we are aware of three other works showing magnification theorems in our sense.…”

Section: Further Remarks and Related Workmentioning

confidence: 99%

“…3 The interesting aspect of this result compared to [15] is that lower bounds against bounded-depth Frege systems are known (but not for the same tautologies). Moreover, the tautologies from [17] have been studied in the past, and lower bounds in weaker proof systems have been obtained for the same class of tautologies. While our results are incomparable to their work in proof complexity, it served as a source of inspiration for our investigations.…”

Section: Introduction a Overviewmentioning

confidence: 99%

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Hardness Magnification for Natural Problems

Oliveira

Santhanam

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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“…This result can be seen as an analogue in circuit complexity of the hardness magnification phenomenon observed in proof complexity by Müller and Pich [17,Proposition 4.14].…”

Section: B Results and Techniquesmentioning

confidence: 55%

Section: Introduction a Overviewmentioning

confidence: 94%

“…Related work on magnification. In addition to the papers [15] and [17] mentioned above, we are aware of three other works showing magnification theorems in our sense.…”

Section: Further Remarks and Related Workmentioning

confidence: 99%

Section: Introduction a Overviewmentioning

confidence: 99%

See 2 more Smart Citations

Hardness Magnification for Natural Problems

Oliveira

Santhanam

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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“…Additional Related Works on Hardness Magnification. Several other similar hardness magnification phenomena have been established: for n 1−ε approximation to CLIQUE [58], sublinear-depth circuit lower bounds for P [44], proof complexity [47], and lower bounds for non-commutative arithmetic circuits [15].…”

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

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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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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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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Feasibly constructive proofs of succinct weak circuit lower bounds

Cited by 19 publications

References 55 publications

Hardness Magnification for Natural Problems

Hardness Magnification for Natural Problems

Sharp threshold results for computational complexity

Hardness Magnification for all Sparse NP Languages

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