Feasibly constructive proofs of succinct weak circuit lower bounds

Müller, Moritz; Pich, Ján

doi:10.1016/j.apal.2019.102735

Cited by 22 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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show abstract

“…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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Localizability of the approximation method

Pich

2024

comput. complex.

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We use the approximation method of Razborov to analyze the locality barrier which arose from the investigation of the hardness magnification approach to complexity lower bounds. Adapting a limitation of the approximation method obtained by Razborov, we show that in many cases it is not possible to combine the approximation method with typical (localizable) hardness magnification theorems to derive strong circuit lower bounds. In particular, one cannot use the approximation method to derive an extremely strong constant-depth circuit lower bound and then magnify it to an$$\textsf{NC}^{1}$$NC1lower bound for an explicit function. To prove this, we show that lower bounds obtained by the approximation method are in many cases localizable in the sense that they imply lower bounds for circuits which are allowed to use arbitrarily powerful oracles with small fan-in.

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Feasibly constructive proofs of succinct weak circuit lower bounds

Cited by 22 publications

References 55 publications

Hardness Magnification for Natural Problems

Hardness Magnification for Natural Problems

Sharp threshold results for computational complexity

Localizability of the approximation method

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