Strong co-nondeterministic lower bounds for NP cannot be proved feasibly

Pich, Ján; Santhanam, Rahul

doi:10.1145/3406325.3451117

Cited by 9 publications

(32 citation statements)

References 54 publications

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“…Krajíček's result significantly strengthens a similar but much simpler proof of the validity of Razborov's conjecture for proof systems with feasible interpolation [34]. The method has been also used to show a conditional hardness of generating hard tautologies [19], a conditional unprovability of p-size circuit lower bounds for SAT in theories of bounded arithmetic below Cook's theory PV 1 [35] and an unconditional unprovability of strong nondeterministic lower bounds in Jeřábek's theory of approximate counting APC 1 [37]. We take advantage of its unique way of exploiting the NW generator: it gives us a reconstruction algorithm which after breaking the NW-generator in a particular interactive fashion allows us to approximately compute the function on which the generator is based.…”

Section: Our Contributionsupporting

confidence: 59%

Learning algorithms from circuit lower bounds

Pich¹

2020

Preprint

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We revisit known constructions of efficient learning algorithms from various notions of constructive circuit lower bounds such as distinguishers breaking pseudorandom generators or efficient witnessing algorithms which find errors of small circuits attempting to compute hard functions. As our main result we prove that if it is possible to find efficiently, in a particular interactive way, errors of many p-size circuits attempting to solve hard problems, then p-size circuits can be PAC learned over the uniform distribution with membership queries by circuits of subexponential size. The opposite implication holds as well. This provides a new characterisation of learning algorithms and extends the natural proofs barrier of Razborov and Rudich. The proof is based on a method of exploiting Nisan-Wigderson generators introduced by Krajíček (2010) and used to analyze complexity of circuit lower bounds in bounded arithmetic.An interesting consequence of known constructions of learning algorithms from circuit lower bounds is a learning speedup of Oliveira and Santhanam (2016). We present an alternative proof of this phenomenon and discuss its potential to advance the program of hardness magnification.

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Section: Our Contributionsupporting

confidence: 59%

Learning algorithms from circuit lower bounds

Pich¹

2020

Preprint

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“…Finally, can we establish the independence of some natural question in circuit complexity from a theory such as VPV 1 ? In particular, can we narrow the gap between our results and the recent result from [PS21] on the unprovability of strong circuit lower bounds in VPV 1 ?…”

Section: Improve the Consequence Of Theorem 14 (Item 3) By Proving Thatmentioning

confidence: 67%

LEARN-Uniform Circuit Lower Bounds and Provability in Bounded Arithmetic

Carmosino¹,

Kabanets²,

Kolokolova³

et al. 2022

2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)

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“…This however seems to be very difficult. We refer to [33,Introduction] or [23, for a description of the resulting research program, and to [30] for a recent result.…”

Section: Introductionmentioning

confidence: 99%

On the Consistency of Circuit Lower Bounds for Non-deterministic Time

Atserias

Buss²,

Müller

2023

Proceedings of the 55th Annual ACM Symposium on Theory of Computing

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We prove the first unconditional consistency result for superpolynomial circuit lower bounds with a relatively strong theory of bounded arithmetic. Namely, we show that the theory V 0 2 is consistent with the conjecture that NEXP ̸ ⊆ P/poly, i.e., some problem that is solvable in non-deterministic exponential time does not have polynomial size circuits. We suggest this is the best currently available evidence for the truth of the conjecture. Additionally, we establish a magnification result on the hardness of proving circuit lower bounds.

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Strong co-nondeterministic lower bounds for NP cannot be proved feasibly

Cited by 9 publications

References 54 publications

Learning algorithms from circuit lower bounds

Learning algorithms from circuit lower bounds

LEARN-Uniform Circuit Lower Bounds and Provability in Bounded Arithmetic

On the Consistency of Circuit Lower Bounds for Non-deterministic Time

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