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Circuit Lower Bounds for MCSP from Local Pseudorandom Generators

Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, Dimitrios Myrisiotis
2020 ACM Transactions on Computation Theory  
We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a PRG is called local if its output bit strings, when viewed as the truth table of a Boolean function  ...  We get new and improved lower bounds for MCSP that almost match the best-known lower bounds against several circuit models.  ...  Then, by applying our "MCSP circuit lower bounds from local PRGs" framework, we get the desired lower bounds. MCSP lower bounds against AC 0 .  ... 
doi:10.1145/3404860 fatcat:eeqmy7heabbsxcosw7yb7kjq6u

One-Tape Turing Machine and Branching Program Lower Bounds for MCSP [article]

Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, Yuichi Yoshida
2020 Electronic colloquium on computational complexity  
These results are the first non-trivial lower bounds for MCSP and MKTP against one-tape Turing machines and non-deterministic branching programs, and essentially match the best-known lower bounds for any  ...  A recent line of work exhibited "hardness magnification" phenomena for MCSP: A very weak lower bound for MCSP implies a breakthrough result in complexity theory.  ...  Our techniques Local HSGs for MCSP lower bounds For a circuit class C, a general approach for obtaining a C-lower bound for MCSP is by constructing a "local" hitting set generator (or a pseudorandom  ... 
dblp:journals/eccc/CheraghchiHMY20 fatcat:xq7z7ngnzzezzhicyiuq2aon6i

Circuit minimization problem

Valentine Kabanets, Jin-Yi Cai
2000 Proceedings of the thirty-second annual ACM symposium on Theory of computing - STOC '00  
We also argue that proving this problem to be NP-complete (if it is indeed true) would imply proving strong circuit lower bounds for the class E, which appears beyond the currently known techniques.  ...  We study the complexity of the circuit minimization problem: given the truth table of a Boolean function f and a parameter s, decide whether f can be realized by a Boolean circuit of size at most s.  ...  , and Charles Racko for his helpful remarks.  ... 
doi:10.1145/335305.335314 dblp:conf/stoc/KabanetsC00 fatcat:njj2yvmrwjdu5i7ojbfqrfbfdm

Beyond Natural Proofs: Hardness Magnification and Locality

Lijie Chen, Shuichi Hirahara, Igor C. Oliveira, Ján Pich, Ninad Rajgopal, Rahul Santhanam
2022 Journal of the ACM  
– Can we adapt known lower bound techniques to establish the desired lower bound for Q ?  ...  circuit size problem \({\sf MCSP} \) imply the non-existence of natural proofs.  ...  Oliveira received support from the Royal Society University Research Fellowship URF\R1\191059. 18  ... 
doi:10.1145/3538391 fatcat:66hzv4sx7bgkxjer4od2pxpcai

Beyond Natural Proofs: Hardness Magnification and Locality [article]

Lijie Chen, Shuichi Hirahara, Igor C. Oliveira, Jan Pich, Ninad Rajgopal, Rahul Santhanam
2019 arXiv   pre-print
Hardness magnification reduces major complexity separations (such as EXP⊈NC^1) to proving lower bounds for some natural problem Q against weak circuit models.  ...  Can we adapt known lower bound techniques to establish the desired lower bound for Q?  ...  Acknowledgements Part of this work was completed while some of the authors were visiting the Simons Institute for the Theory of Computing. We are grateful to the Simons Institute for their support.  ... 
arXiv:1911.08297v1 fatcat:zfitgvmtzngl7elbcqyfvr5m6i

Beyond Natural Proofs: Hardness Magnification and Locality

Lijie Chen, Shuichi Hirahara, Igor C. Oliveira, Ján Pich, Ninad Rajgopal, Rahul Santhanam, Michael Wagner
2020 Innovations in Theoretical Computer Science  
Can we adapt known lower bound techniques to establish the desired lower bound for Q?  ...  circuit size problem MCSP imply the non-existence of natural proofs.  ...  Since each restriction from ρ can be computed by a poly(n)-size circuit, x * has a circuit of poly(n) · t = poly(n) ≤ n c size (here we set c). Let S be the set of input variables that F depends on.  ... 
doi:10.4230/lipics.itcs.2020.70 dblp:conf/innovations/ChenHOPRS20 fatcat:pwksfk26pfdtpkxaoamhb3qmgu

Non-Disjoint Promise Problems from Meta-Computational View of Pseudorandom Generator Constructions

Shuichi Hirahara, Shubhangi Saraf
2020 Computational Complexity Conference  
Applying the principle to complexity-theoretic pseudorandom generators, we introduce a family of Meta-computational Circuit Lower-bound Problems (MCLPs), which are problems of distinguishing the truth  ...  Our results generalize the hardness versus randomness framework and identify problems whose circuit lower bounds characterize the existence of hitting set generators.  ...  Acknowledgements I thank Rahul Santhanam for helpful discussion and anonymous reviewers for helpful comments.  ... 
doi:10.4230/lipics.ccc.2020.20 dblp:conf/coco/Hirahara20 fatcat:wc4333kzavfqjmrssm6obwgkgu

The exact complexity of pseudorandom functions and the black-box natural proof barrier for bootstrapping results in computational complexity

Zhiyuan Fan, Jiatu Li, Tianqi Yang
2022 Symposium on the Theory of Computing  
We resolve the exact complexity of PRFs by proving tight upper and lower bounds for various circuit models. • PRFs can be constructed in 2𝑛 + 𝑜 (𝑛) size general circuits assuming the existence of polynomial-size  ...  We also present an 𝑛 1+Ω (𝑐 −𝑑 ) wire complexity lower bound against depth-𝑑 TC 0 circuits for some 𝑐 > 1.61.  ...  Oliveira for organizing a virtual seminar and valuable comments. We are thankful to Tianyi Zhang for addressing a typo in an earlier draft and Yiding Zhang for his help in improving some writing.  ... 
doi:10.1145/3519935.3520010 dblp:conf/stoc/FanL022 fatcat:axkbbap6tbf3bd7456k7lgsahq

Quantum Meets the Minimum Circuit Size Problem [article]

Nai-Hui Chia, Chi-Ning Chou, Jiayu Zhang, Ruizhe Zhang
2021 arXiv   pre-print
Finally, we systematically generalize results known for classical MCSP to the quantum setting (including quantum cryptography, quantum learning theory, quantum circuit lower bounds, and quantum fine-grained  ...  In this work, we initiate the study of the Minimum Circuit Size Problem (MCSP) in the quantum setting. MCSP is a problem to compute the circuit complexity of Boolean functions.  ...  Circuit lower bounds The classical MCSP is tightly connected to circuit lower bounds.  ... 
arXiv:2108.03171v3 fatcat:bgmake6gnrdt3jvrywagxgn4rq

Computational Complexity of Discrete Problems (Dagstuhl Seminar 17121)

Anna Gál, Michal Koucký, Oded Regev, Till Tantau, Marc Herbstritt
2017 Dagstuhl Reports  
Many circuit lower bounds we currently have fit into the Natural Proof framework, e.g., lower bounds for AC 0 [p] circuits.  ...  Along with the known lower bounds for real monotone circuits for the Clique function, it yields an exponential lower bound for very simple false QBFs based on Clique.  ...  Tight lower bounds are known for most ranges of m. In the case that m = n O(1) , Dietz, Seiferas and Zhang [5] proved an Ω(n log(n)) lower bound.  ... 
doi:10.4230/dagrep.7.3.45 dblp:journals/dagstuhl-reports/GalK0T17 fatcat:og5bioyq4zaszfljjecwfrpjgq

On the (Non) NP-Hardness of Computing Circuit Complexity

Cody D. Murray, R. Ryan Williams, Marc Herbstritt
2015 Computational Complexity Conference  
We prove that the NP-hardness of MCSP under (logtime-uniform) AC0 reductions would imply extremely strong lower bounds: NP ⊂ P /poly and E ⊂ i.o.  ...  Unlike many problems of its kind, MCSP is not known to be NP-hard, yet an efficient algorithm for this problem also seems very unlikely: for example, MCSP ∈ P would imply there are no pseudorandom functions  ...  We thank Greg Bodwin and Brynmor Chapman for their thoughtful discussions on these results. We also thank Eric Allender and Oded Goldreich for helpful comments.  ... 
doi:10.4230/lipics.ccc.2015.365 dblp:conf/coco/MurrayW15 fatcat:6fqqwrfaybfeho2yg4dccg75ay

Conspiracies between Learning Algorithms, Circuit Lower Bounds and Pseudorandomness [article]

Igor C. Oliveira, Rahul Santhanam
2016 arXiv   pre-print
We prove several results giving new and stronger connections between learning, circuit lower bounds and pseudorandomness.  ...  , consequences of non-trivial learning for circuit lower bounds, Karp-Lipton theorems for probabilistic exponential time, and NC^1-hardness for the Minimum Circuit Size Problem.  ...  Lower Bounds from Nontrivial Algorithms Theorem 12 (Circuit lower bounds from nontrivial learning algorithms). Let C be any typical circuit class.  ... 
arXiv:1611.01190v1 fatcat:bx2ikuueefafdjtqq4lv2sgpmy

Learning algorithms from circuit lower bounds [article]

Ján Pich
2020 arXiv   pre-print
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  ...  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.  ...  This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodovska-Curie grant agreement No 890220.  ... 
arXiv:2012.14095v1 fatcat:6rluc3znffcm7lvjwdrwcrvuuu

Algebraic Methods in Computational Complexity (Dagstuhl Seminar 18391)

Markus Bläser, Valentine Kabanets, Jacobo Torán, Christopher Umans, Michael Wagner
2019 Dagstuhl Reports  
There have been significant recent advances in algebraic circuit lower bounds, and the so-called chasm at depth 4 suggests that the restricted models now being considered are not so far from ones that  ...  The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some  ...  It is still open if similar circuit lower bounds hold for MCSP.  ... 
doi:10.4230/dagrep.8.9.133 dblp:journals/dagstuhl-reports/BlaserKTU18 fatcat:bqddvcedazgqngwix6ptk7syuq

Circuit Lower Bounds from NP-Hardness of MCSP Under Turing Reductions

Michael Saks, Rahul Santhanam, Shubhangi Saraf
2020 Computational Complexity Conference  
We show that NP-hardness of MCSP under these kinds of Turing-reductions imply superpolynomial circuit lower bounds for exponential time.  ...  Kabanets and Cai [Kabanets and Cai, 2000] showed that if MCSP is NP-hard under "natural" m-reductions, superpolynomial circuit lower bounds for exponential time would follow.  ...  Moreover, [16] do not derive a circuit lower bound from this assumption, but the weaker statement that EXP = ZPP. 26:6 Circuit Lower Bounds from NP-Hardness of MCSP Under Turing Reductions 2 Preliminaries  ... 
doi:10.4230/lipics.ccc.2020.26 dblp:conf/coco/SaksS20 fatcat:r4iw3psklzexdifagu353l43be
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