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Amplifying Circuit Lower Bounds against Polynomial Time with Applications

Richard J. Lipton, Ryan Williams
2012 2012 IEEE 27th Conference on Computational Complexity  
This implies unconditional polynomial-time uniform circuit lower bounds for solving QBF. We also prove that QBF does not have n c -time uniform NC circuits, for all c < 2.  ...  All of them have the form: "an n 1+ε lower bound against constant-depth class C implies arbitrary polynomial lower bounds against the same constant-depth class C." * A preliminary version of this paper  ...  The amplifying reduction of Theorem 1.1 has applications to proving unconditional lower bounds.  ... 
doi:10.1109/ccc.2012.44 dblp:conf/coco/LiptonW12 fatcat:6wgwlinj7ber5fsqnw3b2l2vdu

Amplifying circuit lower bounds against polynomial time, with applications

Richard J. Lipton, Ryan Williams
2013 Computational Complexity  
This implies unconditional polynomial-time uniform circuit lower bounds for solving QBF. We also prove that QBF does not have n c -time uniform NC circuits, for all c < 2.  ...  All of them have the form: "an n 1+ε lower bound against constant-depth class C implies arbitrary polynomial lower bounds against the same constant-depth class C." * A preliminary version of this paper  ...  The amplifying reduction of Theorem 1.1 has applications to proving unconditional lower bounds.  ... 
doi:10.1007/s00037-013-0069-5 fatcat:ngha5frjwfecfla2kyq26pwtia

An Average-Case Lower Bound Against $$\mathsf {ACC}^0$$ ACC 0 [chapter]

Ruiwen Chen, Igor C. Oliveira, Rahul Santhanam
2018 Lecture Notes in Computer Science  
We also show that learning algorithms for quasi-polynomial size ACC 0 circuits running in time 2 n /n ω (1) imply lower bounds for the randomised exponential time classes RE (randomized time 2 O(n) with  ...  one-sided error) and ZPE/1 (zero-error randomized time 2 O(n) with 1 bit of advice) against polynomial size ACC 0 circuits.  ...  Williams' result gives us a worst-case lower bound against polynomial-size ACC 0 circuits. Can we use hardness amplification to derive an average-case lower bound from this?  ... 
doi:10.1007/978-3-319-77404-6_24 fatcat:wdjooay4yjhe3attx4mgssb2km

Efficient Learning Algorithms Yield Circuit Lower Bounds [chapter]

Lance Fortnow, Adam R. Klivans
2006 Lecture Notes in Computer Science  
bound against C .  ...  Our approach uses the framework of the breakthrough result due to Kabanets and Impagliazzo showing that derandomizing BPP yields non-trivial circuit lower bounds.  ...  [28] ), yet we are unaware of applications from learning theory to circuit lower bounds.  ... 
doi:10.1007/11776420_27 fatcat:2doc46s7mba7bk4dzhjxaznsti

Efficient learning algorithms yield circuit lower bounds

Lance Fortnow, Adam R. Klivans
2009 Journal of computer and system sciences (Print)  
bound against C .  ...  Our approach uses the framework of the breakthrough result due to Kabanets and Impagliazzo showing that derandomizing BPP yields non-trivial circuit lower bounds.  ...  [28] ), yet we are unaware of applications from learning theory to circuit lower bounds.  ... 
doi:10.1016/j.jcss.2008.07.006 fatcat:tb56jbg74zazfcb2vfviik4wgy

Typically-correct derandomization

Ronen Shaltiel
2010 ACM SIGACT News  
A fundamental question in complexity theory is whether every randomized polynomial time algorithm can be simulated by a deterministic polynomial time algorithm (that is, whether BPP=P).  ...  use this opportunity to also provide a brief overview to some results and research directions in "classical derandomization". * This survey article appeared as SIGACT News complexity theory column 66 with  ...  As a consequence, using hardness versus randomness tradeoffs with known lower bounds against AC 0 , it is possible to obtain polynomial time typically-correct simulation of randomized AC 0 .  ... 
doi:10.1145/1814370.1814389 fatcat:unfdkttcq5arrixtomcacrgq2u

On Medium-Uniformity and Circuit Lower Bounds

Rahul Santhanam, Ryan Williams
2013 2013 IEEE Conference on Computational Complexity  
Lower Bounds Against Medium-Uniform Circuits.  ...  That is, for all k there is a language L k ∈ P that does not have O(n k )-size circuits constructible in polynomial time.  ...  Lower bounds by amplifying uniformity: In the first part of the paper, we prove new lower bounds against "medium-uniform" circuits.  ... 
doi:10.1109/ccc.2013.40 dblp:conf/coco/SanthanamW13 fatcat:mm4ls445rjd37mkuqv5sg2lqiy

On Uniformity and Circuit Lower Bounds

Rahul Santhanam, Ryan Williams
2014 Computational Complexity  
On uniformity and circuit lower bounds 179 Lower bounds by amplifying uniformity. In the first part of the paper, we prove new lower bounds against "medium-uniform" circuits.  ...  Lower bounds against medium-uniform circuits.  ...  R.S. was supported in part by the 2013 ERC Consolidator Grant ALUnif: "Algorithms and Lower Bounds: A Unified Approach."  ... 
doi:10.1007/s00037-014-0087-y fatcat:4fy5xs7jrfbtzm37apd7wq2qza

The Large-Error Approximate Degree of AC^0

Mark Bun, Justin Thaler, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We prove two new results about the inability of low-degree polynomials to uniformly approximate constant-depth circuits, even to slightly-better-than-trivial error.  ...  Additional applications in learning theory, communication complexity, and cryptography are described.  ...  (Threshold-of-Majority circuits represent the most powerful class of threshold circuits against which we can prove superpolynomial lower bounds.)  ... 
doi:10.4230/lipics.approx-random.2019.55 dblp:conf/approx/BunT19 fatcat:6anhs6xduraivebmr7pevk5qtm

Agnostic Learning from Tolerant Natural Proofs

Marco L. Carmosino, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, Marc Herbstritt
2017 International Workshop on Approximation Algorithms for Combinatorial Optimization  
For the ideal case, a natural proof of strongly exponential correlation circuit lower bounds against a circuit class C containing AC 0 [2] (i.e., circuits of size exp(Ω(n)) cannot compute some n-variate  ...  Our algorithm runs in randomized quasi-polynomial time, uses membership queries, and outputs a circuit for a given boolean function f : {0, 1} n → {0, 1} that agrees with f on all but at most (poly log  ...  proof of a circuit lower bound, then we can learn f using membership queries in time dependent on the strength of the circuit lower bound.  ... 
doi:10.4230/lipics.approx-random.2017.35 dblp:conf/approx/CarmosinoIKK17 fatcat:vewhdm7sjzhsjmn2ljblst2ezi

Limits on Representing Boolean Functions by Linear Combinations of Simple Functions: Thresholds, ReLUs, and Low-Degree Polynomials

Richard Ryan Williams, Marc Herbstritt
2018 Computational Complexity Conference  
circuit lower bounds).  ...  We also obtain "fixed-polynomial" lower bounds for functions in NP, for the first three representation classes.  ...  Intuitively, an extremely efficient counting algorithm should imply lower bounds for functions in polynomial time against linear combinations of quadratic F 2 -polynomials, perhaps even lower bounds against  ... 
doi:10.4230/lipics.ccc.2018.6 dblp:conf/coco/Williams18 fatcat:h7e7ouo3erddlptuntewi37ojm

The Circuit-Input Game, Natural Proofs, and Testing Circuits With Data

Brynmor Chapman, Ryan Williams
2015 Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science - ITCS '15  
We give two new applications of these classical results to circuit complexity: Natural properties useful against self-checking circuits are equivalent to circuit lower bounds.  ...  Our message is that one should search for lower bound methods that are designed to succeed (only) against circuits with "one-sided error." Circuit Complexity versus Testing Circuits With Data.  ...  Natural Properties Equivalent to Circuit Lower Bounds In this section, we prove that natural properties useful against self-checking circuits are equivalent to circuit lower bounds in some important settings  ... 
doi:10.1145/2688073.2688115 dblp:conf/innovations/ChapmanW15 fatcat:dxgjg4xysnebvjhabo7uss6caa

Limits on representing Boolean functions by linear combinations of simple functions: thresholds, ReLUs, and low-degree polynomials [article]

R. Ryan Williams
2018 arXiv   pre-print
threshold circuit lower bounds. ∙ Depth-two neural networks with ReLU activation function. ∙ R-linear combinations of O(1)-degree F_p-polynomials, for every prime p (related to problems regarding Higher-Order  ...  For example, we show there are functions in nondeterministic quasi-polynomial time that require super-polynomial size: ∙ Depth-two neural networks with sign activation function, a special case of depth-two  ...  I am also grateful to Brynmor Chapman for his proofreading, and patience with my explanations regarding this paper.  ... 
arXiv:1802.09121v1 fatcat:bafti3acane2xlhulsftxilt6y

NEXP Does Not Have Non-uniform Quasipolynomial-Size ACC Circuits of o(loglogn) Depth [chapter]

Fengming Wang
2011 Lecture Notes in Computer Science  
We show that there exists a language in non-deterministic exponential time which can not be computed by any non-uniform family of ACC m circuits of quasipolynomial size and o(log log n) depth, where m  ...  ACC m circuits are circuits consisting of unbounded fan-in AND, OR and MOD m gates and unary NOT gates, where m is a fixed integer.  ...  The main technical difficulty which prevents us from obtaining a depth lower bound of order Ω( log n log log n ) is that each application of modulus-amplifying polynomials creates extra AND gates of large  ... 
doi:10.1007/978-3-642-20877-5_17 fatcat:lxkowswvgvc2bbzj6dgnf232qu

Computational Complexity of Discrete Problems (Dagstuhl Seminar 17121)

Anna Gál, Michal Koucký, Oded Regev, Till Tantau, Marc Herbstritt
2017 Dagstuhl Reports  
However, proving such lower bounds seems exceedingly hard. Avishay Tal presented a new method of amplifying formula size lower bounds from non-approximability lower bounds.  ...  As an application, the class AC 0 [p], for any prime p, is (agnostically) learnable in quasi-polynomial time.  ...  The same authors with Babka and Čunát [1] proved a Ω(n log(n)/(log log(m) − log log(n))) lower bound, that holds for all n ≤ m ≤ 2 n .  ... 
doi:10.4230/dagrep.7.3.45 dblp:journals/dagstuhl-reports/GalK0T17 fatcat:og5bioyq4zaszfljjecwfrpjgq
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