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Luby--Veličković--Wigderson revisited: Improved correlation bounds and pseudorandom generators for depth-two circuits [article]

Rocco A. Servedio, Li-Yang Tan
2018 arXiv   pre-print
We study correlation bounds and pseudorandom generators for depth-two circuits that consist of a SYM-gate (computing an arbitrary symmetric function) or THR-gate (computing an arbitrary linear threshold  ...  The above PRG is actually a special case of a more general PRG which we establish for constant-depth circuits containing multiple SYM or THR gates, including as a special case {SYM,THR}∘AC^0 circuits.  ...  In 1993 Luby, Veličković, and Wigderson [LVW93] gave the first pseudorandom generators for these depth-2 circuits.  ... 
arXiv:1803.04553v1 fatcat:ynglyw7775fs5ho4mlu5rbrnwy

Luby-Velickovic-Wigderson Revisited: Improved Correlation Bounds and Pseudorandom Generators for Depth-Two Circuits

Rocco A. Servedio, Li-Yang Tan, Michael Wagner
2018 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We study correlation bounds and pseudorandom generators for depth-two circuits that consist of a SYM-gate (computing an arbitrary symmetric function) or THR-gate (computing an arbitrary linear threshold  ...  The above PRG is actually a special case of a more general PRG which we establish for constant-depth circuits containing multiple SYM or THR gates, including as a special case {SYM, THR} • AC 0 circuits  ...  In this work we focus on pseudorandom generators for these {SYM, THR} • AND circuits. In 1993 Luby, Veličković, and Wigderson [31] gave the first pseudorandom generators for these depth-2 circuits.  ... 
doi:10.4230/lipics.approx-random.2018.56 dblp:conf/approx/ServedioT18 fatcat:knjpbzpf3zg2zpel4wrheeaqe4

Satisfiability and Derandomization for Small Polynomial Threshold Circuits

Valentine Kabanets, Zhenjian Lu, Michael Wagner
2018 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We show how the classical Nisan-Wigderson (NW) generator (JCSS, 1994) yields a nontrivial pseudorandom generator for PTF circuits (of unrestricted depth) with sub-linearly many gates.  ...  A polynomial threshold function (PTF) is defined as the sign of a polynomial p : {0, 1} n → R. A PTF circuit is a Boolean circuit whose gates are PTFs.  ...  Acknowledgements We thank Suguru Tamaki for suggesting to us to consider sparse PTFs in the case of satisfiability, and for clarifying the MAX-k-SAT algorithm in [20] for us.  ... 
doi:10.4230/lipics.approx-random.2018.46 dblp:conf/approx/KabanetsL18 fatcat:ch4gewg3ybelzjjwutioqml6yq

Derandomizing Arthur-Merlin Games using Hitting Sets

Peter Bro Miltersen, Vinodchandran N. Variyam
1999 BRICS Report Series  
of size less than 2^(epsilon n) or the existence<br />of a language in NE intersection coNE which requires oracle circuits of size 2^(epsilon n)<br />with oracle gates for SAT (satisfiability).  ...  <br />The previous results on derandomizing AM were based on pseudorandom<br />generators.  ...  The authors would like to thank Dieter van Melkebeek and Luca Trevisan for very helpful discussions.  ... 
doi:10.7146/brics.v6i47.20117 fatcat:2ecf3j6i55cqnfvfernj4dxguy

Derandomizing Arthur–Merlin Games using Hitting Sets

Peter Bro Miltersen, N. V. Vinodchandran
2005 Computational Complexity  
The previous results on derandomizing AM were based on pseudorandom generators.  ...  This differs from Impagliazzo and Wigderson's theorem "only" by replacing deterministic circuits with nondeterministic ones.  ...  The authors would like to thank Dieter van Melkebeek and Luca Trevisan for very helpful discussions.  ... 
doi:10.1007/s00037-005-0197-7 fatcat:3lhitagdmvgnjczgwfpolu532e

McCulloch-Pitts Brains and Pseudorandom Functions

Vašek Chvátal, Mark Goldsmith, Nan Yang
2016 Neural Computation  
We show that they cannot build weak pseudorandom functions.  ...  We are grateful to Péter Gács for helpful comments on a draft of this note and to Avi Wigderson for telling us about  ...  Acknowledgments This research was undertaken in part with funding from the Canada Research Chairs program and from the Natural Sciences and Engineering Research Council of Canada.  ... 
doi:10.1162/neco_a_00841 pmid:27136970 fatcat:uluu3btnrbcmnnrrg4mqpa7foa

Depth Reduction for Circuits of Unbounded Fan-in

E. Allender, U. Hertrampf
1994 Information and Computation  
We prove that constant depth circuits of size n log O(1) n over the basis AND, OR, PARITY are no more powerful than circuits of this size with depth four.  ...  Similar techniques are used to obtain several other depth reduction theorems; in particular, we show every set in AC 0 can be recognized by a family of depththree threshold circuits of size n log O(1)  ...  Acknowledgments The rst author acknowledges discussions with Ravi Boppana, Seinosuke Toda, David Barrington, Denis Th erien and Roman Smolensky; thanks are also due to 25 Pierre McKenzie and Denis Th erien  ... 
doi:10.1006/inco.1994.1057 fatcat:ofxkaojqq5dzberra5iaegzt4q

Circuit complexity before the dawn of the new millennium [chapter]

Eric Allender
1996 Lecture Notes in Computer Science  
The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity.  ...  This paper is a necessarily incomplete survey of the state of circuit complexity a s w e a wait the dawn of the new millennium.  ...  This enabled Nisan and Wigderson NW94 t o construct, for any k, a pseudorandom generator that is a computable in AC 0 , and b takes log O1 n bits of input and produces n bits of output, and c is secure  ... 
doi:10.1007/3-540-62034-6_33 fatcat:lqnxje6dkne3nnuqp2dsn37s7y

Algorithms for Circuits and Circuits for Algorithms

Ryan Williams
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
Circuits for algorithms refers to the modeling of uniform algorithms with non-uniform circuit families (or proving such modeling is impossible).  ...  For instance, the NEXP versus P/poly question asks whether nondeterministic exponential-time algorithms can be simulated using non-uniform circuit families of polynomial size.  ...  Babai, Fortnow, Nisan, and Wigderson [BFNW93] showed that EXP ⊂ P/poly implies the existence of pseudorandom generators that can deterministically simulate any randomized polynomial-time algorithm in  ... 
doi:10.1109/ccc.2014.33 dblp:conf/coco/Williams14 fatcat:t2irb2av2bcb5hzfi2dxkrsvqu

On derandomization and average-case complexity of monotone functions

George Karakostas, Jeff Kinne, Dieter van Melkebeek
2012 Theoretical Computer Science  
against general circuits, and that an average-case hard function for monotone circuits is also hard with somewhat weaker parameters for general circuits.  ...  We prove similar results in the settings of pseudorandom generators and average-case hard functions -that a pseudorandom generator secure against monotone circuits is also secure with somewhat weaker parameters  ...  Proof of Theorem 5 We follow the standard proof from the general setting and keep track of monotonicity to verify the final circuit is monotone or anti-monotone.  ... 
doi:10.1016/j.tcs.2012.02.017 fatcat:6mk5ovihtzaxvhrexnd2vt5tci

General Pseudo-random Generators from Weaker Models of Computation [chapter]

George Karakostas
2009 Lecture Notes in Computer Science  
More specifically, we show how PRGs that fool monotone circuits could lead to derandomization for general complexity classes, and how the Nisan-Wigderson construction could use hardness results for monotone  ...  The construction of pseudo-random generators (PRGs) has been based on strong assumptions like the existence of one-way functions or exponential lower bounds for the circuit complexity of Boolean functions  ...  Galesi for helpful discussions, and V. Kabanets for pointing out [8] , [6] .  ... 
doi:10.1007/978-3-642-10631-6_110 fatcat:nslqybzdjjhchlee4dgix2tlra

A Quest for Structure in Complexity

Vikraman Arvind, Meena Mahajan
2017 Bulletin of the European Association for Theoretical Computer Science  
The Nisan-Wigderson method of constructing a pseudorandom generator (prg) from f works as follows. The generator G f : {0, 1} m → {0, 1} n takes an m-bit seed and stretches it n bits.  ...  The proofs turn out to be quite simple, building on the following beautiful insight in [8] about the design of pseudorandom generators from hard computational problems in the seminal work of Nisan-Wigderson  ... 
dblp:journals/eatcs/ArvindM17 fatcat:axk2knwgpbh6nfdz3bespdpqom

My favorite ten complexity theorems of the past decade [chapter]

Lance Fortnow
1994 Lecture Notes in Computer Science  
Lokam,Dieter Van Melkebeek and Sophie Laplante for their comments and help on this paper.  ...  Razborov Raz87] and Smolensky Smo87] show that for primes p and q, p 6 = q, the MOD q function requires an exponential number of gates for boundeddepth circuits with AND, OR and MOD p gates.  ...  Razborov Raz90] uses more general matrix methods to prove similar lower bounds for other problems. Raz and Wigderson RW92] show that monotone circuits for matching require linear depth.  ... 
doi:10.1007/3-540-58715-2_130 fatcat:oqj4nco6ozaxpclwtipoor2egi

Page 3933 of Mathematical Reviews Vol. , Issue 93g [page]

1993 Mathematical Reviews  
Schmidt, Circuit size relative to pseudorandom oracles (95-120); Yishay Mansour, Noam Nisan and Prasoon Tiwari, The computational complexity of universal hashing (121-133); Noam Nisan, On read-once vs.  ...  With semi-unbounded fan-in circuits, the OR gates may have unbounded fan-in, but the AND gates must have bounded fan-in.  ... 

Page 1642 of Mathematical Reviews Vol. , Issue 94c [page]

1994 Mathematical Reviews  
(RS-AOS; Moscow); Wigderson, Avi (IL-HEBR; Jerusalem) n%(o8”) Jower bounds on the size of depth-3 threshold circuits with AND gates at the bottom. (English summary) Inform. Process.  ...  Joseph Ja’Ja’ (1-MD-E; College Park, MD) 94c:68073 68Q15 65C10 65Y20 Nisan, Noam (IL-HEBR-C; Jerusalem) Pseudorandom generators for space-bounded computation.  ... 
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