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The Complexity of Satisfiability of Small Depth Circuits
[chapter]

2009
*
Lecture Notes in Computer Science
*

We show an improved randomized algorithm for

doi:10.1007/978-3-642-11269-0_6
fatcat:wip2jerrsndgtjmbz2nzv26r7e
*the**satisfiability*problem for*circuits**of*constant*depth*d and a linear number*of*gates cn: for each d and c,*the*running time is 2 (1−δ)n where*the*improvement ...*The*algorithm can be adjusted for use with Grover's algorithm to achieve a run time*of*2 1−δ 2 n on a quantum computer. ... Thus, we are primarily left with*the*question*of**the**complexity**of**the**satisfiability*problem for linear-size*circuits*if SETH holds. ...##
###
Connecting SAT Algorithms and Complexity Lower Bounds
[chapter]

2011
*
Lecture Notes in Computer Science
*

I will describe prior and current work on connecting

doi:10.1007/978-3-642-21581-0_1
fatcat:3bpt4mnhbbg5fjmpjtda5ymtca
*the*art*of*finding good*satisfiability*algorithms with*the*art*of*proving*complexity*lower bounds: proofs*of*limitations on what problems can be solved ... Surprisingly, even minor algorithmic progress on solving*the**circuit**satisfiability*problem faster than exhaustive search can be applied to prove strong*circuit**complexity*lower bounds. ... It is anticipated that further progress in*complexity*lower bounds will be made by studying*the**complexity**of**satisfiability*. ...##
###
A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits
[article]

2013
*
arXiv
*
pre-print

Our second motivation is to explore

arXiv:1212.4548v2
fatcat:hqi6f7nz4bhvhcr3rcnx4i5wi4
*the*connection between*the*expressive power*of**the**circuits*and*the**complexity**of**the*corresponding*circuit**satisfiability*problem. ... We believe that this is*the*first nontrivial*satisfiability*algorithm for cn-wire threshold*circuits**of**depth*two. ... threshold*circuits**of**depth*2. ...##
###
A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits

2013
*
2013 IEEE 54th Annual Symposium on Foundations of Computer Science
*

Our second motivation is to explore

doi:10.1109/focs.2013.58
dblp:conf/focs/ImpagliazzoPS13
fatcat:6x3as7ngizcydac4mpgx7hybve
*the*connection between*the*expressive power*of**the**circuits*and*the**complexity**of**the*corresponding*circuit**satisfiability*problem. ... We give a nontrivial algorithm for*the**satisfiability*problem for threshold*circuits**of**depth*two with a linear number*of*wires which improves over exhaustive search by an exponential factor. ... Any opinions, findings and conclusions or recommendations expressed in this material are those*of**the*authors and do not necessarily reflect*the*views*of**the*National Science Foundation ...##
###
Deterministic simulation of probabilistic constant depth circuits

1985
*
26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
*

Moreover,

doi:10.1109/sfcs.1985.19
dblp:conf/focs/AjtaiW85
fatcat:5qb2lcvhabhznmf7msgi262rga
*the*functions $f_{n,\varepsilon}$ themselves can be computed by uniform polynomial size, constant*depth**circuits*. Some (interrelated) consequences*of*this result are given below. ... property: for a random seed,*the*pseudorandom output "looks random" to any polynomial size, constant*depth*, unbounded fan-in*circuit*. ... We also present two results for*the*special case*of**depth*2*circuits*. They deal, respectively, with finding an assignment and approximately counting*the*number*of*assignments. ...##
###
On Medium-Uniformity and Circuit Lower Bounds

2013
*
2013 IEEE Conference on Computational Complexity
*

We explore relationships between

doi:10.1109/ccc.2013.40
dblp:conf/coco/SanthanamW13
fatcat:mm4ls445rjd37mkuqv5sg2lqiy
*circuit**complexity*,*the**complexity**of*generating*circuits*, and algorithms for analyzing*circuits*. Our results can be divided into two parts: 1. ...*of*n O(1) size, output yes when C is unsatisfiable, and output no when C has at least 2 n−2*satisfying*assignments. ...*The**circuits*simulate C n on x implicitly, running*the**small*-*depth**circuit*for L succ to retrieve any bit*of*C n that is required. ...##
###
Satisfiability and Derandomization for Small Polynomial Threshold Circuits

2018
*
International Workshop on Approximation Algorithms for Combinatorial Optimization
*

We study

doi:10.4230/lipics.approx-random.2018.46
dblp:conf/approx/KabanetsL18
fatcat:ch4gewg3ybelzjjwutioqml6yq
*the*problems*of*exact and (promise) approximate counting for PTF*circuits**of*constant*depth*.*Satisfiability*(#SAT ). ... This extends*the*recent result*of*Tell (STOC, 2018) for constant-*depth*LTF*circuits**of*super-linear wire*complexity*. Pseudorandom generators. ... 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. ...##
###
Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits

2016
*
Computational Complexity Conference
*

Siu, Roychowdhury, and Kailath [45] considered majority

doi:10.4230/lipics.ccc.2016.1
dblp:conf/coco/ChenSS16
fatcat:ju424xsqsrhhnalgoo3uq4x7ti
*circuits**of*bounded*depth*and*small*gate*complexity*. They showed that Parity can be computed by*depth*-d*circuits*with O(dn 1/(d 1) ) gates. ... We give*satisfiability*algorithms beating brute force search for*depth*-d threshold*circuits*with a superlinear number*of*wires. These are*the*first such algorithms for*depth*greater than 2. ... Fact 35 (*Small*wire-*complexity*to*small*number*of*reads). Let C be any threshold*circuit*on n variables with wire*complexity*at most cn. ...##
###
On Uniformity and Circuit Lower Bounds

2014
*
Computational Complexity
*

We explore relationships between

doi:10.1007/s00037-014-0087-y
fatcat:4fy5xs7jrfbtzm37apd7wq2qza
*circuit**complexity*,*the**complexity**of*generating*circuits*, and algorithms for analyzing*circuits*. Our results can be divided into two parts: 1. ... Our key insight is that if we assume that a "medium*complexity*" class has*small*medium-uniform*circuits*, this assumption can be applied in multiple ways: not only is it applicable to a language L in*the*... Any opinions, findings and conclusions or recommendations expressed in this material are those*of**the*authors and do not necessarily reflect*the*views*of**the*NSF. ...##
###
Algorithms for Circuits and Circuits for Algorithms

2014
*
2014 IEEE 29th Conference on Computational Complexity (CCC)
*

*The*title

*of*this paper is meant to highlight an emerging duality between two fundamental topics in algorithms and

*complexity*theory. ... Algorithms for

*circuits*refers to

*the*design

*of*interesting algorithms which can perform non-trivial

*circuit*analysis

*of*some kind, on either a

*circuit*or a Boolean function given as a truth table. ... ACKNOWLEDGMENT

*The*author acknowledges support by

*the*Alfred P. Sloan foundation, Microsoft Research, and

*the*NSF under grant CCF-1212372. ...

##
###
A Satisfiability Algorithm for Some Class of Dense Depth Two Threshold Circuits

2015
*
IEICE transactions on information and systems
*

One

doi:10.1587/transinf.2014edp7127
fatcat:ocjj6xlifbasvee6rtghqrgene
*of*our motivations is to consider*the*relationship between*the*various*circuit*classes and*the**complexity**of**the*corresponding*circuit**satisfiability*problem*of*these classes. ... Recently, Impagliazzo et al. constructed a nontrivial algorithm for*the**satisfiability*problem for sparse threshold*circuits**of**depth*two which is a class*of**circuits*with cn wires. ...*depth*two*circuits*with*small*number*of*gates. ...##
###
On the Arithmetic Complexity of Euler Function
[chapter]

2011
*
Lecture Notes in Computer Science
*

*The*theorem can be strengthened to show that permanent polynomial requires size s(n) where s is any function

*satisfying*s(s(n)) = 2 o(n) . ...

*The*theorem can be strengthened to show that permanent polynomial requires size s(n) where s is any function

*satisfying*s(s(n)) = 2 o(n) . ... Dedekind Eta Function η(z) = e πiz 12 E (e 2πiz ). η(z) is defined on

*the*upper half

*of*

*the*

*complex*plane and

*satisfies*many interesting properties: η(z + 1) = e πi 12 η(z). η(− 1 z ) = √ −izη(z). ...

##
###
Exponential Complexity of Satisfiability Testing for Linear-Size Boolean Formulas
[chapter]

2013
*
Lecture Notes in Computer Science
*

*The*exponential

*complexity*

*of*

*the*

*satisfiability*problem for a given class

*of*Boolean

*circuits*is defined to be

*the*infimum

*of*constants α such that

*the*problem can be solved in time poly(m) 2 αn , where ...

*The*first one is about a similar "interweaving" between linearsize

*circuits*

*of*constant

*depth*and k-CNFs [SS12]. ... ; s ∞ = sup k {s k }; r d c is

*the*exponential

*complexity*

*of*

*the*

*satisfiability*problem for

*circuits*

*of*

*depth*at most d and size at most cn; r d ∞ = sup c {r d c }; f c is

*the*exponential

*complexity*

*of*...

##
###
One-way functions and circuit complexity

1987
*
Information and Computation
*

We will show that

doi:10.1016/0890-5401(87)90022-8
fatcat:b3p6uastine7nmhipvy4cwoqle
*the*predicates Qi can be used to construct*small**circuits*for g. ... RECEIVED November 1985; ACCEPTED February 17. 1987*circuit**complexity**of*f by C(f) = min{size(C): C computes f }, and*the**circuit**complexity**of*inverting such a function by C-'(f) = min{ C(g):g is an ...##
###
One-way functions and circuit complexity
[chapter]

1986
*
Lecture Notes in Computer Science
*

We will show that

doi:10.1007/3-540-16486-3_89
fatcat:2wixutunrjf2no27fku3cltg6m
*the*predicates Qi can be used to construct*small**circuits*for g. ... RECEIVED November 1985; ACCEPTED February 17. 1987*circuit**complexity**of*f by C(f) = min{size(C): C computes f }, and*the**circuit**complexity**of*inverting such a function by C-'(f) = min{ C(g):g is an ...
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