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Uniform constant-depth threshold circuits for division and iterated multiplication

William Hesse, Eric Allender, David A. Mix Barrington
2002 Journal of computer and system sciences (Print)  
1 2 HESSE, ALLENDER AND BARRINGTON It has been known since the mid-1980's [15, 46, 47] that integer division can be performed by poly-time uniform constant-depth circuits of Majority gates; equivalently  ...  Recently this was improved to L-uniform TC 0 [19], but it remained unknown whether division can be performed by DLOGTIME-uniform TC 0 circuits.  ...  We also thank Dieter van Melkebeek, Samir Datta, Michal Koucký, Rüdiger Reischuk, and Sambuddha Roy for helpful discussions.  ... 
doi:10.1016/s0022-0000(02)00025-9 fatcat:vwqdpnv7njghzi2326rxqprtca

On Threshold Circuits and Polynomial Computation

John H. Reif, Stephen R. Tate
1992 SIAM journal on computing (Print)  
A surprising relationship is uncovered between Threshold Circuits and another class of unbounded fanin circuits which are denoted Finite Field Z P (n)  ...  This paper investigates the computational power of Threshold Circuits.  ...  Pippenger has given a constant depth threshold circuit for multiplication, and the method used is the straight-forward reduction to iterated sum (i.e., the \gradeschool method" of multiplication) 17].  ... 
doi:10.1137/0221053 fatcat:4sagnkbzy5fq5csptjebulmvu4

Division Is In Uniform TC0 [chapter]

William Hesse
2001 Lecture Notes in Computer Science  
Integer division has been known since 1986 [4, 13, 12] to be in slightly non-uniform TC 0 , i.e., computable by polynomial-size, constant depth threshold circuits.  ...  This has been perhaps the outstanding natural problem known to be in a standard circuit complexity class, but not known to be in its uniform version. We show that indeed division is in uniform TC 0 .  ...  In this paper, we construct uniform constant depth circuits for division and iterated multiplication.  ... 
doi:10.1007/3-540-48224-5_9 fatcat:vu6dhiadb5drjduq47va4sfobq

Threshold Circuits for Iterated Matrix Product and Powering

Carlo Mereghetti, Beatrice Palano
2000 RAIRO - Theoretical Informatics and Applications  
As computational model, we use threshold circuits [12] . We are interested in solving problems by using threshold circuits of constant depth.  ...  We recall that TC^ is the class of probiems solvable by families of (unbounded fan-in) threshold circuits of polynomial weights and size, and constant depth d.  ... 
doi:10.1051/ita:2000105 fatcat:qf4stv4pnraxhgq7kdbrazfv6q

Root finding with threshold circuits

Emil Jeřábek
2012 Theoretical Computer Science  
uniform TC^0 algorithm (a uniform family of constant-depth polynomial-size threshold circuits).  ...  We show that for any constant d, complex roots of degree d univariate rational (or Gaussian rational) polynomials---given by a list of coefficients in binary---can be computed to a given accuracy by a  ...  Acknowledgments I am grateful to Paul Beame and Yuval Filmus for useful discussions, and to anonymous referees for helpful suggestions.  ... 
doi:10.1016/j.tcs.2012.09.001 fatcat:hs2f33kamzhm5ajwshdx4ymdpq

Page 5971 of Mathematical Reviews Vol. , Issue 94j [page]

1994 Mathematical Reviews  
Summary: “We investigate small-depth threshold circuits for iter- ated multiplication and related problems.  ...  This can be compared to the best known construction, which uses four levels of threshold gates (but no AC°-circuitry). Similarly, we design small-depth circuits for powering, division and logarithm.  ... 

On Defining Integers And Proving Arithmetic Circuit Lower Bounds

Peter Bürgisser
2009 Computational Complexity  
X k and n k=1 1 k X k of exp and log, respectively, can be computed by arithmetic circuits of size polynomial in log n (allowing divisions).  ...  We prove that if there are arithmetic circuits of size polynomial in n for computing the permanent of n by n matrices, then τ (n!) is polynomially bounded in log n.  ...  I thank them, as well as Emmanuel Jeandel and Emanuele Viola, for useful comments.  ... 
doi:10.1007/s00037-009-0260-x fatcat:lla4pzno45hgzm5pgz6cceqb44

On defining integers in the counting hierarchy and proving lower bounds in algebraic complexity [article]

Peter Bürgisser
2006 Electronic colloquium on computational complexity  
circuits of size polynomial in log n (allowing divisions).  ...  We prove that if there are arithmetic circuits for computing the permanent of n by n matrices having size polynomial in n, then τ (n!) is polynomially bounded in log n.  ...  [14] on uniform bounded-depth threshold circuits for division and iterated multiplication of integers. Proof.  ... 
dblp:journals/eccc/Burgisser06 fatcat:xnllua7gnnhe5bopn4gfk5dule

Quantum Circuits with Unbounded Fan-out [article]

Peter Hoyer
2004 arXiv   pre-print
, mod[q], And, Or, majority, threshold[t], exact[q], and Counting.  ...  Constant-depth polynomial-size quantum circuits with bounded fan-in and unbounded fan-out over a fixed basis (denoted by QNCf^0) can approximate with polynomially small error the following gates: parity  ...  Acknowledgements I would like to thank Harry Buhrman, Hartmut Klauck, and Hein Röhrig from Centrum voor Wiskunde en Informatica in Amsterdam, and Frederic Green from Clark University in Worcester for plenty  ... 
arXiv:quant-ph/0208043v3 fatcat:j3b7tuwqnzaarn63bfnfnorpxy

On the Complexity of Some Problems on Groups Input as Multiplication Tables

David Mix Barrington, Peter Kadau, Klaus-Jörn Lange, Pierre McKenzie
2001 Journal of computer and system sciences (Print)  
Finally, we examine the implications of our results for the complexity of iterated multiplication, powering, and division of integers in the context of the recent results of Chiu, Davida, and Litow and  ...  cyclicity and nilpotency can each be tested in FOLL 5 L.  ...  In addition, the authors thank Eric Allender and Bill Hesse for permission to allude to results from the forthcoming [1, 2, 21] and for several helpful discussions.  ... 
doi:10.1006/jcss.2001.1764 fatcat:7sicpde6rzf7heo3s5gvnksywu

Permanent does not have succinct polynomial size arithmetic circuits of constant depth

Maurice Jansen, Rahul Santhanam
2013 Information and Computation  
We show that over fields of characteristic zero there does not exist a polynomial p(n) and a constant-free succinct arithmetic circuit family {Φn} using division by constants 1 , where Φn has size at most  ...  p(n) and depth O(1), such that Φn computes the n × n permanent.  ...  We thank Pavel Hrubeš for pointing out to us that without division gates a lower bound can be obtained for succinct circuits of constant depth by a reduction to the Razborov-Smolensky lower bound.  ... 
doi:10.1016/j.ic.2012.10.013 fatcat:t6v6fd4v2zdwxbpxy3m3rf3uuq

Permanent Does Not Have Succinct Polynomial Size Arithmetic Circuits of Constant Depth [chapter]

Maurice Jansen, Rahul Santhanam
2011 Lecture Notes in Computer Science  
We show that over fields of characteristic zero there does not exist a polynomial p(n) and a constant-free succinct arithmetic circuit family {Φn} using division by constants 1 , where Φn has size at most  ...  p(n) and depth O(1), such that Φn computes the n × n permanent.  ...  We thank Pavel Hrubeš for pointing out to us that without division gates a lower bound can be obtained for succinct circuits of constant depth by a reduction to the Razborov-Smolensky lower bound.  ... 
doi:10.1007/978-3-642-22006-7_61 fatcat:tjfzx3b6sngjzcyhqpmrdwcmqi

On TC0, AC0, and Arithmetic Circuits

Manindra Agrawal, Eric Allender, Samir Datta
2000 Journal of computer and system sciences (Print)  
One way to define #AC 0 is as the class of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates.  ...  by NSF grants CCR-9509603 and CCR-9734918.  ...  Acknowledgments We would like to thank Richard Bumby, David Mix Barrington, Pierre McKenzie, Denis Therien, Noam Nisan and Dieter van Melkebeek for discussions and suggestions on the material.  ... 
doi:10.1006/jcss.1999.1675 fatcat:7wrl4jiwqrh37j5lbitrok3wgq

Low-Depth Uniform Threshold Circuits and the Bit-Complexity of Straight Line Programs [chapter]

Eric Allender, Nikhil Balaji, Samir Datta
2014 Lecture Notes in Computer Science  
We present improved uniform TC 0 circuits for division, matrix powering, and related problems, where the improvement is in terms of "majority depth" (as studied by Maciel and Thérien).  ...  As a corollary, we obtain improved bounds on the complexity of certain problems involving arithmetic circuits, which are known to lie in the counting hierarchy.  ...  Acknowledgments The first author acknowledges the support of NSF grants CCF-0832787 and CCF-1064785.  ... 
doi:10.1007/978-3-662-44465-8_2 fatcat:wh23sqqb35dolkokrmqkxc5m24

Lower Bounds Against Weakly-Uniform Threshold Circuits

Ruiwen Chen, Valentine Kabanets, Jeff Kinne
2013 Algorithmica  
This strengthens the results by Allender [All99] (for uniform TC 0 ) and by Jansen and Santhanam [JS11] (for weakly-uniform arithmetic circuits of constant depth).  ...  The main result of [JS11] is that Permanent does not have succinct polynomial-size arithmetic circuits of constant depth, where arithmetic circuits have unbounded fan-in addition and multiplication gates  ...  We also thank the reviewers for comments and suggestions that improved the exposition of the paper.  ... 
doi:10.1007/s00453-013-9823-y fatcat:qhysc73akvgzjgzdxl5d5ymula
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