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On Defining Integers in the Counting Hierarchy and Proving Arithmetic Circuit Lower Bounds
[chapter]

*
STACS 2007
*

We

doi:10.1007/978-3-540-70918-3_12
dblp:conf/stacs/Burgisser07
fatcat:iixkq2cgnvbrxbzz2q2luljaty
*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. ...*The*constant-free Valiant model An*arithmetic**circuit*over*the*field Q is an acyclic finite digraph, where all nodes except*the*input nodes have fan-*in*2*and*are labelled by +, −, × or /.*The**circuit*...*Integers**definable**in**the**counting**hierarchy*We consider sequences of*integers*a(n, k)*defined*for n, k ∈ N*and*0 ≤ k ≤ q(n), where q is polynomially*bounded*, such that ∀n > 1 ∀k ≤ q(n) |a(n, k)| ≤ 2 n ...##
###
Shallow Circuits with High-Powered Inputs
[article]

2010
*
arXiv
*
pre-print

*In*this third version of our paper we show that

*the*same

*lower*

*bound*would follow even if

*one*could only

*prove*a slightly superpolynomial upper

*bound*

*on*

*the*number of real roots. ... We also show that an even weaker

*bound*

*on*

*the*number of real roots would suffice to obtain a

*lower*

*bound*

*on*

*the*size of depth 4

*circuits*computing

*the*permanent. ...

*The*

*counting*

*hierarchy*contains all

*the*polynomial

*hierarchy*PH

*and*is contained

*in*PSPACE.

*The*

*arithmetic*

*circuit*classes

*defined*

*in*Section 2.1 are nonuniform. ...

##
###
Arithmetic Constant-Depth Circuit Complexity Classes
[chapter]

2003
*
Lecture Notes in Computer Science
*

*One*reason for interest

*in*these classes is that they contain

*the*boundary marking

*the*limits of current

*lower*

*bound*technology: such technology exists for AC 0

*and*some of

*the*classes AC 0 [m], while ... Continuing a line of research originating from Valiant's work

*on*

*the*

*counting*class ♯P ,

*the*

*arithmetic*

*circuit*complexity classes ♯AC 0

*and*♯N C 1 have recently been studied. ...

*The*author would like to thank Eric Allender for many interesting discussions. Riccardo Pucella deserves thanks for comments

*on*a draft of this paper. ...

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

2013
*
Information and Computation
*

From this we obtain

doi:10.1016/j.ic.2012.10.013
fatcat:t6v6fd4v2zdwxbpxy3m3rf3uuq
*the**lower**bound*by explicitly constructing a hitting set against*arithmetic**circuits**in**the*polynomial*hierarchy*. 5 It is possible to give a uniform upper of E NP RP for L. 6*One*of ... To obtain this result we develop a novel technique that further strengthens*the*connection between black-box derandomization of polynomial identity testing*and**lower**bounds*for*arithmetic**circuits*. ... 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*. ...##
###
Permanent Does Not Have Succinct Polynomial Size Arithmetic Circuits of Constant Depth
[chapter]

2011
*
Lecture Notes in Computer Science
*

From this we obtain

doi:10.1007/978-3-642-22006-7_61
fatcat:tjfzx3b6sngjzcyhqpmrdwcmqi
*the**lower**bound*by explicitly constructing a hitting set against*arithmetic**circuits**in**the*polynomial*hierarchy*. 5 It is possible to give a uniform upper of E NP RP for L. 6*One*of ... To obtain this result we develop a novel technique that further strengthens*the*connection between black-box derandomization of polynomial identity testing*and**lower**bounds*for*arithmetic**circuits*. ... 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*. ...##
###
On Defining Integers And Proving Arithmetic Circuit Lower Bounds

2009
*
Computational Complexity
*

We

doi:10.1007/s00037-009-0260-x
fatcat:lla4pzno45hgzm5pgz6cceqb44
*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. ... 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). ...*The*author was partially supported by DFG grant BU 1371*and**the*Paderborn Institute for Scientific Computation (PaSCo). ...##
###
The complexity of two problems on arithmetic circuits

2007
*
Theoretical Computer Science
*

This gives a coNP Mod k P algorithm for deciding an upper

doi:10.1016/j.tcs.2007.08.008
fatcat:q2oaf5j5zbhcbnysvnpgbgjlii
*bound**on**the*degree of a polynomial given by a*circuit**in*fields of characteristic k > 0. ... By using*arithmetic**circuits*, encoding multivariate polynomials may be drastically more efficient than writing down*the*list of monomials. ... Acknowledgements*The*authors thank Erich Kaltofen for*the*suggestion of encoding d*in*unary*in**the*language DEG b as well as*the*anonymous referees for useful remarks. ...##
###
Interpolation in Valiant's theory
[article]

2007
*
arXiv
*
pre-print

Our proof method relies

arXiv:0710.0360v1
fatcat:yazhjh5cejduhfujxeocvqnt6m
*on*Lagrange interpolation*and**on*recent results connecting*the*(boolean)*counting**hierarchy*to algebraic complexity classes. ... We investigate*the*following question: if a polynomial can be evaluated at rational points by a polynomial-time boolean algorithm, does it have a polynomial-size*arithmetic**circuit*? ... Acknowledgments We would like to thank Erich Kaltofen*and*Christos Papadimitriou for sharing their thoughts*on*question (*). ...##
###
Marginal hitting sets imply super-polynomial lower bounds for permanent

2012
*
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference on - ITCS '12
*

We

doi:10.1145/2090236.2090275
dblp:conf/innovations/JansenS12
fatcat:nyq3xjitdvhmlmujyxrlojz22e
*prove*that*the*hypothesis implies that Permanent does not have polynomial size constant-free*arithmetic**circuits*. ... Suppose f is a univariate polynomial of degree r = r(n) that is computed by a size n*arithmetic**circuit*. ... Assuming τ (per n ) = n O(1) , for a first compression step,*one*uses*the*relation between*the**counting**hierarchy*CH*and*TC 0 to get*the*coefficients of fn "weakly-*definable*"*in*CH. ...##
###
Monomials in Arithmetic Circuits: Complete Problems in the Counting Hierarchy

2014
*
Computational Complexity
*

We consider

doi:10.1007/s00037-013-0079-3
fatcat:yl5o25ecmngt5lq47rsg6xqoza
*the*complexity of two questions*on*polynomials given by*arithmetic**circuits*: testing whether a monomial is present*and**counting**the*number of monomials. ... We show that these problems are complete for subclasses of*the**counting**hierarchy*which had few or no known natural complete problems before. ...*The*results of this paper were conceived while*the*third author was visiting*the*Équipe de Logique Mathématique at Université Paris Diderot Paris 7. ...##
###
Interpolation in Valiant's Theory

2011
*
Computational Complexity
*

Our proof method relies

doi:10.1007/s00037-011-0002-8
fatcat:up5ivyvkfvefddjsqwj4vsuhgm
*on*Lagrange interpolation*and**on*recent results connecting*the*(boolean)*counting**hierarchy*to algebraic complexity classes. ... Answering it negatively would indeed imply that*the*constant-free versions of*the*algebraic complexity classes VP*and*VNP*defined*by Valiant are different. ... Acknowledgements We would like to thank Erich Kaltofen*and*Christos Papadimitriou for sharing their thoughts*on*question (*),*and**the*anonymous referees for their useful remarks. ...##
###
Bounded Arithmetic, Cryptography and Complexity

2008
*
Theoria
*

2 can

doi:10.1111/j.1755-2567.1997.tb00745.x
fatcat:kof42mxifrdqfdqaqbcyapg7yq
*prove**the*polynomial time*hierarchy*collapses*in*a strong way [14, 6, 23] . ... This survey discusses theories of*bounded**arithmetic*, growth rates of*definable*functions, natural proofs, interpolation theorems, connections to cryptography,*and**the*difficulty of obtaining independence ... Impagliazzo for pointing out*the*construction of section 4.4*and*J. Krajíček, P. Pudlák*and*C. Pollett for comments*on*a preliminary version of this paper. ...##
###
Symmetric Computation (Invited Talk)

2020
*
Annual Conference for Computer Science Logic
*

This is at once a rich class of problems

doi:10.4230/lipics.csl.2020.2
dblp:conf/csl/Dawar20
fatcat:jdwa32g6rnfxhh5ed7mnjayrwe
*and**one*for which we have methods for*proving**lower**bounds*. ...*The*mismatch between algorithms working*on*high-level data structures*and*complexity*defined**in*terms of low-level machines is*one*of*the*central concerns of*the*field of descriptive complexity, which ...*The**lower**bound**on**counting*width established*in*Theorem 5 has interesting consequences for lift-*and*-project*hierarchies*. ...##
###
Finite and Algorithmic Model Theory (Dagstuhl Seminar 17361)

2018
*
Dagstuhl Reports
*

This report documents

doi:10.4230/dagrep.7.9.1
dblp:journals/dagstuhl-reports/DawarGKS17
fatcat:teyilwvdgrd6tch2gdb4d4dd5m
*the*program*and**the*outcomes of Dagstuhl Seminar 17361 "Finite*and*Algorithmic Model Theory". ... We formalize agent-based models as stochastic processes whose states are metafinite models,*and*we*define*a notion of abstraction. ... Our main results are conditions that imply an abstraction is sound,*and*further conditions that imply it preserves*the*Markov property. ...##
###
Monomials in arithmetic circuits: Complete problems in the counting hierarchy
[article]

2012
*
arXiv
*
pre-print

We consider

arXiv:1110.6271v2
fatcat:hagknlo6pfcx5lilpmlnaksyvm
*the*complexity of two questions*on*polynomials given by*arithmetic**circuits*: testing whether a monomial is present*and**counting**the*number of monomials. ... We show that these problems are complete for subclasses of*the**counting**hierarchy*which had few or no known natural complete problems. ...*The*results of this paper were conceived while*the*third author was visiting theÉquipe de Logique Mathématique at Université Paris Diderot Paris 7. He would like to thank Arnaud Durand ...
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