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The Power of Depth 2 Circuits over Algebras
[article]

2009
*
arXiv
*
pre-print

We study

arXiv:0904.2058v1
fatcat:up42y4zzb5fsrfevtuef2kk7iq
*the*problem*of*polynomial identity testing (PIT) for*depth**2*arithmetic*circuits**over*matrix*algebra*. ... We show that identity testing*of**depth*3 (Sigma-Pi-Sigma) arithmetic*circuits**over*a field F is polynomial time equivalent to identity testing*of**depth**2*(Pi-Sigma) arithmetic*circuits**over*U_2(F),*the*... And finally thanks to V Vinay for many useful comments on*the*first draft*of*this paper. ...##
###
The Power of Depth 2 Circuits over Algebras

2009
*
Foundations of Software Technology and Theoretical Computer Science
*

We study

doi:10.4230/lipics.fsttcs.2009.2333
dblp:conf/fsttcs/SahaSS09
fatcat:gn3ossdnyfch5lb7not57ooxmq
*the*problem*of*polynomial identity testing (PIT) for*depth**2*arithmetic*circuits**over*matrix*algebra*. ... We show that identity testing*of**depth*3 (ΣΠΣ) arithmetic*circuits**over*a field F is polynomial time equivalent to identity testing*of**depth**2*(ΠΣ) arithmetic*circuits**over*U*2*(F),*the**algebra**of*upper-triangular ... And finally thanks to V Vinay for many useful comments on*the*first draft*of*this paper. ...##
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CC-circuits and the expressive power of nilpotent algebras
[article]

2022
*
arXiv
*
pre-print

We show that CC-

arXiv:1911.01479v9
fatcat:tw3lcx6syvgwhipifs4cyf6q5u
*circuits**of*bounded*depth*have*the*same expressive*power*as*circuits**over*finite nilpotent*algebras*from congruence modular varieties. ... On*the*other hand, if AND is computable by uniform CC-*circuits**of*bounded*depth*and polynomial size, we can construct a nilpotent*algebra*in which checking identities is coNP-complete, and solving equations ... Acknowledgments I would like to thank*the*anonymous reviewers for their many helpful remarks, in particular for pointing out several inaccurate references. ...##
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Boolean circuits versus arithmetic circuits

1991
*
Information and Computation
*

*Over*finite lields

*of*small characteristic,

*the*two models are equally

*powerful*when size is considered, but Boolean

*circuits*are exponentially more

*powerful*than arithmetic

*circuits*with respect to

*depth*... We compare

*the*two computational models

*of*Boolean

*circuits*and arithmetic

*circuits*in cases where they both apply, namely

*the*computation

*of*polynomials

*over*

*the*rational numbers or

*over*finite fields ... Let K= [F, , f E K[x]

*of*degree e < aq, and a an arithmetic

*circuit*

*of*

*depth*d

*over*K value-computing

*the*function K -+ K given by f: Then d>min{loge,log(aq-e+l),logq-logq-

*2*) If q/

*2*> e

*2*,/$ and q

*2*...

##
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Progress on Polynomial Identity Testing-II
[chapter]

2014
*
Perspectives in Computational Complexity
*

We survey

doi:10.1007/978-3-319-05446-9_7
fatcat:nakjcs6t6vf7pmd4qwjxmqarga
*the*area*of**algebraic*complexity theory; with*the*focus being on*the*problem*of*polynomial identity testing (PIT). ... We discuss*the*key ideas that have gone into*the*results*of**the*last few years. (2010) . Primary 68Q25, 68W30; Secondary 12Y05, 13P25. Mathematics Subject Classification ...*The*width-*2*chasm Here we look at*circuits**over*a matrix*algebra*. ...##
###
Progress on Polynomial Identity Testing - II
[article]

2014
*
arXiv
*
pre-print

We survey

arXiv:1401.0976v1
fatcat:wrc3gfl2ajbmbhl2reb6lm722m
*the*area*of**algebraic*complexity theory; with*the*focus being on*the*problem*of*polynomial identity testing (PIT). ... We discuss*the*key ideas that have gone into*the*results*of**the*last few years. ... Perhaps surprisingly, [SSS09] showed that: A polynomial computed by a*depth*-3*circuit*(*over*a field) can as well be computed by a D*over*a*2*×*2*matrix*algebra*. ...##
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Inversion in finite fields using logarithmic depth

1990
*
Journal of symbolic computation
*

Litow & Davida (1988) show that inverses in large finite fields

doi:10.1016/s0747-7171(08)80028-4
fatcat:gj25oy3ygvai3dcsf5lnm6gii4
*of*small characteristic p, say p =*2*, can be computed by Boolean*circuits**of*(order-optimal) logarithmic*depth*. ... We note that their numerical approach can also be implemented purely*algebraically*, and that*the*resulting much simpler algorithm yields, also for large p, both arithmetic and Boolean reductions*of*inversion ... Acknowledgment It is a pleasure to acknowledge*the*many discussions with Allan Borodin about*the*subject. I also thank an anonymous referee for several valuable suggestions. ...##
###
Diagonal Circuit Identity Testing and Lower Bounds
[chapter]

2008
*
Lecture Notes in Computer Science
*

Suppose we are given a

doi:10.1007/978-3-540-70575-8_6
fatcat:xngyprlwmrcdnlzww2v7ur7ynq
*depth*-3*circuit*(*over*any field F)*of**the*form: where,*the*i,j 's are linear functions living in F[x 1 , . . . , x n ]. ... In this paper we give*the*first deterministic polynomial time algorithm for testing whether a diagonal*depth*-3*circuit*C(x 1 , . . . , x n ) (i.e. C is a sum*of**powers**of*linear functions) is zero. ... Acknowledgements I would like to thank Harry Buhrman and Ronald de Wolf for various useful discussions during*the*course*of*this work. ...##
###
On top fan-in vs formal degree for depth-3 arithmetic circuits
[article]

2018
*
arXiv
*
pre-print

We show that

arXiv:1804.03303v1
fatcat:k7p42vt6vvcnlj2ga2qfufboxi
*over**the*field*of*complex numbers, every homogeneous polynomial*of*degree d can be approximated (in*the*border complexity sense) by a*depth*-3 arithmetic*circuit**of*top fan-in at most d+1. ... This is quite surprising since there exist homogeneous polynomials P on n variables*of*degree*2*, such that any*depth*-3 arithmetic*circuit*computing P must have top fan-in at least Ω(n). ... Another recent example*of**the**power*non-homogeneous*depth*-3*circuits*is a beautiful result*of*Gupta, Kamath, Kayal and Saptharishi [GKKS13] , who showed that*over**the*field C, there is a*circuit**of*formal ...##
###
Simple Hard Instances for Low-Depth Algebraic Proofs
[article]

2022
*
arXiv
*
pre-print

We prove super-polynomial lower bounds on

arXiv:2205.07175v1
fatcat:5rpfawkfqrco7bkbbkwcao4mgu
*the*size*of*propositional proof systems operating with constant-*depth**algebraic**circuits**over*fields*of*zero characteristic. ... with small*depth*-*2**circuits*. ... Thus,*the*minimal product-*depth*-∆*circuit*-size*of*g(x) lower bounds*the*minimal product-*depth*-∆*circuit*-size*of*C(x, y, w). ...##
###
Circuit Complexity, Proof Complexity, and Polynomial Identity Testing

2014
*
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
*

Proving super-polynomial lower bounds on AC 0 [p]-Frege implies VNP Fp does not have polynomial-size

doi:10.1109/focs.2014.20
dblp:conf/focs/GrochowP14
fatcat:mhmjd72cr5gmfdgtfl4zgsfhsi
*circuits**of**depth*d-a notoriously open question for any d ≥ 4-thus explaining*the*difficulty*of*lower ... Finally, using*the**algebraic*structure*of*our proof system, we propose a novel way to extend techniques from*algebraic**circuit*complexity to prove lower bounds in proof complexity. ... We thank Pascal Koiran for providing*the*second half*of**the*proof*of*Proposition 2.4. We thank Iddo Tzameret for useful discussions that led to Proposition 2.2. ...##
###
Circuit complexity, proof complexity, and polynomial identity testing
[article]

2014
*
arXiv
*
pre-print

Using

arXiv:1404.3820v1
fatcat:2nerc6uatvdh5pqwtrhelt6jvq
*the**algebraic*structure*of*our proof system, we propose a novel way to extend techniques from*algebraic**circuit*complexity to prove lower bounds in proof complexity. ...*the*difficulty*of*lower bounds on AC^0[p]-Frege, or b) AC^0[p]-Frege cannot efficiently prove*the**depth*d PIT axioms, and hence we have a lower bound on AC^0[p]-Frege. ... We thank Pascal Koiran for providing*the*second half*of**the*proof*of*Proposition 2.4. We thank Iddo Tzameret for useful discussions that led to Proposition 2.2. ...##
###
Algebraic independence over positive characteristic: New criterion and applications to locally low-algebraic-rank circuits

2018
*
Computational Complexity
*

Currently, even

doi:10.1007/s00037-018-0167-5
fatcat:7ki7km76ynbulnnbmjavvebuku
*the*case*of*two bivariate*circuits**over*F*2*is open. We come up with a natural generalization*of*Jacobian criterion, that works*over*all characteristic. ...*depth*-4*circuits*). ... We thank Markus Bläser and*the*anonymous reviewers for*the*elaborate suggestions to improve*the*draft. NS acknowledges*the*support from DST/SJF/MSA-01/2013-14 and SB/FTP/ETA-177/2013. ...##
###
Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing
[chapter]

2013
*
Lecture Notes in Computer Science
*

Finally, we consider

doi:10.1007/978-3-642-40328-6_37
fatcat:jd4yqz6a7jgdfewml4inacgsvm
*the**depth*-3 diagonal*circuit*model as defined by Saxena [Sax08], as PIT algorithms for this model also have implications in Mulmuley's work. ... We then observe that*the*weaker conjecture has recently been nearly settled by*the*authors ([FS12]), who gave quasipolynomial size hitting sets for*the*class*of*read-once oblivious*algebraic*branching ... As matrix multiplication and trace both have small*algebraic**circuits*, it follows that traces*of*matrix*powers*have small*circuits*. ...##
###
Lower Bounds for Matrix Factorization
[article]

2019
*
arXiv
*
pre-print

A_d where

arXiv:1904.01182v1
fatcat:7rswr2curbacxlx37ua53awwji
*the*total sparsity*of*A_1,...,A_d is less than n^1+1/(2d). In other words, any*depth*-d linear*circuit*computing*the*linear transformation M_n· x has size at least n^1+Ω(1/d). ... In addition to being a natural mathematical question on its own, this problem appears in various incarnations in computer science;*the*most significant being in*the*context*of*lower bounds for*algebraic*... Acknowledgements We thank Swastik Kopparty for an insightful discussion on explicit construction*of*Sidon sets*over*finite fields. ...
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