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The Power of Depth 2 Circuits over Algebras [article]

Chandan Saha, Ramprasad Saptharishi, Nitin Saxena
2009 arXiv   pre-print
We study 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.  ... 
arXiv:0904.2058v1 fatcat:up42y4zzb5fsrfevtuef2kk7iq

The Power of Depth 2 Circuits over Algebras

Chandan Saha, Ramprasad Saptharishi, Nitin Saxena, Marc Herbstritt
2009 Foundations of Software Technology and Theoretical Computer Science  
We study 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.  ... 
doi:10.4230/lipics.fsttcs.2009.2333 dblp:conf/fsttcs/SahaSS09 fatcat:gn3ossdnyfch5lb7not57ooxmq

CC-circuits and the expressive power of nilpotent algebras [article]

Michael Kompatscher
2022 arXiv   pre-print
We show that CC-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.  ... 
arXiv:1911.01479v9 fatcat:tw3lcx6syvgwhipifs4cyf6q5u

Boolean circuits versus arithmetic circuits

Joachim von zur Gathen, Gadiel Seroussi
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  ... 
doi:10.1016/0890-5401(91)90078-g fatcat:aydwnfiodfgy3bizzgauso2sei

Progress on Polynomial Identity Testing-II [chapter]

Nitin Saxena
2014 Perspectives in Computational Complexity  
We survey 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.  ... 
doi:10.1007/978-3-319-05446-9_7 fatcat:nakjcs6t6vf7pmd4qwjxmqarga

Progress on Polynomial Identity Testing - II [article]

Nitin Saxena
2014 arXiv   pre-print
We survey 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.  ... 
arXiv:1401.0976v1 fatcat:wrc3gfl2ajbmbhl2reb6lm722m

Inversion in finite fields using logarithmic depth

Joachim von zur Gathen
1990 Journal of symbolic computation  
Litow & Davida (1988) show that inverses in large finite fields 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.  ... 
doi:10.1016/s0747-7171(08)80028-4 fatcat:gj25oy3ygvai3dcsf5lnm6gii4

Diagonal Circuit Identity Testing and Lower Bounds [chapter]

Nitin Saxena
2008 Lecture Notes in Computer Science  
Suppose we are given a 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.  ... 
doi:10.1007/978-3-540-70575-8_6 fatcat:xngyprlwmrcdnlzww2v7ur7ynq

On top fan-in vs formal degree for depth-3 arithmetic circuits [article]

Mrinal Kumar
2018 arXiv   pre-print
We show that 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  ... 
arXiv:1804.03303v1 fatcat:k7p42vt6vvcnlj2ga2qfufboxi

Simple Hard Instances for Low-Depth Algebraic Proofs [article]

Nashlen Govindasamy, Tuomas Hakoniemi, Iddo Tzameret
2022 arXiv   pre-print
We prove super-polynomial lower bounds on 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).  ... 
arXiv:2205.07175v1 fatcat:5rpfawkfqrco7bkbbkwcao4mgu

Circuit Complexity, Proof Complexity, and Polynomial Identity Testing

Joshua A. Grochow, Toniann Pitassi
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 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.  ... 
doi:10.1109/focs.2014.20 dblp:conf/focs/GrochowP14 fatcat:mhmjd72cr5gmfdgtfl4zgsfhsi

Circuit complexity, proof complexity, and polynomial identity testing [article]

Joshua A. Grochow, Toniann Pitassi
2014 arXiv   pre-print
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.  ...  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.  ... 
arXiv:1404.3820v1 fatcat:2nerc6uatvdh5pqwtrhelt6jvq

Algebraic independence over positive characteristic: New criterion and applications to locally low-algebraic-rank circuits

Anurag Pandey, Nitin Saxena, Amit Sinhababu
2018 Computational Complexity  
Currently, even 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.  ... 
doi:10.1007/s00037-018-0167-5 fatcat:7ki7km76ynbulnnbmjavvebuku

Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing [chapter]

Michael A. Forbes, Amir Shpilka
2013 Lecture Notes in Computer Science  
Finally, we consider 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.  ... 
doi:10.1007/978-3-642-40328-6_37 fatcat:jd4yqz6a7jgdfewml4inacgsvm

Lower Bounds for Matrix Factorization [article]

Mrinal Kumar, Ben Lee Volk
2019 arXiv   pre-print
A_d where 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.  ... 
arXiv:1904.01182v1 fatcat:7rswr2curbacxlx37ua53awwji
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