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Lower Bounds for Symmetric Circuits for the Determinant [article]

Anuj Dawar, Gregory Wilsenach
2021 arXiv   pre-print
We develop a general framework for proving lower bounds for symmetric circuits with restricted symmetries, based on a new support theorem and new two-player restricted bijection games.  ...  Dawar and Wilsenach (ICALP 2020) introduce the model of symmetric arithmetic circuits and show an exponential separation between the sizes of symmetric circuits for computing the determinant and the permanent  ...  Acknowledgements: We are grateful to Albert Atserias for useful discussions on the construction in Section 5.2.  ... 
arXiv:2107.10986v2 fatcat:crh6oaw5ybfftkd4wrc7mcmoqi

Lower Bounds for Symmetric Circuits for the Determinant

Anuj Dawar, Gregory Wilsenach, Mark Braverman
2022
We develop a general framework for proving lower bounds for symmetric circuits with restricted symmetries, based on a new support theorem and new two-player restricted bijection games.  ...  Dawar and Wilsenach (ICALP 2020) introduce the model of symmetric arithmetic circuits and show an exponential separation between the sizes of symmetric circuits for computing the determinant and the permanent  ...  Lower Bound for the Permanent We previously established in [11] lower bounds on symmetric circuits for the permanent showing that there are no subexponential square-symmetric circuits computing the permanent  ... 
doi:10.4230/lipics.itcs.2022.52 fatcat:6o5qc6iunfdwlg7545ytzkzzhq

Symmetric Arithmetic Circuits [article]

Anuj Dawar, Gregory Wilsenach
2021 arXiv   pre-print
We establish unconditional exponential lower bounds on the size of any symmetric circuit for computing the permanent.  ...  In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characteristic zero.  ...  Their lower bound is for the equivariant determinantal complexity of the permanent.  ... 
arXiv:2002.06451v2 fatcat:unqswydznrdilitkbpza5gfsdu

Symmetric Arithmetic Circuits

Anuj Dawar, Gregory Wilsenach, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming  
We establish unconditional, nearly exponential, lower bounds on the size of any symmetric circuit for computing the permanent over any field of characteristic other than 2.  ...  In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characteristic zero.  ...  Their lower bound is for the equivariant determinantal complexity of the permanent.  ... 
doi:10.4230/lipics.icalp.2020.36 dblp:conf/icalp/DawarW20 fatcat:hjjh3sbiwbcuzdakr2cak5v44m

Observations on Symmetric Circuits [article]

Christian Engels
2020 arXiv   pre-print
Their result showed an exponential lower bound of the permanent computed by symmetric circuits.  ...  We study symmetric arithmetic circuits and improve on lower bounds given by Dawar and Wilsenach (ArXiv 2020).  ...  The authors showed that for circuits that have to be symmetric for certain permutation groups, the permanent and determinant have an exponential separation.  ... 
arXiv:2007.07496v3 fatcat:r2nl6u3de5hd3mduu4pbtxo2va

Page 567 of Mathematical Reviews Vol. , Issue 2003A [page]

2003 Mathematical Reviews  
Upper bounds are derived for the elementary symmetric functions. Next, we summarize key results for symmetric functions and determinants. Symmetric function lower bounds.  ...  A circuit all of whose non-output nodes have fan-out | is said to be a formula. Lower bounds are derived for circuits, and upper bounds are derived for the computationally weaker formulas.  ... 

Symmetric Computation (Invited Talk)

Anuj Dawar, Michael Wagner
2020 Annual Conference for Computer Science Logic  
This is at once a rich class of problems and one for which we have methods for proving lower bounds.  ...  We discuss a recent convergence of notions of symmetric computation arising in the theory of linear programming, in logic and in circuit complexity.  ...  This game can be used directly as a lower bound method for symmetric circuits (see [19] for an exposition).  ... 
doi:10.4230/lipics.csl.2020.2 dblp:conf/csl/Dawar20 fatcat:jdwa32g6rnfxhh5ed7mnjayrwe

Affine projections of symmetric polynomials

Amir Shpilka
2002 Journal of computer and system sciences (Print)  
We prove two non-trivial linear lower bounds for our model. The first lower bound is for computing the determinant, and the second is for computing the sum of two monomials.  ...  prove a super-linear lower bound for the symmetric model.  ...  Acknowledgments I thank Avi Wigderson for suggesting this line of research, and for his unstoppable encouragement in the last couple of years.  ... 
doi:10.1016/s0022-0000(02)00021-1 fatcat:mm7gujcfbja23mddrnk3z4gu5e

Lower bounds on arithmetic circuits via partial derivatives

Noam Nisan, Avi Wigderson
1996 Computational Complexity  
We use the technique to obtain new lower bounds for computing symmetric polynomials and iterated matrix prod-  ...  In this paper we describe a new technique for obtaining lower bounds on restriced classes of nonmonotone arithmetic circuits.  ...  Acknowledgements We are greatful to Jiri Sgall for careful reading and many comments on an earlier version of this paper.  ... 
doi:10.1007/bf01294256 fatcat:ro7foiksize4lb33vidkglz6oy

Special issue "Conference on Computational Complexity 2014" Guest Editor's Foreword

Michael E. Saks
2015 Computational Complexity  
The paper "Quantum algorithms for learning symmetric juntas via the adversary method" by Aleksandrs Belovs obtains efficient quantum algorithms for determining the set of relevant variables  ...  They were invited for submission and evaluated through the standard refereeing process of the journal.  ...  The quest for circuit lower bounds, i.e., proving that explicit functions cannot be computed within a given circuit class C, involves finding a "weakness" in the class C that limits its ability to compute  ... 
doi:10.1007/s00037-015-0101-z fatcat:2yaxvxycuzg7xolnd57va7sh4a

Concrete Multiplicative Complexity of Symmetric Functions [chapter]

Joan Boyar, René Peralta
2006 Lecture Notes in Computer Science  
They are powerful enough to give exact multiplicative complexities for several classes of symmetric functions.  ...  The multiplicative complexity of a Boolean function f is defined as the minimum number of binary conjunction (AND) gates required to construct a circuit representing f , when only exclusive-or, conjunction  ...  The technique of hyperplane restrictions yields lower bounds on multiplicative complexity which are better than the degree lower bound for many symmetric functions, including all with degree less than  ... 
doi:10.1007/11821069_16 fatcat:iwebwgfrvbfsbfcufwmnnhvie4

Page 506 of Mathematical Reviews Vol. , Issue 2004a [page]

2004 Mathematical Reviews  
The first lower bound is for computing the determinant, and the second is for computing the sum of two monomials.  ...  These are the best lower bounds known for depth-3 circuits over fields of characteristic zero.”  ... 

On Symmetric and Choiceless Computation [chapter]

Anuj Dawar
2016 Lecture Notes in Computer Science  
Having established a super-polynomial lower bound for symmetric threshold circuits for one (artificial) problem, we are able to tranfer such lower bounds to other problems by means of reductions.  ...  Lower Bounds for Symmetric Circuits It is clear that symmetric circuits are necessarily invariant and it is not difficult to come up with examples that show that the converse is not true.  ... 
doi:10.1007/978-3-319-28678-5_2 fatcat:p6ku7i4gcjfefbbjbgiygmhzmq

Algebraic and Combinatorial Methods in Computational Complexity (Dagstuhl Seminar 12421)

Manindra Agrawal, Thomas Thierauf, Christopher Umans, Marc Herbstritt
2013 Dagstuhl Reports  
Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved algebraically.  ...  These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting.  ...  lower bounds for arithmetics circuits.  ... 
doi:10.4230/dagrep.2.10.60 dblp:journals/dagstuhl-reports/AgrawalTU12 fatcat:dg7ithf6xfgadkzwkxjtjhy7ge

Represent MOD function by low degree polynomial with unbounded one-sided error [article]

Chris Beck, Yuan Li
2013 arXiv   pre-print
For example, if the immunity of f over F_p is lower bounded by n/2 - o(√(n)), and |1_f| = Ω(2^n), then f requires circuit of exponential size to compute.  ...  Our result improves the previous bound n/2(q-1) by Green. We observe how immunity over F_p is related to circuit lower bound.  ...  Acknowledgment Yuan Li would like to thank Sasha Razborov for illuminating discussions.  ... 
arXiv:1304.0713v1 fatcat:uorkenf7fvernkxzjnzp6pybgu
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