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Succinct Hitting Sets and Barriers to Proving Algebraic Circuits Lower Bounds [article]

Michael A. Forbes, Amir Shpilka, Ben Lee Volk
2018 arXiv   pre-print
Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known.  ...  Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits.  ...  We also thank the anonymous reviewers for their careful reading of this paper and for many useful comments.  ... 
arXiv:1701.05328v2 fatcat:ueg2mlf5l5febo7hojpqri67dq

Succinct hitting sets and barriers to proving algebraic circuits lower bounds

Michael A. Forbes, Amir Shpilka, Ben Lee Volk
2017 Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017  
Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits.  ...  Just as with the natural proofs notion of Razborov and Rudich [RR97] for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known.  ...  Acknowledgements We thank Scott Aaronson, Andy Drucker, Josh Grochow, Mrinal Kumar, Shubhangi Saraf and Dor Minzer for useful conversations regarding this work.  ... 
doi:10.1145/3055399.3055496 dblp:conf/stoc/ForbesSV17 fatcat:amglevgewvdlpozsxppzhdqify

Succinct Hitting Sets and Barriers to Proving Lower Bounds for Algebraic Circuits

Michael A. Forbes, Amir Shpilka, Ben Lee Volk
2018 Theory of Computing  
However, unlike in the Boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little  ...  Following a similar result of Williams (2016) in the Boolean setting, we show that the existence of an algebraic natural proofs barrier is equivalent to the existence of succinct derandomization of the  ...  We also thank the anonymous reviewers for their careful reading of this paper and for many useful comments.  ... 
doi:10.4086/toc.2018.v014a018 dblp:journals/toc/ForbesSV18 fatcat:xca443ndhzfxfjnclcmx6y57py

Towards an algebraic natural proofs barrier via polynomial identity testing [article]

Joshua A. Grochow and Mrinal Kumar and Michael Saks and Shubhangi Saraf
2017 arXiv   pre-print
We observe that a certain kind of algebraic proof - which covers essentially all known algebraic circuit lower bounds to date - cannot be used to prove lower bounds against VP if and only if what we call  ...  succinct hitting sets exist for VP.  ...  We thank Amir Shpilka for conversations related to Section 4.1 and for his encouragement to publicize our thinking, even in light of the results of [14] which (independently) supercede ours.  ... 
arXiv:1701.01717v1 fatcat:tfx54wjxcrdlpkm4kdkou3qthm

If VNP is hard, then so are equations for it [article]

Mrinal Kumar, C. Ramya, Ramprasad Saptharishi, Anamay Tengse
2020 arXiv   pre-print
In a recent work of Chatterjee and the authors (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic  ...  Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations.  ...  We also thank Prerona Chatterjee and Ben Lee Volk for helpful discussions at various stages of this work.  ... 
arXiv:2012.07056v1 fatcat:5szqqp2uvrfhna2lpwwf7ahyum

On the Existence of Algebraically Natural Proofs [article]

Prerona Chatterjee, Mrinal Kumar, C. Ramya, Ramprasad Saptharishi, Anamay Tengse
2021 arXiv   pre-print
Thus, in this setting of polynomials with small integer coefficients, this provides evidence against a natural proof like barrier for proving algebraic circuit lower bounds, a framework for which was proposed  ...  Our proofs are elementary and rely on the existence of (non-explicit) hitting sets for VP (and VNP) to show that there are efficiently constructible, low degree equations for these classes.  ...  Mrinal thanks Rahul Santhanam and Ben Lee Volk for many insightful conversations about algebraic natural proofs and succinct hitting sets.  ... 
arXiv:2004.14147v2 fatcat:4ikspq4ebjeuxnscucjccfx4by

Computational Complexity of Discrete Problems (Dagstuhl Seminar 17121)

Anna Gál, Michal Koucký, Oded Regev, Till Tantau, Marc Herbstritt
2017 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 17121 "Computational Complexity of Discrete Problems".  ...  The first section gives an overview of the topics covered and the organization of the meeting. Section 2 lists the talks given in alphabetical order.  ...  Succinct Hitting Sets and Barriers to Proving Algebraic Circuits Lower Bounds Ben Lee Volk (Tel Aviv University, IL) This talk presents a framework of "algebraically natural lower bounds" for algebraic  ... 
doi:10.4230/dagrep.7.3.45 dblp:journals/dagstuhl-reports/GalK0T17 fatcat:og5bioyq4zaszfljjecwfrpjgq

Why are Proof Complexity Lower Bounds Hard?

Jan Pich, Rahul Santhanam
2019 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)  
Perhaps the hope was that circuits are simpler and more 'combinatorial' objects, and therefore more amenable to lower bounds via combinatorial and algebraic techniques.  ...  Circuit Lower Bounds Program: Separate NP and P by proving super-polynomial lower bounds for functions in NP against more and more expressive classes of Boolean circuits.  ...  Let Q be an arbitrary proof system and Q be a proof system simulating both Q and EF. Since Q is not optimal, there is a proof system P such that Q does not have p-size proofs of Ref P .  ... 
doi:10.1109/focs.2019.00080 dblp:conf/focs/PichS19 fatcat:4of45zsirzd6ldkebom27aeaqy

Barriers for Rank Methods in Arithmetic Complexity [article]

Klim Efremenko and Ankit Garg and Rafael Oliveira and Avi Wigderson
2017 arXiv   pre-print
Rank methods (or flattenings) are also in wide use in algebraic geometry for proving tensor rank and symmetric tensor rank lower bounds. Our main results are barriers to these methods.  ...  Despite many successes and rapid progress, however, challenges like proving super-polynomial lower bounds on circuit or formula size for explicit polynomials, or super-linear lower bounds on explicit 3  ...  Acknowledgments The authors would like to thanks Mrinal Kumar  ... 
arXiv:1710.09502v1 fatcat:q77slcdltfa7hh5awrcko2wmbq

On the Symmetries of and Equivalence Test for Design Polynomials

Nikhil Gupta, Chandan Saha, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science  
The family of polynomials N W := {NW d,k : d is a prime} and close variants of it have been used as hard explicit polynomial families in several recent arithmetic circuit lower bound proofs.  ...  This is proved by completely characterizing the Lie algebra of NW d,k , and using an interplay between the Hessian of NW d,k and the evaluation dimension.  ...  Input: Black-box access to f ∈ F[x]. Output: Black-box access to g ∈ F[x] such that if f is BD-PS equivalent to NW then g is scaling equivalent to NW.  ... 
doi:10.4230/lipics.mfcs.2019.53 dblp:conf/mfcs/GuptaS19 fatcat:46awkhafpfghhnr5p47z6kc4gq

Iterated lower bound formulas: a diagonalization-based approach to proof complexity

Rahul Santhanam, Iddo Tzameret
2021 Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing  
Our proof of this fact uses the implication from IPS lower bounds to algebraic complexity lower bounds due to Grochow and Pitassi together with a diagonalization argument: the formulas φ n themselves assert  ...  This provides the first natural equivalence between proof complexity lower bounds and standard algebraic complexity lower bounds.  ...  Acknowledgements We wish to thank Jan Pich for very helpful discussions during the work on this paper.  ... 
doi:10.1145/3406325.3451010 fatcat:n7c3szaru5awtkpbe4jqni7jwy

Natural proofs versus derandomization

Ryan Williams
2013 Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13  
We study connections between Natural Proofs, derandomization, and the problem of proving "weak" circuit lower bounds such as NEXP ⊂ TC 0 , which are still wide open.  ...  We prove: • Constructivity is unavoidable, even for NEXP lower bounds.  ...  the hitting set, then use the hitting set and the NP oracle to simulate the AM computation. 2.  ... 
doi:10.1145/2488608.2488612 dblp:conf/stoc/Williams13 fatcat:ecvxel2o2vdw3aafixugvf4xzu

Natural Proofs versus Derandomization

R. Ryan Williams
2016 SIAM journal on computing (Print)  
We study connections between Natural Proofs, derandomization, and the problem of proving "weak" circuit lower bounds such as NEXP ⊂ TC 0 , which are still wide open.  ...  We prove: • Constructivity is unavoidable, even for NEXP lower bounds.  ...  the hitting set, then use the hitting set and the NP oracle to simulate the AM computation. 2.  ... 
doi:10.1137/130938219 fatcat:gxa2yzaxkfgo7pojgjz7bkkcui

Natural Proofs Versus Derandomization [article]

Ryan Williams
2015 arXiv   pre-print
We study connections between Natural Proofs, derandomization, and the problem of proving "weak" circuit lower bounds such as NEXP⊂ TC^0.  ...  We prove: ∙ Constructivity is unavoidable, even for NEXP lower bounds.  ...  For the other direction, suppose NEXP = BPP and BPP ⊆ io-HeuristicZPTIME[2 n ε ]/n ε for all ε > 0. We wish to prove a contradiction.  ... 
arXiv:1212.1891v3 fatcat:pks6jxag7jhmvbjypvqxylqbwu

Unifying Known Lower Bounds via Geometric Complexity Theory

Joshua A. Grochow
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
We show that most algebraic circuit lower bounds and relations between lower bounds naturally fit into the representationtheoretic framework suggested by geometric complexity theory (GCT), including: the  ...  ), the connected components technique (Ben-Or, Steele-Yao), depth 3 algebraic circuit lower bounds over finite fields (Grigoriev-Karpinski), lower bounds on permanent versus determinant (Mignon-Ressayre  ...  Acknowledgements The author would like to thank Scott Aaronson, Eric Allender, Saugata Basu, Tom Church, Klim Efremenko, Kaveh Ghasemloo, J. M.  ... 
doi:10.1109/ccc.2014.35 dblp:conf/coco/Grochow14 fatcat:z5eytgp64jh2pclri4tlmky53e
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