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Monotone Classes Beyond VNP

Authors Prerona Chatterjee , Kshitij Gajjar , Anamay Tengse

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Author Details

Prerona Chatterjee
  • Blavatnik School of Computer Science, Tel Aviv University, Israel
Kshitij Gajjar
  • Indian Institute of Technology Jodhpur, Rajasthan, India
Anamay Tengse
  • Department of Computer Science, University of Haifa, Israel

Funding

  • Chatterjee, Prerona: Azrieli International Postdoctoral Fellowship, the Israel Science Foundation (grant number 514/20) and the Len Blavatnik and the Blavatnik Family foundation. Parts of this work was done while I was a PhD student at TIFR Mumbai, where I was partially supported by a Google PhD Fellowship and also while I was a postdoctoral researcher at the Czech Academy of Sciences, where I was supported by Grant GX19-27871X of the Czech Science Foundation.
  • Gajjar, Kshitij: Part of this work was done as a postdoc in NUS, where I was funded by NUS ODPRT Grant WBS No. R-252-000-A94-133.
  • Tengse, Anamay: Grant 716/20 of the Israel Science Foundation.

Acknowledgements

We would like to thank the anonymous reviewers for the helpful comments. We are also grateful to Meena Mahajan for suggestions that greatly improved the presentation of the paper.

Cite AsGet BibTex

Prerona Chatterjee, Kshitij Gajjar, and Anamay Tengse. Monotone Classes Beyond VNP. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.FSTTCS.2023.11

Abstract

In this work, we study the natural monotone analogues of various equivalent definitions of VPSPACE: a well studied class (Poizat 2008, Koiran & Perifel 2009, Malod 2011, Mahajan & Rao 2013) that is believed to be larger than VNP. We observe that these monotone analogues are not equivalent unlike their non-monotone counterparts, and propose monotone VPSPACE (mVPSPACE) to be defined as the monotone analogue of Poizat’s definition. With this definition, mVPSPACE turns out to be exponentially stronger than mVNP and also satisfies several desirable closure properties that the other analogues may not. Our initial goal was to understand the monotone complexity of transparent polynomials, a concept that was recently introduced by Hrubeš & Yehudayoff (2021). In that context, we show that transparent polynomials of large sparsity are hard for the monotone analogues of all the known definitions of VPSPACE, except for the one due to Poizat.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Algebraic Complexity
  • Monotone Computation
  • VPSPACE
  • Transparent Polynomials

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References

  1. Bruno Pasqualotto Cavalar, Mrinal Kumar, and Benjamin Rossman. Monotone circuit lower bounds from robust sunflowers. In Yoshiharu Kohayakawa and Flávio Keidi Miyazawa, editors, LATIN 2020: Theoretical Informatics - 14th Latin American Symposium, São Paulo, Brazil, January 5-8, 2021, Proceedings, volume 12118 of Lecture Notes in Computer Science, pages 311-322. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-61792-9_25.
  2. Prerona Chatterjee, Kshitij Gajjar, and Anamay Tengse. Monotone classes beyond VNP. CoRR, abs/2202.13103, 2022. URL: https://arxiv.org/abs/2202.13103.
  3. Arkadev Chattopadhyay, Rajit Datta, Utsab Ghosal, and Partha Mukhopadhyay. Monotone complexity of spanning tree polynomial re-visited. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, volume 215 of LIPIcs, pages 39:1-39:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.39.
  4. Arkadev Chattopadhyay, Rajit Datta, and Partha Mukhopadhyay. Lower bounds for monotone arithmetic circuits via communication complexity. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 786-799. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451069.
  5. Arkadev Chattopadhyay, Utsab Ghosal, and Partha Mukhopadhyay. Robustly separating the arithmetic monotone hierarchy via graph inner-product. In Anuj Dawar and Venkatesan Guruswami, editors, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2022, December 18-20, 2022, IIT Madras, Chennai, India, volume 250 of LIPIcs, pages 12:1-12:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2022.12.
  6. S. B. Gashkov and I. S. Sergeev. A method for deriving lower bounds for the complexity of monotone arithmetic circuits computing real polynomials. Sbornik. Mathematics, 203(10), 2012. Google Scholar
  7. S.B. Gashkov. The complexity of monotone computations of polynomials. Moscow University Math Bulletin, 42(5):1-8, 1987. Google Scholar
  8. Joshua A. Grochow. Monotone projection lower bounds from extended formulation lower bounds. Theory of Computing, 13(18):1-15, 2017. URL: https://doi.org/10.4086/toc.2017.v013a018.
  9. Pavel Hrubes and Amir Yehudayoff. On isoperimetric profiles and computational complexity. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 89:1-89:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.89.
  10. Pavel Hrubeš and Amir Yehudayoff. Formulas are exponentially stronger than monotone circuits in non-commutative setting. In Proceedings of the 28th Conference on Computational Complexity, CCC 2013, K.lo Alto, California, USA, 5-7 June, 2013, pages 10-14. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/CCC.2013.11.
  11. Pavel Hrubeš and Amir Yehudayoff. Shadows of newton polytopes. In Valentine Kabanets, editor, 36th Computational Complexity Conference, CCC 2021, July 20-23, 2021, Toronto, Ontario, Canada (Virtual Conference), volume 200 of LIPIcs, pages 9:1-9:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.9.
  12. Mark Jerrum and Marc Snir. Some exact complexity results for straight-line computations over semirings. Journal of the ACM, 29(3):874-897, 1982. URL: https://doi.org/10.1145/322326.322341.
  13. O. M. Kasim-Zade. Arithmetic complexity of monotone polynomials. Theoretical Problems in Cybernetics. Abstracts of lectures, pages 68-69, 1986. Google Scholar
  14. Pascal Koiran and Sylvain Perifel. VPSPACE and a transfer theorem over the reals. Computational Complexity, 18(4):551-575, 2009. URL: https://doi.org/10.1007/s00037-009-0269-1.
  15. Pascal Koiran and Sylvain Perifel. Vpspace and a transfer theorem over the complex field. Theoretical Computer Science, 410(50):5244-5251, 2009. Mathematical Foundations of Computer Science (MFCS 2007). URL: https://doi.org/10.1016/j.tcs.2009.08.026.
  16. Pascal Koiran, Natacha Portier, Sébastien Tavenas, and Stéphan Thomassé. A τ-conjecture for newton polygons. Foundations of Computational Mathematics, 15:185-197, 2015. URL: https://doi.org/10.1007/s10208-014-9216-x.
  17. Meena Mahajan and B. V. Raghavendra Rao. Small space analogues of valiant’s classes and the limitations of skew formulas. Computational Complexity, 22(1):1-38, 2013. URL: https://doi.org/10.1007/s00037-011-0024-2.
  18. Guillaume Malod. Succinct algebraic branching programs characterizing non-uniform complexity classes. In Fundamentals of Computation Theory - 18th International Symposium, FCT 2011, Oslo, Norway, August 22-25, 2011. Proceedings, pages 205-216, 2011. URL: https://doi.org/10.1007/978-3-642-22953-4_18.
  19. Bruno Poizat. A la recherche de la definition de la complexite d'espace pour le calcul des polynomes a la maniere de valiant. J. Symb. Log., 73(4):1179-1201, 2008. URL: https://doi.org/10.2178/jsl/1230396913.
  20. Ran Raz and Amir Yehudayoff. Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors. Journal of Computer and System Sciences, 77(1):167-190, 2011. URL: https://doi.org/10.1016/j.jcss.2010.06.013.
  21. Claus-Peter Schnorr. A lower bound on the number of additions in monotone computations. Theor. Comput. Sci., 2(3):305-315, 1976. URL: https://doi.org/10.1016/0304-3975(76)90083-9.
  22. Eli Shamir and Marc Snir. Lower bounds on the number of multiplications and the number of additions in monotone computations. IBM Thomas J. Watson Research Division, 1977. Google Scholar
  23. Eli Shamir and Marc Snir. On the depth complexity of formulas. Math. Syst. Theory, 13:301-322, 1980. URL: https://doi.org/10.1007/BF01744302.
  24. Srikanth Srinivasan. Strongly exponential separation between monotone VP and monotone VNP. ACM Transactions on Computing Theory, 12(4):23:1-23:12, 2020. URL: https://doi.org/10.1145/3417758.
  25. Leslie G. Valiant. Negation can be exponentially powerful. Theor. Comput. Sci., 12:303-314, 1980. URL: https://doi.org/10.1016/0304-3975(80)90060-2.
  26. Amir Yehudayoff. Separating monotone VP and VNP. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 425-429. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316311.
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