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Monotone Classes Beyond VNP

Authors Prerona Chatterjee , Kshitij Gajjar , Anamay Tengse

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

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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.

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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

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.

References

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