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Lower bounds for Sum and Sum of Products of Read-once Formulas

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Published:02 April 2019Publication History
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Abstract

We study limitations of polynomials computed by depth-2 circuits built over read-once formulas (ROFs). In particular:

• We prove a 2Ω(n) lower bound for the sum of ROFs computing the 2n-variate polynomial in VP defined by Raz and Yehudayoff [21].

• We obtain a 2Ω(√n) lower bound on the size of Σ Π[n1/15] arithmetic circuits built over restricted ROFs of unbounded depth computing the permanent of an n × n matrix (superscripts on gates denote bound on the fan-in). The restriction is that the number of variables with + gates as a parent in a proper sub formula of the ROF has to be bounded by √n. This proves an exponential lower bound for a subclass of possibly non-multilinear formulas of unbounded depth computing the permanent polynomial.

• We also show an exponential lower bound for the above model against a polynomial in VP.

• Finally, we observe that the techniques developed yield an exponential lower bound on the size of ΣΠ[N1/30] arithmetic circuits built over syntactically multi-linear ΣΠΣ[N1/4] arithmetic circuits computing a product of variable disjoint linear forms on N variables, where the superscripts on gates denote bound on the fan-in.

Our proof techniques are built on the measure developed by Kumar et al. [14] and are based on a non-trivial analysis of ROFs under random partitions. Further, our results exhibit strengths and provide more insight into the lower bound techniques introduced by Raz [19].

References

  1. Manindra Agrawal and V. Vinay. 2008. Arithmetic circuits: A chasm at depth four. In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS’08). 67--75. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Walter Baur and Volker Strassen. 1983. The complexity of partial derivatives. Theor. Comput. Sci. 22 (1983), 317--330.Google ScholarGoogle ScholarCross RefCross Ref
  3. Peter Bürgisser. 2013. Completeness and Reduction in Algebraic Complexity Theory. Vol. 7. Springer Science 8 Business Media.Google ScholarGoogle Scholar
  4. Devdatt Dubhashi and Alessandro Panconesi. 2009. Concentration of Measure for the Analysis of Randomized Algorithms (1st ed.). Cambridge University Press, New York, NY. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Michael Forbes. 2014. Polynomial Identity Testing of Read-once Oblivious Algebraic Branching Programs. PhD Thesis, Massachusetts Institute of Technology.Google ScholarGoogle Scholar
  6. Michael A. Forbes and Amir Shpilka. 2013. Quasipolynomial-time identity testing of non-commutative and read-once oblivious algebraic branching programs. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS’13). 243--252. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Dima Grigoriev and Marek Karpinski. 1998. An exponential lower bound for depth-3 arithmetic circuits. In Proceedings of the Annual ACM Symposium on Theory of Computing (STOC’98). 577--582. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. 2014. Approaching the chasm at depth four. J. ACM 61, 6 (2014), 33:1--33:16. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Neeraj Kayal. 2012. An exponential lower bound for the sum of powers of bounded degree polynomials. Electron. Colloq. Comput. Complex. 19 (2012), 81.Google ScholarGoogle Scholar
  10. Neeraj Kayal, Nutan Limaye, Chandan Saha, and Srikanth Srinivasan. 2014. Super-polynomial lower bounds for depth-4 homogeneous arithmetic formulas. In Proceedings of the Annual ACM Symposium on Theory of Computing (STOC’14). 119--127. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Neeraj Kayal and Chandan Saha. 2015. Multi-k-ic depth three circuit lower bound. In Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS’15). 527--539.Google ScholarGoogle Scholar
  12. Neeraj Kayal, Chandan Saha, and Sébastien Tavenas. 2016. An almost cubic lower bound for depth three arithmetic circuits. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP’16). 33:1--33:15.Google ScholarGoogle Scholar
  13. Pascal Koiran. 2012. Arithmetic circuits: The chasm at depth four gets wider. Theor. Comput. Sci. 448 (2012), 56--65. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Mrinal Kumar, Gaurav Maheshwari, and Jayalal Sarma. 2016. Arithmetic circuit lower bounds via maximum-rank of partial derivative matrices. Trans. Comput. Theory 8, 3 (2016), 8. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Mrinal Kumar and Shubhangi Saraf. 2014. On the power of homogeneous depth-4 arithmetic circuits. In Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS’14). 364--373. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Meena Mahajan and Anuj Tawari. 2018. Sums of read-once formulas: How many summands are necessary? Theor. Comput. Sci. 708 (2018), 34--45.Google ScholarGoogle ScholarCross RefCross Ref
  17. Noam Nisan. 1991. Lower bounds for non-commutative computation (extended abstract). In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing (STOC’91). 410--418. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. Noam Nisan and Avi Wigderson. 1997. Lower bounds on arithmetic circuits via partial derivatives. Comput. Complex. 6, 3 (1997), 217--234. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Ran Raz. 2004. Multi-linear formulas for permanent and determinant are of super-polynomial size. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC’04). 633--641. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Ran Raz. 2009. Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM 56, 2 (2009). Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Ran Raz and Amir Yehudayoff. 2008. Balancing syntactically multilinear arithmetic circuits. Comput. Complex. 17, 4 (2008), 515--535. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Ran Raz and Amir Yehudayoff. 2009. Lower bounds and separations for constant depth multilinear circuits. Comput. Complex. 18, 2 (2009), 171--207.Google ScholarGoogle ScholarCross RefCross Ref
  23. Amir Shpilka and Ilya Volkovich. 2015. Read-once polynomial identity testing. Comput. Complex. 24, 3 (2015), 477--532. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Amir Shpilka and Avi Wigderson. 2001. Depth-3 arithmetic circuits over fields of characteristic zero. Comput. Complex. 10, 1 (2001), 1--27. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Sébastien Tavenas. 2015. Improved bounds for reduction to depth-4 and depth-3. Info. Comput. 240 (2015), 2--11. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Iddo Tzameret. 2008. Studies in Algebraic and Propositional Proof Complexity. Ph.D Thesis (2008), 33. Retrieved from http://www.cs.rhul.ac.uk/home/tzameret/Iddo-PhD-thesis.pdf.Google ScholarGoogle Scholar
  27. Leslie G. Valiant. 1979. Completeness classes in algebra. In Proceedings of the Annual ACM Symposium on Theory of Computing (STOC’79). 249--261. Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. J. H. van Lint and R. M. Wilson. 2001. A Course in Combinatorics (2nd ed.). Cambridge University Press.Google ScholarGoogle Scholar
  29. Ilya Volkovich. 2016. Characterizing arithmetic read-once formulae. Trans. Comput. Theory 8, 1 (2016), 2. Google ScholarGoogle ScholarDigital LibraryDigital Library

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 11, Issue 2
        June 2019
        169 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3312746
        Issue’s Table of Contents

        Copyright © 2019 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 2 April 2019
        • Accepted: 1 December 2018
        • Revised: 1 June 2018
        • Received: 1 December 2016
        Published in toct Volume 11, Issue 2

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