Document Open Access Logo

Functional Lower Bounds for Arithmetic Circuits and Connections to Boolean Circuit Complexity

Authors Michael A. Forbes, Mrinal Kumar, Ramprasad Saptharishi

PDF
Thumbnail PDF

File

LIPIcs.CCC.2016.33.pdf
  • Filesize: 0.6 MB
  • 19 pages

Document Identifiers

Author Details

Michael A. Forbes
Mrinal Kumar
Ramprasad Saptharishi

Cite AsGet BibTex

Michael A. Forbes, Mrinal Kumar, and Ramprasad Saptharishi. Functional Lower Bounds for Arithmetic Circuits and Connections to Boolean Circuit Complexity. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 33:1-33:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.CCC.2016.33

Abstract

We say that a circuit C over a field F {functionally} computes a polynomial P in F[x_1, x_2, ..., x_n] if for every x in {0,1}^n we have that C(x) = P(x). This is in contrast to syntactically computing P, when C = P as formal polynomials. In this paper, we study the question of proving lower bounds for homogeneous depth-3 and depth-4 arithmetic circuits for functional computation. We prove the following results: 1. Exponential lower bounds for homogeneous depth-3 arithmetic circuits for a polynomial in VNP. 2. Exponential lower bounds for homogeneous depth-4 arithmetic circuits with bounded individual degree for a polynomial in VNP. Our main motivation for this line of research comes from our observation that strong enough functional lower bounds for even very special depth-4 arithmetic circuits for the Permanent imply a separation between #P and ACC0. Thus, improving the second result to get rid of the bounded individual degree condition could lead to substantial progress in boolean circuit complexity. Besides, it is known from a recent result of Kumar and Saptharishi [Kumar/Saptharishi, ECCC 2015] that over constant sized finite fields, strong enough {average case} functional lower bounds for homogeneous depth-4 circuits imply superpolynomial lower bounds for homogeneous depth-5 circuits. Our proofs are based on a family of new complexity measures called shifted evaluation dimension, and might be of independent interest.
Keywords
  • boolean circuits
  • arithmetic circuits
  • lower bounds
  • functional computation

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Sanjeev Arora and Boaz Barak. Computational Complexity: A Modern Approach. Cambridge University Press, New York, NY, USA, 1st edition, 2009. Google Scholar
  2. Richard Beigel and Jun Tarui. On acc. Computational Complexity, 4(4):350-366, 1994. URL: http://dx.doi.org/10.1007/BF01263423.
  3. Michael A. Forbes, Amir Shpilka, Iddo Tzameret, and Avi Wigderson. Proof complexity lower bounds from algebraic circuit complexity. Manuscript, 2015. Google Scholar
  4. Hervé Fournier, Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. Lower bounds for depth 4 formulas computing iterated matrix multiplication. In Proceedings of the \nth\intcalcSub20141968 Annual ACM Symposium on Theory of Computing (STOC 2014), pages 128-135, 2014. URL: http://dx.doi.org/10.1145/2591796.2591824.
  5. Joshua A. Grochow and Toniann Pitassi. Circuit complexity, proof complexity, and polynomial identity testing. In Proceedings of the \ifnumcomp2014<1966 \nth\intcalcSub20141959 Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 2014) \ifnumcomp2014<1975 \nth\intcalcSub20141959 Annual Symposium on Switching and Automata Theory (SWAT 2014) \nth\intcalcSub20141959 Annual IEEE Symposium on Foundations of Computer Science (FOCS 2014), pages 110-119, 2014. Full version at http://arxiv.org/abs/abs/1404.3820. URL: http://dx.doi.org/10.1109/FOCS.2014.20.
  6. Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Approaching the chasm at depth four. Journal of the ACM, 61(6):33:1-33:16, 2014. Preliminary version in the \ifnumcomp2013<1996 \nth\intcalcSub20131985 Annual Structure in Complexity Theory Conference (Structures 2013) \ifnumcomp2013<2015 \nth\intcalcSub20131985 Annual IEEE Conference on Computational Complexity (CCC 2013) \nth\intcalcSub20131985 Annual Computational Complexity Conference (CCC 2013). URL: http://dx.doi.org/10.1145/2629541.
  7. Neeraj Kayal, Nutan Limaye, Chandan Saha, and Srikanth Srinivasan. An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Circuits. In Proceedings of the \ifnumcomp2014<1966 \nth\intcalcSub20141959 Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 2014) \ifnumcomp2014<1975 \nth\intcalcSub20141959 Annual Symposium on Switching and Automata Theory (SWAT 2014) \nth\intcalcSub20141959 Annual IEEE Symposium on Foundations of Computer Science (FOCS 2014), 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.15.
  8. Neeraj Kayal, Chandan Saha, and Sébastien Tavenas. On the size of homogeneous and of depth four formulas with low individual degree. Electronic Colloquium on Computational Complexity (ECCC), 2015. http://eccc.hpi-web.de/report/\ifnumcomp15>93192015/181/. URL: http://eccc.hpi-web.de/report/2015/181/.
  9. Mrinal Kumar and Ramprasad Saptharishi. An exponential lower bound for homogeneous depth-5 circuits over finite fields. Electronic Colloquium on Computational Complexity (ECCC), 2015. http://eccc.hpi-web.de/report/\ifnumcomp15>93192015/109/. URL: http://eccc.hpi-web.de/report/2015/109/.
  10. Mrinal Kumar and Shubhangi Saraf. On the power of homogeneous depth 4 arithmetic circuits. In Proceedings of the \ifnumcomp2014<1966 \nth\intcalcSub20141959 Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 2014) \ifnumcomp2014<1975 \nth\intcalcSub20141959 Annual Symposium on Switching and Automata Theory (SWAT 2014) \nth\intcalcSub20141959 Annual IEEE Symposium on Foundations of Computer Science (FOCS 2014), 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.46.
  11. Noam Nisan and Avi Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Computational Complexity, 6(3):217-234, 1997. URL: http://dx.doi.org/10.1007/BF01294256.
  12. Ran Raz. Separation of multilinear circuit and formula size. Theory of Computing, 2(1):121-135, 2006. Preliminary version in the \ifnumcomp2004<1966 \nth\intcalcSub20041959 Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 2004) \ifnumcomp2004<1975 \nth\intcalcSub20041959 Annual Symposium on Switching and Automata Theory (SWAT 2004) \nth\intcalcSub20041959 Annual IEEE Symposium on Foundations of Computer Science (FOCS 2004). URL: http://dx.doi.org/10.4086/toc.2006.v002a006.
  13. Ran Raz. Multi-linear formulas for permanent and determinant are of super-polynomial size. Journal of the ACM, 56(2), 2009. Preliminary version in the \nth\intcalcSub20041968 Annual ACM Symposium on Theory of Computing (STOC 2004). URL: http://dx.doi.org/10.1145/1502793.1502797.
  14. Ran Raz and Amir Yehudayoff. Lower bounds and separations for constant depth multilinear circuits. Computational Complexity, 18(2):171-207, 2009. Preliminary version in the \ifnumcomp2008<1996 \nth\intcalcSub20081985 Annual Structure in Complexity Theory Conference (Structures 2008) \ifnumcomp2008<2015 \nth\intcalcSub20081985 Annual IEEE Conference on Computational Complexity (CCC 2008) \nth\intcalcSub20081985 Annual Computational Complexity Conference (CCC 2008). URL: http://dx.doi.org/10.1007/s00037-009-0270-8.
  15. Alexander A. Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematical notes of the Academy of Sciences of the USSR, 41(4):333-338, 1987. URL: http://dx.doi.org/10.1007/BF01137685.
  16. Roman Smolensky. Algebraic methods in the theory of lower bounds for boolean circuit complexity. In Proceedings of the \nth\intcalcSub19871968 Annual ACM Symposium on Theory of Computing (STOC 1987), pages 77-82, 1987. URL: http://dx.doi.org/10.1145/28395.28404.
  17. Ryan Williams. Non-uniform ACC Circuit Lower Bounds. In Proceedings of the \ifnumcomp2011<1996 \nth\intcalcSub20111985 Annual Structure in Complexity Theory Conference (Structures 2011) \ifnumcomp2011<2015 \nth\intcalcSub20111985 Annual IEEE Conference on Computational Complexity (CCC 2011) \nth\intcalcSub20111985 Annual Computational Complexity Conference (CCC 2011), pages 115-125, 2011. Google Scholar
  18. Andrew Chi-Chih Yao. Separating the polynomial-time hierarchy by oracles. In Proceedings of the \ifnumcomp1985<1966 \nth\intcalcSub19851959 Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 1985) \ifnumcomp1985<1975 \nth\intcalcSub19851959 Annual Symposium on Switching and Automata Theory (SWAT 1985) \nth\intcalcSub19851959 Annual IEEE Symposium on Foundations of Computer Science (FOCS 1985), pages 1-10, Oct 1985. URL: http://dx.doi.org/10.1109/SFCS.1985.49.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023 Schloss Dagstuhl - LZI GmbH Imprint Privacy Contact