Document Open Access Logo

Lower Bounds for DeMorgan Circuits of Bounded Negation Width

Authors Stasys Jukna, Andrzej Lingas



PDF
Thumbnail PDF

File

LIPIcs.STACS.2019.41.pdf
  • Filesize: 496 kB
  • 17 pages

Document Identifiers

Author Details

Stasys Jukna
  • Institute of Computer Science, Goethe University Frankfurt, Frankfurt am Main, Germany
  • Institute of Data Science and Digital Technologies, Vilnius University, Lithuania
Andrzej Lingas
  • Department of Computer Science, Lund University, Box 118, 22100 Lund, Sweden

Cite AsGet BibTex

Stasys Jukna and Andrzej Lingas. Lower Bounds for DeMorgan Circuits of Bounded Negation Width. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 41:1-41:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.STACS.2019.41

Abstract

We consider Boolean circuits over {or, and, neg} with negations applied only to input variables. To measure the "amount of negation" in such circuits, we introduce the concept of their "negation width". In particular, a circuit computing a monotone Boolean function f(x_1,...,x_n) has negation width w if no nonzero term produced (purely syntactically) by the circuit contains more than w distinct negated variables. Circuits of negation width w=0 are equivalent to monotone Boolean circuits, while those of negation width w=n have no restrictions. Our motivation is that already circuits of moderate negation width w=n^{epsilon} for an arbitrarily small constant epsilon>0 can be even exponentially stronger than monotone circuits. We show that the size of any circuit of negation width w computing f is roughly at least the minimum size of a monotone circuit computing f divided by K=min{w^m,m^w}, where m is the maximum length of a prime implicant of f. We also show that the depth of any circuit of negation width w computing f is roughly at least the minimum depth of a monotone circuit computing f minus log K. Finally, we show that formulas of bounded negation width can be balanced to achieve a logarithmic (in their size) depth without increasing their negation width.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • Boolean circuits
  • monotone circuits
  • lower bounds

Metrics

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

References

  1. N. Alon and R. Boppana. The monotone circuit complexity of boolean functions. Combinatorica, 7(1):1-22, 1987. Google Scholar
  2. K. Amano and A. Maruoka. The Potential of the Approximation Method. SIAM J. Comput., 33(2):433-447, 2004. Google Scholar
  3. E. Blais, C.L. Canonne, I.C. Oliveira, R.A. Servedio, and L.Y. Tan. Learning Circuits with few Negations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, volume 40 of LIPIcs, pages 512-527, 2015. Google Scholar
  4. R.P. Brent, D.J. Kuck, and K. Maruyama. The parallel evaluation of arithmetic expressions without divisions,. IEEE Trans. Computers, C-22:523-534, 1973. Google Scholar
  5. Y. Crama and P.L. Hammer, editors. Boolean Functions: Theory, Algorithms, and Applications, volume 142 of Encyclopedia of Mathematics and Its Applications. Cambridge University Pess, 2011. Google Scholar
  6. P.E. Dunne. Relationship between monotone and non-monotone network complexity. In M.S. Paterson, editor, Boolean Function Complexity, volume 169 of London Math. Soc. Lect. Note Series, pages 1-24. Cambridge University Press, 1992. Google Scholar
  7. F. Le Gall. Powers of Tensors and Fast Matrix Multiplication. In Proc. of 39th Int. Symp. on Symbolic and Algebraic Computation, pages 296-303, 2014. Google Scholar
  8. M. Grötschel, L. Lovász, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1(2):169-197, 1981. Google Scholar
  9. S. Guo, T. Malkin, I.C. Oliveira, and A. Rosen. The Power of Negations in Cryptography. In Proc. of 12th Theory of Cryptography Conference, TCC, volume 9014 of Lect. Notes in Comput. Sci., pages 36-65. Springer, 2015. Google Scholar
  10. D. Harnik and R. Raz. Higher lower bounds on monotone size. In Proc. 32nd Ann. ACM Symp. on Theory of Computing, pages 378-387, 2000. Google Scholar
  11. J.E. Hopcroft and R.M. Karp. An n^5/2 algorithm for maximum matching in bipartite graphs. SIAM J. Comput., 2:225-231, 1973. Google Scholar
  12. S. Jukna. Combinatorics of monotone computations. Combinatorica, 19(1):65-85, 1999. Preliminary versions in: ECCC Report Nr. 26, 1996, and in Proc. of 12th Ann. IEEE Conf. on Comput. Complexity. 1997, pp. 223-238. Google Scholar
  13. S. Jukna. Boolean Function Complexity: Advances and Frontiers. Springer-Verlag, 2012. Google Scholar
  14. V.M. Khrapchenko. On a relation between the complexity and the depth of formulas. In Methods of Discrete Analysis in Synthesis of Control Systems, volume 32, pages 76-94. Institute of Mathematics. Novosibirsk, 1978. (In Russian). Google Scholar
  15. M. Kochol. Efficient monotone circuits for threshold functions. Inf. Process. Lett., 32:121-122, 1989. Google Scholar
  16. S. Koroth and J. Sarma. Depth Lower Bounds against Circuits with Sparse Orientation. Fundam. Inform., 152(2):123-144, 2017. Google Scholar
  17. A. Lingas. Small Normalized Boolean Circuits for Semi-disjoint Bilinear Forms Require Logarithmic Conjunction-depth. In Proc. of 33rd Comput. Complexity Conf., volume 102 of LIPIcs, pages 26:1-26:10, 2018. Extended version in: ECCC Report Nr. 108, 2018. Google Scholar
  18. E.A. Okol'nishnikova. On the influence of one type of restrictions to the complexity of combinational circuits. Diskrete Analysis, 36:46-58, 1981. (In Russian). Google Scholar
  19. T. Pitassi and R. Robere. Strongly exponential lower bounds for monotone computation. In Proc. 49th Ann. ACM Symp. on Theory of Computing, STOC, pages 1246-1255, 2017. Google Scholar
  20. A. Potechin. Bounds on Monotone Switching Networks for Directed Connectivity. J. ACM, 64(4):29:1-29:48, 2017. Google Scholar
  21. R. Raz and Wigderson. Probabilistic Communication Complexity of Boolean Relations. In Proc. of 30th Ann. Symp. on Foundations of Computer Sci., FOCS, pages 562-567, 1989. Google Scholar
  22. R. Raz and A. Wigderson. Monotone circuits for matching require linear depth. JACM, 39(3):736-744, 1992. Google Scholar
  23. A.A. Razborov. Lower bounds for the monotone complexity of some boolean functions. Soviet Math. Dokl., 31:354-357, 1985. Google Scholar
  24. A.A. Razborov. Lower bounds on monotone complexity of the logical permanent. Math. Notes of the Acad. of Sci. of the USSR, 37(6):485-493, 1985. Google Scholar
  25. A.A. Razborov. Applications of matrix methods to the theory of lower bounds in computational complexity. Combinatorica, 10(1):81-93, 1990. Google Scholar
  26. B. Rossman. Correlation Bounds Against Monotone NC^1. In Proc. of 30th Comput. Complexity Conf., volume 33 of LIPIcs, pages 392 - -411, 2015. Google Scholar
  27. P.M. Spira. On time-hardware complexity tradeoffs for Boolean functions. In Proc. of 4th Hawaii Symp. on System Sciences, pages 525-527. Western Periodicals Company, North Hollywood, 1971. Google Scholar
  28. V. Strassen. Gaussian elimination is not optimal. Numer. Math., 13:354-356, 1969. Google Scholar
  29. É. Tardos. The gap between monotone and non-monotone circuit complexity is exponential. Combinatorica, 7(4):141-142, 1987. Google Scholar
  30. I. Wegener. Relating monotone formula size and monotone depth of Boolean functions. Infrom. Process. Letters, 16:41-42, 1983. Google Scholar
  31. I. Wegener. The complexity of Boolean functions. Wiley-Teubner, 1987. Google Scholar
  32. V. Vassilevska Williams. Multiplying matrices faster than Coppersmith-Winograd. In Proc. of 44th Symp. on Theory of Comput., STOC, pages 887-898, 2012. Google Scholar
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