Learning Circuits with few Negations

Authors Eric Blais, Clément L. Canonne, Igor C. Oliveira, Rocco A. Servedio, Li-Yang Tan



PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2015.512.pdf
  • Filesize: 0.53 MB
  • 16 pages

Document Identifiers

Author Details

Eric Blais
Clément L. Canonne
Igor C. Oliveira
Rocco A. Servedio
Li-Yang Tan

Cite As Get BibTex

Eric Blais, Clément L. Canonne, Igor C. Oliveira, Rocco A. Servedio, and Li-Yang Tan. Learning Circuits with few Negations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 512-527, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.512

Abstract

Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions.  We study this generalization of monotonicity from the vantage point of learning theory, establishing nearly matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A.A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions).

Subject Classification

Keywords
  • Boolean functions
  • monotonicity
  • negations
  • PAC learning

Metrics

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

References

  1. K. Amano and A. Maruoka. On learning monotone Boolean functions under the uniform distribution. In International Conference on Algorithmic Learning Theory (ALT), pages 57-68, 2002. Google Scholar
  2. K. Amano and A. Maruoka. A Superpolynomial Lower Bound for a Circuit Computing the Clique Function with At Most (1/6) log log n Negation Gates. SIAM Journal on Computing, 35(1):201-216, 2005. Google Scholar
  3. D. Angluin. Queries and concept learning. Machine Learning, 2:319-342, 1988. Google Scholar
  4. E. Blais and L-Y. Tan. Approximating Boolean functions with depth-2 circuits. In Conference on Computational Complexity (CCC), pages 74-85, 2013. Google Scholar
  5. A. Blum, C. Burch, and J. Langford. On learning monotone Boolean functions. In Symposium on Foundations of Computer Science (FOCS), pages 408-415, 1998. Google Scholar
  6. N. Bshouty and C. Tamon. On the Fourier spectrum of monotone functions. Journal of the ACM, 43(4):747-770, 1996. Google Scholar
  7. V. Feldman, H. K. Lee, and R. A. Servedio. Lower bounds and hardness amplification for learning shallow monotone formulas. Journal of Machine Learning Research - Proceedings Track, 19:273-292, 2011. Google Scholar
  8. O. Goldreich and R. Izsak. Monotone circuits: One-way functions versus pseudorandom generators. Theory of Computing, 8(1):231-238, 2012. Google Scholar
  9. S. Guo and I. Komargodski. Negation-limited formulas. Technical Report 22(26), Electronic Colloquium on Computational Complexity (ECCC), 2015. Google Scholar
  10. S. Guo, T. Malkin, I. C. Oliveira, and A. Rosen. The power of negations in cryptography. In Theory of Cryptography Conference (TCC), pages 36-65, 2015. Google Scholar
  11. M. Kearns and L. Valiant. Cryptographic limitations on learning Boolean formulae and finite automata. Journal of the ACM, 41(1):67-95, 1994. Google Scholar
  12. A. D. Korshunov. Monotone Boolean functions. Russian Mathematical Surveys (Uspekhi Matematicheskikh Nauk), 58(5):929-1001, 2003. Google Scholar
  13. N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform and learnability. Journal of the ACM, 40(3):607-620, 1993. Google Scholar
  14. A. A. Markov. On the inversion complexity of systems of functions. Doklady Akademii Nauk SSSR, 116:917-919, 1957. English translation in [Markov, 1958]. Google Scholar
  15. A. A. Markov. On the inversion complexity of a system of functions. Journal of the ACM, 5(4):331-334, October 1958. Google Scholar
  16. H. Morizumi. Limiting negations in formulas. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 701-712, 2009. Google Scholar
  17. H. Morizumi. Limiting negations in non-deterministic circuits. Theoretical Computer Science, 410(38-40):3988-3994, 2009. Google Scholar
  18. E. Mossel and R. O'Donnell. On the noise sensitivity of monotone functions. Random Structures and Algorithms, 23(3):333-350, 2003. Google Scholar
  19. R. O'Donnell. Computational applications of noise sensitivity. PhD thesis, MIT, June 2003. Google Scholar
  20. R. O'Donnell and R. Servedio. Learning monotone decision trees in polynomial time. SIAM Journal on Computing, 37(3):827-844, 2007. Google Scholar
  21. R. O'Donnell and K. Wimmer. KKL, Kruskal-Katona, and monotone nets. SIAM Journal on Computing, 42(6):2375-2399, 2013. Google Scholar
  22. Ran Raz and Avi Wigderson. Monotone circuits for matching require linear depth. Journal of the ACM, 39(3):736-744, 1992. Google Scholar
  23. A. Razborov. Lower bounds on the monotone complexity of some Boolean functions. Doklady Akademii Nauk SSSR, 281:798-801, 1985. English translation in: Soviet Mathematics Doklady 31:354-357, 1985. Google Scholar
  24. B. Rossman. Correlation bounds against monotone NC¹. In Conference on Computational Complexity (CCC), 2015. Google Scholar
  25. M. Santha and C. Wilson. Limiting negations in constant depth circuits. SIAM Journal on Computing, 22(2):294-302, 1993. Google Scholar
  26. R. Servedio. On learning monotone DNF under product distributions. Information and Computation, 193(1):57-74, 2004. Google Scholar
  27. S. Sung and K. Tanaka. Limiting Negations in Bounded-Depth Circuits: an Extension of Markov’s Theorem. In International Symposium on Algorithms and Computation (ISAAC), pages 108-116, 2003. Google Scholar
  28. M. Talagrand. How much are increasing sets positively correlated? Combinatorica, 16(2):243-258, 1996. 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