Skip to main content
SpringerLink
Log in

On the thresholds in linear and nonlinear Boolean equations

  • Interdisciplinary Physics
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

We introduce a generalized XORSAT model, named as Massive Algebraic System (Hereafter abbreviated as MAS) consisting of linear and nonlinear Boolean equations. Through adjusting the proportion of nonlinear equations, denoted by p, this MAS model smoothly interpolates between XORSAT (p = 0) and MAS-nonlinear (p = 1). We conduct a systematic and complete study about a series of phase transitions in the space of solutions at given p and also present how the phase diagram evolves with the increase of p. First of all, using the probabilistic method and energetic 1RSB cavity method, we compute the satisfiability thresholds for any given p ∈ [0,1) and determine a region where the satisfaction of problem all depends on its subproblem MAS-nonlinear. Furthermore, we locate three important non-satisfiability transitions, i.e. clustering, condensation and freezing, using entropic 1RSB cavity method, and find the space of solution undergoing different phase transition processes with the increase of p.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S.A. Cook, in Proceedings of the 3rd annual ACM Symposium on the Theory of Computing (ACM, New York, 1971), p. 151

  2. D.G. Mitchell, B. Selman, H.J. Levesque, in Proceedings of the 10th National Conference on Artificial Intelligence (AAAI Press, Menlo Park California, 1992), p. 45

  3. S. Kirkpatrick, B. Selman, Science 264, 1297 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  4. R. Monasson, R. Zecchina, S. Kirpatrick, B. Selman, Nature 400, 133 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  5. N. Creignou, H. Daudé, Discr. Math. 309, 2085 (2009)

    Article  MATH  Google Scholar 

  6. E. Friedgut, J. Amer. Math. Soc. 12, 1017 (1999). An appendix by J. Bourgain

    Article  MATH  MathSciNet  Google Scholar 

  7. W. Barthel, A.K. Hartmann, M. Weigt, Phys. Rev. E 67, 066104 (2003)

    Article  ADS  Google Scholar 

  8. G. Semerjian, R. Monasson, Phys. Rev. E 67, 066103 (2003)

    Article  ADS  Google Scholar 

  9. J. Ardelius, E. Aurell, Phys. Rev. E 74, 037702 (2006)

    Article  ADS  Google Scholar 

  10. F. Altarelli, e-print arXiv:0801.2858v1

  11. D. Achlioptas, C. Moore, in Proceedings of the 43rd annual IEEE Symposium on Foundations of Computer Science (2002), p. 779

  12. D. Achlioptas, Y. Peres, J. Am. Math. Soc. 17, 947 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. D. Achlioptas, A. Naor, Y. Peres, Nature 435, 759 (2005)

    Article  ADS  Google Scholar 

  14. D. Achlioptas, C. Moore, SIAM J. Comput. 36, 740 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. M.T. Chao, J. Franco, SIAM J. Comput. 15, 1106 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  16. A.M. Frieze, S. Suen, J. Algorithms 20, 312 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. D. Achlioptas, Theor. Comp. Sci. 265, 159 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. V. Kalapala, C. Moore, CJTCS 2008, 5 (2008)

    MathSciNet  Google Scholar 

  19. K. Xu, W. Li, Journal of Artifitial Intelligence Research 12, 93 (2000)

    MATH  Google Scholar 

  20. D. Achlioptas, L.M. Kirousis, E. Kranakis, D. Krizanc, Theor. Comp. Sci. 265, 109 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. J. Raymond, A. Sportiello, L. Zdeboroví, Phys. Rev. E 76, 011101 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  22. D. Achlioptas, A.D. Chtcherba, G. Istrate, C. Moore, SODA 721 (2001)

  23. M. Mézard, G. Parisi, Eur. Phys. J. B 20, 217 (2001)

    Article  ADS  Google Scholar 

  24. M. Mézard, G. Parisi, J. Stat. Phys. 111, 1 (2003)

    Article  MATH  Google Scholar 

  25. G. Biroli, R. Monasson, M. Weight, Eur. Phys. J. B 14, 551 (2000)

    Article  ADS  Google Scholar 

  26. M. Mézard, G. Parisi, R. Zecchina, Science 297, 812 (2002)

    Article  ADS  Google Scholar 

  27. M. Mézard, R. Zecchina, Phys. Rev. E 66, 056126 (2002)

    Article  ADS  Google Scholar 

  28. S. Mertens, M. Mézard, R. Zecchina, Random Struct. Algorithms 28, 340 (2006)

    Article  MATH  Google Scholar 

  29. A. Braunstein, M. Mézard, R. Zecchina, Random Struct. Algorithms 27, 201 (2005)

    Article  MATH  Google Scholar 

  30. F. Krzakala, A. Montanari, F. Ricci-Tersenghi, G. Semerjian, L. Zdeboroví, Proc. Natl. Acad. Sci. 104, 10318 (2007)

    Article  MATH  ADS  Google Scholar 

  31. L. Zdeboroví, F. Krzakala, Phys. Rev. E 76, 031131 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  32. A. Montanari, F. Ricci-Tersenghi, G. Semerjian, J. Stat. Mech. P04004 (2008)

  33. M. Mézard, M. Palassini, O. Rivoire, Phys. Rev. Lett. 95, 200202 (2005)

    Article  Google Scholar 

  34. M. Mézard, F. Ricci-Tersenghi, R. Zecchina, J. Stat. Phys. 111, 505 (2003)

    Article  MATH  Google Scholar 

  35. S. Cocco, O. Dubois, J. Mandler, R. Monasson, Phys. Rev. Lett. 90, 047205 (2003)

    Article  ADS  Google Scholar 

  36. T. Mora, M. Mézard, J. Stat. Mech. P10007 (2006)

  37. W. Wei, B.H. Guo, Z.M. Zheng, in Proceeding of MACIS (Paris, France, 2007)

  38. W. Wei, B.H. Guo, Z.M. Zheng, J. Stat. Mech. P02010 (2009)

  39. E.N. Maneva, E. Mossel, M.J. Wainwritght, J. ACM 54, 2007

  40. L. Zdeboroví, Acta Physica Slovaca 59, 169 (2009)

    Google Scholar 

  41. R.C. Robinson, in An Introduction to Dynamical System: Continuous and Discrete (Pearson Education, 2004)

  42. F. Altarelli, A. Braunstein, J. Realpe-Gomez, R. Zecchina, J. Stat. Mech. P07002 (2009)

  43. M. Weight, H.J. Zhou, Phys. Rev. E 74, 046110 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  44. A. Montanari, G. Parisi, F. Ricci-Tersenghi, J. Phys. A: Math. Gen. 37, 2073 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  45. A. Montanari, F. Ricci-Tersenghi, Eur. Phys. J. B 33, 339 (2003)

    Article  ADS  Google Scholar 

  46. M. Mézard, A. Montanari, in Information, Physics and Computation (Oxford University Press, Oxford, 2009)

  47. L. Zdeboroví, M. Mézard, Phys. Rev. Lett. 101, 078702 (2008)

    Article  ADS  Google Scholar 

  48. L. Zdeboroví, M. Mézard, J. Stat. Mech. P12004 (2008)

  49. A. Montanari, G. Semerjian, Phys. Rev. Lett. 94, 247201 (2005)

    Article  ADS  Google Scholar 

  50. G. Semerjian, J. Stat. Phys. 130, 251 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  51. A. Montanari, G. Semerjian, J. Stat. Phys. 124, 103 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  52. B.H. Guo, W. Wei, Y.F. Sun, Z.M. Zheng, In preparation

  53. M. Zito, Ann. Appl. Probab. 5, 1217 (1995)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LMIB and School of Mathematics and Systems Science, Beihang University, 100191, Beijing, P.R. China

    Y. F. Sun, B. H. Guo, W. Wei & Z. M. Zheng

Authors
  1. Y. F. Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. H. Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. W. Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Z. M. Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Y. F. Sun.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sun, Y., Guo, B., Wei, W. et al. On the thresholds in linear and nonlinear Boolean equations. Eur. Phys. J. B 76, 123–146 (2010). https://doi.org/10.1140/epjb/e2010-00173-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjb/e2010-00173-7

Keywords

Search

Navigation