A Continuous-Discontinuous Second-Order Transition in the Satisfiability of Random Horn-SAT Formulas

  • Cristopher Moore
  • Gabriel Istrate
  • Demetrios Demopoulos
  • Moshe Y. Vardi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3624)


We compute the probability of satisfiability of a class of random Horn-SAT formulae, motivated by a connection with the nonemptiness problem of finite tree automata. In particular, when the maximum clause length is three, this model displays a curve in its parameter space along which the probability of satisfiability is discontinuous, ending in a second-order phase transition where it becomes continuous. This is the first case in which a phase transition of this type has been rigorously established for a random constraint satisfaction problem.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Achlioptas, D.: Lower Bounds for Random 3-SAT via Differential Equations. Theoretical Computer Science 265(1-2), 159–185 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Achlioptas, D., Chtcherba, A., Istrate, G., Moore, C.: The phase transition in 1-in-k SAT and NAE 3-SAT. In: Proc. 12th ACM-SIAM Symp. on Discrete Algorithms, pp. 721–722 (2001)Google Scholar
  3. 3.
    Achlioptas, D., Kirousis, L.M., Kranakis, E., Krizanc, D.: Rigorous results for random (2 + p)-SAT. Theor. Comput. Sci. 265(1-2), 109–129 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bench-Capon, T., Dunne, P.: A sharp threshold for the phase transition of a restricted satisfiability problem for Horn clauses. Journal of Logic and Algebraic Programming 47(1), 1–14 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Binney, J.J., Dowrick, N.J., Fisher, A.J., Newman, M.E.J.: The Theory of Critical Phenomena. Oxford University Press, Oxford (1992)zbMATHGoogle Scholar
  6. 6.
    Bollobás, B.: Random Graphs. Academic Press, London (1985)zbMATHGoogle Scholar
  7. 7.
    Cocco, S., Dubois, O., Mandler, J., Monasson, R.: Rigorous decimation-based construction of ground pure states for spin glass models on random lattices. Phys. Rev. Lett. 90 (2003)Google Scholar
  8. 8.
    Chvátal, V., Reed, B.: Mick gets some (the odds are on his side). In: Proc. 33rd IEEE Symp. on Foundations of Computer Science, pp. 620–627. IEEE Comput. Soc. Press, Los Alamitos (1992)CrossRefGoogle Scholar
  9. 9.
    Crawford, J.M., Auton, L.D.: Experimental results on the crossover point in random 3-SAT. Artificial Intelligence 81(1-2), 31–57 (1996)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Dowling, W.F., Gallier, J.H.: Linear-time algorithms for testing the satisfiability of propositional Horn formulae. Logic Programming (USA) 1(3), 267–284 (1984) ISSN: 0743-1066zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dubois, O., Boufkhad, Y., Mandler, J.: Typical random 3-SAT formulae and the satisfiability theshold. In: Proc. 11th ACM-SIAM Symp. on Discrete Algorithms, pp. 126–127 (2000)Google Scholar
  12. 12.
    Dubois, O., Mandler, J.: The 3-XORSAT threshold. In: Proc. 43rd IEEE Symp. on Foundations of Computer Science, pp. 769–778 (2002)Google Scholar
  13. 13.
    Darling, R., Norris, J.R.: Structure of large random hypergraphs. Annals of Applied Probability 15(1A) (2005)Google Scholar
  14. 14.
    Demopoulos, D., Vardi, M.: The phase transition in random 1-3 Hornsat problems. In: Percus, A., Istrate, G., Moore, C. (eds.) Computational Complexity and Statistical Physics. Santa Fe Institute Lectures in the Sciences of Complexity. Oxford University Press, Oxford (2005), available at Google Scholar
  15. 15.
    Erdös, P., Rényi, A.: On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Science 5, 17–61 (1960)zbMATHGoogle Scholar
  16. 16.
    Friedgut, E.: Necessary and sufficient conditions for sharp threshold of graph properties and the k-SAT problem. J. Amer. Math. Soc. 12, 1054–1917 (1999)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Goerdt, A.: A threshold for unsatisfiability. J. Comput. System Sci. 53(3), 469–486 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hajiaghayi, M., Sorkin, G.B.: The satisfiability threshold for random 3-SAT is at least 3.52. IBM Technical Report (2003)Google Scholar
  19. 19.
    Hogg, T., Williams, C.P.: The hardest constraint problems: A double phase transition. Artificial Intelligence 69(1-2), 359–377 (1994)zbMATHCrossRefGoogle Scholar
  20. 20.
    Istrate, G.: The phase transition in random Horn satisfiability and its algorithmic implications. Random Structures and Algorithms 4, 483–506 (2002)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Istrate, G.: On the satisfiability of random k-Horn formulae. In: Winkler, P., Nesetril, J. (eds.) Graphs, Morphisms and Statistical Physics. AMS-DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 64, pp. 113–136 (2004)Google Scholar
  22. 22.
    Karp, R.: The transitive closure of a random digraph. Random Structures and Algorithms 1, 73–93 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kaporis, L.M.: Kirousis, and E. Lalas. Selecting complementary pairs of literals. In: Proceedings of LICS 2003 Workshop on Typical Case Complexity and Phase Transitions (June 2003)Google Scholar
  24. 24.
    Makowsky, J.A.: Why Horn formulae matter in Computer Science: Initial structures and generic examples. JCSS 34(2-3), 266–292 (1987)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Mézard, M., Zecchina, R.: Random k-satisfiability problem: from an analytic solution to an efficient algorithm. Phys. Rev. E 66, 56126 (2002)CrossRefGoogle Scholar
  26. 26.
    Mézard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297, 812–815 (2002)CrossRefGoogle Scholar
  27. 27.
    Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyansky, L.: 2+p-SAT:Relation of typical-case complexity to the nature of the phase transition. Random Structures and Algorithms 15(3-4), 414–435 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Selman, B., Mitchell, D.G., Levesque, H.J.: Generating hard satisfiability problems. Artificial Intelligence 81(1-2), 17–29 (1996)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Selman, B., Kirkpatrick, S.: Critical behavior in the computational cost of satisfiability testing. Artificial Intelligence 81(1-2), 273–295 (1996)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Vardi, M.Y., Wolper, P.: Automata-Theoretic Techniques for Modal Logics of Programs J. Computer and System Science 32(2), 181–221 (1986)MathSciNetGoogle Scholar
  31. 31.
    Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification (preliminary report). In: Proc. 1st IEEE Symp. on Logic in Computer Science, pp. 332–344 (1986)Google Scholar
  32. 32.
    Wormald, N.: Differential equations for random processes and random graphs. Annals of. Applied Probability 5(4), 1217–1235 (1995)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Cristopher Moore
    • 1
  • Gabriel Istrate
    • 2
  • Demetrios Demopoulos
    • 3
  • Moshe Y. Vardi
    • 4
  1. 1.Department of Computer ScienceUniversity of New MexicoAlbuquerqueUSA
  2. 2.CCS-5, Los Alamos National LaboratoryLos AlamosUSA
  3. 3.Archimedean AcademyMiamiUSA
  4. 4.Department of Computer ScienceRice UniversityHoustonUSA

Personalised recommendations