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

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Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques (APPROX 2005, RANDOM 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3624))

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Abstract

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.

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References

  1. Achlioptas, D.: Lower Bounds for Random 3-SAT via Differential Equations. Theoretical Computer Science 265(1-2), 159–185 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  5. Binney, J.J., Dowrick, N.J., Fisher, A.J., Newman, M.E.J.: The Theory of Critical Phenomena. Oxford University Press, Oxford (1992)

    MATH  Google Scholar 

  6. Bollobás, B.: Random Graphs. Academic Press, London (1985)

    MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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: 0743-1066

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Darling, R., Norris, J.R.: Structure of large random hypergraphs. Annals of Applied Probability 15(1A) (2005)

    Google Scholar 

  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 http://www.cs.rice.edu/~vardi/papers/

    Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  17. Goerdt, A.: A threshold for unsatisfiability. J. Comput. System Sci. 53(3), 469–486 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  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. Hogg, T., Williams, C.P.: The hardest constraint problems: A double phase transition. Artificial Intelligence 69(1-2), 359–377 (1994)

    Article  MATH  Google Scholar 

  20. Istrate, G.: The phase transition in random Horn satisfiability and its algorithmic implications. Random Structures and Algorithms 4, 483–506 (2002)

    Article  MathSciNet  Google Scholar 

  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. Karp, R.: The transitive closure of a random digraph. Random Structures and Algorithms 1, 73–93 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  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. Makowsky, J.A.: Why Horn formulae matter in Computer Science: Initial structures and generic examples. JCSS 34(2-3), 266–292 (1987)

    MATH  MathSciNet  Google Scholar 

  25. Mézard, M., Zecchina, R.: Random k-satisfiability problem: from an analytic solution to an efficient algorithm. Phys. Rev. E 66, 56126 (2002)

    Article  Google Scholar 

  26. Mézard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297, 812–815 (2002)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  28. Selman, B., Mitchell, D.G., Levesque, H.J.: Generating hard satisfiability problems. Artificial Intelligence 81(1-2), 17–29 (1996)

    Article  MathSciNet  Google Scholar 

  29. Selman, B., Kirkpatrick, S.: Critical behavior in the computational cost of satisfiability testing. Artificial Intelligence 81(1-2), 273–295 (1996)

    Article  MathSciNet  Google Scholar 

  30. Vardi, M.Y., Wolper, P.: Automata-Theoretic Techniques for Modal Logics of Programs J. Computer and System Science 32(2), 181–221 (1986)

    MathSciNet  Google Scholar 

  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. Wormald, N.: Differential equations for random processes and random graphs. Annals of. Applied Probability 5(4), 1217–1235 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of New Mexico, Albuquerque, NM, USA

    Cristopher Moore

  2. CCS-5, Los Alamos National Laboratory, Los Alamos, NM, USA

    Gabriel Istrate

  3. Archimedean Academy, 12425 SW Sunset Drive, Miami, FL, USA

    Demetrios Demopoulos

  4. Department of Computer Science, Rice University, Houston, TX, USA

    Moshe Y. Vardi

Authors
  1. Cristopher Moore
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  2. Gabriel Istrate
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  3. Demetrios Demopoulos
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  4. Moshe Y. Vardi
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Editor information

Editors and Affiliations

  1. Dept. of Computer Science, University of Illinois, 61801, Urbana, IL

    Chandra Chekuri

  2. Institute for Computer Science, University of Kiel, Olshausenstrasse 40, 24118, Kiel, Germany

    Klaus Jansen

  3. Battelle Bâtiment A, Centre Universitaire d’Informatique, Route de Drize 7, 1227, Carouge, Geneva, Switzerland

    José D. P. Rolim

  4. UC Berkeley,  

    Luca Trevisan

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© 2005 Springer-Verlag Berlin Heidelberg

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Moore, C., Istrate, G., Demopoulos, D., Vardi, M.Y. (2005). A Continuous-Discontinuous Second-Order Transition in the Satisfiability of Random Horn-SAT Formulas. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_35

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  • DOI: https://doi.org/10.1007/11538462_35

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-28239-6

  • Online ISBN: 978-3-540-31874-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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