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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Achlioptas, D.: Lower Bounds for Random 3-SAT via Differential Equations. Theoretical Computer Science 265(1-2), 159–185 (2001)
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)
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)
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)
Binney, J.J., Dowrick, N.J., Fisher, A.J., Newman, M.E.J.: The Theory of Critical Phenomena. Oxford University Press, Oxford (1992)
Bollobás, B.: Random Graphs. Academic Press, London (1985)
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)
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)
Crawford, J.M., Auton, L.D.: Experimental results on the crossover point in random 3-SAT. Artificial Intelligence 81(1-2), 31–57 (1996)
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
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)
Dubois, O., Mandler, J.: The 3-XORSAT threshold. In: Proc. 43rd IEEE Symp. on Foundations of Computer Science, pp. 769–778 (2002)
Darling, R., Norris, J.R.: Structure of large random hypergraphs. Annals of Applied Probability 15(1A) (2005)
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/
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)
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)
Goerdt, A.: A threshold for unsatisfiability. J. Comput. System Sci. 53(3), 469–486 (1996)
Hajiaghayi, M., Sorkin, G.B.: The satisfiability threshold for random 3-SAT is at least 3.52. IBM Technical Report (2003)
Hogg, T., Williams, C.P.: The hardest constraint problems: A double phase transition. Artificial Intelligence 69(1-2), 359–377 (1994)
Istrate, G.: The phase transition in random Horn satisfiability and its algorithmic implications. Random Structures and Algorithms 4, 483–506 (2002)
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)
Karp, R.: The transitive closure of a random digraph. Random Structures and Algorithms 1, 73–93 (1990)
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)
Makowsky, J.A.: Why Horn formulae matter in Computer Science: Initial structures and generic examples. JCSS 34(2-3), 266–292 (1987)
Mézard, M., Zecchina, R.: Random k-satisfiability problem: from an analytic solution to an efficient algorithm. Phys. Rev. E 66, 56126 (2002)
Mézard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297, 812–815 (2002)
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)
Selman, B., Mitchell, D.G., Levesque, H.J.: Generating hard satisfiability problems. Artificial Intelligence 81(1-2), 17–29 (1996)
Selman, B., Kirkpatrick, S.: Critical behavior in the computational cost of satisfiability testing. Artificial Intelligence 81(1-2), 273–295 (1996)
Vardi, M.Y., Wolper, P.: Automata-Theoretic Techniques for Modal Logics of Programs J. Computer and System Science 32(2), 181–221 (1986)
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)
Wormald, N.: Differential equations for random processes and random graphs. Annals of. Applied Probability 5(4), 1217–1235 (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
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)