Abstract
We consider the problem of estimating the model count (number of solutions) of Boolean formulas, and present two techniques that compute estimates of these counts, as well as either lower or upper bounds with different trade-offs between efficiency, bound quality, and correctness guarantee. For lower bounds, we use a recent framework for probabilistic correctness guarantees, and exploit message passing techniques for marginal probability estimation, namely, variations of Belief Propagation (BP). Our results suggest that BP provides useful information even on structured loopy formulas. For upper bounds, we perform multiple runs of the MiniSat SAT solver with a minor modification, and obtain statistical bounds on the model count based on the observation that the distribution of a certain quantity of interest is often very close to the normal distribution. Our experiments demonstrate that our model counters based on these two ideas, BPCount and MiniCount, can provide very good bounds in time significantly less than alternative approaches.
Research supported by IISI, Cornell University (AFOSR grant FA9550-04-1-0151), DARPA (REAL Grant FA8750-04-2-0216), and NSF (Grant 0514429).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bacchus, F., Dalmao, S., Pitassi, T.: Algorithms and complexity results for #SAT and Bayesian inference. In: 44nd FOCS, pp. 340–351 (October 2003)
Bayardo Jr., R.J., Pehoushek, J.D.: Counting models using connected components. In: 17th AAAI, Austin, TX, July 2000, pp. 157–162 (2000)
Darwiche, A.: New advances in compiling CNF into decomposable negation normal form. In: 16th ECAI, Valencia, Spain, August 2004, pp. 328–332 (2004)
Darwiche, A.: The quest for efficient probabilistic inference. In: IJCAI 2005 (July 2005) Invited Talk
Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. CACM 5, 394–397 (1962)
Davis, M., Putnam, H.: A computing procedure for quantification theory. CACM 7, 201–215 (1960)
Eén, N., Sörensson, N.: MiniSat: A SAT solver with conflict-clause minimization. In: 8th SAT, St. Andrews, U.K. (June 2005)
Gogate, V., Dechter, R.: Approximate counting by sampling the backtrack-free search space. In: 22th AAAI, Vancouver, BC, July 2007, pp. 198–203 (2007)
Gomes, C.P., Hoffmann, J., Sabharwal, A., Selman, B.: From sampling to model counting. In: 20th IJCAI, Hyderabad, India, January 2007, pp. 2293–2299 (2007)
Gomes, C.P., Sabharwal, A., Selman, B.: Model counting: A new strategy for obtaining good bounds. In: 21th AAAI, Boston, MA, July 2006, pp. 54–61 (2006)
Hsu, E.I., McIlraith, S.A.: Characterizing propagation methods for boolean satisfiabilty. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 325–338. Springer, Heidelberg (2006)
Littman, M.L., Majercik, S.M., Pitassi, T.: Stochastic Boolean satisfiability. J. Auto. Reas. 27(3), 251–296 (2001)
Maneva, E., Mossel, E., Wainwright, M.J.: A new look at survey propagation and its generalizations. J. Assoc. Comput. Mach. 54(4), 17 (2007)
Park, J.D.: MAP complexity results and approximation methods. In: 18th UAI, Edmonton, Canada, August 2002, pp. 388–396 (2002)
Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco (1988)
Pretti, M.: A message-passing algorithm with damping. J. Stat. Mech. P11008 (2005)
Roth, D.: On the hardness of approximate reasoning. AI J. 82(1-2), 273–302 (1996)
Sang, T., Bacchus, F., Beame, P., Kautz, H.A., Pitassi, T.: Combining component caching and clause learning for effective model counting. In: 7th SAT (2004)
Sang, T., Beame, P., Kautz, H.A.: Performing Bayesian inference by weighted model counting. In: 20th AAAI, Pittsburgh, PA, July 2005, pp. 475–482 (2005)
Thode, H.C.: Testing for Normality. CRC Press, Boca Raton (2002)
Valiant, L.G.: The complexity of computing the permanent. Theoretical Comput. Sci. 8, 189–201 (1979)
Wei, W., Erenrich, J., Selman, B.: Towards efficient sampling: Exploiting random walk strategies. In: 19th AAAI, San Jose, CA, July 2004, pp. 670–676 (2004)
Wei, W., Selman, B.: A new approach to model counting. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 324–339. Springer, Heidelberg (2005)
Yedidia, J.S., Freeman, W.T., Weiss, Y.: Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Trans. Inf. Theory 51(7), 2282–2312 (2005)
Yuille, A.L.: CCCP algorithms to minimize the Bethe and Kikuchi free energies: Convergent alternatives to belief prop. Neural Comput. 14(7), 1691–1722 (2002)
Zhou, X.-H., Sujuan, G.: Confidence intervals for the log-normal mean. Statistics In Medicine 16, 783–790 (1997)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kroc, L., Sabharwal, A., Selman, B. (2008). Leveraging Belief Propagation, Backtrack Search, and Statistics for Model Counting. In: Perron, L., Trick, M.A. (eds) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. CPAIOR 2008. Lecture Notes in Computer Science, vol 5015. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68155-7_12
Download citation
DOI: https://doi.org/10.1007/978-3-540-68155-7_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-68154-0
Online ISBN: 978-3-540-68155-7
eBook Packages: Computer ScienceComputer Science (R0)