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Probabilistic Inference and Monadic Second Order Logic

  • Marijke Hans L. Bodlaender
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7604)

Abstract

This paper combines two classic results from two different fields: the result by Lauritzen and Spiegelhalter [21] that the probabilistic inference problem on probabilistic networks can be solved in linear time on networks with a moralization of bounded treewidth, and the result by Courcelle [10] that problems that can be formulated in counting monadic second order logic can be solved in linear time on graphs of bounded treewidth. It is shown that, given a probabilistic network whose moralization has bounded treewidth and a property P of the network and the values of the variables that can be formulated in counting monadic second order logic, one can determine in linear time the probability that P holds.

Keywords

Linear Time Directed Acyclic Graph Order Logic Tree Decomposition Parse Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. Journal of Algorithms 12, 308–340 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25, 1305–1317 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science 209, 1–45 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L., Koster, A.M.C.A.: Treewidth computations I. Upper bounds. Information and Computation 208, 259–275 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L., Koster, A.M.C.A.: Treewidth computations II. Lower bounds. Information and Computation 209, 1103–1119 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bodlaender, H.L., Möhring, R.H.: The pathwidth and treewidth of cographs. SIAM Journal on Discrete Mathematics 6, 181–188 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bodlaender, H.L., van Antwerpen-de Fluiter, B.: Reduction algorithms for graphs of small treewidth. Information and Computation 167, 86–119 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Borie, R.B., Parker, R.G., Tovey, C.A.: Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively constructed graph families. Algorithmica 7, 555–581 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Borie, R.B., Parker, R.G., Tovey, C.A.: Solving problems on recursively constructed graphs. ACM Computing Surveys 41(4) (2008)Google Scholar
  10. 10.
    Courcelle, B.: The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Courcelle, B., Durand, I.A.: Fly-Automata, Their Properties and Applications. In: Bouchou-Markhoff, B., Caron, P., Champarnaud, J.-M., Maurel, D. (eds.) CIAA 2011. LNCS, vol. 6807, pp. 264–272. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    Courcelle, B., Mosbah, M.: Monadic second-order evaluations on tree-decomposable graphs. Theoretical Computer Science 109, 49–82 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Daskalakis, C., Papadimitriou, C.H.: Computing pure Nash equilibria in graphical games via Markov random fields. In: Feigenbaum, J., Chuang, J.C.-I., Pennock, D.M. (eds.) Proceedings 7th ACM Conference on Electronic Commerce EC-2006, pp. 91–99. ACM (2006)Google Scholar
  14. 14.
    Elberfeld, M., Jakoby, A., Tantau, T.: Logspace versions of the theorems of Bodlaender and Courcelle, pp. 143–152 (2010)Google Scholar
  15. 15.
    Fellows, M.R., Langston, M.A.: An analogue of the Myhill-Nerode theorem and its use in computing finite-basis characterizations. In: Proceedings of the 30th Annual Symposium on Foundations of Computer Science, FOCS 1989, pp. 520–525 (1989)Google Scholar
  16. 16.
    Jensen, F.V., Nielsen, T.D.: Bayesian Networks and Decision Graphs, 2nd edn. Information Science and Statistics. Springer (2007)Google Scholar
  17. 17.
    Kabanets, V.: Recognizability Equals Definability for Partial k-paths. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 805–815. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  18. 18.
    Kaller, D.: Definability equals recognizability of partial 3-trees and k-connected partial k-trees. Algorithmica 27, 348–381 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Kloks, T.: Treewidth. Computations and Approximations. LNCS, vol. 842. Springer, Heidelberg (1994)zbMATHGoogle Scholar
  20. 20.
    Koster, A.M.C.A., van Hoesel, S.P.M., Kolen, A.W.J.: Solving partial constraint satisfaction problems with tree decomposition. Networks 40(3), 170–180 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Lauritzen, S.J., Spiegelhalter, D.J.: Local computations with probabilities on graphical structures and their application to expert systems. The Journal of the Royal Statistical Society. Series B (Methodological) 50, 157–224 (1988)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Pearl, J.: Probablistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, Palo Alto (1988)Google Scholar
  23. 23.
    Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. Journal of Algorithms 7, 309–322 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Roth, D.: On the hardness of approximate reasoning. Artificial Intelligence 82, 273–302 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    van der Gaag, L.C.: Probability-Based Models for Plausible Reasoning. PhD thesis, University of Amsterdam (1990)Google Scholar
  26. 26.
    Wimer, T.V.: Linear Algorithms on k-Terminal Graphs. PhD thesis, Dept. of Computer Science, Clemson University (1987)Google Scholar

Copyright information

© IFIP International Federation for Information Processing 2012

Authors and Affiliations

  • Marijke Hans L. Bodlaender
    • 1
  1. 1.Department of Computing SciencesUtrecht UniversityThe Netherlands

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