The Descriptive Complexity of Bayesian Network Specifications

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10369)

Abstract

We adapt the theory of descriptive complexity to Bayesian networks, by investigating how expressive can be specifications based on predicates and quantifiers. We show that Bayesian network specifications that employ first-order quantification capture the complexity class \(\mathsf {PP}\); that is, any phenomenon that can be simulated with a polynomial time probabilistic Turing machine can be also modeled by such a network. We also show that, by allowing quantification over predicates, the resulting Bayesian network specifications capture the complexity class \(\mathsf {PP}^\mathsf {NP}\), a result that does not seem to have equivalent in the literature.

References

  1. 1.
    Abadi, M., Halpern, J.Y.: Decidability and expressiveness for first-order logics of probability. Inf. Comput. 112(1), 1–36 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Cozman, F.G., Mauá, D.D.: Bayesian networks specified using propositional and relational constructs: Combined, data, and domain complexity. In: AAAI Conference on Artificial Intelligence (2015)Google Scholar
  3. 3.
    Cozman, F.G., Polastro, R.B.: Complexity analysis and variational inference for interpretation-based probabilistic description logics. In: Conference on Uncertainty in Artificial Intelligence, pp. 117–125 (2009)Google Scholar
  4. 4.
    Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Springer, Heidelberg (1995)MATHGoogle Scholar
  5. 5.
    Enderton, H.B.: A Mathematical Introduction to Logic. Academic Press, Cambridge (1972)MATHGoogle Scholar
  6. 6.
    Gaifman, H.: Concerning measures on first-order calculi. Isr. J. Math. 2, 1–18 (1964)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Getoor, L., Taskar, B.: Introduction to Statistical Relational Learning. MIT Press, Cambridge (2007)MATHGoogle Scholar
  8. 8.
    Grädel, E.: Finite model theory and descriptive complexity. In: Grädel, E. (ed.) Finite Model Theory and its Applications, pp. 125–229. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Grädel, E.E., Kolaitis, P.G., Libkin, L., Marx, M., Spencer, J., Vardi, M.Y., Venema, Y., Weinstein, S.: Finite Model Theory and its Applications. Springer, Heidelberg (2007)MATHGoogle Scholar
  10. 10.
    Jaeger, M.: Lower complexity bounds for lifted inference. Theor. Pract. Logic Program. 15(2), 246–264 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Koller, D., Friedman, N., Models, P.G.: Probabilistic Graphical Models: Principles and Techniques. MIT Press, Cambridge (2009)Google Scholar
  12. 12.
    Libkin, L.: Elements of Finite Model Theory. Springer, Heidelberg (2004)CrossRefMATHGoogle Scholar
  13. 13.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Boston (1994)MATHGoogle Scholar
  14. 14.
    Poole, D.: Probabilistic programming languages: independent choices and deterministic systems. In: Dechter, R., Geffner, H., Halpern, J.Y. (eds.) Heuristics, Probability and Causality - A Tribute to Judea Pearl, pp. 253–269. College Publications, London (2010)Google Scholar
  15. 15.
    De Raedt, L.: Logical and Relational Learning. Springer, Heidelberg (2008)CrossRefMATHGoogle Scholar
  16. 16.
    De Raedt, L., Frasconi, P., Kersting, K., Muggleton, S.: Probabilistic Inductive Logic Programming. Springer, Heidelberg (2010)CrossRefMATHGoogle Scholar
  17. 17.
    Saluja, S., Subrahmanyam, K.V.: Descriptive complexity of #P functions. J. Comput. Syst. Sci. 50, 493–505 (1995)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Van den Broeck, G.: On the completeness of first-order knowledge compilation for lifted probabilistic inference. In: Neural Processing Information Systems, pp. 1386–1394 (2011)Google Scholar
  19. 19.
    Wagner, K.W.: The complexity of combinatorial problems with succinct input representation. Acta Informatica 23, 325–356 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Escola PolitécnicaUniversidade de São PauloSão PauloBrazil
  2. 2.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil

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