The Complexity of Inferences and Explanations in Probabilistic Logic Programming

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

Abstract

A popular family of probabilistic logic programming languages combines logic programs with independent probabilistic facts. We study the complexity of marginal inference, most probable explanations, and maximum a posteriori calculations for propositional/relational probabilistic logic programs that are acyclic/definite/stratified/normal/ disjunctive. We show that complexity classes \(\varSigma _k\) and \(\mathsf {PP}^{\varSigma _k}\) (for various values of k) and \(\mathsf {NP}^\mathsf {PP}\) are all reached by such computations.

References

  1. 1.
    Apt, K.R., Bezem, M.: Acyclic programs. New Gener. Comput. 9, 335–363 (1991)CrossRefMATHGoogle Scholar
  2. 2.
    Augustin, T., Coolen, F.P.A., de Cooman, G., Troffaes, M.C.M.: Introduction to Imprecise Probabilities. Wiley, USA (2014)CrossRefMATHGoogle Scholar
  3. 3.
    Baral, C., Gelfond, M., Rushton, N.: Probabilistic reasoning with answer sets. Theor. Pract. Logic Program. 9(1), 57–144 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Baral, C., Subrahmanian, V.: Dualities between alternative semantics for logic programming and nonmonotonic reasoning. J. Autom. Reason. 10(3), 399–420 (1993)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ceylan, Í.Í, Lukasiewicz, T., Peñaloza, R.: Complexity results for probabilistic Datalog\(^\pm \). In: European Conference on Artificial Intelligence, pp. 1414–1422 (2016)Google Scholar
  6. 6.
    Cozman, F.G., Mauá, D.D.: The structure and complexity of credal semantics. In: Workshop on Probabilistic Logic Programming, pp. 3–14 (2016)Google Scholar
  7. 7.
    Cozman, F.G., Mauá, D.D.: Probabilistic graphical models specified by probabilistic logic programs: semantics and complexity. In: Conference on Probabilistic Graphical Models – JMLR Proceedings, vol. 52, pp. 110–121 (2016)Google Scholar
  8. 8.
    Cozman, F.G., Mauá. D.D.: The well-founded semantics of cyclic probabilistic logic programs: meaning and complexity. In: Encontro Nacional de Inteligência Artificial e Computacional, pp. 1–12 (2016)Google Scholar
  9. 9.
    Dantsin, E., Eiter, T., Voronkov, A.: Complexity and expressive power of logic programming. ACM Comput. Surv. 33(3), 374–425 (2001)CrossRefGoogle Scholar
  10. 10.
    Darwiche, A.: Modeling and Reasoning with Bayesian Networks, Cambridge (2009)Google Scholar
  11. 11.
    Polpo de Campos, C., Cozman, F.G.: The inferential complexity of Bayesian and credal networks. In: IJCAI, pp. 1313–1318 (2005)Google Scholar
  12. 12.
    Eiter, T., Faber, W., Fink, M., Woltran, S.: Complexity results for answer set programming with bounded predicate arities and implications. Ann. Math. Artif. Intell. 5, 123–165 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Eiter, T., Gottlob, G.: On the computational cost of disjunctive logic programming: propositional case. Ann. Math. Artif. Intell. 15, 289–323 (1995)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Eiter, T., Ianni, G., Krennwallner, T.: Answer set programming: a primer. In: Tessaris, S., Franconi, E., Eiter, T., Gutierrez, C., Handschuh, S., Rousset, M.-C., Schmidt, R.A. (eds.) Reasoning Web 2009. LNCS, vol. 5689, pp. 40–110. Springer, Heidelberg (2009). doi:10.1007/978-3-642-03754-2_2 CrossRefGoogle Scholar
  15. 15.
    Fierens, D., Van den Broeck, G., Renkens, J., Shrerionov, D., Gutmann, B., Janssens, G., de Raedt, L.: Inference and learning in probabilistic logic programs using weighted Boolean formulas. Theor. Pract. Logic Program. 15(3), 358–401 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Goldsmith, J., Hagen, M., Mundhenk, M.: Complexity of DNF minimization and isomorphism testing for monotone formulas. Inf. Comput. 206(6), 760–775 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hadjichristodoulou, S., Warren, D.S.: Probabilistic logic programming with well-founded negation. In: International Symposium on Multiple-Valued Logic, pp. 232–237 (2012)Google Scholar
  18. 18.
    Lukasiewicz, T.: Probabilistic description logic programs. In: Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty, pp. 737–749 (2005)Google Scholar
  19. 19.
    Lukasiewicz, T.: Probabilistic description logic programs. Int. J. Approx. Reason. 45(2), 288–307 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mauá, D.D., Polpo de Campos, C., Cozman, F.G.: The complexity of MAP inference in Bayesian networks specified through logical languages. In: International Joint Conference on Artificial Intelligence (IJCAI), pp. 889–895 (2015)Google Scholar
  21. 21.
    Michels, S., Hommersom, A., Lucas, P.J.F., Velikova, M.: A new probabilistic constraint logic programming language based on a generalised distribution semantics. Artif. Intell. J. 228, 1–44 (2015)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Longman (1994)MATHGoogle Scholar
  23. 23.
    Poole, D.: Probabilistic Horn abduction and Bayesian networks. Artif. Intell. 64, 81–129 (1993)CrossRefMATHGoogle Scholar
  24. 24.
    Poole, D.: The independent choice logic and beyond. In: Raedt, L., Frasconi, P., Kersting, K., Muggleton, S. (eds.) Probabilistic Inductive Logic Programming. LNCS, vol. 4911, pp. 222–243. Springer, Heidelberg (2008). doi:10.1007/978-3-540-78652-8_8 CrossRefGoogle Scholar
  25. 25.
    Sato, T.: A statistical learning method for logic programs with distribution semantics. In: International Conference on Logic Programming, pp. 715–729 (1995)Google Scholar
  26. 26.
    Sato, T., Kameya, Y., Zhou, N.-F.: Generative modeling with failure in PRISM. In: International Joint Conference on Artificial Intelligence, pp. 847–852 (2005)Google Scholar
  27. 27.
    Stockmeyer, L.J.: The polynomial-time hierarchy. Theor. Comput. Sci. 3(1), 1–22 (1976)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Toda, S., Watanabe, O.: Polynomial-time 1-Turing reductions from #PH to #P. Theor. Comput. Sci. 100, 205–221 (1992)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Tóran, J.: Complexity classes defined by counting quantifiers. J. ACM 38(3), 753–774 (1991)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    van Gelder, A., Ross, J.A., Schlipf, J.S.: The well-founded semantics for general logic programs. J. Assoc. Comput. Mach. 38(3), 620–650 (1991)MathSciNetMATHGoogle Scholar
  31. 31.
    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

Personalised recommendations