Revisiting the Semantics of Interval Probabilistic Logic Programs

  • Alex Dekhtyar
  • Michael I. Dekhtyar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3662)


Two approaches to logic programming with probabilities emerged over time: bayesian reasoning and probabilistic satisfiability (PSAT). The attractiveness of the former is in tying the logic programming research to the body of work on Bayes networks. The second approach ties computationally reasoning about probabilities with linear programming, and allows for natural expression of imprecision in probabilities via the use of intervals.

In this paper we construct precise semantics for one PSAT-based formalism for reasoning with inteval probabilities, probabilistic logic programs (p-programs), orignally considered by Ng and Subrahmanian. We show that the probability ranges of atoms and formulas in p-programs cannot be expressed as single intervals. We construct the prescise description of the set of models of p-programs and study the computational complexity if this problem, as well as the problem of consistency of a p-program. We also study the conditions under which our semantics coincides with the single-interval semantics originally proposed by Ng and Subrahmanian for p-programs. Our work sheds light on the complexity of construction of reasoning formalisms for imprecise probabilities and suggests that interval probabilities alone are inadequate to support such reasoning.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boole, G.: The Laws of Thought. Macmillan, London (1854)Google Scholar
  2. 2.
    Baral, C., Gelfond, M., Nelson Rushton, J.: Probabilistic reasoning with answer sets. In: Lifschitz, V., Niemelä, I. (eds.) LPNMR 2004. LNCS (LNAI), vol. 2923, pp. 21–33. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    de Campos, L.M., Huete, J.F., Moral, S.: Probability Intervals: A Tool for Uncertain Reasoning. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems (IJUFKS) 2(2), 167–196 (1994)CrossRefGoogle Scholar
  4. 4.
    Chvátal, V.: Linear Programming. W. Freeman and Co., San Fancisco (1983)MATHGoogle Scholar
  5. 5.
    Dekhtyar, A., Dekhtyar, M.I.: Possible Worlds Semantics for Probabilistic Logic Programs. In: Demoen, B., Lifschitz, V. (eds.) ICLP 2004. LNCS, vol. 3132, pp. 137–148. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Dekhtyar, A., Subrahmanian, V.S.: Hybrid Probabilistic Programs. Journal of Logic Programming 43(3), 187–250 (2000)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fagin, R., Halpern, J., Megiddo, N.: A logic for reasoning about probabilities. Information and Computation 87(1, 2), 78–128 (1990)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Georgakopoulos, G., Kavvadias, D., Papadimitriou, C.H.: Probabilistic Satisfiability. Journal of Complexity 4, 1–11 (1988)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hailperin, T.: Best Possible Inequalities for the Probability of a Logical Function of Events. American Mathematical Monthly 72, 343–359 (1965)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kyburg Jr., H.E.: Interval-valued Probabilities. In: de Cooman, G., Walley, P., Cozman, F.G. (eds.) Imprecise Probabilities Project (1998),
  11. 11.
    Ng, R., Subrahmanian, V.S.: Probabilistic Logic Programming. Information and Computation 101(2), 150–201 (1993)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Ng, R., Subrahmanian, V.S.: A Semantical Framework for Supporting Subjective and Conditional Probabilities in Deductive Databases. Journal of Automated Reasoning 10(2), 191–235 (1993)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ng, R., Subrahmanian, V.S.: Stable Semantics for Probabilistic Deductive Databases. Information and Computation 101(1), 42–83 (1995)Google Scholar
  14. 14.
    Ngo, L., Haddawy, P.: Probabilistic Logic Programming and Bayesian Networks. In: Kanchanasut, K., Levy, J.-J. (eds.) ACSC 1995. LNCS, vol. 1023, pp. 286–300. Springer, Heidelberg (1995)Google Scholar
  15. 15.
    Nilsson, N.: Probabilistic Logic. AI Journal 28, 71–87 (1986)MATHMathSciNetGoogle Scholar
  16. 16.
    Poole, D.: Probabilistic Horn Abduction and Bayesian Networks. Artificial Intelligence 64(1), 81–129 (1993)MATHCrossRefGoogle Scholar
  17. 17.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, Boca Raton (1991)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Alex Dekhtyar
    • 1
  • Michael I. Dekhtyar
    • 2
  1. 1.Department of Computer ScienceUniversity of Kentucky 
  2. 2.Department of Computer ScienceTver State University 

Personalised recommendations