Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

A semantical framework for supporting subjective and conditional probabilities in deductive databases

  • 53 Accesses

  • 56 Citations


We present a theoretical basis for supporting subjective and conditional probabilities in deductive databases. We design a language that allows a user greater expressive power than classical logic programming. In particular, a user can express the fact thatA is possible (i.e.A has non-zero probability),B is possible, but (AB) as a whole is impossible. A user can also freely specify probability annotations that may contain variables. The focus of this paper is to study the semantics of programs written in such a language in relation to probability theory. Our model theory which is founded on the classical one captures the uncertainty described in a probabilistic program at the level of Herbrand interpretations. Furthermore, we develop a fixpoint theory and a proof procedure for such programs and present soundness and completeness results. Finally we characterize the relationships between probability theory and the fixpoint, model, and proof theory of our programs.

This is a preview of subscription content, log in to check access.


  1. 1.

    Anderson, E. J. and Nash, P.,Linear Programming in Infinite-Dimensional Spaces: Theory and Applications, Wiley (1987).

  2. 2.

    Bacchus, F.,Representing and Reasoning with Probabilistic Knowledge, Research Report CS-88-31, University of Waterloo (1988).

  3. 3.

    Baldwin, J. F., ‘Evidential support logic programming’,J. Fuzzy Sets and Systems 24, 1–26 (1987).

  4. 4.

    Bandler, W. and Kohout L. J., ‘Unified theory of multivalued logical operations in the light of the checklist paradigm’,Proceedings IEEE Trans. Systems, Man Cybernet. (1984).

  5. 5.

    Blair, H. A. and Subrahmanian, V. S., ‘Paraconsistent logic programming’,Theore. Computer Science 68, 35–54 (1987). Preliminary version in:Proc. 7th Conference on Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Computer Science, 287, pp. 340–360, Springer-Verlag.

  6. 6.

    Blair, H. A. and Subrahmanian, V. S., ‘Paraconsistent foundations for logic programming’,J. Non-Classical Logic 5(2) 45–73 (1988).

  7. 7.

    Carnap, R.,The Logical Foundations of Probability, 2nd edn., University of Chicago Press (1962).

  8. 8.

    Cheeseman, P., ‘In defense of probability’,Proc. IJCAI-85, pp. 1002–1009 (1985).

  9. 9.

    da Costa, N. C. A., Abe, J. M., and Subrahmanian, V. S., ‘Remarks on annotated logic’,Z. f. Math. Logik u. Grundlagen der Mathematik 37 (1991).

  10. 10.

    Dempster, A. P., ‘A generalization of Bayesian inference’,J. Royal Statistical Soc., Ser. B 30, 205–247 (1968).

  11. 11.

    Duda, R. O., Hart P. E., and Nilsson, N. J. ‘Subjective Bayesian methods for rule-based inference systems’,Proc. National Computer Conference, pp. 1075–1082 (1976).

  12. 12.

    Fagin, R. and Halpern, J. ‘Uncertainty, belief and probability’,Proc. IJCAI-89, Morgan Kauffman (1988).

  13. 13.

    Fagin, R., Halpern, J. Y., and Megiddo, N., ‘A logic for reasoning about probabilities’,Information and Computation (1989).

  14. 14.

    Fenstad, J. E., ‘The structure of probabilities defined on first-order languages’,Studies in Inductive Logic and Probabilities, Volume 2 (ed. R. C. Jeffrey), University of California Press, pp. 251–262 (1980).

  15. 15.

    Fitting, M. C., ‘Logic programming on a topological bilattice’,Fundamenta Informatica 11, 209–218 (1988).

  16. 16.

    Fitting, M. C., ‘Bilattices and the semantics of logic programming’,J. Logic Programming (1988).

  17. 17.

    Gaifman, H., ‘Concerning measures in first order calculi’,Israel J. Math. 2, 1–17 (1964).

  18. 18.

    Ginsberg, M., ‘Multivalued logics: A uniform approach to reasoning in artificial intelligence’,Computational Intelligence 4, 265–316 (1988).

  19. 19.

    Gnedenko, B. V. and Khinchin, A. Y.,An Elementary Introduction to the Theory of Probability, Dover Publications (1962).

  20. 20.

    Hailperin, T., ‘Probability logic’,Notre Dame J. Formal Logic 25(3), 198–212 (1984).

  21. 21.

    Khachiyan, L. G. ‘A polynomial algorithm in linear programming’,Doklady Akad. Nauk SSR 244, 1093–1096 (1979). Translated in:Soviet Mathematics — Doklady20, 191–194 (1979).

  22. 22.

    Kifer, M. and Krishnaprasad, T., ‘An evidence based framework for a theory of inheritance’,Proc. 11th International Joint Conf. on Artificial Intelligence, 1093–1098, Morgan-Kaufmann (1989).

  23. 23.

    Kifer, M., Krishnaprasad, T., and Warren, D. S., ‘On the declarative semantics of inheritance networks’,Proc. 11th International Joint Conf. on Artificial Intelligence, Morgan-Kaufmann (1989).

  24. 24.

    Kifer, M. and Li, A., ‘On the semantics of rule-based expert systems with uncertainty’,2nd Int. Conf. on Database Theory (LNCS 326) (eds. M. Gyssens, J. Paredaens, D. Van Gucht), (Springer Verlag) Bruges, Belgium, pp. 102–117 (1988).

  25. 25.

    Kifer, M. and Lozinskii, E., ‘RI: a logic for reasoning with inconsistency’,4th Symposium on Logic in Computer Science, Asilomar, CA, pp. 253–262 (1989).

  26. 26.

    Kifer, M. and Subrahmanian, V. S., ‘Theory of generalized annotated logic programming and its applications,J. Logic Programming 12(4), 335–368 (1992).

  27. 27.

    Kolmogorov, A. N.,Foundations of the Theory of Probability, Chelsea Publishing Co. (1956).

  28. 28.

    Kyburg, H.,The Logical Foundations of Statistical Inference, D. Reidel (1974).

  29. 29.

    Lloyd, J. W.,Foundations of Logic Programming, Springer (1987).

  30. 30.

    Lukasiewicz, J., Logical foundations of probability theory, in:Selected Works of Jan Lukasiewicz (ed. L. Berkowski), North Holland, pp. 16–43 (1970).

  31. 31.

    Martelli, A., and Montanari, U., ‘An efficient unification algorithm’,ACM Trans. Prog. Lang. and Systems 4(2), 258–282 (1982).

  32. 32.

    Morishita, S., A unified approach to semantics of multi-valued logic programs, Tech. Report RT 5006, IBM Tokyo, April 9th (1990).

  33. 33.

    Ng, R. T. and Subrahmanian, V. S., ‘Probabilistic logic programming’,Information and Computation (1992). Preliminary version inProc. 5th International Symposium on Methodologies for Intelligent Systems, pp. 9–16 (19xx).

  34. 34.

    Nilsson, N., ‘Probabilistic logic’,AI Journal 28, 71–87 (1986).

  35. 35.

    Pearl, J.,Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann (1988).

  36. 36.

    Schrijver, A.,Theory of Linear and Integer Programming, Wiley (1986).

  37. 37.

    Scott, D. S. and Krauss, P., ‘Assigning probabilities to logical formulas’,Aspects of Inductive Logic (ed. J. Hintikka and P. Suppes), North-Holland (1966).

  38. 38.

    Shafer, G.,A Mathematical Theory of Evidence, Princeton University Press (1976).

  39. 39.

    Shapiro, E., ‘Logic programs with uncertainties: A tool for implementing expert systems’,Proc. IJCAI '83, William Kauffman, pp. 529–532 (1983).

  40. 40.

    Shoenfield, J.,Mathematical Logic, Addison-Wesley (1967).

  41. 41.

    Subrahmanian, V. S., ‘On the semantics of quantitative logic programs’,Proc. 4th IEEE Symposium on Logic Programming, Computer Society Press, Washington DC, pp. 173–182 (1987).

  42. 42.

    Subrahmanian, V. S., ‘Mechanical proof procedures for many valued lattice based logic programming’, to appear.

  43. 43.

    Subrahmanian, V. S., ‘Paraconsistent disjunctive deductive databases’,Theor. Computer Sci. Vol. 93, pp. 115–141 (1992).

  44. 44.

    van Emden, M. H., ‘Quantitative deduction and its fixpoint theory’,J. Logic Programming,4(1), 37–53 (1986).

  45. 45.

    Zadeh, L. A., ‘Fuzzy sets’,Information and Control 8, 338–353 (1965).

  46. 46.

    Zadeh, L. A., ‘Fuzzy algorithms’,Information and Control 12, 94–102 (1968).

Download references

Author information

Correspondence to V. S. Subrahmanian.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ng, R., Subrahmanian, V.S. A semantical framework for supporting subjective and conditional probabilities in deductive databases. J Autom Reasoning 10, 191–235 (1993). https://doi.org/10.1007/BF00881836

Download citation

Key words

  • Subjective and conditional probabilities
  • deductive databases
  • probability theory