Dempster-Shafer Logic Programs and Stable Semantics

  • Raymond Ng
  • V. S. Subrahmanian
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 12)


Many researchers (e.g. Baldwin [2] and Ishizuka [19]) have observed that the Dempster-Shafer rule of combination, which is at the heart of Dempster-Shafer theory, exhibits non-monotonic behaviour. However, as they focus entirely on operational concerns, two important issues remain unresolved. In [13], Fitting observes that developing a declarative semantics for logic programs based on the Dempster-Shafer rule is an open problem. More importantly, it is unclear how the mode of non-monotonicity demonstrated by the Dempster-Shafer rule is related to well-understood nonmonotonic logics.

In this paper we study Dempster-Shafer logic programs (DS-programs for short). We first develop a declarative semantics for such logic programs. This task alone is complicated by the non-monotonic nature of the Dempster-Shafer rule. Then, given a DS-program P, we transform P to a program P whose clauses may contain non-monotonic negations in their bodies. We proceed to present a stable semantics for P, which is a quantitative extension of the stable semantics for classical logic programs with negations. The major result of this paper is that the meaning of a class of DS-program P, as defined by the declarative semantics based on the Dempster-Shafer rule, is identical to the meaning of P, as defined by the stable semantics. This equivalence links the Dempster-Shafer mode of non-monotonicity very firmly to the stable semantics, and thus to other non-monotonic rule systems due to the results provided by Marek et al [27, 28,


Bark Doyle 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bacchus, F. [1988], Representing and Reasoning with Probabilistic Knowledge. Research Report CS-88-31, University of Waterloo.Google Scholar
  2. [2]
    Baldwin, J.F. [1987], Evidential Support Logic Programming. Journal of Fuzzy Sets and Systems, 24, 1–26.MATHCrossRefGoogle Scholar
  3. [3]
    Baral, C. and V.S. Subrahmanian [1990], Stable and Extension Class Theory for Logic Programs and Default Logics. To appear in Journal of Automated Reasoning.. Preliminary version in: Proc. 1990 Intl. Workshop on Non-Monotonic Reasoning, Lake Tahoe, June 1990.Google Scholar
  4. [4]
    Baral, C. and V.S. Subrahmanian [1991], Dualities between Alternative Semantics for Logic Programming and Non-Monotonic Reasoning. In: Proc. 1991 Intl. Workshop on Logic Programming and Non-Monotonic Reasoning (eds. A. Nerode, W. Marek and V.S. Subrahmanian), MIT Press.Google Scholar
  5. [5]
    Blair, H.A. and V.S. Subrahmanian [1987], Paraconsistent Logic Programming. Theoretical Computer Science, 68, 35–54.MathSciNetGoogle Scholar
  6. [6]
    Buntine, W. [1990], Modelling Default and Likelihood Reasoning as Probabilistic. Technical Report FIA-90-09-11-01. NASA Ames Research Center.Google Scholar
  7. [7]
    Cheeseman, P. [1985], In Defense of Probability. In: Proc. IJCAI-85, 1002–1009.Google Scholar
  8. [8]
    Dempster, A.P. [1968], A Generalization of Bayesian Inference. J. of the Royal Statistical Soc, Series B, 30, 205–247.MathSciNetMATHGoogle Scholar
  9. [9]
    Deutsch-McLeish, M. [1990], A Model for Non-monotonic Reasoning Using Dempster’s Rule. Proc. of Sixth Conference on Uncertainty in Artificial Intelligence, 518–528.Google Scholar
  10. [10]
    Dubois, D. and H. Prade [1988], Default Reasoning and Possibility Theory. Artificial Intelligence, 35, 243–257.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Fagin, R. and J. Halpern [1988], Uncertainty, Belief and Probability. In: Proc. IJCAI-89, Morgan-Kauffman.Google Scholar
  12. [12]
    Fagin, R., J.Y. Halpern and N. Meggido [1989], A Logic for Reasoning About Probabilities. To appear in: Information and Computation.Google Scholar
  13. [13]
    Fitting, M.C. [1988], Logical Programming on a Topological Bilattice. Fundamenta Informaticae, 11, 209–218.MathSciNetMATHGoogle Scholar
  14. [14]
    Fitting, M.C. [1988], Bilattices and the Semantics of Logic Programming. To appear in: Journal of Logic Programming.Google Scholar
  15. [15]
    Geffner, H. [1989], Default Reasoning: Causal and Conditional Theories. Technical Report 137, Cognitive Systems Laboratory, University of California, Los Angeles.Google Scholar
  16. [16]
    Gelfond, M. and V. Lifschitz [1988], The Stable Model Semantics for Logic Programming. In: Proc. 5th Intl. Conference and Symposium on Logic Programming, ed. R.A. Kowalski and K.A. Bowen, 1070–1080.Google Scholar
  17. [17]
    Ginsberg, M. [1984], Non-monotonic Reasoning Using Dempster’s Rule. Proc. AAAI-84, 126–129.Google Scholar
  18. [18]
    Halpern, T. [1984], Probability Logic. Notre Dame J. of Formal Logic, 25, 3, 198–212.CrossRefGoogle Scholar
  19. [19]
    Ishizuka, M. [1983], Inference Methods Based on Extended Dempster-Shafer Theory for Problems with Uncertainty/Fuzziness. New Generation Computing, 1, 2, 159–168.CrossRefGoogle Scholar
  20. [20]
    Kifer, M. and A. Li [1988], On the Semantics of Rule-Based Expert Systems with Uncertainty. 2nd Intl. Conf. on Database Theory. Springer Verlag LNCS 326 (eds. M. Gyssens, J. Paredaens, D. van Gucht), Bruges, Belgium, 102–117.Google Scholar
  21. [21]
    Kifer, M. and E. Lozinskii [1989], RI: A Logic for Reasoning with Inconsistency. 4th Symp. on Logic in Computer Science. Asilomar, CA, 253–262. Full version to appear in: Journal of Automated Reasoning.Google Scholar
  22. [22]
    Kifer, M. and V.S. Subrahmanian [1991], Theory of Generalized Annotated Logic Programming and its Applications. To appear in: Journal of Logic Programming. Preliminary version in: Proc. 1989 North American Conf on Logic Programming (eds. E. Lusk and R. Overbeek), MIT Press.Google Scholar
  23. [23]
    Kolmogorov, A.N. [1956], Foundations of the Theory of Probability. Chelsea Publishing Co.Google Scholar
  24. [24]
    Kyburg, H. [1987], Bayesian and non-Bayesian Evidential Updating. Artificial Intelligence, 31, 271–293.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    Laskey, K.B. and P.E. Lehner [1989], Assumptions, Beliefs and Probabilities, Artificial Intelligence, 41, 65–77.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Lifschitz, V. [1987], Pointwise Circumscription. In: Readings in Nonmonotonic Reasoning, ed. M. Ginsberg, 179–193, Morgan Kaufmann.Google Scholar
  27. [27]
    Marek, W. and M. Truszczynski [1988], Stable Semantics for Logic Programs and Default Theories. Proc. of 1989 North American Conf. on Logic Programming (eds. E. Lusk and R. Overbeek), MIT Press, 243–256.Google Scholar
  28. [28]
    Marek, W. and M. Truszczynski [1988], Relating Autoepistemic and Default Logic. In: Principles of Knowledge Representation and Reasoning (eds. R. Brachman, H. Levesque and R. Reiter), Morgan Kauffman, 276–288.Google Scholar
  29. [29]
    Marek, W., A. Nerode and J. Remmel [1990], A Theory of Non-Monotonic Rule Systems, Part 1. To appear in: Annals of Math, and AI. Prelim, version in LICS-90.Google Scholar
  30. [30]
    Martelli, A. and U. Montanari [1982], An Efficient Unification Algorithm ACM Trans, on Prog. Lang, and Systems 4, 2, 258–282.MATHCrossRefGoogle Scholar
  31. [31]
    McCarthy, J. [1980], Circumscription — a Form of Non-monotonic Reasoning Artificial Intelligence, 13, 27–39.MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    McDermott, D. and J. Doyle [1980], Non-monotonic Logic I, Artificial Intelligence, 13, 41–72.MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Morishita, S. [1989], A Unified Approach to Semantics of Multi-Valued Logic Programs. Tech. Report RT 5006, IBM Tokyo, April 9, 1990.Google Scholar
  34. [34]
    Ng, R.T. and V.S. Subrahmanian [1989], Probabilistic Logic Programming. To appear in: Information and Computation (1992). Prelim, version in: Proc. 5th Intl. Symposium on Methodologies for Intelligent Systems, 9–16.Google Scholar
  35. [35]
    Ng, R.T. and V.S. Subrahmanian [1990], A Semantical Framework for Supporting Subjective and Conditional Probabilities in Deductive Databases. To appear in: Journal of Automated Reasoning (1993). Prelim, version in: Proc. 1991 Intl. Conference of Logic Programming, 565–580.Google Scholar
  36. [36]
    Ng, R.T. and V.S. Subrahmanian [1991], Stable Semantics for Probabilistic Deductive Databases. To appear in: Information and Computation (1993). Prelim, version in: Proc. 1991 Intl. Conference on Uncertainty in AI, 249–256.Google Scholar
  37. [37]
    Nilsson, N. [1986], Probabilistic Logic. Artificial Intelligence, 28, 71–87.MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    Pearl, J. [1988], Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann.Google Scholar
  39. [39]
    Reiter, R. [1980], A Logic for Default Reasoning. Artificial Intelligence, 13, 81–132.MathSciNetMATHCrossRefGoogle Scholar
  40. [40]
    Rich. E. and K. Knight [1991], Artificial Intelligence. McGraw-Hill.Google Scholar
  41. [41]
    Shafer, G. [1976], A Mathematical Theory of Evidence. Princeton University Press.Google Scholar
  42. [42]
    Shafer, G. Dempster’s Rule of Combination. Unpublished manuscript.Google Scholar
  43. [43]
    Shapiro, E. [1983], Logic Programs with Uncertainties: A Tool for Implementing Expert Systems. Proc. IJCAI’ 83, William Kaufmann, 529–532.Google Scholar
  44. [44]
    Smets, P. and Y.T. Hsia [1990], Default Reasoning and the Transferable Belief Model. Proc. of Sixth Conf. on Uncertainty in Artificial Intelligence, 529–537.Google Scholar
  45. [45]
    Thomason, R., J. Horty and D. Touretzky [1987], A Calculus for Inheritance in Monotonic Semantic Nets. Proc. 2nd Intl. Symposium on Methodologies for Intelligent Systems, 280–287.Google Scholar
  46. [46]
    Touretzky, D.S. [1986], The Mathematics of Inheritance Systems. Pitman and Morgan Kaufmann.Google Scholar
  47. [47]
    van Emden, M.H. [1986], Quantitative Deduction and its Fixpoint Theory. Journal of Logic Programming, 4, 1, 37–53.CrossRefGoogle Scholar
  48. [48]
    Yager, R. [1987], On the Dempster-Shafer Framework and New Combination Rules. Information Sciences, 41, 91–137.Google Scholar
  49. [49]
    Zadeh, L.A. [1965], Fuzzy Sets. Information and Control, 8, 338–353.MathSciNetMATHCrossRefGoogle Scholar
  50. [50]
    Zadeh, L.A. [1986], A Simple View of the Dempster-Shafer Theory of Evidence and Its Implications for the Rule of Combination. AI Magazine, summer 1986, 85–90.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Raymond Ng
    • 1
  • V. S. Subrahmanian
    • 2
  1. 1.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Computer ScienceUniversity of MarylandCollege ParkUSA

Personalised recommendations