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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)

Abstract

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,

Keywords

Logic Program Ground Instance Nonmonotonic Logic Formula Function Stable Semantic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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.zbMATHCrossRefGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.zbMATHCrossRefGoogle Scholar
  31. [31]
    McCarthy, J. [1980], Circumscription — a Form of Non-monotonic Reasoning Artificial Intelligence, 13, 27–39.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    McDermott, D. and J. Doyle [1980], Non-monotonic Logic I, Artificial Intelligence, 13, 41–72.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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

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