Logical Methods pp 654-704 | Cite as

# Dempster-Shafer Logic Programs and Stable Semantics

- 159 Downloads

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Bacchus, F. [1988], Representing and Reasoning with Probabilistic Knowledge.
*Research Report CS-88-31*, University of Waterloo.Google Scholar - [2]Baldwin, J.F. [1987], Evidential Support Logic Programming.
*Journal of Fuzzy Sets and Systems*,*24*, 1–26.zbMATHCrossRefGoogle Scholar - [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]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]Blair, H.A. and V.S. Subrahmanian [1987], Paraconsistent Logic Programming.
*Theoretical Computer Science*,*68*, 35–54.MathSciNetGoogle Scholar - [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]Cheeseman, P. [1985], In Defense of Probability. In:
*Proc. IJCAI-85*, 1002–1009.Google Scholar - [8]Dempster, A.P. [1968], A Generalization of Bayesian Inference.
*J. of the Royal Statistical Soc*,*Series B*,*30*, 205–247.MathSciNetzbMATHGoogle Scholar - [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]Dubois, D. and H. Prade [1988], Default Reasoning and Possibility Theory.
*Artificial Intelligence*,*35*, 243–257.MathSciNetzbMATHCrossRefGoogle Scholar - [11]Fagin, R. and J. Halpern [1988], Uncertainty, Belief and Probability. In:
*Proc. IJCAI-89*, Morgan-Kauffman.Google Scholar - [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]Fitting, M.C. [1988], Logical Programming on a Topological Bilattice.
*Fundamenta Informaticae*,*11*, 209–218.MathSciNetzbMATHGoogle Scholar - [14]Fitting, M.C. [1988], Bilattices and the Semantics of Logic Programming. To appear in:
*Journal of Logic Programming*.Google Scholar - [15]Geffner, H. [1989], Default Reasoning: Causal and Conditional Theories.
*Technical Report 137*, Cognitive Systems Laboratory, University of California, Los Angeles.Google Scholar - [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]Ginsberg, M. [1984], Non-monotonic Reasoning Using Dempster’s Rule.
*Proc. AAAI-84*, 126–129.Google Scholar - [18]Halpern, T. [1984], Probability Logic.
*Notre Dame J. of Formal Logic*,*25, 3*, 198–212.CrossRefGoogle Scholar - [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]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]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]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]Kolmogorov, A.N. [1956],
*Foundations of the Theory of Probability*. Chelsea Publishing Co.Google Scholar - [24]Kyburg, H. [1987], Bayesian and non-Bayesian Evidential Updating.
*Artificial Intelligence*,*31*, 271–293.MathSciNetzbMATHCrossRefGoogle Scholar - [25]Laskey, K.B. and P.E. Lehner [1989], Assumptions, Beliefs and Probabilities,
*Artificial Intelligence*,*41*, 65–77.MathSciNetCrossRefGoogle Scholar - [26]Lifschitz, V. [1987], Pointwise Circumscription. In:
*Readings in Nonmonotonic Reasoning*, ed. M. Ginsberg, 179–193, Morgan Kaufmann.Google Scholar - [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]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]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]Martelli, A. and U. Montanari [1982], An Efficient Unification Algorithm
*ACM Trans, on Prog. Lang, and Systems 4, 2*, 258–282.zbMATHCrossRefGoogle Scholar - [31]McCarthy, J. [1980], Circumscription — a Form of Non-monotonic Reasoning
*Artificial Intelligence*,*13*, 27–39.MathSciNetzbMATHCrossRefGoogle Scholar - [32]McDermott, D. and J. Doyle [1980], Non-monotonic Logic I,
*Artificial Intelligence*,*13*, 41–72.MathSciNetzbMATHCrossRefGoogle Scholar - [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]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]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]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]Nilsson, N. [1986], Probabilistic Logic.
*Artificial Intelligence*,*28*, 71–87.MathSciNetzbMATHCrossRefGoogle Scholar - [38]Pearl, J. [1988],
*Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference*. Morgan Kaufmann.Google Scholar - [39]Reiter, R. [1980], A Logic for Default Reasoning.
*Artificial Intelligence*,*13*, 81–132.MathSciNetzbMATHCrossRefGoogle Scholar - [40]Rich. E. and K. Knight [1991],
*Artificial Intelligence*. McGraw-Hill.Google Scholar - [41]Shafer, G. [1976],
*A Mathematical Theory of Evidence*. Princeton University Press.Google Scholar - [42]Shafer, G. Dempster’s Rule of Combination. Unpublished manuscript.Google Scholar
- [43]Shapiro, E. [1983], Logic Programs with Uncertainties: A Tool for Implementing Expert Systems.
*Proc. IJCAI’ 83*, William Kaufmann, 529–532.Google Scholar - [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]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]Touretzky, D.S. [1986],
*The Mathematics of Inheritance Systems*. Pitman and Morgan Kaufmann.Google Scholar - [47]van Emden, M.H. [1986], Quantitative Deduction and its Fixpoint Theory.
*Journal of Logic Programming*,*4, 1*, 37–53.CrossRefGoogle Scholar - [48]Yager, R. [1987], On the Dempster-Shafer Framework and New Combination Rules.
*Information Sciences*,*41*, 91–137.Google Scholar - [49]Zadeh, L.A. [1965], Fuzzy Sets.
*Information and Control*,*8*, 338–353.MathSciNetzbMATHCrossRefGoogle Scholar - [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