Skip to main content
SpringerLink
Log in

Probabilistic Lexicographic Entailment under Variable-Strength Inheritance with Overriding

  • Conference paper
Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2003)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2711))

Abstract

In previous work, I have presented approaches to nonmonotonic probabilistic reasoning, which is a probabilistic generalization of default reasoning from conditional knowledge bases. In this paper, I continue this exciting line of research. I present a new probabilistic generalization of Lehmann’s lexicographic entailment, called lex λ -entailment, which is parameterized through a value λ ∈ [0,1] that describes the strength of the inheritance of purely probabilistic knowledge. Roughly, the new notion of entailment is obtained from logical entailment in model-theoretic probabilistic logic by adding (i) the inheritance of purely probabilistic knowledge of strength λ, and (ii) a mechanism for resolving inconsistencies due to the inheritance of logical and purely probabilistic knowledge. I also explore the semantic properties of lex λ -entailment.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amarger, S., Dubois, D., Prade, H.: Constraint propagation with imprecise conditional probabilities. In: Proceedings UAI 1991, pp. 26–34. Morgan Kaufmann, San Francisco (1991)

    Google Scholar 

  2. Benferhat, S., Cayrol, C., Dubois, D., Lang, J., Prade, H.: Inconsistency management and prioritized syntax-based entailment. In: Proceedings IJCAI 1993, pp. 640–645 (1993)

    Google Scholar 

  3. Benferhat, S., Dubois, D., Prade, H.: Representing default rules in possibilistic logic. In: Proceedings KR 1992, pp. 673–684. Morgan Kaufmann, San Francisco (1992)

    Google Scholar 

  4. Benferhat, S., Dubois, D., Prade, H.: Nonmonotonic reasoning, conditional objects and possibility theory. Artif. Intell. 92(1-2), 259–276 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  5. Biazzo, V., Gilio, A.: Ageneralization of the fundamental theorem of de Finetti for imprecise conditional probability assessments. Int. J. Approx. Reasoning 24, 251–272 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Biazzo, V., Gilio, A., Lukasiewicz, T., Sanfilippo, G.: Probabilistic logic under coherence: Complexity and algorithms. In: Proceedings ISIPTA 2001, pp. 51–61 (2001)

    Google Scholar 

  7. Biazzo, V., Gilio, A., Lukasiewicz, T., Sanfilippo, G.: Probabilistic logic under coherence, model-theoretic probabilistic logic, and default reasoning in System P. Journal of Applied Non-Classical Logics 12(2), 189–213 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Boole, G.: An Investigation of the Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probabilities. Walton and Maberley, London (1854) (Reprint: Dover Publications, NewYork, 1958)

    Google Scholar 

  9. Dubois, D., Prade, H.: Possibilistic logic, preferential models, non-monotonicity and related issues. In: Proceedings IJCAI 1991, pp. 419–424. Morgan Kaufmann, San Francisco (1991)

    Google Scholar 

  10. Dubois, D., Prade, H.: Conditional objects as nonmonotonic consequence relationships. IEEE Trans. Syst. Man Cybern. 24(12), 1724–1740 (1994)

    Article  MathSciNet  Google Scholar 

  11. Eiter, T., Lukasiewicz, T.: Default reasoning from conditional knowledge bases: Complexity and tractable cases. Artif. Intell. 124(2), 169–241 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fagin, R., Halpern, J.Y., Megiddo, N.: A logic for reasoning about probabilities. Inf. Comput. 87, 78–128 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Friedman, N., Halpern, J.Y.: Plausibility measures and default reasoning. J. ACM 48(4), 648–685 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Frisch, A.M., Haddawy, P.: Anytime deduction for probabilistic logic. Artif. Intell. 69, 93–122 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Geffner, H.: Default Reasoning: Causal and Conditional Theories. MIT Press, Cambridge (1992)

    Google Scholar 

  16. Gilio, A.: Probabilistic consistency of conditional probability bounds. In: Bouchon-Meunier, B., Yager, R.R., Zadeh, L.A. (eds.) IPMU 1994. LNCS, vol. 945, pp. 200–209. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  17. Gilio, A.: Probabilistic reasoning under coherence in System P. Ann. Math. Artif. Intell. 34(1-3), 5–34 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Giugno, R., Lukasiewicz, T.: P-\(\mathcal{SHOQ}(D)\): A probabilistic extension of \(\mathcal{SHOQ}(D)\) for probabilistic ontologies in the semantic web. In: Flesca, S., Greco, S., Leone, N., Ianni, G. (eds.) JELIA 2002. LNCS (LNAI), vol. 2424, pp. 86–97. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  19. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 14(1), 167–207 (1990)

    Article  MathSciNet  Google Scholar 

  20. Kyburg Jr., H.E.: The Logical Foundations of Statistical Inference. D. Reidel, Dordrecht (1974)

    MATH  Google Scholar 

  21. Kyburg Jr., H.E.: The reference class. Philos. Sci. 50, 374–397 (1983)

    Article  MathSciNet  Google Scholar 

  22. Lamarre, P.: Apromenade from monotonicity to non-monotonicity following a theorem prover. In: Proceedings KR 1992, pp. 572–580. Morgan Kaufmann, San Francisco (1992)

    Google Scholar 

  23. Lehmann, D.: What does a conditional knowledge base entail? In: Proceedings KR 1989, pp. 212–222. Morgan Kaufmann, San Francisco (1989)

    Google Scholar 

  24. Lehmann, D.: Another perspective on default reasoning. Ann. Math. Artif. Intell. 15(1), 61–82 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lukasiewicz, T.: Local probabilistic deduction from taxonomic and probabilistic knowledgebases over conjunctive events. Int. J. Approx. Reasoning 21(1), 23–61 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lukasiewicz, T.: Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res. 10, 199–241 (1999)

    MATH  MathSciNet  Google Scholar 

  27. Lukasiewicz, T.: Probabilistic logic programming under inheritance with overriding. In: Proceedings UAI 2001, pp. 329–336. Morgan Kaufmann, San Francisco (2001)

    Google Scholar 

  28. Lukasiewicz, T.: Probabilistic logic programming with conditional constraints. ACM Trans. on Computational Logic (TOCL) 2(3), 289–339 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lukasiewicz, T.: Nonmonotonic probabilistic logics between model-theoretic probabilistic logic and probabilistic logic under coherence. In: Proceedings NMR 2002, pp. 265–274 (2002)

    Google Scholar 

  30. Lukasiewicz, T.: Probabilistic default reasoning with conditional constraints. Ann. Math. Artif. Intell. 34(1-3), 35–88 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lukasiewicz, T.: Nonmonotonic probabilistic reasoning under variable-strength inheritance with overriding. Technical Report INFSYS RR-1843-03-02, Institut für Informationssysteme, TU Wien (2003)

    Google Scholar 

  32. Nilsson, N.J.: Probabilistic logic. Artif. Intell. 28(1), 71–88 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  33. Pearl, J.: System Z: A natural ordering of defaults with tractable applications to default reasoning. In: Proceedings TARK 1990, pp. 121–135. Morgan Kaufmann, San Francisco (1990)

    Google Scholar 

  34. Pollock, J.L.: Nomic Probabilities and the Foundations of Induction. Oxford University Press, Oxford (1990)

    Google Scholar 

  35. Reichenbach, H.: Theory of Probability. University of California Press, Berkeley (1949)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, Via Salaria 113, I-00198, Rome, Italy

    Thomas Lukasiewicz

Authors
  1. Thomas Lukasiewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Aalborg University, Fredrik Bajersvej 7E, 9220, Aalborg, Denmark

    Thomas Dyhre Nielsen

  2. Department of Computer Science, Hong Kong University of Science and Technology, ClearWater Bay Road, Kowloon, Hong Kong, China,

    Nevin Lianwen Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lukasiewicz, T. (2003). Probabilistic Lexicographic Entailment under Variable-Strength Inheritance with Overriding. In: Nielsen, T.D., Zhang, N.L. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2003. Lecture Notes in Computer Science(), vol 2711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45062-7_47

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-45062-7_47

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40494-1

  • Online ISBN: 978-3-540-45062-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation