Symbolic Possibilistic Logic: Completeness and Inference Methods

  • Claudette CayrolEmail author
  • Didier Dubois
  • Fayçal Touazi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9161)


This paper studies the extension of possibilistic logic to the case when weights attached to formulas are symbolic and stand for variables that lie in a totally ordered scale, and only partial knowledge is available on the relative strength of these weights. A proof of the soundness and the completeness of this logic according to the relative certainty semantics in the sense of necessity measures is provided. Based on this result, two syntactic inference methods are presented. The first one calculates the necessity degree of a possibilistic formula using the notion of minimal inconsistent sub-base. A second method is proposed that takes inspiration from the concept of ATMS. Notions introduced in that area, such as nogoods and labels, are used to calculate the necessity degree of a possibilistic formula. A comparison of the two methods is provided, as well as a comparison with the original version of symbolic possibilistic logic.


  1. 1.
    Dubois, D., Lang, J., Prade, H.: Possibilistic logic. In: Gabbay, D., Hogger, C., Robinson, J., Nute, D. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3, pp. 439–513. Oxford University Press, Oxford (1994)Google Scholar
  2. 2.
    Dubois, D., Prade, H.: Possibility theory: qualitative and quantitative aspects. In: Smets, P. (ed.) Handbook on Defeasible Reasoning and Uncertainty Management Systems. Quantified Representation of Uncertainty and Imprecision, vol. 1, pp. 169–226. Kluwer Academic Publ., Dordrecht (1998)Google Scholar
  3. 3.
    Dubois, D., Prade, H.: Possibilistic logic - an overview. In: Gabbay, D., Siekmann, J., Woods, J., (eds.): Computational Logic. Handbook of the History of Logic. vol. 9, pp. 283–342. Elsevier (2014)Google Scholar
  4. 4.
    Benferhat, S., Prade, H.: Encoding formulas with partially constrained weights in a possibilistic-like many-sorted propositional logic. In: Kaelbling, L.P., Saffiotti, A., (eds.): IJCAI, pp. 1281–1286 Professional Book Center (2005)Google Scholar
  5. 5.
    Halpern, J.Y.: Defining relative likelihood in partially-ordered preferential structures. J. Artif. Intell. Res. 7, 1–24 (1997)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Benferhat, S., Lagrue, S., Papini, O.: A possibilistic handling of partially ordered information. In: Proceedings 19th Conference on Uncertainty in Artificial Intelligence UAI 2003, pp. 29–36 (2003)Google Scholar
  7. 7.
    Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32, 57–95 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dubois, D., Fargier, H., Prade, H.: Ordinal and probabilistic representations of acceptance. J. Artif. Intell. Res. (JAIR) 22, 23–56 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Benferhat, S., Dubois, D., Prade, H.: Nonmonotonic reasoning, conditional objects and possibility theory. Artif. Intell. 92, 259–276 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    McAreavey, K., Liu, W., Miller, P.: Computational approaches to finding and measuring inconsistency in arbitrary knowledge bases. IJAR 55, 1659–1693 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Liffiton, M.H., Sakallah, K.A.: On finding all minimally unsatisfiable subformulas. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 173–186. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    De Kleer, J.: An assumption-based tms. Artif. Intell. 28, 127–162 (1986)CrossRefGoogle Scholar
  13. 13.
    De Kleer, J.: A general labeling algorithm for assumption-based truth maintenance. AAAI 88, 188–192 (1988)Google Scholar
  14. 14.
    Dubois, D., Prade, H., Touazi, F.: Conditional Preference Nets and Possibilistic Logic. In: van der Gaag, L.C. (ed.) ECSQARU 2013. LNCS, vol. 7958, pp. 181–193. Springer, Heidelberg (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Claudette Cayrol
    • 1
    Email author
  • Didier Dubois
    • 1
  • Fayçal Touazi
    • 1
  1. 1.IRITUniversity of ToulouseToulouseFrance

Personalised recommendations