Skip to main content
Log in

Compilation of static and evolving conditional knowledge bases for computing induced nonmonotonic inference relations

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

Several different semantics have been proposed for conditional knowledge bases \(\mathcal {R}\) containing qualitative conditionals of the form “If A, then usually B”, leading to different nonmonotonic inference relations induced by \(\mathcal {R}\). For the notion of c-representations which are a subclass of all ranking functions accepting \(\mathcal {R}\), a skeptical inference relation, called c-inference and taking all c-representations of \(\mathcal {R}\) into account, has been suggested. In this article, we develop a 3-phase compilation scheme for both knowledge bases and skeptical queries to constraint satisfaction problems. In addition to skeptical c-inference, we show how also credulous and weakly skeptical c-inference can be modelled as constraint satisfaction problems, and that the compilation scheme can be extended to such queries. We further extend the compilation approach to knowledge bases evolving over time. The compiled form of \(\mathcal {R}\) is reused for incrementally compiling extensions, contractions, and updates of \(\mathcal {R}\). For each compilation step, we prove its soundness and completeness, and demonstrate significant efficiency benefits when querying the compiled version of \(\mathcal {R}\). These findings are also supported by experiments with the software system InfOCF that employs the proposed compilation scheme.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, E.: Probability and the logic of conditionals. In: Hintikka, J., Suppes, P. (eds.) Aspects of Inductive Logic, pp 265–316, North-Holland (1966)

  2. Adams, E.: The Logic of Conditionals. D. Reidel, Dordrecht (1975)

    Book  MATH  Google Scholar 

  3. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, United Kingdom (1998)

    Book  MATH  Google Scholar 

  4. Beierle, C., Eichhorn, C., Kern-Isberner, G.: Skeptical inference based on c-representations and its characterization as a constraint satisfaction problem. In: FoIKS-2016, Volume 9616 of LNCS, pp 65–82. Springer (2016)

  5. Beierle, C., Eichhorn, C., Kern-Isberner, G., Kutsch, S.: Skeptical, weakly skeptical, and credulous inference based on preferred ranking functions. In: Kaminka, G.A., Fox, M., Bouquet, P., Hüllermeier, E., Dignum, V., Dignum, F., van Harmelen, F. (eds.) Proceedings 22nd European Conference on Artificial Intelligence, ECAI-2016, Volume 285 of Frontiers in Artificial Intelligence and Applications, pp 1149–1157. IOS Press (2016)

  6. Beierle, C., Eichhorn, C., Kern-Isberner, G., Kutsch, S.: Properties of skeptical c-inference for conditional knowledge bases and its realization as a constraint satisfaction problem. Ann. Math. Artif. Intell. 83(3-4), 247–275 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beierle, C., Eichhorn, C., Kutsch, S.: A practical comparison of qualitative inferences with preferred ranking models. KI – Künstliche Intelligenz 31(1), 41–52 (2017)

    Article  Google Scholar 

  8. Beierle, C., Kern-Isberner, G.: Semantical investigations into nonmonotonic and probabilistic logics. Ann. Math. Artif. Intell. 65(2-3), 123–158 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Beierle, C., Kern-Isberner, G., Sauerwald, K., Bock, T., Ragni, M.: Towards a general framework for kinds of forgetting in common-sense belief management. KI – Künstliche Intelligenz 33(1), 57–68 (2019)

    Article  Google Scholar 

  10. Beierle, C., Kern-Isberner, G., Södler, K.: A declarative approach for computing ordinal conditional functions using constraint logic programming. In: Tompits, H., Abreu, S., Oetsch, J., Pührer, J., Seipel, D., Umeda, M., Wolf, A. (eds.) Applications of Declarative Programming and Knowledge Management, Volume 7773 of LNAI, pp 175–192. Springer (2013)

  11. Beierle, C., Kutsch, S.: Comparison of inference relations defined over different sets of ranking functions. In: Antonucci, A., Cholvy, L., Papini, O. (eds.) S.mbolic and Quantitative Approaches to Reasoning with Uncertainty - 14th European Conference, ECSQARU 2017, Lugano, Switzerland, July 10–14, 2017, Proceedings, Volume 10369 of Lecture Notes in Computer Science, pp 225–235 (2017)

  12. Beierle, C., Kutsch, S.: Computation and comparison of nonmonotonic skeptical inference relations induced by sets of ranking models for the realization of intelligent agents. Appl. Intell. 49(1), 28–43 (2019)

    Article  Google Scholar 

  13. Beierle, C., Kutsch, S., Sauerwald, K.: Compilation of conditional knowledge bases for computing c-inference relations. In: Ferrarotti, F., Woltran, S. (eds.) Foundations of Information and Knowledge Systems - 10th International Symposium, FoIKS 2018, Budapest, Hungary, May 14-18, 2018, Proceedings, Volume 10833 of LNCS, pp 34–54. Springer (2018)

  14. Ben Amor, N., Dubois, D., Gouider, H., Prade, H.: Preference modeling with possibilistic networks and symbolic weights: A theoretical study. In: Kaminka, G.A., Fox, M., Bouquet, P., Hüllermeier, E., Dignum, V., Dignum, F., van Harmelen, F. (eds.) ECAI 2016 - 22nd European Conference on Artificial Intelligence, 29 August-2 September 2016, The Hague, The Netherlands - Including Prestigious Applications of Artificial Intelligence (PAIS 2016), Volume 285 of Frontiers in Artificial Intelligence and Applications, pp 1203–1211. IOS Press (2016)

  15. Benferhat, S., Dubois, D., Prade, H.: Possibilistic and standard probabilistic semantics of conditional knowledge bases. J. Log. Comput. 9(6), 873–895 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Benferhat, S., Dubois, D., Prade, H.: Possibilistic and standard probabilistic semantics of conditional knowledge bases. J. Logic Comput. 9(6), 873–895 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Byrne, R.M.: Suppressing valid inferences with conditionals. Cognition 31, 61–83 (1989)

    Article  Google Scholar 

  18. Carlsson, M., Ottosson, G., Carlson, B.: An open-ended finite domain constraint solver. In: PLILP’97, Volume 1292 of LNCS, pp 191–206. Springer (1997)

  19. Cayrol, C., Dubois, D., Touazi, F.: Symbolic possibilistic logic: Completeness and inference methods. J. Log. Comput. 28(1), 219–244 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Darwiche, A., Marquis, P.: A knowledge compilation map. J. Artif. Intell. Res. 17, 229–264 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Darwiche, A., Marquis, P.: Compiling propositional weighted bases. Artif. Intell. 157(1–2), 81–113 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. de Finetti, B.: La prévision, ses lois logiques et ses sources subjectives. Ann. Inst. H. Poincaré 7(1), 1–68 (1937). English translation in Studies in Subjective Probability, ed. H. Kyburg and H.E. Smokler, 1974, 93–158. New York: Wiley & Sons

    MathSciNet  MATH  Google Scholar 

  23. Dubois, D., Prade, H.: Conditional objects as nonmonotonic consequence relationships. Special Issue on Conditional Event Algebra, IEEE Transactions on Systems Man and Cybernetics 24(12), 1724–1740 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dubois, D., Prade, H.: Possibility theory and its applications: Where do we stand? In: Kacprzyk, J., Pedrycz, W. (eds.) Springer Handbook of Computational Intelligence, pp 31–60. Springer, Berlin (2015)

    Chapter  MATH  Google Scholar 

  25. Dupin de Saint-Cyr, F., Lang, J., Schiex, T.: Penalty logic and its link with dempster-shafer theory. In: de Mántaras, R.L., Poole, D. (eds.) UAI ’94: Proceedings of the Tenth Annual Conference on Uncertainty in Artificial Intelligence, Seattle, Washington, USA, July 29-31, 1994, pp 204–211. Morgan Kaufmann (1994)

  26. Eichhorn, C., Kern-Isberner, G., Ragni, M.: Rational inference patterns based on conditional logic. In: McIlraith, S., Weinberger, K. (eds.) Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI-18), pp 1827–1834. AAAI Press (2018)

  27. Eiter, T., Lukasiewicz, T.: Complexity results for structure-based causality. Artif Intell. 142(1), 53–89 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Finthammer, M., Beierle, C.: A two-level approach to maximum entropy model computation for relational probabilistic logic based on weighted conditional impacts. In: Straccia, U., Calì, A. (eds.) Scalable Uncertainty Management - 8th International Conference, SUM 2014, Oxford, UK, September 15-17, 2014. Proceedings, Volume 8720 of LNAI, pp 162–175. Springer (2014)

  29. Gärdenfors, P.: Knowledge in Flux: Modeling the Dynamics of Epistemic States. MIT Press, Cambridge (1988)

    MATH  Google Scholar 

  30. Goldszmidt, M., Pearl, J.: On the Relation Between Rational Closure and System-Z. UCLA Computer Science Department (1991)

  31. Goldszmidt, M., Pearl, J.: Qualitative probabilities for default reasoning, belief revision, and causal modeling. Artif. Intell. 84(1–2), 57–112 (1996)

    Article  MathSciNet  Google Scholar 

  32. Isberner, M., Kern-Isberner, G.: Plausible reasoning and plausibility monitoring in language comprehension. Int. J Approx. Reas. 88, 53–71 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kern-Isberner, G.: Conditionals in Nonmonotonic Reasoning and Belief Revision, Volume 2087 of LNAI. Springer (2001)

  34. Kern-Isberner, G.: A thorough axiomatization of a principle of conditional preservation in belief revision. Ann. of Math. and Artif. Intell. 40(1-2), 127–164 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kern-Isberner, G., Eichhorn, C.: Structural inference from conditional knowledge bases. Stud. Logica. 102(4), 751–769 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Knuth, D.E., Bendix, P.B.: Simple word problems in universal algebra. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp 263–297. Pergamon Press (1970)

  37. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 44, 167–207 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lehmann, D.J., Magidor, M.: What does a conditional knowledge base entail? Artif. Intell. 55(1), 1–60 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  39. Leopold, T., Kern-Isberner, G., Peters, G.: Belief revision with reinforcement learning for interactive object recognition. In: Proceedings 18th European Conference on Artificial Intelligence, ECAI’08 (2008)

  40. Lewis, D.: Counterfactuals. Harvard University Press, Cambridge (1973)

    MATH  Google Scholar 

  41. Lukasiewicz, T.: Weak nonmonotonic probabilistic logics. Artif. Intell. 168(1-2), 119–161 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Makinson, D.: General patterns in nonmonotonic reasoning. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3, pp 35–110. Oxford University Press (1994)

  43. Marquis, P.: Consequence finding algorithms. In: Handbook of Defeasible Reasoning and Uncertainty Management Systems, pp 41–145. Springer (2000)

  44. Minock, M., Kraus, H.: Z-log: Applying system-Z. In: Proceedings of the 8th European Conference on Logics in AI, JELIA 2002, Volume 2424 of LNAI, pp 545–548. Springer (2002)

  45. Nute, D.: Topics in Conditional Logic. D. Reidel Publishing Company, Dordrecht (1980)

    Book  MATH  Google Scholar 

  46. Paris, J.: The Uncertain Reasoner’S Companion – A Mathematical Perspective. Cambridge University Press (1994)

  47. Pearl, J.: System Z: A natural ordering of defaults with tractable applications to nonmonotonic reasoning. In: Parikh, R. (ed.) Proceedings of the 3rd Conference on Theoretical Aspects of Reasoning About Knowledge (TARK1990), pp 121–135. Morgan Kaufmann Publishers Inc., San Francisco (1990)

  48. Pinkas, G.: Reasoning, nonmonotonicity and learning in connectionist networks that capture propositional knowledge. Artif. Intell. 77(2), 203–247 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  49. Sperner, E.: Ein Satz über Untermengen einer endlichen Menge. Math. Z. 27 (1), 544–548 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  50. Spohn, W.: Ordinal conditional functions: A dynamic theory of epistemic states. In: Harper, W., Skyrms, B. (eds.) Causation in Decision, Belief Change, and Statistics, II, pp 105–134. Kluwer Academic Publishers (1988)

  51. Spohn, W.: The Laws of Belief: Ranking Theory and Its Philosophical Applications. Oxford University Press, Oxford (2012)

    Book  Google Scholar 

  52. Wason, P.C.: Reasoning about a rule. Q. J. Exper. Psychol. 20(3), 273–281 (1968)

    Article  Google Scholar 

  53. Wilhelm, M., Kern-Isberner, G., Finthammer, M., Beierle, C.: A generalized iterative scaling algorithm for maximum entropy model computations respecting probabilistic independencies. In: Ferrarotti, F., Woltran, S. (eds.) Foundations of Information and Knowledge Systems - 10th International Symposium, FoIKS 2018, Budapest, Hungary, May 14-18, 2018, Proceedings, Volume 10833 of LNCS, pp 379–399. Springer (2018)

Download references

Acknowledgements

This work was supported by DFG Grant BE 1700/9-1 given to Christoph Beierle as part of the priority program “Intentional Forgetting in Organizations” (SPP 1921). Kai Sauerwald is supported by this Grant. We thank the anonymous reviewers for their valuable hints and comments that helped us to improve the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Christoph Beierle.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beierle, C., Kutsch, S. & Sauerwald, K. Compilation of static and evolving conditional knowledge bases for computing induced nonmonotonic inference relations. Ann Math Artif Intell 87, 5–41 (2019). https://doi.org/10.1007/s10472-019-09653-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-019-09653-7

Keywords

Mathematics Subject Classification (2010)

Navigation