Belief Shadowing

  • Łukasz Białek
  • Barbara Dunin-Kęplicz
  • Andrzej SzałasEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11375)


Adapting beliefs to new circumstances, like belief change, update, revision or merging, typically requires deep and/or complex adjustments of belief bases even when adaptations happen to be transient. We present a novel, lightweight and tractable approach to a new kind of beliefs’ interference which we call belief shadowing. Put simply, it is a transient swap of beliefs when part of one belief base is to be shadowed by another belief base representing new observations and/or beliefs of superior agents/teams. In this case no changes to belief bases are needed. This substantially improves the performance of systems based on doxastic reasoning. We ensure tractability of our formal framework, what makes it suitable for real-world applications.

The presented approach is based on a carefully chosen four-valued paraconsistent logic with truth values representing truth, falsity, incompleteness and inconsistency. Moreover, potentially undesired or forbidden conclusions are prevented by integrity constrains together with their shadowing machinery.

As an implementation environment we use \(4\hbox {QL}^{\mathrm{Bel}}\), a recently developed four-valued query language based on the same underlying logic and providing necessary reasoning tools. Importantly, the shadowing techniques are general enough to be embedded in any reasoning environment addressing related phenomena.


  1. 1.
    Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases: The Logical Level. Addison-Wesley, Boston (1995)Google Scholar
  2. 2.
    Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symb. Log. 50(2), 510–530 (1985)MathSciNetCrossRefGoogle Scholar
  3. 3.
    de Amo, S., Pais, M.: A paraconsistent logic approach for querying inconsistent databases. Int. J. Approx. Reason. 46, 366–386 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bertossi, L., Hunter, A., Schaub, T. (eds.): Inconsistency Tolerance. LNCS, vol. 3300. Springer, Heidelberg (2005). Scholar
  5. 5.
    Béziau, J.Y., Carnielli, W., Gabbay, D. (eds.): Handbook of Paraconsistency. College Publications, London (2007)zbMATHGoogle Scholar
  6. 6.
    Białek, Ł., Dunin-Kęplicz, B., Szałas, A.: Rule-based reasoning with belief structures. In: Kryszkiewicz, M., Appice, A., Ślęzak, D., Rybinski, H., Skowron, A., Raś, Z.W. (eds.) ISMIS 2017. LNCS (LNAI), vol. 10352, pp. 229–239. Springer, Cham (2017). Scholar
  7. 7.
    Bochman, A.: A Logical Theory of Nonmonotonic Inference and Belief Change. Springer, Heidelberg (2001). Scholar
  8. 8.
    Dechter, R.: Constraint Processing. Morgan Kaufmann, Burlington (2003)zbMATHGoogle Scholar
  9. 9.
    Dunin-Kęplicz, B., Strachocka, A.: Paraconsistent argumentation schemes. Web Intell. 14(1), 43–65 (2016)CrossRefGoogle Scholar
  10. 10.
    Dunin-Kęplicz, B., Strachocka, A., Szałas, A., Verbrugge, R.: Perceiving speech acts under incomplete and inconsistent information. In: Barbucha, D., et al. (eds.) Proceedings of the 7th KES AMSTA Conference, pp. 255–264. IOS Press (2013)Google Scholar
  11. 11.
    Dunin-Kęplicz, B., Szałas, A.: Taming complex beliefs. In: Nguyen, N.T. (ed.) Transactions on Computational Collective Intelligence XI. LNCS, vol. 8065, pp. 1–21. Springer, Heidelberg (2013). Scholar
  12. 12.
    Dunin-Kęplicz, B., Szałas, A.: Indeterministic belief structures. In: Jezic, G., Kusek, M., Lovrek, I., J. Howlett, R., Jain, L.C. (eds.) Agent and Multi-Agent Systems: Technologies and Applications. AISC, vol. 296, pp. 57–66. Springer, Cham (2014). Scholar
  13. 13.
    Dunin-Kęplicz, B., Verbrugge, R.: Teamwork in Multi-Agent Systems: A Formal Approach. Wiley, Hoboken (2010)CrossRefGoogle Scholar
  14. 14.
    Fagin, R., Halpern, J., Moses, Y., Vardi, M.: Reasoning About Knowledge. The MIT Press, Cambridge (2003)zbMATHGoogle Scholar
  15. 15.
    Fermé, E., Hansson, S.O.: AGM 25 years: twenty-five years of research in belief change. J. Philosophical Logic 40(2), 295–331 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gärdenfors, P.: Conditionals and changes of belief. Acta Philos. Fenn. 30, 381–404 (1978)zbMATHGoogle Scholar
  17. 17.
    Hadley, R.F.: The many uses of ‘belief’ in AI. Mind. Mach. 1(1), 55–73 (1991)Google Scholar
  18. 18.
    Hansson, S.O.: Taking belief bases seriously. In: Prawitz, D., Westerståhl, D. (eds.) Logic and Philosophy of Science in Uppsala. Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science), vol. 236, pp. 13–28. Springer, Dordrecht (1994). Scholar
  19. 19.
    Hansson, S.O.: Belief change. In: Dubois, D., Prade, H. (eds.) Revision of Belief Sets and Belief Bases. Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 3, pp. 17–75. Springer, Dordrecht (1998). Scholar
  20. 20.
    Hansson, S.O.: A Textbook of Belief Dynamics: Theory Change and Database Updating. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar
  21. 21.
    Herzig, A., Rifi, O.: Propositional belief base update and minimal change. Artif. Intell. 115(1), 107–138 (1999)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hewitt, C., Woods, J. (eds.): Inconsistency Robustness. College Publications, London (2015)zbMATHGoogle Scholar
  23. 23.
    van Hoeve, W.J., Katriel, I.: Global constraints. Found. AI 2, 169–208 (2006)Google Scholar
  24. 24.
    Huber, F.: Formal representations of belief. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Stanford University, Palo Alto (2016)Google Scholar
  25. 25.
    Konieczny, S., Lang, J., Marquis, P.: \(\text{ DA }^{2}\) merging operators. Artif. Intell. 157(1–2), 49–79 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Konieczny, S., Pérez, R.P.: Merging with integrity constraints. In: Hunter, A., Parsons, S. (eds.) ECSQARU 1999. LNCS (LNAI), vol. 1638, pp. 233–244. Springer, Heidelberg (1999). Scholar
  27. 27.
    Konieczny, S., Pino Pérez, R.: Merging information under constraints: a logical framework. J. Log. Comput. 12(5), 773–808 (2002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lang, J.: Belief update revisited. In: Proceedings of the 20th IJCAI, pp. 2517–2522. Morgan Kaufmann (2007)Google Scholar
  29. 29.
    Liberatore, P.: The complexity of belief update. Artif. Intell. 119(1), 141–190 (2000)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Małuszyński, J., Szałas, A.: Living with inconsistency and taming nonmonotonicity. In: de Moor, O., Gottlob, G., Furche, T., Sellers, A. (eds.) Datalog 2.0 2010. LNCS, vol. 6702, pp. 384–398. Springer, Heidelberg (2011). Scholar
  31. 31.
    Małuszyński, J., Szałas, A.: Partiality and inconsistency in agents’ belief bases. In: Barbucha, D., et al. (eds.) Proceedings of the 7th KES AMSTA Conference, pp. 3–17. IOS Press (2013)Google Scholar
  32. 32.
    Marchi, J., Bittencourt, G., Perrussel, L.: A syntactical approach to belief update. In: Gelbukh, A., de Albornoz, Á., Terashima-Marín, H. (eds.) MICAI 2005. LNCS (LNAI), vol. 3789, pp. 142–151. Springer, Heidelberg (2005). Scholar
  33. 33.
    Meseguer, P., Rossi, F., Schiex, T.: Soft constraints. In: Rossi, et al. [38], pp. 281–328CrossRefGoogle Scholar
  34. 34.
    Meyer, J.J.C., van der Hoek, W.: Epistemic Logic for AI and Theoretical Computer Science. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  35. 35.
    Peppas, P.: Belief revision. In: van Harmelen, F., Lifschitz, V., Porter, B. (eds.) Handbook of KR, pp. 317–359. Elsevier, Amsterdam (2008)Google Scholar
  36. 36.
    Pigozzi, G.: Belief merging and judgment aggregation. In: Zalta, E. (ed.) The Stanford Encyclopedia of Philosophy. Stanford University, Palo Alto (2016)Google Scholar
  37. 37.
    Priest, G.: Paraconsistent belief revision. Theoria 67(3), 214–228 (2001)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Rossi, F., van Beek, P., Walsh, T. (eds.): Handbook of Constraint Programming, Foundations of AI, vol. 2, Supplement C. Elsevier, Amsterdam (2006)Google Scholar
  39. 39.
    Santos, Y.D., Ribeiro, M.M., Wassermann, R.: Between belief bases and belief sets: Partial meet contraction. In: Proceedings of the 2015 International Conference on Defeasible and Ampliative Reasoning. DARe 2015, vol. 1423, pp. 50–56. (2015)Google Scholar
  40. 40.
    Shepherdson, J.: Negation in logic programming. In: Minker, J. (ed.) Foundations of Deductive Databases and Logic Programming, pp. 19–88. Morgan Kaufmann, Burlington (1988)CrossRefGoogle Scholar
  41. 41.
    Szałas, A.: How an agent might think. Log. J. IGPL 21(3), 515–535 (2013)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Testa, R.R., Coniglio, M.E., Ribeiro, M.M.: Paraconsistent belief revision based on a formal consistency operator. CLE E-Prints 15(8), 01–11 (2015)Google Scholar
  43. 43.
    Testa, R.R., Coniglio, M.E., Ribeiro, M.M.: AGM-like paraconsistent belief change. Log. J. IGPL 25(4), 632–672 (2017)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Vitória, A., Małuszyński, J., Szałas, A.: Modeling and reasoning with paraconsistent rough sets. Fundam. Inform. 97(4), 405–438 (2009)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Wooldridge, M.: Reasoning About Rational Agents. MIT Press, Cambridge (2000)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Łukasz Białek
    • 1
  • Barbara Dunin-Kęplicz
    • 1
  • Andrzej Szałas
    • 1
    • 2
    Email author
  1. 1.Institute of InformaticsUniversity of WarsawWarsawPoland
  2. 2.Department of Computer and Information ScienceLinköping UniversityLinköpingSweden

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