Learning in a Changing World, an Algebraic Modal Logical Approach

  • Prakash Panangaden
  • Mehrnoosh Sadrzadeh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6486)


We develop an algebraic modal logic that combines epistemic and dynamic modalities with a view to modelling information acquisition (learning) by automated agents in a changing world. Unlike most treatments of dynamic epistemic logic, we have transitions that “change the state” of the underlying system and not just the state of knowledge of the agents. The key novel feature that emerges is the need to have a way of “inverting transitions” and distinguishing between transitions that “really happen” and transitions that are possible.

Our approach is algebraic, rather than being based on a Kripke-style semantics. The semantics are given in terms of quantales. We study a class of quantales with the appropriate inverse operations and prove properties of the setting. We illustrate the ideas with toy robot-navigation problems. These illustrate how an agent learns information by taking actions.


Real Action Robot Navigation Epistemic Logic Epistemic Modality Kripke Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AV93]
    Abramsky, S., Vickers, S.: Quantales observational logic and process semantics. Mathematical Structures in Computer Science 3, 161–227 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [Auch07]
    Aucher, G., Herzig, A.: From DEL to EDL: Exploring the Power of Converse Events. In: Mellouli, K. (ed.) ECSQARU 2007. LNCS (LNAI), vol. 4724, pp. 199–209. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. [BCS07]
    Baltag, A., Coecke, B., Sadrzadeh, M.: Epistemic actions as resources. Journal of Logic and Computation 17, 555–585 (2007) (arXiv:math/0608166)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [DMS06]
    Desharnais, J., Müller, B., Struth, G.: Kleene algebra with domain. ACM Trans. Comput. Log. 7, 798–833 (2006)MathSciNetCrossRefGoogle Scholar
  5. [Dun05]
    Dunn, M.: Positive modal logic. Studia Logica 55, 301–317 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [FHMV95]
    Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. MIT Press, Cambridge (1995)zbMATHGoogle Scholar
  7. [vanDit05]
    van Ditmarsch, H.P., van der Hoek, W., Kooi, B.P.: Dynamic Epistemic Logic with Assignment. In: Proceedings of AAMAS, pp. 141–148 (2005)Google Scholar
  8. [GNV05]
    Gehrke, M., Nagahashi, H., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Annals of Pure and Applied Logic 131, 65–102 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [HKT00]
    Harel, D., Kozen, D., Tiuryn, J.: Propositional Dynamic Logic. MIT Press, Cambridge (2000)zbMATHGoogle Scholar
  10. [HM84]
    Halpern, J.Y., Moses, Y.: Knowledge and common knowledge in a distributed environment. In: Proceedings of the Third ACM Symposium on Principles of Distributed Computing, pp. 50–61 (1984); A revised version appears as IBM Research Report RJ 4421 (August 1987)Google Scholar
  11. [HM90]
    Halpern, J., Moses, Y.: Knowledge and common knowledge in a distributed environment. JACM 37, 549–587 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Kri63]
    Kripke, S.: Semantical analysis of modal logic. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik 9, 67–96 (1963)CrossRefzbMATHGoogle Scholar
  13. [PS10]
    Panangaden, P., Sadrzadeh, M.: Learning in a changing world via algebraic modal logic, and
  14. [Par78]
    Parikh, R.: The Completeness of Propositional Dynamic Logic. In: Winkowski, J. (ed.) MFCS 1978. LNCS, vol. 64, pp. 403–415. Springer, Heidelberg (1978)CrossRefGoogle Scholar
  15. [Phi09]
    Phillips, C.: An algebraic approach to dynamic epistemic logic. Master’s thesis, School of Computer Sciecne. McGill University (2009)Google Scholar
  16. [Sad06]
    Sadrzadeh, M.: Actions and Resources in Epistemic Logic. PhD thesis, Université du Québec à Montréal (2006)Google Scholar
  17. [SD09]
    Sadrzadeh, M., Dyckhoff, R.: Positive logic with adjoint modalities: Proof theory, semantics and reasoning about information. ENTCS 23, 211–225 (2009)zbMATHGoogle Scholar
  18. [vDvdHK08]
    van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Synthese Library, vol. 337. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  19. [vK98]
    von Karger, B.: Temporal algebras. Mathematical Structures in Computer Science 8, 277–320 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [Win09]
    Winskel, G.: Prime algebraicity. Theoretical Computer Science 410, 4160–4168 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Prakash Panangaden
    • 1
  • Mehrnoosh Sadrzadeh
    • 2
  1. 1.School of Computer ScienceMcGill UniversityMontréalCanada
  2. 2.Computing LaboratoryOxford UniversityOxfordUK

Personalised recommendations