Advertisement

On the Definability of Simulability and Bisimilarity by Finite Epistemic Models

  • Hans van Ditmarsch
  • David Fernández-Duque
  • Wiebe van der Hoek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6814)

Abstract

We explore when finite epistemic models are definable up to simulability or bisimulation, either over the basic multi-agent epistemic language \(\mathsf L\) or over its extension \(\mathsf L^C\) with common knowledge operators. Our negative results are that: simulability is not definable in general in \(\mathsf L^C\), and finite epistemic states (i.e., pointed models) are not definable up to bisimulation in \(\mathsf L\). Our positive results are that: finite epistemic states are definable up to bisimulation by model validity of \(\mathsf L\)-formulas, and there is a class of epistemic models we call well-multifounded for which simulability is definable over \(\mathsf L\). From our method it also follows that finite epistemic models (i.e., not-pointed models) are definable up to bisimulation using model validity in \(\mathsf L\). Our results may prove useful for the logical specification of multi-agent systems, as it provides justification for the ubiquitous but often unjustified claims of the form ‘suppose action a can only be performed in state s’: we show when such preconditions exist. An application are characteristic formulae for interpreted systems. They have a special form wherein factual knowledge, positive knowledge, and ignorance can be separated.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aczel, P.: Non-Well-Founded Sets. CSLI Lecture Notes, vol. 14. CSLI Publications, Stanford (1988)zbMATHGoogle Scholar
  2. 2.
    Barwise, J., Moss, L.S.: Vicious Circles. CSLI Publications, Stanford (1996)zbMATHGoogle Scholar
  3. 3.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  4. 4.
    Browne, M., Clarke, E., Grümberg, O.: Characterizing Kripke structures in temporal logic. In: Ehrig, H., Levi, G., Montanari, U., Kowalski, R. (eds.) CAAP 1987 and TAPSOFT 1987. LNCS, vol. 249, pp. 256–270. Springer, Heidelberg (1987)Google Scholar
  5. 5.
    Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. MIT Press, Cambridge (1995)zbMATHGoogle Scholar
  6. 6.
    Fernández-Duque, D.: On the modal definability of simulability by finite transitive models. In: Studia Logica (forthcoming, 2011)Google Scholar
  7. 7.
    Lomuscio, A.R., Ryan, M.D.: On the relation between interpreted systems and kripke models. In: Wobcke, W.R., Pagnucco, M., Zhang, C. (eds.) Agents and Multi-Agent Systems Formalisms, Methodologies, and Applications. LNCS (LNAI), vol. 1441, pp. 46–59. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  8. 8.
    Meyer, J.-J.C., van der Hoek, W.: Epistemic Logic for AI and Computer Science. Cambridge Tracts in Theoretical Computer Science, vol. 41. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  9. 9.
    van Benthem, J.: Dynamic odds and ends. ILLC Technical Report ML (1998)Google Scholar
  10. 10.
    van Benthem, J.: ‘One is a lonely number’: on the logic of communication. In: Chatzidakis, Z., Koepke, P., Pohlers, W. (eds.) Logic Colloquium 2002. Lecture Notes in Logic, vol. 27. Association for Symbolic Logic (2002)Google Scholar
  11. 11.
    van Ditmarsch, H., van der Hoek, W., Kooi, B.: Dynamic Epistemic Logic. Springer, Berlin (2007)CrossRefzbMATHGoogle Scholar
  12. 12.
    van Ditmarsch, H., van der Hoek, W., Kooi, B.P.: Descriptions of game states. In: Mints, G., Muskens, R. (eds.) Logic, Games, and Constructive Sets. CSLI Lecture Notes, vol. 161, pp. 43–58. CSLI Publications, Stanford (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Hans van Ditmarsch
    • 1
  • David Fernández-Duque
    • 1
  • Wiebe van der Hoek
    • 2
  1. 1.University of SevillaSpain
  2. 2.University of LiverpoolUnited Kingdom

Personalised recommendations