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Synthese

, Volume 139, Issue 2, pp 165–224 | Cite as

Logics for Epistemic Programs

  • Alexandru Baltag
  • Lawrence S. Moss
Article

Abstract

We construct logical languages which allow one to represent a variety of possible types of changes affecting the information states of agents in a multi-agent setting. We formalize these changes by defining a notion of epistemic program. The languages are two-sorted sets that contain not only sentences but also actions or programs. This is as in dynamic logic, and indeed our languages are not significantly more complicated than dynamic logics. But the semantics is more complicated. In general, the semantics of an epistemic program is what we call aprogram model. This is a Kripke model of ‘actions’,representing the agents' uncertainty about the current action in a similar way that Kripke models of ‘states’ are commonly used in epistemic logic to represent the agents' uncertainty about the current state of the system. Program models induce changes affecting agents' information, which we represent as changes of the state model, called epistemic updates. Formally, an update consists of two operations: the first is called the update map, and it takes every state model to another state model, called the updated model; the second gives, for each input state model, a transition relation between the states of that model and the states of the updated model.

Each variety of epistemic actions, such as public announcements or completely private announcements to groups, gives what we call an action signature, and then each family of action signatures gives a logical language. The construction of these languages is the main topic of this paper. We also mention the systems that capture the valid sentences of our logics. But we defer to a separate paper the completeness proof.

The basic operation used in the semantics is called the update product. A version of this was introduced in Baltag et al. (1998), and the presentation here improves on the earlier one. The update product is used to obtain from any program model the corresponding epistemic update, thus allowing us to compute changes of information or belief. This point is of interest independently of our logical languages. We illustrate the update product and our logical languages with many examples throughout the paper.

Keywords

State Model Program Model Action Signature Input State Basic Operation 
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.

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REFERENCES

  1. Baltag, Alexandru: 1999, ‘A Logic of Epistemic Actions’, (Electronic) Proceedings of the FACAS workshop, held at ESSLLI'99, Utrecht University, Utrecht.Google Scholar
  2. Baltag, Alexandru: 2001, ‘Logics for Insecure Communication’, in J. van Bentham (ed.) Proceedings of the Eighth Conference on Rationality and Knowledge (TARK'01), Morgan Kaufmann, Los Altos, pp. 111–122.Google Scholar
  3. Baltag, Alexandru: 2002, ‘A Logic for Suspicious Players: Epistemic Actions and Belief Updates in Games’, Bulletin Of Economic Research 54(1), 1–46.CrossRefGoogle Scholar
  4. Baltag, Alexandru: 2003, ‘A Coalgebraic Semantics for Epistemic Programs’, in Proceedings of CMCS'03, Electronic Notes in Theoretical Computer Science 82(1), 315–335.Google Scholar
  5. Baltag, Alexandru: 2003, Logics for Communication: Reasoning about Information Flow in Dialogue Games. Course presented at NASSLLI'03. Available at http://www.indiana.edu/~nasslli.Google Scholar
  6. Baltag, Alexandru, Lawrence S. Moss, and Sławomir Solecki: 1998, ‘The Logic of Common Knowledge, Public Announcements, and Private Suspicions’, in I. Gilboa (ed.), Proceedings of the 7th Conference on Theoretical Aspects of Rationality and Knowledge (TARK'98), pp. 43–56.Google Scholar
  7. Baltag, Alexandru, Lawrence S. Moss, and Sławomir Solecki: 2003, ‘The Logic of Epistemic Actions: Completeness, Decidability, Expressivity’, manuscript.Google Scholar
  8. Fagin, Ronald, Joseph Y. Halpern, Yoram Moses, and Moshe Y. Vardi: 1996, Reasoning About Knowledge, MIT Press.Google Scholar
  9. Fischer, Michael J. and Richard E. Ladner: 1979, ‘Propositional Modal Logic of Programs’, J. Comput. System Sci. 18(2), 194–211.CrossRefGoogle Scholar
  10. Gerbrandy, Jelle: 1999a, ‘Dynamic Epistemic Logic’, in Lawrence S. Moss, et al (eds), Logic, Language, and Information, Vol. 2, CSLI Publications, Stanford University.Google Scholar
  11. Gerbrandy, Jelle: 1999b, Bisimulations on Planet Kripke, Ph.D. dissertation, University of Amsterdam.Google Scholar
  12. Gerbrandy, Jelle and Willem Groeneveld: 1997, ‘Reasoning about Information Change', J. Logic, Language, and Information 6, 147–169.Google Scholar
  13. Gochet, P. and P. Gribomont: 2003, ‘Epistemic Logic’, manuscript.Google Scholar
  14. Kooi, Barteld P.: 2003, Knowledge, Chance, and Change, Ph.D. dissertation, University of Groningen.Google Scholar
  15. Meyer, J.-J. and W. van der Hoek: 1995, Epistemic Logic for AI and Computer Science, Cambridge University Press, Cambridge.Google Scholar
  16. Miller, Joseph S. and Lawrence S. Moss: 2003, ‘The Undecidability of Iterated Modal Relativization’, Indiana University Computer Science Department Technical Report 586.Google Scholar
  17. Moss, Lawrence S.: 1999, ‘From Hypersets to Kripke Models in Logics of Announcements’, in J. Gerbrandy et al. (eds), JFAK. Essays Dedicated to Johan van Benthem on the Occasion of his 60th Birthday, Vossiuspers, Amsterdam University Press.Google Scholar
  18. Plaza, Jan: 1989, ‘Logics of Public Communications’, Proceedings, 4th International Symposium on Methodologies for Intelligent Systems.Google Scholar
  19. Pratt, Vaughn R.: 1976, ‘Semantical Considerations on Floyd-Hoare Logic’, in 7th Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc., Long Beach, CA, pp. 109–121.Google Scholar
  20. van Benthem, Johan: 2000, ‘Update Delights’, manuscript.Google Scholar
  21. van Benthem, Johan: 2002, ‘Games in Dynamic Epistemic Logic’, Bulletin of Economic Research 53(4), 219–248.Google Scholar
  22. van Benthem, Johan: 2003, ‘Logic for Information update’, in J. van Bentham (ed.) Proceedings of the Eighth Conference on Rationality and Knowledge (TARK'01), Morgan Kaufmann, Los Altos, pp. 51–68.Google Scholar
  23. van Ditmarsch, Hans P.: 2000, ‘Knowledge Games’, Ph.D. dissertation, University of Groningen.Google Scholar
  24. van Ditmarsch, Hans P.: 2001, ‘Knowledge Games’, Bulletin of Economic Research 53(4), 249–273.CrossRefGoogle Scholar
  25. van Ditmarsch, Hans P., W. van der Hoek, and B. P. Kooi: 2003, in V. F. Hendricks et al. (eds), Concurrent Dynamic Epistemic Logic, Synt. Lib. vol. 322, Kluwer Academic Publishers.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Alexandru Baltag
    • 1
  • Lawrence S. Moss
    • 2
  1. 1.Computing LaboratoryOxford UniversityOxfordU.K.
  2. 2.Mathematics DepartmentIndiana UniversityBloomingtonU.S.A.

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