Studia Logica

, Volume 79, Issue 3, pp 373–407 | Cite as

The Undecidability of Iterated Modal Relativization

  • Joseph S. MillerEmail author
  • Lawrence S. Moss


In dynamic epistemic logic and other fields, it is natural to consider relativization as an operator taking sentences to sentences. When using the ideas and methods of dynamic logic, one would like to iterate operators. This leads to iterated relativization. We are also concerned with the transitive closure operation, due to its connection to common knowledge. We show that for three fragments of the logic of iterated relativization and transitive closure, the satisfiability problems are fi1 Σ11–complete. Two of these fragments do not include transitive closure. We also show that the question of whether a sentence in these fragments has a finite (tree) model is fi0 Σ01–complete. These results go via reduction to problems concerning domino systems.


Dynamic epistemic logic iterated relativization modal logic undecidability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    BALTAG, ALEXANDRU, and LAWRENCE S. MOSS, ‘Logics for epistemic programs’, Synthese 139, no. 2, (2004), 165–224.CrossRefGoogle Scholar
  2. [2]
    BALTAG, ALEXANDRU, LAWRENCE S. MOSS, and SŁLAWOMIR SOLECKI, ‘The logic of common knowledge, public announcements, and private suspicions’, Proceedings of TARK–VII (Theoretical Aspects of Rationality and Knowledge), 1998.Google Scholar
  3. [3]
    BALTAG, ALEXANDRU, LAWRENCE S. MOSS, and SŁLAWOMIR SOLECKI, Revised version of [2], to appear.Google Scholar
  4. [4]
    BERGER, ROBERT, The Undecidability of the Domino Problem, Memoirs of the American Mathematical Society No. 66, 1966.Google Scholar
  5. [5]
    BLACKBURN, PATRICK, MAARTEN DE RIJKE, and YDE VENEMA, Modal Logic, Cambridge Tracts in Theoretical Computer Science, vol. 53, Cambridge University Press, 2001.Google Scholar
  6. [6]
    BÖRGER, EGON, ERICH GRÄDEL, and YURI GUREVICH, The Classical Decision Problem, Springer-Verlag, Berlin, 1997.Google Scholar
  7. [7]
    DAWAR, ANUJ, ERICH GRÄDEL, and STEPHAN KREUTZER, ‘Infiationary fixed points in modal logic’, ACM Transactions on Computational Logic, to appear.Google Scholar
  8. [8]
    DERSHOWITZ, NACHUM, ‘Orderings for term-rewriting systems’, Theoretical Computer Science 17 (1982), 279–301.CrossRefGoogle Scholar
  9. [9]
    FLUM, JORG, ‘On the (infinite) model theory of fixed-point logics’, in Models, Algebras, and Proofs (Bogot, 199), Lecture Notes in Pure and Applied Mathematics, vol. 203, Marcel Dekker, New York, 1999, pp. 67–75.Google Scholar
  10. [10]
    GERBRANDY, JELLE, ‘Dynamic epistemic logic’, in Lawrence S. Moss, et al., (eds.), Logic, Language, and Information, vol. 2, CSLI Publications, Stanford University, 1999.Google Scholar
  11. [11]
    GERBRANDY, JELLE, Bisimulations on Planet Kripke, Ph.D. dissertation, University of Amsterdam, 1999.Google Scholar
  12. [12]
    GERBRANDY, JELLE, and WILLEM GROENEVELD, ‘Reasoning about information change’, Journal of Logic, Language, and Information 6 (1997), 147–169.Google Scholar
  13. [13]
    GRÄDEL, ERICH, and STEPHAN KREUTZER, ‘Will defiation lead to depletion?’, On non-monotone fixed-point inductions, IEEE Conference on Logic in Computer Science (LICS’03), pp. 158–167.Google Scholar
  14. [14]
    HAREL, DAVID, ‘Recurring dominoes: making the highly undecidable highly understandable’, Topics in the Theory of Computation (Borgholm, 1983), North-Holland Math. Stud., 102, North-Holland, Amsterdam, 1985, pp. 51–71.Google Scholar
  15. [15]
    HAREL, DAVID, DEXTER C. KOZEN, and J. TIURYN, Dynamic Logic, MIT Press, Cambridge Mass., 2000.Google Scholar
  16. [16]
    LIN, YU CAI, ‘The decision problems about the periodic solutions of the domino problems’, Chinese Annals of Mathematics, Series B 5, no. 4, (1984), 721–726.Google Scholar
  17. [17]
    MOSCHOVAKIS, YIANNIS N., ‘On nonmonotone inductive definability’, Fundamenta Mathematicae 82 (1974–75), 39–83.Google Scholar
  18. [18]
    PLAISTED, DAVID A., ‘Termination orderings’, in D. Gabbay, et al. (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, vol. I, pp. 273–364.Google Scholar
  19. [19]
    PLAZA, JAN, ‘Logics of public communications’, Proceedings, 4th International Symposium on Methodologies for Intelligent Systems, 1989.Google Scholar
  20. [20]
    VAN BENTHEM, JOHAN, ‘ ‘One is a lonely number’: on the logic of communication’, ILLC Research Report PP–2003–07, Amsterdam, 2003.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA

Personalised recommendations