Rank-1 Modal Logics Are Coalgebraic

  • Lutz Schröder
  • Dirk Pattinson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)


Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatised in rank 1. Here we establish the converse, i.e. every rank 1 modal logic has a sound and strongly complete coalgebraic semantics. As a consequence, recent results on coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, become applicable to arbitrary rank 1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of these results. As an extended example, we apply our framework to recently defined deontic logics.


Modal Logic Natural Transformation Canonical Model Propositional Variable Deontic Logic 
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  1. 1.
    Adámek, J., Porst, H.-E.: On tree coalgebras and coalgebra presentations. Theoret. Comput. Sci. 311, 257–283 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Barr, M.: Terminal coalgebras in well-founded set theory. Theoret. Comput. Sci. 114, 299–315 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Chellas, B.: Modal Logic. Cambridge (1980)Google Scholar
  4. 4.
    Fine, K.: In so many possible worlds. Notre Dame J. Formal Logic 13, 516–520 (1972)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Goble, L.: A proposal for dealing with deontic dilemmas. In: Lomuscio, A., Nute, D. (eds.) DEON 2004. LNCS (LNAI), vol. 3065, pp. 74–113. Springer, Heidelberg (2004)Google Scholar
  6. 6.
    Gumm, H.-P.: Functors for coalgebras. Algebra Universalis 45, 135–147 (2001)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Heifetz, A., Mongin, P.: Probabilistic logic for type spaces. Games and Economic Behavior 35, 31–53 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Hilpinen, R.: Deontic logic. In: Goble, L. (ed.) The Blackwell Guide to Philosophical Logic, Blackwell, Malden (2001)Google Scholar
  9. 9.
    Jacobs, B.: Towards a duality result in the modal logic of coalgebras. In: Coalgebraic Methods in Computer Science. ENTCS, vol. 33, Elsevier, Amsterdam (2000)Google Scholar
  10. 10.
    Kupke, C., Kurz, A., Pattinson, D.: Ultrafilter extensions for coalgebras. In: Fiadeiro, J.L., et al. (eds.) CALCO 2005. LNCS, vol. 3629, pp. 263–277. Springer, Heidelberg (2005)Google Scholar
  11. 11.
    Kurz, A.: Specifying coalgebras with modal logic. Theoret. Comput. Sci. 260, 119–138 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Inform. Comput. 94, 1–28 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Pacuit, E., Salame, S.: Majority logic. In: Principles of Knowledge Representation and Reasoning, KR 04, pp. 598–605. AAAI Press, Menlo Park (2004)Google Scholar
  14. 14.
    Pattinson, D.: Semantical principles in the modal logic of coalgebras. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 514–526. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Pattinson, D.: Expressive logics for coalgebras via terminal sequence induction. Notre Dame J. Formal Logic 45, 19–33 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Pauly, M.: A modal logic for coalitional power in games. J. Logic Comput. 12, 149–166 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Rößiger, M.: Coalgebras and modal logic. In: Coalgebraic Methods in Computer Science. ENTCS, vol. 33, Elsevier, Amsterdam (2000)Google Scholar
  18. 18.
    Rutten, J.: Universal coalgebra: A theory of systems. Theoret. Comput. Sci. 249, 3–80 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Schröder, L.: Expressivity of coalgebraic modal logic: the limits and beyond. In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 440–454. Springer, Heidelberg (Extended version to appear in Theoret. Comput. Sci.) (2005)Google Scholar
  20. 20.
    Schröder, L.: A finite model construction for coalgebraic modal logic. In: Aceto, L., Ingólfsdóttir, A. (eds.) FOSSACS 2006 and ETAPS 2006. LNCS, vol. 3921, pp. 157–171. Springer, Heidelberg (Extended version to appear in J. Logic Algebraic Programming) (2006)CrossRefGoogle Scholar
  21. 21.
    Schröder, L., Pattinson, D.: PSPACE reasoning for rank-1 modal logics. In: Logic in Computer Science, pp. 231–240. IEEE Computer Society Press, Los Alamitos (2006), Presentation slides available under Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Lutz Schröder
    • 1
  • Dirk Pattinson
    • 2
  1. 1.Department of Computer Science, University of Bremen, and DFKI Lab Bremen 
  2. 2.Department of Computing, Imperial College London 

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