Characterizing Relative Frame Definability in Team Semantics via the Universal Modality

  • Katsuhiko Sano
  • Jonni VirtemaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9803)


Let Open image in new window denote the fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. We characterise the relative definability of Open image in new window relative to finite transitive frames in the spirit of the well-known Goldblatt–Thomason theorem. We show that a class \(\mathbb {F}\) of finite transitive frames is definable in Open image in new window relative to finite transitive frames if and only if \(\mathbb {F}\) is closed under taking generated subframes and bounded morphic images. In addition, we study modal definability in team-based logics. We study (extended) modal dependence logic, (extended) modal inclusion logic, and modal team logic. With respect to global model definability we obtain a trichotomy and with respect to frame definability a dichotomy. As a corollary we obtain relative Goldblatt–Thomason -style theorems for each of the logics listed above.


  1. 1.
    van Benthem, J.: Notes on modal definability. Notre Dame J. Formal Log. 30(1), 20–35 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, New York (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Durand, A., Kontinen, J., Vollmer, H.: Expressivity and Complexity of Dependence Logic. Springer (2016) (In Press)Google Scholar
  4. 4.
    Ebbing, J., Hella, L., Meier, A., Müller, J.-S., Virtema, J., Vollmer, H.: Extended modal dependence logic \(\cal EMDL\). In: Libkin, L., Kohlenbach, U., de Queiroz, R. (eds.) WoLLIC 2013. LNCS, vol. 8071, pp. 126–137. Springer, Heidelberg (2013)Google Scholar
  5. 5.
    Galliani, P.: Inclusion and exclusion dependencies in team semantics - on some logics of imperfect information. Ann. Pure Appl. Log. 163(1), 68–84 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gargov, G., Goranko, V.: Modal logic with names. J. Philos. Log. 22, 607–636 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Goldblatt, R.I., Thomason, S.K.: Axiomatic classes in propositional modal logic. In: Crossley, J.N. (ed.) Algebra and Logic, pp. 163–73. Springer, Heidelberg (1975)CrossRefGoogle Scholar
  8. 8.
    Goranko, V., Otto, M.: Model theory of modal logic. In: Blackburn, P., Van Benthem, J., Wolter, F. (eds.) Handbook of Modal Logic. Studies in Logic and Practical Reasoning, vol. 3, pp. 249–329. Elsevier, Amsterdam (2007)CrossRefGoogle Scholar
  9. 9.
    Goranko, V., Passy, S.: Using the universal modality: gains and questions. J. Log. Comput. 2(1), 5–30 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grädel, E., Väänänen, J.: Dependence and independence. Stud. Logica 101(2), 399–410 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hella, L., Luosto, K., Sano, K., Virtema, J.: The expressive power of modal dependence logic. In: AiML 2014 (2014)Google Scholar
  12. 12.
    Hella, L., Stumpf, J.: The expressive power of modal logic with inclusion atoms. In: GandALF 2015 (2015)Google Scholar
  13. 13.
    Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. J. ACM 32(1), 137–161 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hintikka, J., Sandu, G.: Informational independence as a semantical phenomenon. In: Fenstad, J.E., et al. (eds.) Logic, methodology and philosophy of science, VIII (Moscow, 1987). Studies in Logic and the Foundations of Mathematics, vol. 126, pp. 571–589. North-Holland, Amsterdam (1989)Google Scholar
  15. 15.
    Hodges, W.: Some strange quantifiers. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science. LNCS, vol. 1261, pp. 51–65. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  16. 16.
    Kontinen, J., Müller, J.-S., Schnoor, H., Vollmer, H.: Modal independence logic. In: AiML 2014 (2014)Google Scholar
  17. 17.
    Kontinen, J., Müller, J.-S., Schnoor, H., Vollmer, H.: A van Benthem theorem for modal team semantics. In: 24th EACSL Annual Conference on Computer Science Logic (2015)Google Scholar
  18. 18.
    Sano, K., Virtema, J.: Characterizing frame definability in team semantics via the universal modality. In: de Paiva, V., de Queiroz, R., Moss, L.S., Leivant, D., de Oliveira, A. (eds.) WoLLIC 2015. LNCS, vol. 9160, pp. 140–155. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  19. 19.
    Väänänen, J.: Dependence Logic - A New Approach to Independence Friendly Logic. London Mathematical Society student texts, vol. 70. Cambridge University Press, New York (2007)CrossRefzbMATHGoogle Scholar
  20. 20.
    Väänänen, J.: Modal dependence logic. In: Apt, K.R., van Rooij, R. (eds.) New Perspectives on Games and Interaction. Texts in Logic and Games, vol. 4, pp. 237–254. Amsterdam University Press, Amsterdam (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Japan Advanced Institute of Science and TechnologyNomiJapan
  2. 2.University of HelsinkiHelsinkiFinland
  3. 3.Leibniz Universität HannoverHanoverGermany

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