# Characterizing Frame Definability in Team Semantics via the Universal Modality

## Abstract

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 characterize the definability of Open image in new window in the spirit of the well-known Goldblatt–Thomason theorem. We show that an elementary class \({\mathbb {F}}\) of Kripke frames is definable in Open image in new window if and only if \({\mathbb {F}}\) is closed under taking generated subframes and bounded morphic images, and reflects ultrafilter extensions and finitely generated subframes. In addition, we initiate the study of modal frame definability in team-based logics. We show that, with respect to frame definability, the logics Open image in new window , modal logic with intuitionistic disjunction, and (extended) modal dependence logic all coincide. Thus we obtain Goldblatt–Thomason -style theorems for each of the logics listed above.

## References

- 1.Areces, C., ten Cate, B.: Hybrid logics. In: Blackburn, P., van Benthem, J., Wolter, F. (eds.) Handbook of Modal Logic, pp. 821–868. Elsevier (2007)Google Scholar
- 2.van Benthem, J.: Modal frame classes revisited. Fundamenta Informaticae
**18**, 303–317 (1993)Google Scholar - 3.Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, New York (2001)CrossRefzbMATHGoogle Scholar
- 4.ten Cate, B.: Model theory for extended modal languages. Ph.D. thesis, University of Amsterdam, Institute for Logic, Language and Computation (2005)Google Scholar
- 5.Chang, C.C., Keisler, H.J.: Model Theory, 3rd edn. North-Holland Publishing Company, Amsterdam (1990)zbMATHGoogle Scholar
- 6.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) CrossRefGoogle Scholar
- 7.Ebbing, J., Lohmann, P., Yang, F.: Model checking for modal intuitionistic dependence logic. In: Bezhanishvili, G., Löbner, S., Marra, V., Richter, F. (eds.) Logic, Language, and Computation. LNCS, vol. 7758, pp. 231–256. Springer, Heidelberg (2013) CrossRefGoogle Scholar
- 8.Gargov, G., Goranko, V.: Modal logic with names. J. Philosophical Logic
**22**, 607–636 (1993)CrossRefMathSciNetzbMATHGoogle Scholar - 9.Goldblatt, R.I., Thomason, S.K.: Axiomatic classes in propositional modal logic. In: Crossley, J.N. (ed.) Algebra and Logic. Lecture Notes in Mathematics, vol. 450, pp. 163–173. Springer, Heidelberg (1975) CrossRefGoogle Scholar
- 10.Goranko, V., Passy, S.: Using the universal modality: gains and questions. J. Log. Comput.
**2**(1), 5–30 (1992)CrossRefMathSciNetzbMATHGoogle Scholar - 11.Hella, L., Luosto, K., Sano, K., Virtema, J.: The expressive power of modal dependence logic. In: AiML 2014 (2014)Google Scholar
- 12.Kontinen, J., Müller, J.-S., Schnoor, H., Vollmer, H.: Modal independence logic. In: AiML 2014 (2014)Google Scholar
- 13.Kontinen, J., Müller, J.-S., Schnoor, H., Vollmer, H.: A van Benthem theorem for modal team semantics (2014). arXiv:1410.6648
- 14.Lohmann, P., Vollmer, H.: Complexity results for modal dependence logic. Stud. Logica.
**101**(2), 343–366 (2013)CrossRefMathSciNetzbMATHGoogle Scholar - 15.Müller, J.-S., Vollmer, H.: Model checking for modal dependence logic: an approach through post’s lattice. In: Libkin, L., Kohlenbach, U., de Queiroz, R. (eds.) WoLLIC 2013. LNCS, vol. 8071, pp. 238–250. Springer, Heidelberg (2013) CrossRefGoogle Scholar
- 16.Sano, K., Ma, M.: Goldblatt-Thomason-style theorems for graded modal language. In: Beklemishev, L., Goranko, V., Shehtman, V. (eds.) Advances in Modal Logic 2010. pp. 330–349 (2010)Google Scholar
- 17.Sano, K., Virtema, J.: Axiomatizing propositional dependence logics (2014). arXiv:1410.5038
- 18.Sevenster, M.: Model-theoretic and computational properties of modal dependence logic. J. Log. Comput.
**19**(6), 1157–1173 (2009)CrossRefMathSciNetzbMATHGoogle Scholar - 19.Virtema, J.: Complexity of validity for propositional dependence logics. In: GandALF 2014 (2014)Google Scholar
- 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 (2008)Google Scholar
- 21.Yang, F.: On Extensions and Variants of Dependence Logic. Ph.D. thesis, University of Helsinki (2014)Google Scholar