Characterising Modal Definability of Team-Based Logics via the Universal Modality

Abstract

We study model and frame definability of various modal logics. Let ML(A+) denote the fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. We show that a class of Kripke models is definable in ML(A+) if and only if the class is elementary and closed under disjoint unions and surjective bisimulations. We also characterise the definability of ML(A+) in the spirit of the well-known Goldblatt--Thomason theorem. We show that an elementary class F of Kripke frames is definable in ML(A+) if and only if F is closed under taking generated subframes and bounded morphic images, and reflects ultrafilter extensions and finitely generated subframes. In addition we study frame definability relative to finite transitive frames and give an analogous characterisation of ML(A+)-definability relative to finite transitive frames. Finally, we initiate the study of model and frame definability in team-based logics. We study (extended) modal dependence logic, (extended) modal inclusion logic, and modal team logic. We establish strict linear hierarchies with respect to model definability and frame definability, respectively. We show that, with respect to model and frame definability, the before mentioned team-based logics, except modal dependence logic, either coincide with ML(A+) or plain modal logic ML. Thus as a corollary we obtain model theoretic characterisation of model and frame definability for the team-based logics.