Studies in the Extension of Standard Modal Logic with an Infinite Modality.
Ignacio Bellas Acosta

Abstract:
We consider the modal logic ML^∞, the extension of standard modal logic where the modality ♦^∞ is added to the signature. Interpreted using Kripke semantics, the ♦^∞ modality captures the distinction between finite and infinite. We first provide a collection of results on the model theoretic aspects of this logic. Introducing an alternative definition of bisimulation, we establish a collection of invariance results as well as a characterization of ML^∞ in terms of this new notion of bisimulation. Furthermore we adapt the Hennessy-Milner property to the ML^∞ framework and characterize a collection of frames that enjoy this property.
In a second line of research we establish some positive results on the finite axiomatization of ML^∞ . We introduce the ML^∞ logics K^∞ and S5^∞ and we show that they are, respectively, sound and weakly complete with respect to the class of Kripke frames and the class of equivalent Kripke frames.