Many-valued modal logics generalize the Kripke frame semantics of classical modal logic to allow a many-valued semantics at each world based on an algebra with a complete lattice reduct, where the accessibility relation may also take values in this algebra. Such logics can be designed to model modal notions such as necessity, belief, and spatio-temporal relations in the presence of uncertainty, possibility, or vagueness, and also provide a basis for defining fuzzy description logics. More generally, many-valued modal logics provide a first foray into investigating useful and computationally feasible fragments of corresponding first-order logics. The aim of my talk will be to describe recent axiomatization, decidability, and complexity results for many-valued modal logics based on algebras over (infinite) sets of real numbers. In particular, I will report on joint work with X. Caicedo, R. Rodriguez, and J. Rogger establishing decidability and complexity results for a family o!

 f order-based modal logics, using an alternative semantics admitting the finite model property. I will also present an axiomatization, obtained in joint work with D. Diaconescu and L. Schnueriger, for a many-valued modal logic equipped with the usual group operations over the real numbers that provides a first step towards solving an open axiomatization problem for a Lukasiewicz modal logic with continuous operations.