Complexity of {\L}ukasiewicz Modal Probabilistic Logics

Abstract

Modal probabilistic logics provide a framework for reasoning about probability in modal contexts, involving notions such as knowledge, belief, time, and action. In this paper, we study a particular family of these logics, extending the modal {\L}ukasiewicz many-valued logic. These logics are shown to be capable of expressing nuanced probabilistic concepts, including upper and lower probabilities. Our main contribution is a PSPACE-completeness result for two variants of the local consequence problem, providing a precise computational characterisation.