The Alternation Hierarchy of the μ-calculus over Weakly Transitive Frames

It is known that the μ -calculus collapses to its alternation-free fragment over transitive frames and to modal logic over equivalence relations. We adapt a proof by D’Agostino and Lenzi to show that the μ -calculus collapses to its alternation-free fragment over weakly transitive frames. As a consequence, we show that the μ -calculus with derivative topological semantics collapses to its alternation-free fragment. We also study the collapse over frames of S 4.2 , S 4.3 , S 4.3 . 2 , S 4.4 and KD 45 , logics important for Epistemic Logic. At last, we use the μ -calculus to define degrees of ignorance on Epistemic Logic and study the implications of μ -calculus’s collapse over the logics above.