On the expressive completeness of the propositional mu-calculus with respect to monadic second order logic

Bisimulation and Language Equivalence

“…An interesting question is the relationship between µM and 2OL. Van Benthem's result was generalised by Janin and Walukiewicz [12] as follows.…”

Expressiveness Modulo Bisimilarity: A Coalgebraic Perspective

“…It follows from Proposition 4.3(2) that V f = V • f , and from this it is immediate that (T f )σ ∼ V,V f σ . But then (13) follows from the one-step T -invariance of ϕ.…”

“…Here a key example is the theorem by Janin & Walukiewicz [13], characterizing the modal µ-calculus as the bisimulation-invariant fragment of monadic second-order logic.…”

Johan van Benthem on Logic and Information Dynamics

“…We only mention here two such results. First, Janin and Walukievicz [160] gave a similar characterization of modal mu-calculus (obtained from modal logic by adding fixed points of monotonic operators definable by positive modal formulas): mu-calculus is the largest ''dynamic'' fragment of monadic second-order logic. Second, the analogue of van Benthem's theorem for the guarded fragment was proved by Andréka et al [1], and this was later extended by Graedel, Hirsch and Otto to an analogue of the Janin-Walukievicz theorem for the fixed-point extension of the guarded fragment [141].…”