Relating Levels of the Mu-Calculus Hierarchy and Levels of the Monadic Hierarchy

As already known [14], the mu-calculus [17] is as expressive as the bisimulation invariant fragment of monadic second order Logic (MSO). In this paper, we relate the expressiveness of levels of the fixpoint alternation depth hierarchy of the mu-calculus (the mu-calculus hierarchy) with the expressiveness of the bisimulation invariant fragment of levels of the monadic quantifiers alternation-depth hierarchy (the monadic hierarchy). From van Benthem’s result [3], we know already that the fixpoint free fragment of the mu-calculus (i.e. polymodal Logic) is as expressive as the bisimulation invariant fragment of monadic 0 (i.e. first order logic). We show here that the -level (resp. the -level) of the mu-calculus hierarchy is as expressive as the bisimulation invariant fragment of monadic 1 (resp. monadic 2) and we show that no other level k for k > 2 of the monadic hierarchy can be related similarly with any other level of the mu-calculus hierarchy. The possible inclusion of all the mu-calculus in some level k of the monadic hierarchy, for some k > 2, is also discussed.