Relating levels of the mu-calculus hierarchy and levels of the monadic hierachy

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, ive 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 Sigma (0) (i.e. first order logic). We show here that the v-level (resp. the nu mu -level) of the mu-calculus hierarchy is as expressive cis the bisimulation invariant fragment of monadic Sigma (1) (resp. monadic Sigma (2)) and we show that no other level Sigma (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 Sigma (k) of the monadic hierarchy, for some k > 2, is also discussed.