Relating levels of the mu-calculus hierarchy and levels of the monadic hierarchy
Proceedings 16th Annual IEEE Symposium on Logic in Computer Science
As already known  , the mu-calculus  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  , we know already that the
... t free fragment of the mu-calculus (i.e. polymodal Logic) is as expressive as the bisimulation invariant fragment of monadic ¦ ¼ (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 ¦ ½ (resp. monadic ¦ ¾ ) and we show that no other level ¦ for ¾ 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 ¦ of the monadic hierarchy, for some ¾, is also discussed.