Generalizing standard monadic second-order logic for Kripke models, we introduce monadic second-order logic interpreted over coalgebras for an arbitrary set functor. We then consider invariance under behavioral equivalence of MSO-formulas. More specifically, we investigate whether the coalgebraic mu-calculus is the bisimulation-invariant fragment of the monadic second-order language for a given functor. Using automatatheoretic techniques and building on recent results by the third author, we show that in order to provide such a characterization result it suffices to find what we call an adequate uniform construction for the coalgebraic type functor. As direct applications of this result we obtain a partly new proof of the Janin-Walukiewicz Theorem for the modal mu-calculus, avoiding the use of syntactic normal forms, and bisimulation invariance results for the bag functor (graded modal logic) and all exponential polynomial functors (including the "game functor"). As a more involved application, involving additional non-trivial ideas, we also derive a characterization theorem for the monotone modal mu-calculus, with respect to a natural monadic second-order language for monotone neighborhood models.Comment: arXiv admin note: substantial text overlap with arXiv:1501.07215

@article{EnqvistSV17,
  title = {An expressive completeness theorem for coalgebraic modal mu-calculi},
  author = {Sebastian Enqvist and Fatemeh Seifan and Yde Venema},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 13, Issue 2},
  year = {2017},
  doi = {10.23638/lmcs-13(2:14)2017}
}