An expressive completeness theorem for coalgebraic modal mu-calculi

• Published in: Logical methods in computer science 2017-01, Vol.13 (2)
• Main Authors: Enqvist, Sebastian, Seifan, Fatemeh, Venema, Yde
• Format: Article
• Language: English
• Subjects: computer science - logic in computer science
• ISSN: 1860-5974; 1860-5974
• DOI: 10.23638/LMCS-13(2:14)2017

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 automata theoretic 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.