This paper presents a logical characterization of coalgebraic behavioral equivalence. The characterization is given in terms of coalgebraic modal logic, an abstract framework for reasoning about, and specifying properties of, coalgebras, for an endofunctor on the category of sets. Its main feature is the use of predicate liftings which give rise to the interpretation of modal operators on coalgebras. We show that coalgebraic modal logic is adequate for reasoning about coalgebras, that is, behaviorally equivalent states cannot be distinguished by formulas of the logic. Subsequently, we isolate properties which also ensure expressiveness of the logic, that is, logical and behavioral equivalence coincide.
"Expressive Logics for Coalgebras via Terminal Sequence Induction." Notre Dame J. Formal Logic 45 (1) 19 - 33, 2004. https://doi.org/10.1305/ndjfl/1094155277