Coalgebraic Expressions

We show that certain fixpoint expressions used to describe finite Kripke polynomial coalgebras can be seen as coalgebraic modal fixpoint formulae. Both the synthesis of a coalgebra from its expression and the ability to check behavioural equivalence follow from the same tableau construction. There is an associated complete equational logic, analogous to Kleene algebra, which may now be seen as an equational presentation of a fragment of the coalgebraic μ-calculus. These expressions include the regular expressions, the free Kleene algebra with tests and fragments of CCS and Linear Temporal Logic.