Equational Coalgebraic Logic
Raul Leal, Alexander Kurz
Abstract:
Coalgebra develops a general theory of transition systems,
parametric in a functor $T$; the functor $T$ specifies the possible
one-step behaviours of the system. A fundamental question in this
area is how to obtain, for an arbitrary functor $T$, a logic for
$T$-coalgebras. We compare two existing proposals, Moss's
coalgebraic logic and the logic of all predicate liftings, by
providing one-step translations between them, extending the results
in \cite{leal:cmcs08} by making systematic use of Stone duality. Our
main contribution then is a novel coalgebraic logic, which can be
seen as an equational axiomatization of Moss's logic. The three
logics are equivalent for a natural but restricted class of
functors. We give examples showing that the logics fall apart in
general. Finally, we argue that the quest for a generic logic for
$T$-coalgebras is still open in the general case.