06039193

j

06039193 Kurz, Alexander Leal, Raul Modalities in the Stone age: a comparison of coalgebraic logics. Theor. Comput. Sci. 430, 88-116 (2012). 2012 Elsevier Science Publishers, Amsterdam EN coalgebra coalgebraic logic Stone duality predicate liftings Moss-modality nabla-modality

doi:10.1016/j.tcs.2012.03.027

Summary: 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 [{\it R. A. Leal}, Electron. Notes Theor. Comput. Sci. 203, No. 5, 195--220 (2008)] by making systematic use of Stone duality. Our main contribution then is a novel coalgebraic logic, which can be seen as an equational axiomatisation of Moss's logic. The three logics are equivalent for a natural but restricted class of functors. We give examples showing that the logics differ in general. Finally, we argue that the quest for a generic logic for $T$-coalgebras is still open in the general case.