Which Semantics for Neighbourhood Semantics?

In this article we discuss two alternative proposals for neighbourhood semantics (which we call strict and loose neighbourhood semantics, NSS and NSL respectively) that have been previously introduced in the literature. Our main tools are suitable notions of bisimulation. While an elegant notion of bisimulation exists for NSL, the required bisimulation for NSS is rather involved. We propose a simple extension of NSS with a universal modality that we call NSS(E), which comes together with a natural notion of bisimulation. We also investigate the complexity of the satisfiability problem for NSL and NSS(E).