Which Semantics for Neighbourhood Semantics?

In this article we discuss two alternative proposals for neighbourhood semantics (which we call strict and loose neighbourhood semantics, N = and N⊆ 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 N⊆, the required bisimulation for N = is rather involved. We propose a simple extension of N = with a universal modality that we call N=(E), which comes together with a natural notion of bisimulation. We also investigate the complexity of the satisfiability problem for N ⊆ and N=(E). 1 Epistemic Logic and Neighbourhood Semantics Epistemic logic, the logic that studies notions like agents’s beliefs and knowledge, is an important and long-standing area of research in artificial intelligence [Fagin et al., 1995]. In epistemic logic, the formula [α]ϕ is used to represent that agent α believes or knows that ϕ is the case. When the agent α is understood by context, or when we are not interested on modelling the behaviour of different agents at the same time, we will usually write []ϕ instead of [α]ϕ. In the rest of this article we will discuss the case for a single agent. By adding the []operator to classical propositional logic, we can already express a number of interesting properties. For example, the formula [](ϕ∧ψ) → ( []ϕ ∧ []ψ) intuitively says that if an agent believes or knows the conjunction of two facts ϕ and ψ, then it also knows both ϕ and ψ. Epistemic logic is usually considered a member of the large family of modal logics [Blackburn et al., 2001], and as we will discuss in this article, it shares with them many of their properties (e.g., characterizations in terms of bisimulations, good computational behavior, etc.). But, as it is well known (see [Vardi, 1986]), semantics specified over standard Kripke models in terms of possible worlds and accessibility relations [Blackburn et al., 2001] have some undesired epistemic properties. The reason is that even the weakest logic definable in terms of Kripke models (i.e., the modal logic K defined as Work partially done while Master’s student at Departamento d