Section: New Results
Neighbourhood and Topological Semantics for Hybrid Languages
The most popular semantics for modal languages nowadays is relational semantics. It has a generalisation introduced by Dana Scott in the 1970s, called neighbourhood semantics. A particular case of neighbourhood structures are topological spaces, which means that modal languages can be used for spatial reasoning. Hybrid languages, being extensions of modal languages, can be used for that purpose too, offering additional expressive power. Some natural model-theoretic questions arise.
Recently, members in LED had started a semantic exploration of neighbourhood semantics. Two different semantics proposals have been introduced in the literature and we have found characterizations, via appropriate notions of bisimulations, for them. We also proposed extensions that lead to more natural languages. Finally, we have also investigated the computational complexity of these logics, extending the work of Vardi et al.  . This work will soon be submitted for publication.
LED has also investigated the issue of definability for hybrid languages in topological semantics. In a paper  , we prove a theorem that provides necessary and sufficient conditions for a topological property to be definable in hybrid languages (this is a topological analogue of Goldblatt-Thomason theorem).
More recently an investigation has been done on the computational properties of logics of different classes of topological spaces. It has been proven that basic modal logics of separation axioms T0 , T1 and T2 coincide and are decidable. The hybrid logic of T1 has also been proven decidable.