RESULTS

The original results of this book split into two parts. This division reflects the division of the book itself. We can say that the first type of results concern model-theoretic and complexity properties of hybrid logics. Since hybrid logics which we call standard are quite well investigated, our efforts focused on hybrid logics referred to as non-standard in this book. By non-standard hybrid logics, we understand modal logics with global counting operators (M(En)) whose expressive power matches the expressive power of binder-free standard hybrid logics. The relevant results comprise:

1. Establishing a sound and complete axiomatization for the modal logic K with global counting operators (MK(En)), which can be easily extended onto other frame classes,

2. Establishing tight complexity bounds, namely NExpTime-completeness for the modal logic with global counting operators defined over the classes of arbitrary, reflexive, symmetric, serial and transitive frames (MK(En), MT(En), MD(En), MB(En), MK4(En)) with numerical subscripts coded in binary. Establishing the exponential-size model property for logics defined over the classes of Euclidean and equivalential frames (MK5(En), MS5(En)). In the second case, we currently lack complexity bounds which are tight. We only conjecture that the satisfiability problem for the modal logic with global counting operators over Euclidean and equivalential frames is NP-complete. Nevertheless, the computational complexity of non-standard hybrid logics turned out to be rather fixed and less dependant on the frame class it is defined over than standard hybrid logics.

Results of the second type consist of designing concrete deductive (tableau and sequent) systems for standard and non-standard hybrid logics. More precisely, they include:

1. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for a rich class of binder-free standard hybrid logics.

2. Devising a prefixed and an internalized tableau calculi which are sound, complete and terminating for non-standard hybrid logics.