Completeness of tableau calculi for two-dimensional hybrid logics

Abstract Hybrid logic is one of the extensions of modal logic. The many-dimensional product of hybrid logic is called hybrid product logic (HPL). We construct a sound and complete tableau calculus for two-dimensional HPL. Also, we made a tableau calculus for hybrid dependent product logic (HdPL), where one dimension depends on the other. In addition, we add a special rule to the tableau calculus for HdPL and show that it is still sound and complete. All of them lack termination, however.