In the first place, let's recall and briefly recapitulate some basic facts about hybrid logics that we use in the present chapter. Various issues related to hybrid logics are discussed more thoroughly in Chapter 4. Hybrid logics are powerful extensions of modal logics which allow for referring to particular states of a model without using meta-language. In order to achieve it, the language of standard modal logics is enriched with the countably infinite set of propositional expressions called nominals (we fix the notation nom = {i, j, k, …} to stand for the set of nominals), disjoint from the set of propositional variables prop. Each nominal is true at exactly one world and therefore can serve both as a label and as a formula. Supplying a language with nominals significantly strengthens its expressive power. In the present chapter we consider a suitably modified hybrid logic obtained by adding the satisfaction operators, the universal modality, the difference modality and the inverse modality. As indicated in Chapter 4, the satisfaction operators of the form @i allow for stating that a particular formula holds at a world labelled by i, the universal modality E expresses the fact that there exists a world in a domain, at which a particular formula holds, the difference modality D stands for the fact that a particular formula holds at a world different from the current one and, eventually, the inverse modality allows us to “jump back” to a predecessor-world along the accessibility relation edges.

As we mention in Section 4.2 of the preceding chapter, some hybrid logics additionally contain a different sort of expressions, the state variables, which allow quantifying over worlds, and additional operators like the ↓ operator or ∃. However, these logics are proven to be undecidable ([3]) so, in principle, they cannot be subjected to a terminating tableau-based decision procedure. We therefore hold our word from Chapter 4.3 and confine ourselves only to the foregoing decidable hybrid logic.