HyLoRes 1.0: Direct Resolution for Hybrid Logics

Abstract

Hybrid languages are modal languages that allow direct reference to the elements of a model. Even the basic hybrid language ${\text{(}}\mathcal{H}{\text{(@))}}$, which only extends the basic modal language with the addition of nominals (i,j, k,...) and satisfiability operators (@ _i , @ _j , @ _k ,...), increases the expressive power: it can explicitly check whether the point of evaluation is a specific, named point in the model (w ⊩ i), and whether a named point satisfies a given formula (w ⊩ @ _iϕ ). The extended expressivity allows one to define elegant decision algorithms, where nominals and @ play the role of labels, or prefixes, which are usually needed during the construction of proofs in the modal setup [5,3]. Note that they do so inside the object language. All these features we get with no increase in complexity: the complexity of the satisfiability problem for $\mathcal{H}{\text{(@)}}$ is the same as for the basic modal language, PSPACE [2]. When we move to very expressive hybrid languages containing binders, we obtain an impressive boost in expressivity, but usually we also move beyond the boundaries of decidability. Classical binders like ∀ and ∃ (together with @) make the logic as expressive as first-order logic (FOL) while adding the more “modal” binder ↓ gives a logic weaker than FOL [1]. We refer to the Hybrid Logic site at http://www.hylo.net for a broad on-line bibliography.