Constraint Tableaux for Hybrid Logics Draft

Hybrid logics are modal logics with names for worlds. Tableau systems for various hybrid logics were proposed in [6, 7, 20]. We present an improved tableau system for hybrid logic that handles equality for nominals by substitution and generation of inequality constraints. This compiles out the equalities while storing the inequalities, thus allowing for efficient equality reasoning. The proof procedure based on the tableau calculus —the constraint proof engine — uses two other kinds of constraints: box constraints and inverse box constraints. Completeness of the system follows in the usual way from fairness of the proof procedure, together with a model generation argument. Next, calculus and proof engine are extended to incorporate the universal modality. Among other things, universal modalities allow us to tune the proof engine to specific frame classes. This leads to a general completeness result for the calculus, for all frame classes that can be described by a Aφ formula. We also propose rules for minimal model generation, and a compression algorithm for generating minimal models from complete open tableau nodes. Finally, we focus on some fragments of hybrid logic that are decided by the constraint proof engine. A theorem prover for hybrid logic based on the constraint proof engine, HyLoTab, has been implemented. A companion paper [11] contains the full code of of this implementation in Haskell [15], in ‘literate programming’ style [16]. This documented code can be found at http://www.cwi.nl/~jve/hylotab.