Tableaux with Substitution for Hybrid Logic with the Global and Converse Modalities

This work provides the full proofs of the properties of the tableaux calculus for hybrid logic with the global and converse modalities presented in [3], which focuses on the HL(@) fragment of the calculus. While such a fragment terminates without loop checks, when the converse and global modalities are added to the language, and the corresponding rules to the system, termination is achieved by means of a loop checking mechanism. The peculiarity of the system is the treatment of nominal equalities by means of a substitution rule. The main advantage of such a rule, compared with other approaches, is its efficiency, that has been experimentally verified for the HL(@) fragment. Such an advantage should persist in the extended calculus. In this work we give the detailed termination and completeness proofs for the entire calculus. Although the main guidelines are the same as the corresponding proofs for HL(@), the proofs for the extended calculus conceal many subtleties that have to be handled with care.