Modal Logics, Description Logics and Arithmetic Reasoning

Forthcoming in the Journal of Aritificial Intelligence We introduce mathematical programming and atomic decomposition as the basic modal (T-Box) inference techniques for a large class of modal and description logics. The class of description logics suitable for the proposed methods is strong on the arithmetical side. In particular there may be complex arithmetical conditions on sets of accessible worlds (role fillers). The atomic decomposition technique can deal with set constructors for modal parameters (role terms) and parameter (role) hierarchies specified in full propositional logic. Besides the standard modal operators, a number of other constructors can be added in a relatively straightforward way. Examples are graded modalities (qualified number restrictions) and also generalized quantifiers like ‘most’, ‘n%’, ‘more’ and ‘many’.