Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics

We establish a generic upper bound ExpTime for reasoning with global assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier results of this kind, our bound does not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that potentially avoids building the entire exponential-sized space of candidate states, and thus offers a basis for practical reasoning. This algorithm still involves frequent fixpoint computations; we show how these can be handled efficiently in a concrete algorithm modelled on Liu and Smolka’s linear-time fixpoint algorithm. Finally, we show that the upper complexity bound is preserved under adding nominals to the logic, i.e., in coalgebraic hybrid logic.

Design of the Tableau Reasoner TGC2 for Description Logics

Ontologies have been applied in a wide range of practical domains. They play a key role in data modeling, information integration, and the creation of semantic web services, intelligent web sites and intelligent software agents. The Web ontology language OWL, recommended by W3C, is based on description logics (DLs). Automated reasoning in DLs is very important for the success of OWL, as it provides support for visualization, debugging, and querying of ontologies. The existing ontology reasoners are not yet satisfactory, especially when dealing with qualified number restrictions and large ontologies. In this paper, we present the design of our new reasoner TGC2, which uses tableaux with global caching for reasoning in E xpTime-complete DLs. The characteristic of TGC2 is that it is based on our tableau methods with the optimal ( ExpTime) complexity, while the existing well-known tableau-based reasoners for DLs have a non-optimal complexity (at least NExpTime). We briefly describe the tableau methods used by TGC2. We then provide the design principles of TGC2 and some important optimization techniques for increasing the efficiency of this reasoner. We also present preliminary evaluation results of TGC2. They show that TGC2 deals with qualified number restrictions much better than the other existing reasoners.