Reasoning in many dimensions: uncertainty and products of modal logics

Description Logics (DLs) are a popular family of knowledge representation languages. They are fragments of first-order logic (FO) that combine high expressiveness with reasonable computational properties; in particular, most DLs are decidable. However, being based on first-order logic, they share also the shortcomings. One of these shortcomings is that DLs do not have built-in means to capture uncertainty, a feature that is commonly needed in many applications. This problem has been addressed in many different ways; one of the most recent proposals is the introduction of Probabilistic Description Logics (ProbDLs) which relate to Probabilistic first-order logic (ProbFO) in the same way as DLs relate to standard FO. In order to capture the uncertainty, ProbFO and thus ProbDLs adopt a possible world semantics. More specifically, a ProbDL or ProbFO knowledge base describes a family of distributions over possible worlds. These logics constitute the scope of the first part of the thesis. We investigate the following settings: • Reasoning in full ProbFO is highly undecidable and standard restrictions like the guarded fragment do not lead to decidability. We identify a fragment, monodic ProbFO, that shows several nice properties: the validity problem is recursively enumerable and decidability of FO fragments carries over to the corresponding monodic ProbFO fragment; • In order to identify well-behaved, in best-case tractable ProbDLs, we study the complexity landscape for different fragments of ProbEL; amongst others, we are able to identify a tractable fragment. • We then turn our attention to the recently popular reasoning problem of ontological query answering, but apply it to probabilistic data. More precisely, we define the framework of ontology-based access to probabilistic data and study the computational complexity therein. The main results here are dichotomy theorems between PTime and #P. Probabilistic logics as described above can be viewed as instances of the framework of many-dimensional logics, one dimension being classical logic and the other being reasoning with probabilities. In the final part of the thesis, we remain in this framework and study the complexity of the satisfiability problem in the two-dimensional modal logic K×K. Particularly, we are able to close a gap that has been open for more than ten years.