Tractable Reasoning via Approximation

Problems in logic are well-known to be hard to solve in the worst case. Twodi erent strategies for dealing with this aspect are known from the literature:language restriction and theory approximation.In this paper we are concerned with the second strategy. Our main goalis to de ne a semantically well-founded logic for approximate reasoning, whichis justi able from the intuitive point of view, and to provide fast algorithmsfor dealing with it even when using expressive languages. We also want ourlogic to be useful to perform approximate reasoning in di erent contexts. Wede ne a method for the approximation of decision reasoning problems basedon multivalued logics. Our work expands and generalizes in several directionsideas presented by other researchers. The major features of our technique are:1) approximate answers give semantically clear information about the problemat hand; 2) approximate answers are easier to compute than answers to theoriginal problem; 3) approximate answers can be improved, and eventually theyconverge to the right answer; 4) both sound approximations and complete onesare described.The method we propose is exible enough to be applied to a wide range of rea-soning problems. In our research we considered approximation of several decid-able problems with di erent worst-case complexity, involving both propositionaland rst-order languages. In particular we de ned approximation techniques for:propositional logic, fragments of rst order logic (concept description languages)and modal logic. In our research we also addressed the issue of representing theknowledge of a reasoner with limited resources and how to use such a knowledgefor approximate reasoning purposes.2