Tractable Reasoning via Approximation

Abstract Problems in logic are well known to be hard to solve in the worst case. Two different 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 goal is to define a semantically well-founded logic for approximate reasoning, which is justifiable from the intuitive point of view, and to provide fast algorithms for dealing with it even when using expressive languages. We also want our logic to be useful to perform approximate reasoning in different contexts. We define a method for the approximation of decision reasoning problems based on multivalued logics. Our work expands and generalizes, in several directions, ideas presented by other researchers. The major features of our technique are: (1) approximate answers give semantically clear information about the problem at hand; (2) approximate answers are easier to compute than answers to the original problem; (3) approximate answers can be improved, and eventually they converge to the right answer; (4) both sound approximations and complete ones are described. The method we propose is flexible enough to be applied to a wide range of reasoning problems. In our research we considered approximation of several decidable problems with different worstcase complexity, involving both propositional and first-order languages. In particular we defined approximation techniques for: propositional logic, fragments of first-order logic (concept description languages) and modal logic. In our research we also addressed the issue of representing the knowledge of a reasoner with limited resources and how to use such a knowledge for approximate reasoning purposes.