Approximation of relations by propositional formulas: Complexity and semantics

Abstract

Selman and Kautz introduced the notion of approximation of a theory and showed its usefulness for knowledge compilation and on-line reasoning. We study here the complexity of the main computational problems related to the approximation of relations (sets of possible worlds) by propositional formulas, and the semantics of reasoning with these approximations (deduction and abduction). The classes of formulas that we consider are those of (reverse-)Horn, bijunctive and affine formulas, which are the most interesting for storing knowledge. Concerning the computation of approximations, we survey and complete the results that can be found in the literature, trying to adopt a unified point of view. On the contrary, as far as we know this paper is the first real attempt to study the semantics of abduction with the bounds of a theory.