Abstract

In this paper we study fixpoints of operators on lattices and bilattices in a systematic and principled way. The key concept is that of an approximating operator, a monotone operator on the product bilattice, which gives approximate information on the original operator in an intuitive and well-defined way. With any given approximating operator our theory associates several different types of fixpoints, including the Kripke-Kleene fixpoint, stable fixpoints and the well-founded fixpoint, and relates them to fixpoints of operators being approximated. Compared to our earlier work on approximation theory, the contribution of this paper is that we provide an alternative, more intuitive and better motivated construction of the well-founded and stable fixpoints. In addition, we study the space of approximating operators by means of a precision ordering and show that each lattice operator O has a unique most precise — we call it ultimate — approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We then discuss applications of these results in logic programming.