Approximation, solution operators and quantale-valued metrics

A generalized solution operator is a mapping abstractly describing a computational problem and its approximate solutions. It assigns a set of ε-approximations of a solution to the problem instance f and accuracy of approximation ε. In this paper we study generalized solution operators for which the accuracy of approximation is described by elements of a complete lattice equipped with a compatible monoid structure, namely, a quantale. We provide examples of computational problems for which the accuracy of approximation of a solution is measured by such objects. We show that the sets of ε-approximations are, roughly, closed balls with radii ε with respect to a certain family of quantale-valued generalized metrics induced by a generalized solution operator.