δ-Decidability over the Reals∗

Given any collection F of computable functions over the reals, we show that there exists an algorithm that, given any LF -sentence φ containing only bounded quantifiers, and any positive rational number δ, decides either “φ is true”, or “a δ-strengthening of φ is false”’. Under mild assumptions, for a C-computable signature F , the δ-decision problem for bounded Σk-sentences in LF resides in (ΣPk ) C . The results stand in sharp contrast to the well-known undecidability results, and serve as a theoretical basis for the use of numerical methods in decision procedures for nonlinear first-order theories over the reals.