  Given any collection F of computable functions over the reals, we show that
there exists an algorithm that, given any L_F-sentence \varphi containing only
bounded quantifiers, and any positive rational number \delta, decides either
"\varphi is true", or "a \delta-strengthening of \varphi is false". Under mild
assumptions, for a C-computable signature F, the \delta-decision problem for
bounded \Sigma_k-sentences in L_F resides in (\Sigma_k^P)^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.
