  Let $\RR_S$ denote the expansion of the real ordered field by a family of
real-valued functions $S$, where each function in $S$ is defined on a compact
box and is a member of some quasianalytic class which is closed under the
operations of function composition, division by variables, and implicitly
defined functions. It is shown that the first order theory of $\RR_S$ is
decidable if and only if two oracles, called the approximation and precision
oracles for $S$, are decidable. Loosely stated, the approximation oracle for
$S$ allows one to approximate any partial derivative of any function in $S$ to
within any given error, and the precision oracle for $S$ allows one to decide
when a manifold $M\subseteq\RR^n$ is contained in a coordinate hyperplane
$\{x\in\RR^n : x_i = 0\}$ when one is given $i\in\{1,\ldots,n\}$ and a system
of equations which defines $M$ nonsingularly, where the functions occurring in
the equations are rational polynomials of the coordinate variables $x =
(x_1,\ldots,x_n)$ and the partial derivatives of the functions in $S$. A key
component of the proof is the development of a local resolution of
singularities procedure which is effective in the approximation and precision
oracles for $\S$, and in the course of proving our main theorem, numerous
theorems about the model theory of such structures $\RR_S$ are also proven.
