Paper detail

Characterizing decidability in a quasianalytic setting

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 $§$, and in the course of proving our main theorem, numerous theorems about the model theory of such structures $\RR_S$ are also proven.